1 \input texinfo @c -*-texinfo-*-
2 @comment %**start of header (This is for running Texinfo on a region.)
4 @setfilename ../../info/calc
6 @settitle GNU Emacs Calc Manual
8 @comment %**end of header (This is for running Texinfo on a region.)
10 @include emacsver.texi
12 @c The following macros are used for conditional output for single lines.
14 @c `foo' will appear only in TeX output
16 @c `foo' will appear only in non-TeX output
18 @c @expr{expr} will typeset an expression;
19 @c $x$ in TeX, @samp{x} otherwise.
24 @alias infoline=comment
38 @alias texline=comment
39 @macro infoline{stuff}
58 % Suggested by Karl Berry <karl@@freefriends.org>
59 \gdef\!{\mskip-\thinmuskip}
62 @c Fix some other things specifically for this manual.
65 @mathcode`@:=`@: @c Make Calc fractions come out right in math mode
67 \gdef\coloneq{\mathrel{\mathord:\mathord=}}
69 \gdef\beforedisplay{\vskip-10pt}
70 \gdef\afterdisplay{\vskip-5pt}
71 \gdef\beforedisplayh{\vskip-25pt}
72 \gdef\afterdisplayh{\vskip-10pt}
74 @newdimen@kyvpos @kyvpos=0pt
75 @newdimen@kyhpos @kyhpos=0pt
76 @newcount@calcclubpenalty @calcclubpenalty=1000
79 @newtoks@calcoldeverypar @calcoldeverypar=@everypar
80 @everypar={@calceverypar@the@calcoldeverypar}
81 @ifx@ninett@undefinedzzz@font@ninett=cmtt9@fi
82 @catcode`@\=0 \catcode`\@=11
84 \catcode`\@=0 @catcode`@\=@active
90 This file documents Calc, the GNU Emacs calculator.
93 This file documents Calc, the GNU Emacs calculator, included with
94 GNU Emacs @value{EMACSVER}.
97 Copyright @copyright{} 1990-1991, 2001-2012 Free Software Foundation, Inc.
100 Permission is granted to copy, distribute and/or modify this document
101 under the terms of the GNU Free Documentation License, Version 1.3 or
102 any later version published by the Free Software Foundation; with the
103 Invariant Sections being just ``GNU GENERAL PUBLIC LICENSE'', with the
104 Front-Cover texts being ``A GNU Manual,'' and with the Back-Cover
105 Texts as in (a) below. A copy of the license is included in the section
106 entitled ``GNU Free Documentation License.''
108 (a) The FSF's Back-Cover Text is: ``You have the freedom to copy and
109 modify this GNU manual. Buying copies from the FSF supports it in
110 developing GNU and promoting software freedom.''
114 @dircategory Emacs misc features
116 * Calc: (calc). Advanced desk calculator and mathematical tool.
121 @center @titlefont{Calc Manual}
123 @center GNU Emacs Calc
126 @center Dave Gillespie
127 @center daveg@@synaptics.com
130 @vskip 0pt plus 1filll
143 @node Top, Getting Started, (dir), (dir)
144 @chapter The GNU Emacs Calculator
147 @dfn{Calc} is an advanced desk calculator and mathematical tool
148 written by Dave Gillespie that runs as part of the GNU Emacs environment.
150 This manual, also written (mostly) by Dave Gillespie, is divided into
151 three major parts: ``Getting Started,'' the ``Calc Tutorial,'' and the
152 ``Calc Reference.'' The Tutorial introduces all the major aspects of
153 Calculator use in an easy, hands-on way. The remainder of the manual is
154 a complete reference to the features of the Calculator.
158 For help in the Emacs Info system (which you are using to read this
159 file), type @kbd{?}. (You can also type @kbd{h} to run through a
160 longer Info tutorial.)
166 * Getting Started:: General description and overview.
168 * Interactive Tutorial::
170 * Tutorial:: A step-by-step introduction for beginners.
172 * Introduction:: Introduction to the Calc reference manual.
173 * Data Types:: Types of objects manipulated by Calc.
174 * Stack and Trail:: Manipulating the stack and trail buffers.
175 * Mode Settings:: Adjusting display format and other modes.
176 * Arithmetic:: Basic arithmetic functions.
177 * Scientific Functions:: Transcendentals and other scientific functions.
178 * Matrix Functions:: Operations on vectors and matrices.
179 * Algebra:: Manipulating expressions algebraically.
180 * Units:: Operations on numbers with units.
181 * Store and Recall:: Storing and recalling variables.
182 * Graphics:: Commands for making graphs of data.
183 * Kill and Yank:: Moving data into and out of Calc.
184 * Keypad Mode:: Operating Calc from a keypad.
185 * Embedded Mode:: Working with formulas embedded in a file.
186 * Programming:: Calc as a programmable calculator.
188 * Copying:: How you can copy and share Calc.
189 * GNU Free Documentation License:: The license for this documentation.
190 * Customizing Calc:: Customizing Calc.
191 * Reporting Bugs:: How to report bugs and make suggestions.
193 * Summary:: Summary of Calc commands and functions.
195 * Key Index:: The standard Calc key sequences.
196 * Command Index:: The interactive Calc commands.
197 * Function Index:: Functions (in algebraic formulas).
198 * Concept Index:: General concepts.
199 * Variable Index:: Variables used by Calc (both user and internal).
200 * Lisp Function Index:: Internal Lisp math functions.
204 @node Getting Started, Interactive Tutorial, Top, Top
207 @node Getting Started, Tutorial, Top, Top
209 @chapter Getting Started
211 This chapter provides a general overview of Calc, the GNU Emacs
212 Calculator: What it is, how to start it and how to exit from it,
213 and what are the various ways that it can be used.
217 * About This Manual::
218 * Notations Used in This Manual::
219 * Demonstration of Calc::
221 * History and Acknowledgments::
224 @node What is Calc, About This Manual, Getting Started, Getting Started
225 @section What is Calc?
228 @dfn{Calc} is an advanced calculator and mathematical tool that runs as
229 part of the GNU Emacs environment. Very roughly based on the HP-28/48
230 series of calculators, its many features include:
234 Choice of algebraic or RPN (stack-based) entry of calculations.
237 Arbitrary precision integers and floating-point numbers.
240 Arithmetic on rational numbers, complex numbers (rectangular and polar),
241 error forms with standard deviations, open and closed intervals, vectors
242 and matrices, dates and times, infinities, sets, quantities with units,
243 and algebraic formulas.
246 Mathematical operations such as logarithms and trigonometric functions.
249 Programmer's features (bitwise operations, non-decimal numbers).
252 Financial functions such as future value and internal rate of return.
255 Number theoretical features such as prime factorization and arithmetic
256 modulo @var{m} for any @var{m}.
259 Algebraic manipulation features, including symbolic calculus.
262 Moving data to and from regular editing buffers.
265 Embedded mode for manipulating Calc formulas and data directly
266 inside any editing buffer.
269 Graphics using GNUPLOT, a versatile (and free) plotting program.
272 Easy programming using keyboard macros, algebraic formulas,
273 algebraic rewrite rules, or extended Emacs Lisp.
276 Calc tries to include a little something for everyone; as a result it is
277 large and might be intimidating to the first-time user. If you plan to
278 use Calc only as a traditional desk calculator, all you really need to
279 read is the ``Getting Started'' chapter of this manual and possibly the
280 first few sections of the tutorial. As you become more comfortable with
281 the program you can learn its additional features. Calc does not
282 have the scope and depth of a fully-functional symbolic math package,
283 but Calc has the advantages of convenience, portability, and freedom.
285 @node About This Manual, Notations Used in This Manual, What is Calc, Getting Started
286 @section About This Manual
289 This document serves as a complete description of the GNU Emacs
290 Calculator. It works both as an introduction for novices and as
291 a reference for experienced users. While it helps to have some
292 experience with GNU Emacs in order to get the most out of Calc,
293 this manual ought to be readable even if you don't know or use Emacs
296 This manual is divided into three major parts: the ``Getting
297 Started'' chapter you are reading now, the Calc tutorial, and the Calc
300 @c This manual has been printed in two volumes, the @dfn{Tutorial} and the
301 @c @dfn{Reference}. Both volumes include a copy of the ``Getting Started''
304 If you are in a hurry to use Calc, there is a brief ``demonstration''
305 below which illustrates the major features of Calc in just a couple of
306 pages. If you don't have time to go through the full tutorial, this
307 will show you everything you need to know to begin.
308 @xref{Demonstration of Calc}.
310 The tutorial chapter walks you through the various parts of Calc
311 with lots of hands-on examples and explanations. If you are new
312 to Calc and you have some time, try going through at least the
313 beginning of the tutorial. The tutorial includes about 70 exercises
314 with answers. These exercises give you some guided practice with
315 Calc, as well as pointing out some interesting and unusual ways
318 The reference section discusses Calc in complete depth. You can read
319 the reference from start to finish if you want to learn every aspect
320 of Calc. Or, you can look in the table of contents or the Concept
321 Index to find the parts of the manual that discuss the things you
324 @c @cindex Marginal notes
325 Every Calc keyboard command is listed in the Calc Summary, and also
326 in the Key Index. Algebraic functions, @kbd{M-x} commands, and
327 variables also have their own indices.
329 @c @infoline In the printed manual, each
330 @c paragraph that is referenced in the Key or Function Index is marked
331 @c in the margin with its index entry.
333 @c [fix-ref Help Commands]
334 You can access this manual on-line at any time within Calc by pressing
335 the @kbd{h i} key sequence. Outside of the Calc window, you can press
336 @kbd{C-x * i} to read the manual on-line. From within Calc the command
337 @kbd{h t} will jump directly to the Tutorial; from outside of Calc the
338 command @kbd{C-x * t} will jump to the Tutorial and start Calc if
339 necessary. Pressing @kbd{h s} or @kbd{C-x * s} will take you directly
340 to the Calc Summary. Within Calc, you can also go to the part of the
341 manual describing any Calc key, function, or variable using
342 @w{@kbd{h k}}, @kbd{h f}, or @kbd{h v}, respectively. @xref{Help Commands}.
345 The Calc manual can be printed, but because the manual is so large, you
346 should only make a printed copy if you really need it. To print the
347 manual, you will need the @TeX{} typesetting program (this is a free
348 program by Donald Knuth at Stanford University) as well as the
349 @file{texindex} program and @file{texinfo.tex} file, both of which can
350 be obtained from the FSF as part of the @code{texinfo} package.
351 To print the Calc manual in one huge tome, you will need the
352 source code to this manual, @file{calc.texi}, available as part of the
353 Emacs source. Once you have this file, type @kbd{texi2dvi calc.texi}.
354 Alternatively, change to the @file{man} subdirectory of the Emacs
355 source distribution, and type @kbd{make calc.dvi}. (Don't worry if you
356 get some ``overfull box'' warnings while @TeX{} runs.)
357 The result will be a device-independent output file called
358 @file{calc.dvi}, which you must print in whatever way is right
359 for your system. On many systems, the command is
372 @c Printed copies of this manual are also available from the Free Software
375 @node Notations Used in This Manual, Demonstration of Calc, About This Manual, Getting Started
376 @section Notations Used in This Manual
379 This section describes the various notations that are used
380 throughout the Calc manual.
382 In keystroke sequences, uppercase letters mean you must hold down
383 the shift key while typing the letter. Keys pressed with Control
384 held down are shown as @kbd{C-x}. Keys pressed with Meta held down
385 are shown as @kbd{M-x}. Other notations are @key{RET} for the
386 Return key, @key{SPC} for the space bar, @key{TAB} for the Tab key,
387 @key{DEL} for the Delete key, and @key{LFD} for the Line-Feed key.
388 The @key{DEL} key is called Backspace on some keyboards, it is
389 whatever key you would use to correct a simple typing error when
390 regularly using Emacs.
392 (If you don't have the @key{LFD} or @key{TAB} keys on your keyboard,
393 the @kbd{C-j} and @kbd{C-i} keys are equivalent to them, respectively.
394 If you don't have a Meta key, look for Alt or Extend Char. You can
395 also press @key{ESC} or @kbd{C-[} first to get the same effect, so
396 that @kbd{M-x}, @kbd{@key{ESC} x}, and @kbd{C-[ x} are all equivalent.)
398 Sometimes the @key{RET} key is not shown when it is ``obvious''
399 that you must press @key{RET} to proceed. For example, the @key{RET}
400 is usually omitted in key sequences like @kbd{M-x calc-keypad @key{RET}}.
402 Commands are generally shown like this: @kbd{p} (@code{calc-precision})
403 or @kbd{C-x * k} (@code{calc-keypad}). This means that the command is
404 normally used by pressing the @kbd{p} key or @kbd{C-x * k} key sequence,
405 but it also has the full-name equivalent shown, e.g., @kbd{M-x calc-precision}.
407 Commands that correspond to functions in algebraic notation
408 are written: @kbd{C} (@code{calc-cos}) [@code{cos}]. This means
409 the @kbd{C} key is equivalent to @kbd{M-x calc-cos}, and that
410 the corresponding function in an algebraic-style formula would
411 be @samp{cos(@var{x})}.
413 A few commands don't have key equivalents: @code{calc-sincos}
416 @node Demonstration of Calc, Using Calc, Notations Used in This Manual, Getting Started
417 @section A Demonstration of Calc
420 @cindex Demonstration of Calc
421 This section will show some typical small problems being solved with
422 Calc. The focus is more on demonstration than explanation, but
423 everything you see here will be covered more thoroughly in the
426 To begin, start Emacs if necessary (usually the command @code{emacs}
427 does this), and type @kbd{C-x * c} to start the
428 Calculator. (You can also use @kbd{M-x calc} if this doesn't work.
429 @xref{Starting Calc}, for various ways of starting the Calculator.)
431 Be sure to type all the sample input exactly, especially noting the
432 difference between lower-case and upper-case letters. Remember,
433 @key{RET}, @key{TAB}, @key{DEL}, and @key{SPC} are the Return, Tab,
434 Delete, and Space keys.
436 @strong{RPN calculation.} In RPN, you type the input number(s) first,
437 then the command to operate on the numbers.
440 Type @kbd{2 @key{RET} 3 + Q} to compute
441 @texline @math{\sqrt{2+3} = 2.2360679775}.
442 @infoline the square root of 2+3, which is 2.2360679775.
445 Type @kbd{P 2 ^} to compute
446 @texline @math{\pi^2 = 9.86960440109}.
447 @infoline the value of `pi' squared, 9.86960440109.
450 Type @key{TAB} to exchange the order of these two results.
453 Type @kbd{- I H S} to subtract these results and compute the Inverse
454 Hyperbolic sine of the difference, 2.72996136574.
457 Type @key{DEL} to erase this result.
459 @strong{Algebraic calculation.} You can also enter calculations using
460 conventional ``algebraic'' notation. To enter an algebraic formula,
461 use the apostrophe key.
464 Type @kbd{' sqrt(2+3) @key{RET}} to compute
465 @texline @math{\sqrt{2+3}}.
466 @infoline the square root of 2+3.
469 Type @kbd{' pi^2 @key{RET}} to enter
470 @texline @math{\pi^2}.
471 @infoline `pi' squared.
472 To evaluate this symbolic formula as a number, type @kbd{=}.
475 Type @kbd{' arcsinh($ - $$) @key{RET}} to subtract the second-most-recent
476 result from the most-recent and compute the Inverse Hyperbolic sine.
478 @strong{Keypad mode.} If you are using the X window system, press
479 @w{@kbd{C-x * k}} to get Keypad mode. (If you don't use X, skip to
483 Click on the @key{2}, @key{ENTER}, @key{3}, @key{+}, and @key{SQRT}
484 ``buttons'' using your left mouse button.
487 Click on @key{PI}, @key{2}, and @tfn{y^x}.
490 Click on @key{INV}, then @key{ENTER} to swap the two results.
493 Click on @key{-}, @key{INV}, @key{HYP}, and @key{SIN}.
496 Click on @key{<-} to erase the result, then click @key{OFF} to turn
497 the Keypad Calculator off.
499 @strong{Grabbing data.} Type @kbd{C-x * x} if necessary to exit Calc.
500 Now select the following numbers as an Emacs region: ``Mark'' the
501 front of the list by typing @kbd{C-@key{SPC}} or @kbd{C-@@} there,
502 then move to the other end of the list. (Either get this list from
503 the on-line copy of this manual, accessed by @w{@kbd{C-x * i}}, or just
504 type these numbers into a scratch file.) Now type @kbd{C-x * g} to
505 ``grab'' these numbers into Calc.
516 The result @samp{[1.23, 1.97, 1.6, 2, 1.19, 1.08]} is a Calc ``vector.''
517 Type @w{@kbd{V R +}} to compute the sum of these numbers.
520 Type @kbd{U} to Undo this command, then type @kbd{V R *} to compute
521 the product of the numbers.
524 You can also grab data as a rectangular matrix. Place the cursor on
525 the upper-leftmost @samp{1} and set the mark, then move to just after
526 the lower-right @samp{8} and press @kbd{C-x * r}.
529 Type @kbd{v t} to transpose this
530 @texline @math{3\times2}
533 @texline @math{2\times3}
535 matrix. Type @w{@kbd{v u}} to unpack the rows into two separate
536 vectors. Now type @w{@kbd{V R + @key{TAB} V R +}} to compute the sums
537 of the two original columns. (There is also a special
538 grab-and-sum-columns command, @kbd{C-x * :}.)
540 @strong{Units conversion.} Units are entered algebraically.
541 Type @w{@kbd{' 43 mi/hr @key{RET}}} to enter the quantity 43 miles-per-hour.
542 Type @w{@kbd{u c km/hr @key{RET}}}. Type @w{@kbd{u c m/s @key{RET}}}.
544 @strong{Date arithmetic.} Type @kbd{t N} to get the current date and
545 time. Type @kbd{90 +} to find the date 90 days from now. Type
546 @kbd{' <25 dec 87> @key{RET}} to enter a date, then @kbd{- 7 /} to see how
547 many weeks have passed since then.
549 @strong{Algebra.} Algebraic entries can also include formulas
550 or equations involving variables. Type @kbd{@w{' [x + y} = a, x y = 1] @key{RET}}
551 to enter a pair of equations involving three variables.
552 (Note the leading apostrophe in this example; also, note that the space
553 in @samp{x y} is required.) Type @w{@kbd{a S x,y @key{RET}}} to solve
554 these equations for the variables @expr{x} and @expr{y}.
557 Type @kbd{d B} to view the solutions in more readable notation.
558 Type @w{@kbd{d C}} to view them in C language notation, @kbd{d T}
559 to view them in the notation for the @TeX{} typesetting system,
560 and @kbd{d L} to view them in the notation for the @LaTeX{} typesetting
561 system. Type @kbd{d N} to return to normal notation.
564 Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @expr{a = 7.5} in these formulas.
565 (That's the letter @kbd{l}, not the numeral @kbd{1}.)
568 @strong{Help functions.} You can read about any command in the on-line
569 manual. Type @kbd{C-x * c} to return to Calc after each of these
570 commands: @kbd{h k t N} to read about the @kbd{t N} command,
571 @kbd{h f sqrt @key{RET}} to read about the @code{sqrt} function, and
572 @kbd{h s} to read the Calc summary.
575 @strong{Help functions.} You can read about any command in the on-line
576 manual. Remember to type the letter @kbd{l}, then @kbd{C-x * c}, to
577 return here after each of these commands: @w{@kbd{h k t N}} to read
578 about the @w{@kbd{t N}} command, @kbd{h f sqrt @key{RET}} to read about the
579 @code{sqrt} function, and @kbd{h s} to read the Calc summary.
582 Press @key{DEL} repeatedly to remove any leftover results from the stack.
583 To exit from Calc, press @kbd{q} or @kbd{C-x * c} again.
585 @node Using Calc, History and Acknowledgments, Demonstration of Calc, Getting Started
589 Calc has several user interfaces that are specialized for
590 different kinds of tasks. As well as Calc's standard interface,
591 there are Quick mode, Keypad mode, and Embedded mode.
595 * The Standard Interface::
596 * Quick Mode Overview::
597 * Keypad Mode Overview::
598 * Standalone Operation::
599 * Embedded Mode Overview::
600 * Other C-x * Commands::
603 @node Starting Calc, The Standard Interface, Using Calc, Using Calc
604 @subsection Starting Calc
607 On most systems, you can type @kbd{C-x *} to start the Calculator.
608 The key sequence @kbd{C-x *} is bound to the command @code{calc-dispatch},
609 which can be rebound if convenient (@pxref{Customizing Calc}).
611 When you press @kbd{C-x *}, Emacs waits for you to press a second key to
612 complete the command. In this case, you will follow @kbd{C-x *} with a
613 letter (upper- or lower-case, it doesn't matter for @kbd{C-x *}) that says
614 which Calc interface you want to use.
616 To get Calc's standard interface, type @kbd{C-x * c}. To get
617 Keypad mode, type @kbd{C-x * k}. Type @kbd{C-x * ?} to get a brief
618 list of the available options, and type a second @kbd{?} to get
621 To ease typing, @kbd{C-x * *} also works to start Calc. It starts the
622 same interface (either @kbd{C-x * c} or @w{@kbd{C-x * k}}) that you last
623 used, selecting the @kbd{C-x * c} interface by default.
625 If @kbd{C-x *} doesn't work for you, you can always type explicit
626 commands like @kbd{M-x calc} (for the standard user interface) or
627 @w{@kbd{M-x calc-keypad}} (for Keypad mode). First type @kbd{M-x}
628 (that's Meta with the letter @kbd{x}), then, at the prompt,
629 type the full command (like @kbd{calc-keypad}) and press Return.
631 The same commands (like @kbd{C-x * c} or @kbd{C-x * *}) that start
632 the Calculator also turn it off if it is already on.
634 @node The Standard Interface, Quick Mode Overview, Starting Calc, Using Calc
635 @subsection The Standard Calc Interface
638 @cindex Standard user interface
639 Calc's standard interface acts like a traditional RPN calculator,
640 operated by the normal Emacs keyboard. When you type @kbd{C-x * c}
641 to start the Calculator, the Emacs screen splits into two windows
642 with the file you were editing on top and Calc on the bottom.
648 --**-Emacs: myfile (Fundamental)----All----------------------
649 --- Emacs Calculator Mode --- |Emacs Calculator Trail
657 --%*-Calc: 12 Deg (Calculator)----All----- --%*- *Calc Trail*
661 In this figure, the mode-line for @file{myfile} has moved up and the
662 ``Calculator'' window has appeared below it. As you can see, Calc
663 actually makes two windows side-by-side. The lefthand one is
664 called the @dfn{stack window} and the righthand one is called the
665 @dfn{trail window.} The stack holds the numbers involved in the
666 calculation you are currently performing. The trail holds a complete
667 record of all calculations you have done. In a desk calculator with
668 a printer, the trail corresponds to the paper tape that records what
671 In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
672 were first entered into the Calculator, then the 2 and 4 were
673 multiplied to get 8, then the 3 and 8 were subtracted to get @mathit{-5}.
674 (The @samp{>} symbol shows that this was the most recent calculation.)
675 The net result is the two numbers 17.3 and @mathit{-5} sitting on the stack.
677 Most Calculator commands deal explicitly with the stack only, but
678 there is a set of commands that allow you to search back through
679 the trail and retrieve any previous result.
681 Calc commands use the digits, letters, and punctuation keys.
682 Shifted (i.e., upper-case) letters are different from lowercase
683 letters. Some letters are @dfn{prefix} keys that begin two-letter
684 commands. For example, @kbd{e} means ``enter exponent'' and shifted
685 @kbd{E} means @expr{e^x}. With the @kbd{d} (``display modes'') prefix
686 the letter ``e'' takes on very different meanings: @kbd{d e} means
687 ``engineering notation'' and @kbd{d E} means ``@dfn{eqn} language mode.''
689 There is nothing stopping you from switching out of the Calc
690 window and back into your editing window, say by using the Emacs
691 @w{@kbd{C-x o}} (@code{other-window}) command. When the cursor is
692 inside a regular window, Emacs acts just like normal. When the
693 cursor is in the Calc stack or trail windows, keys are interpreted
696 When you quit by pressing @kbd{C-x * c} a second time, the Calculator
697 windows go away but the actual Stack and Trail are not gone, just
698 hidden. When you press @kbd{C-x * c} once again you will get the
699 same stack and trail contents you had when you last used the
702 The Calculator does not remember its state between Emacs sessions.
703 Thus if you quit Emacs and start it again, @kbd{C-x * c} will give you
704 a fresh stack and trail. There is a command (@kbd{m m}) that lets
705 you save your favorite mode settings between sessions, though.
706 One of the things it saves is which user interface (standard or
707 Keypad) you last used; otherwise, a freshly started Emacs will
708 always treat @kbd{C-x * *} the same as @kbd{C-x * c}.
710 The @kbd{q} key is another equivalent way to turn the Calculator off.
712 If you type @kbd{C-x * b} first and then @kbd{C-x * c}, you get a
713 full-screen version of Calc (@code{full-calc}) in which the stack and
714 trail windows are still side-by-side but are now as tall as the whole
715 Emacs screen. When you press @kbd{q} or @kbd{C-x * c} again to quit,
716 the file you were editing before reappears. The @kbd{C-x * b} key
717 switches back and forth between ``big'' full-screen mode and the
718 normal partial-screen mode.
720 Finally, @kbd{C-x * o} (@code{calc-other-window}) is like @kbd{C-x * c}
721 except that the Calc window is not selected. The buffer you were
722 editing before remains selected instead. If you are in a Calc window,
723 then @kbd{C-x * o} will switch you out of it, being careful not to
724 switch you to the Calc Trail window. So @kbd{C-x * o} is a handy
725 way to switch out of Calc momentarily to edit your file; you can then
726 type @kbd{C-x * c} to switch back into Calc when you are done.
728 @node Quick Mode Overview, Keypad Mode Overview, The Standard Interface, Using Calc
729 @subsection Quick Mode (Overview)
732 @dfn{Quick mode} is a quick way to use Calc when you don't need the
733 full complexity of the stack and trail. To use it, type @kbd{C-x * q}
734 (@code{quick-calc}) in any regular editing buffer.
736 Quick mode is very simple: It prompts you to type any formula in
737 standard algebraic notation (like @samp{4 - 2/3}) and then displays
738 the result at the bottom of the Emacs screen (@mathit{3.33333333333}
739 in this case). You are then back in the same editing buffer you
740 were in before, ready to continue editing or to type @kbd{C-x * q}
741 again to do another quick calculation. The result of the calculation
742 will also be in the Emacs ``kill ring'' so that a @kbd{C-y} command
743 at this point will yank the result into your editing buffer.
745 Calc mode settings affect Quick mode, too, though you will have to
746 go into regular Calc (with @kbd{C-x * c}) to change the mode settings.
748 @c [fix-ref Quick Calculator mode]
749 @xref{Quick Calculator}, for further information.
751 @node Keypad Mode Overview, Standalone Operation, Quick Mode Overview, Using Calc
752 @subsection Keypad Mode (Overview)
755 @dfn{Keypad mode} is a mouse-based interface to the Calculator.
756 It is designed for use with terminals that support a mouse. If you
757 don't have a mouse, you will have to operate Keypad mode with your
758 arrow keys (which is probably more trouble than it's worth).
760 Type @kbd{C-x * k} to turn Keypad mode on or off. Once again you
761 get two new windows, this time on the righthand side of the screen
762 instead of at the bottom. The upper window is the familiar Calc
763 Stack; the lower window is a picture of a typical calculator keypad.
767 \advance \dimen0 by 24\baselineskip%
768 \ifdim \dimen0>\pagegoal \vfill\eject \fi%
773 |--- Emacs Calculator Mode ---
777 |--%*-Calc: 12 Deg (Calcul
778 |----+----+--Calc---+----+----1
779 |FLR |CEIL|RND |TRNC|CLN2|FLT |
780 |----+----+----+----+----+----|
781 | LN |EXP | |ABS |IDIV|MOD |
782 |----+----+----+----+----+----|
783 |SIN |COS |TAN |SQRT|y^x |1/x |
784 |----+----+----+----+----+----|
785 | ENTER |+/- |EEX |UNDO| <- |
786 |-----+---+-+--+--+-+---++----|
787 | INV | 7 | 8 | 9 | / |
788 |-----+-----+-----+-----+-----|
789 | HYP | 4 | 5 | 6 | * |
790 |-----+-----+-----+-----+-----|
791 |EXEC | 1 | 2 | 3 | - |
792 |-----+-----+-----+-----+-----|
793 | OFF | 0 | . | PI | + |
794 |-----+-----+-----+-----+-----+
798 Keypad mode is much easier for beginners to learn, because there
799 is no need to memorize lots of obscure key sequences. But not all
800 commands in regular Calc are available on the Keypad. You can
801 always switch the cursor into the Calc stack window to use
802 standard Calc commands if you need. Serious Calc users, though,
803 often find they prefer the standard interface over Keypad mode.
805 To operate the Calculator, just click on the ``buttons'' of the
806 keypad using your left mouse button. To enter the two numbers
807 shown here you would click @w{@kbd{1 7 .@: 3 ENTER 5 +/- ENTER}}; to
808 add them together you would then click @kbd{+} (to get 12.3 on
811 If you click the right mouse button, the top three rows of the
812 keypad change to show other sets of commands, such as advanced
813 math functions, vector operations, and operations on binary
816 Because Keypad mode doesn't use the regular keyboard, Calc leaves
817 the cursor in your original editing buffer. You can type in
818 this buffer in the usual way while also clicking on the Calculator
819 keypad. One advantage of Keypad mode is that you don't need an
820 explicit command to switch between editing and calculating.
822 If you press @kbd{C-x * b} first, you get a full-screen Keypad mode
823 (@code{full-calc-keypad}) with three windows: The keypad in the lower
824 left, the stack in the lower right, and the trail on top.
826 @c [fix-ref Keypad Mode]
827 @xref{Keypad Mode}, for further information.
829 @node Standalone Operation, Embedded Mode Overview, Keypad Mode Overview, Using Calc
830 @subsection Standalone Operation
833 @cindex Standalone Operation
834 If you are not in Emacs at the moment but you wish to use Calc,
835 you must start Emacs first. If all you want is to run Calc, you
836 can give the commands:
846 emacs -f full-calc-keypad
850 which run a full-screen Calculator (as if by @kbd{C-x * b C-x * c}) or
851 a full-screen X-based Calculator (as if by @kbd{C-x * b C-x * k}).
852 In standalone operation, quitting the Calculator (by pressing
853 @kbd{q} or clicking on the keypad @key{EXIT} button) quits Emacs
856 @node Embedded Mode Overview, Other C-x * Commands, Standalone Operation, Using Calc
857 @subsection Embedded Mode (Overview)
860 @dfn{Embedded mode} is a way to use Calc directly from inside an
861 editing buffer. Suppose you have a formula written as part of a
875 and you wish to have Calc compute and format the derivative for
876 you and store this derivative in the buffer automatically. To
877 do this with Embedded mode, first copy the formula down to where
878 you want the result to be, leaving a blank line before and after the
893 Now, move the cursor onto this new formula and press @kbd{C-x * e}.
894 Calc will read the formula (using the surrounding blank lines to tell
895 how much text to read), then push this formula (invisibly) onto the Calc
896 stack. The cursor will stay on the formula in the editing buffer, but
897 the line with the formula will now appear as it would on the Calc stack
898 (in this case, it will be left-aligned) and the buffer's mode line will
899 change to look like the Calc mode line (with mode indicators like
900 @samp{12 Deg} and so on). Even though you are still in your editing
901 buffer, the keyboard now acts like the Calc keyboard, and any new result
902 you get is copied from the stack back into the buffer. To take the
903 derivative, you would type @kbd{a d x @key{RET}}.
917 (Note that by default, Calc gives division lower precedence than multiplication,
918 so that @samp{1 / x ln(x)} is equivalent to @samp{1 / (x ln(x))}.)
920 To make this look nicer, you might want to press @kbd{d =} to center
921 the formula, and even @kbd{d B} to use Big display mode.
930 % [calc-mode: justify: center]
931 % [calc-mode: language: big]
939 Calc has added annotations to the file to help it remember the modes
940 that were used for this formula. They are formatted like comments
941 in the @TeX{} typesetting language, just in case you are using @TeX{} or
942 @LaTeX{}. (In this example @TeX{} is not being used, so you might want
943 to move these comments up to the top of the file or otherwise put them
946 As an extra flourish, we can add an equation number using a
947 righthand label: Type @kbd{d @} (1) @key{RET}}.
951 % [calc-mode: justify: center]
952 % [calc-mode: language: big]
953 % [calc-mode: right-label: " (1)"]
961 To leave Embedded mode, type @kbd{C-x * e} again. The mode line
962 and keyboard will revert to the way they were before.
964 The related command @kbd{C-x * w} operates on a single word, which
965 generally means a single number, inside text. It searches for an
966 expression which ``looks'' like a number containing the point.
967 Here's an example of its use (before you try this, remove the Calc
968 annotations or use a new buffer so that the extra settings in the
969 annotations don't take effect):
972 A slope of one-third corresponds to an angle of 1 degrees.
975 Place the cursor on the @samp{1}, then type @kbd{C-x * w} to enable
976 Embedded mode on that number. Now type @kbd{3 /} (to get one-third),
977 and @kbd{I T} (the Inverse Tangent converts a slope into an angle),
978 then @w{@kbd{C-x * w}} again to exit Embedded mode.
981 A slope of one-third corresponds to an angle of 18.4349488229 degrees.
984 @c [fix-ref Embedded Mode]
985 @xref{Embedded Mode}, for full details.
987 @node Other C-x * Commands, , Embedded Mode Overview, Using Calc
988 @subsection Other @kbd{C-x *} Commands
991 Two more Calc-related commands are @kbd{C-x * g} and @kbd{C-x * r},
992 which ``grab'' data from a selected region of a buffer into the
993 Calculator. The region is defined in the usual Emacs way, by
994 a ``mark'' placed at one end of the region, and the Emacs
995 cursor or ``point'' placed at the other.
997 The @kbd{C-x * g} command reads the region in the usual left-to-right,
998 top-to-bottom order. The result is packaged into a Calc vector
999 of numbers and placed on the stack. Calc (in its standard
1000 user interface) is then started. Type @kbd{v u} if you want
1001 to unpack this vector into separate numbers on the stack. Also,
1002 @kbd{C-u C-x * g} interprets the region as a single number or
1005 The @kbd{C-x * r} command reads a rectangle, with the point and
1006 mark defining opposite corners of the rectangle. The result
1007 is a matrix of numbers on the Calculator stack.
1009 Complementary to these is @kbd{C-x * y}, which ``yanks'' the
1010 value at the top of the Calc stack back into an editing buffer.
1011 If you type @w{@kbd{C-x * y}} while in such a buffer, the value is
1012 yanked at the current position. If you type @kbd{C-x * y} while
1013 in the Calc buffer, Calc makes an educated guess as to which
1014 editing buffer you want to use. The Calc window does not have
1015 to be visible in order to use this command, as long as there
1016 is something on the Calc stack.
1018 Here, for reference, is the complete list of @kbd{C-x *} commands.
1019 The shift, control, and meta keys are ignored for the keystroke
1020 following @kbd{C-x *}.
1023 Commands for turning Calc on and off:
1027 Turn Calc on or off, employing the same user interface as last time.
1029 @item =, +, -, /, \, &, #
1030 Alternatives for @kbd{*}.
1033 Turn Calc on or off using its standard bottom-of-the-screen
1034 interface. If Calc is already turned on but the cursor is not
1035 in the Calc window, move the cursor into the window.
1038 Same as @kbd{C}, but don't select the new Calc window. If
1039 Calc is already turned on and the cursor is in the Calc window,
1040 move it out of that window.
1043 Control whether @kbd{C-x * c} and @kbd{C-x * k} use the full screen.
1046 Use Quick mode for a single short calculation.
1049 Turn Calc Keypad mode on or off.
1052 Turn Calc Embedded mode on or off at the current formula.
1055 Turn Calc Embedded mode on or off, select the interesting part.
1058 Turn Calc Embedded mode on or off at the current word (number).
1061 Turn Calc on in a user-defined way, as defined by a @kbd{Z I} command.
1064 Quit Calc; turn off standard, Keypad, or Embedded mode if on.
1065 (This is like @kbd{q} or @key{OFF} inside of Calc.)
1072 Commands for moving data into and out of the Calculator:
1076 Grab the region into the Calculator as a vector.
1079 Grab the rectangular region into the Calculator as a matrix.
1082 Grab the rectangular region and compute the sums of its columns.
1085 Grab the rectangular region and compute the sums of its rows.
1088 Yank a value from the Calculator into the current editing buffer.
1095 Commands for use with Embedded mode:
1099 ``Activate'' the current buffer. Locate all formulas that
1100 contain @samp{:=} or @samp{=>} symbols and record their locations
1101 so that they can be updated automatically as variables are changed.
1104 Duplicate the current formula immediately below and select
1108 Insert a new formula at the current point.
1111 Move the cursor to the next active formula in the buffer.
1114 Move the cursor to the previous active formula in the buffer.
1117 Update (i.e., as if by the @kbd{=} key) the formula at the current point.
1120 Edit (as if by @code{calc-edit}) the formula at the current point.
1127 Miscellaneous commands:
1131 Run the Emacs Info system to read the Calc manual.
1132 (This is the same as @kbd{h i} inside of Calc.)
1135 Run the Emacs Info system to read the Calc Tutorial.
1138 Run the Emacs Info system to read the Calc Summary.
1141 Load Calc entirely into memory. (Normally the various parts
1142 are loaded only as they are needed.)
1145 Read a region of written keystroke names (like @kbd{C-n a b c @key{RET}})
1146 and record them as the current keyboard macro.
1149 (This is the ``zero'' digit key.) Reset the Calculator to
1150 its initial state: Empty stack, and initial mode settings.
1153 @node History and Acknowledgments, , Using Calc, Getting Started
1154 @section History and Acknowledgments
1157 Calc was originally started as a two-week project to occupy a lull
1158 in the author's schedule. Basically, a friend asked if I remembered
1160 @texline @math{2^{32}}.
1161 @infoline @expr{2^32}.
1162 I didn't offhand, but I said, ``that's easy, just call up an
1163 @code{xcalc}.'' @code{Xcalc} duly reported that the answer to our
1164 question was @samp{4.294967e+09}---with no way to see the full ten
1165 digits even though we knew they were there in the program's memory! I
1166 was so annoyed, I vowed to write a calculator of my own, once and for
1169 I chose Emacs Lisp, a) because I had always been curious about it
1170 and b) because, being only a text editor extension language after
1171 all, Emacs Lisp would surely reach its limits long before the project
1172 got too far out of hand.
1174 To make a long story short, Emacs Lisp turned out to be a distressingly
1175 solid implementation of Lisp, and the humble task of calculating
1176 turned out to be more open-ended than one might have expected.
1178 Emacs Lisp didn't have built-in floating point math (now it does), so
1179 this had to be simulated in software. In fact, Emacs integers would
1180 only comfortably fit six decimal digits or so (at the time)---not
1181 enough for a decent calculator. So I had to write my own
1182 high-precision integer code as well, and once I had this I figured
1183 that arbitrary-size integers were just as easy as large integers.
1184 Arbitrary floating-point precision was the logical next step. Also,
1185 since the large integer arithmetic was there anyway it seemed only
1186 fair to give the user direct access to it, which in turn made it
1187 practical to support fractions as well as floats. All these features
1188 inspired me to look around for other data types that might be worth
1191 Around this time, my friend Rick Koshi showed me his nifty new HP-28
1192 calculator. It allowed the user to manipulate formulas as well as
1193 numerical quantities, and it could also operate on matrices. I
1194 decided that these would be good for Calc to have, too. And once
1195 things had gone this far, I figured I might as well take a look at
1196 serious algebra systems for further ideas. Since these systems did
1197 far more than I could ever hope to implement, I decided to focus on
1198 rewrite rules and other programming features so that users could
1199 implement what they needed for themselves.
1201 Rick complained that matrices were hard to read, so I put in code to
1202 format them in a 2D style. Once these routines were in place, Big mode
1203 was obligatory. Gee, what other language modes would be useful?
1205 Scott Hemphill and Allen Knutson, two friends with a strong mathematical
1206 bent, contributed ideas and algorithms for a number of Calc features
1207 including modulo forms, primality testing, and float-to-fraction conversion.
1209 Units were added at the eager insistence of Mass Sivilotti. Later,
1210 Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable
1211 expert assistance with the units table. As far as I can remember, the
1212 idea of using algebraic formulas and variables to represent units dates
1213 back to an ancient article in Byte magazine about muMath, an early
1214 algebra system for microcomputers.
1216 Many people have contributed to Calc by reporting bugs and suggesting
1217 features, large and small. A few deserve special mention: Tim Peters,
1218 who helped develop the ideas that led to the selection commands, rewrite
1219 rules, and many other algebra features;
1220 @texline Fran\c{c}ois
1222 Pinard, who contributed an early prototype of the Calc Summary appendix
1223 as well as providing valuable suggestions in many other areas of Calc;
1224 Carl Witty, whose eagle eyes discovered many typographical and factual
1225 errors in the Calc manual; Tim Kay, who drove the development of
1226 Embedded mode; Ove Ewerlid, who made many suggestions relating to the
1227 algebra commands and contributed some code for polynomial operations;
1228 Randal Schwartz, who suggested the @code{calc-eval} function; Juha
1229 Sarlin, who first worked out how to split Calc into quickly-loading
1230 parts; Bob Weiner, who helped immensely with the Lucid Emacs port; and
1231 Robert J. Chassell, who suggested the Calc Tutorial and exercises as
1232 well as many other things.
1234 @cindex Bibliography
1235 @cindex Knuth, Art of Computer Programming
1236 @cindex Numerical Recipes
1237 @c Should these be expanded into more complete references?
1238 Among the books used in the development of Calc were Knuth's @emph{Art
1239 of Computer Programming} (especially volume II, @emph{Seminumerical
1240 Algorithms}); @emph{Numerical Recipes} by Press, Flannery, Teukolsky,
1241 and Vetterling; Bevington's @emph{Data Reduction and Error Analysis
1242 for the Physical Sciences}; @emph{Concrete Mathematics} by Graham,
1243 Knuth, and Patashnik; Steele's @emph{Common Lisp, the Language}; the
1244 @emph{CRC Standard Math Tables} (William H. Beyer, ed.); and
1245 Abramowitz and Stegun's venerable @emph{Handbook of Mathematical
1246 Functions}. Also, of course, Calc could not have been written without
1247 the excellent @emph{GNU Emacs Lisp Reference Manual}, by Bil Lewis and
1250 Final thanks go to Richard Stallman, without whose fine implementations
1251 of the Emacs editor, language, and environment, Calc would have been
1252 finished in two weeks.
1257 @c This node is accessed by the `C-x * t' command.
1258 @node Interactive Tutorial, Tutorial, Getting Started, Top
1262 Some brief instructions on using the Emacs Info system for this tutorial:
1264 Press the space bar and Delete keys to go forward and backward in a
1265 section by screenfuls (or use the regular Emacs scrolling commands
1268 Press @kbd{n} or @kbd{p} to go to the Next or Previous section.
1269 If the section has a @dfn{menu}, press a digit key like @kbd{1}
1270 or @kbd{2} to go to a sub-section from the menu. Press @kbd{u} to
1271 go back up from a sub-section to the menu it is part of.
1273 Exercises in the tutorial all have cross-references to the
1274 appropriate page of the ``answers'' section. Press @kbd{f}, then
1275 the exercise number, to see the answer to an exercise. After
1276 you have followed a cross-reference, you can press the letter
1277 @kbd{l} to return to where you were before.
1279 You can press @kbd{?} at any time for a brief summary of Info commands.
1281 Press the number @kbd{1} now to enter the first section of the Tutorial.
1287 @node Tutorial, Introduction, Interactive Tutorial, Top
1290 @node Tutorial, Introduction, Getting Started, Top
1295 This chapter explains how to use Calc and its many features, in
1296 a step-by-step, tutorial way. You are encouraged to run Calc and
1297 work along with the examples as you read (@pxref{Starting Calc}).
1298 If you are already familiar with advanced calculators, you may wish
1300 to skip on to the rest of this manual.
1302 @c to skip on to volume II of this manual, the @dfn{Calc Reference}.
1304 @c [fix-ref Embedded Mode]
1305 This tutorial describes the standard user interface of Calc only.
1306 The Quick mode and Keypad mode interfaces are fairly
1307 self-explanatory. @xref{Embedded Mode}, for a description of
1308 the Embedded mode interface.
1310 The easiest way to read this tutorial on-line is to have two windows on
1311 your Emacs screen, one with Calc and one with the Info system. Press
1312 @kbd{C-x * t} to set this up; the on-line tutorial will be opened in the
1313 current window and Calc will be started in another window. From the
1314 Info window, the command @kbd{C-x * c} can be used to switch to the Calc
1315 window and @kbd{C-x * o} can be used to switch back to the Info window.
1316 (If you have a printed copy of the manual you can use that instead; in
1317 that case you only need to press @kbd{C-x * c} to start Calc.)
1319 This tutorial is designed to be done in sequence. But the rest of this
1320 manual does not assume you have gone through the tutorial. The tutorial
1321 does not cover everything in the Calculator, but it touches on most
1325 You may wish to print out a copy of the Calc Summary and keep notes on
1326 it as you learn Calc. @xref{About This Manual}, to see how to make a
1327 printed summary. @xref{Summary}.
1330 The Calc Summary at the end of the reference manual includes some blank
1331 space for your own use. You may wish to keep notes there as you learn
1337 * Arithmetic Tutorial::
1338 * Vector/Matrix Tutorial::
1340 * Algebra Tutorial::
1341 * Programming Tutorial::
1343 * Answers to Exercises::
1346 @node Basic Tutorial, Arithmetic Tutorial, Tutorial, Tutorial
1347 @section Basic Tutorial
1350 In this section, we learn how RPN and algebraic-style calculations
1351 work, how to undo and redo an operation done by mistake, and how
1352 to control various modes of the Calculator.
1355 * RPN Tutorial:: Basic operations with the stack.
1356 * Algebraic Tutorial:: Algebraic entry; variables.
1357 * Undo Tutorial:: If you make a mistake: Undo and the trail.
1358 * Modes Tutorial:: Common mode-setting commands.
1361 @node RPN Tutorial, Algebraic Tutorial, Basic Tutorial, Basic Tutorial
1362 @subsection RPN Calculations and the Stack
1364 @cindex RPN notation
1367 Calc normally uses RPN notation. You may be familiar with the RPN
1368 system from Hewlett-Packard calculators, FORTH, or PostScript.
1369 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1373 Calc normally uses RPN notation. You may be familiar with the RPN
1374 system from Hewlett-Packard calculators, FORTH, or PostScript.
1375 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1379 The central component of an RPN calculator is the @dfn{stack}. A
1380 calculator stack is like a stack of dishes. New dishes (numbers) are
1381 added at the top of the stack, and numbers are normally only removed
1382 from the top of the stack.
1386 In an operation like @expr{2+3}, the 2 and 3 are called the @dfn{operands}
1387 and the @expr{+} is the @dfn{operator}. In an RPN calculator you always
1388 enter the operands first, then the operator. Each time you type a
1389 number, Calc adds or @dfn{pushes} it onto the top of the Stack.
1390 When you press an operator key like @kbd{+}, Calc @dfn{pops} the appropriate
1391 number of operands from the stack and pushes back the result.
1393 Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
1394 @kbd{2 @key{RET} 3 @key{RET} +}. (The @key{RET} key, Return, corresponds to
1395 the @key{ENTER} key on traditional RPN calculators.) Try this now if
1396 you wish; type @kbd{C-x * c} to switch into the Calc window (you can type
1397 @kbd{C-x * c} again or @kbd{C-x * o} to switch back to the Tutorial window).
1398 The first four keystrokes ``push'' the numbers 2 and 3 onto the stack.
1399 The @kbd{+} key ``pops'' the top two numbers from the stack, adds them,
1400 and pushes the result (5) back onto the stack. Here's how the stack
1401 will look at various points throughout the calculation:
1409 C-x * c 2 @key{RET} 3 @key{RET} + @key{DEL}
1413 The @samp{.} symbol is a marker that represents the top of the stack.
1414 Note that the ``top'' of the stack is really shown at the bottom of
1415 the Stack window. This may seem backwards, but it turns out to be
1416 less distracting in regular use.
1418 @cindex Stack levels
1419 @cindex Levels of stack
1420 The numbers @samp{1:} and @samp{2:} on the left are @dfn{stack level
1421 numbers}. Old RPN calculators always had four stack levels called
1422 @expr{x}, @expr{y}, @expr{z}, and @expr{t}. Calc's stack can grow
1423 as large as you like, so it uses numbers instead of letters. Some
1424 stack-manipulation commands accept a numeric argument that says
1425 which stack level to work on. Normal commands like @kbd{+} always
1426 work on the top few levels of the stack.
1428 @c [fix-ref Truncating the Stack]
1429 The Stack buffer is just an Emacs buffer, and you can move around in
1430 it using the regular Emacs motion commands. But no matter where the
1431 cursor is, even if you have scrolled the @samp{.} marker out of
1432 view, most Calc commands always move the cursor back down to level 1
1433 before doing anything. It is possible to move the @samp{.} marker
1434 upwards through the stack, temporarily ``hiding'' some numbers from
1435 commands like @kbd{+}. This is called @dfn{stack truncation} and
1436 we will not cover it in this tutorial; @pxref{Truncating the Stack},
1437 if you are interested.
1439 You don't really need the second @key{RET} in @kbd{2 @key{RET} 3
1440 @key{RET} +}. That's because if you type any operator name or
1441 other non-numeric key when you are entering a number, the Calculator
1442 automatically enters that number and then does the requested command.
1443 Thus @kbd{2 @key{RET} 3 +} will work just as well.
1445 Examples in this tutorial will often omit @key{RET} even when the
1446 stack displays shown would only happen if you did press @key{RET}:
1459 Here, after pressing @kbd{3} the stack would really show @samp{1: 2}
1460 with @samp{Calc:@: 3} in the minibuffer. In these situations, you can
1461 press the optional @key{RET} to see the stack as the figure shows.
1463 (@bullet{}) @strong{Exercise 1.} (This tutorial will include exercises
1464 at various points. Try them if you wish. Answers to all the exercises
1465 are located at the end of the Tutorial chapter. Each exercise will
1466 include a cross-reference to its particular answer. If you are
1467 reading with the Emacs Info system, press @kbd{f} and the
1468 exercise number to go to the answer, then the letter @kbd{l} to
1469 return to where you were.)
1472 Here's the first exercise: What will the keystrokes @kbd{1 @key{RET} 2
1473 @key{RET} 3 @key{RET} 4 + * -} compute? (@samp{*} is the symbol for
1474 multiplication.) Figure it out by hand, then try it with Calc to see
1475 if you're right. @xref{RPN Answer 1, 1}. (@bullet{})
1477 (@bullet{}) @strong{Exercise 2.} Compute
1478 @texline @math{(2\times4) + (7\times9.5) + {5\over4}}
1479 @infoline @expr{2*4 + 7*9.5 + 5/4}
1480 using the stack. @xref{RPN Answer 2, 2}. (@bullet{})
1482 The @key{DEL} key is called Backspace on some keyboards. It is
1483 whatever key you would use to correct a simple typing error when
1484 regularly using Emacs. The @key{DEL} key pops and throws away the
1485 top value on the stack. (You can still get that value back from
1486 the Trail if you should need it later on.) There are many places
1487 in this tutorial where we assume you have used @key{DEL} to erase the
1488 results of the previous example at the beginning of a new example.
1489 In the few places where it is really important to use @key{DEL} to
1490 clear away old results, the text will remind you to do so.
1492 (It won't hurt to let things accumulate on the stack, except that
1493 whenever you give a display-mode-changing command Calc will have to
1494 spend a long time reformatting such a large stack.)
1496 Since the @kbd{-} key is also an operator (it subtracts the top two
1497 stack elements), how does one enter a negative number? Calc uses
1498 the @kbd{_} (underscore) key to act like the minus sign in a number.
1499 So, typing @kbd{-5 @key{RET}} won't work because the @kbd{-} key
1500 will try to do a subtraction, but @kbd{_5 @key{RET}} works just fine.
1502 You can also press @kbd{n}, which means ``change sign.'' It changes
1503 the number at the top of the stack (or the number being entered)
1504 from positive to negative or vice-versa: @kbd{5 n @key{RET}}.
1506 @cindex Duplicating a stack entry
1507 If you press @key{RET} when you're not entering a number, the effect
1508 is to duplicate the top number on the stack. Consider this calculation:
1512 1: 3 2: 3 1: 9 2: 9 1: 81
1516 3 @key{RET} @key{RET} * @key{RET} *
1521 (Of course, an easier way to do this would be @kbd{3 @key{RET} 4 ^},
1522 to raise 3 to the fourth power.)
1524 The space-bar key (denoted @key{SPC} here) performs the same function
1525 as @key{RET}; you could replace all three occurrences of @key{RET} in
1526 the above example with @key{SPC} and the effect would be the same.
1528 @cindex Exchanging stack entries
1529 Another stack manipulation key is @key{TAB}. This exchanges the top
1530 two stack entries. Suppose you have computed @kbd{2 @key{RET} 3 +}
1531 to get 5, and then you realize what you really wanted to compute
1532 was @expr{20 / (2+3)}.
1536 1: 5 2: 5 2: 20 1: 4
1540 2 @key{RET} 3 + 20 @key{TAB} /
1545 Planning ahead, the calculation would have gone like this:
1549 1: 20 2: 20 3: 20 2: 20 1: 4
1554 20 @key{RET} 2 @key{RET} 3 + /
1558 A related stack command is @kbd{M-@key{TAB}} (hold @key{META} and type
1559 @key{TAB}). It rotates the top three elements of the stack upward,
1560 bringing the object in level 3 to the top.
1564 1: 10 2: 10 3: 10 3: 20 3: 30
1565 . 1: 20 2: 20 2: 30 2: 10
1569 10 @key{RET} 20 @key{RET} 30 @key{RET} M-@key{TAB} M-@key{TAB}
1573 (@bullet{}) @strong{Exercise 3.} Suppose the numbers 10, 20, and 30 are
1574 on the stack. Figure out how to add one to the number in level 2
1575 without affecting the rest of the stack. Also figure out how to add
1576 one to the number in level 3. @xref{RPN Answer 3, 3}. (@bullet{})
1578 Operations like @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, and @kbd{^} pop two
1579 arguments from the stack and push a result. Operations like @kbd{n} and
1580 @kbd{Q} (square root) pop a single number and push the result. You can
1581 think of them as simply operating on the top element of the stack.
1585 1: 3 1: 9 2: 9 1: 25 1: 5
1589 3 @key{RET} @key{RET} * 4 @key{RET} @key{RET} * + Q
1594 (Note that capital @kbd{Q} means to hold down the Shift key while
1595 typing @kbd{q}. Remember, plain unshifted @kbd{q} is the Quit command.)
1597 @cindex Pythagorean Theorem
1598 Here we've used the Pythagorean Theorem to determine the hypotenuse of a
1599 right triangle. Calc actually has a built-in command for that called
1600 @kbd{f h}, but let's suppose we can't remember the necessary keystrokes.
1601 We can still enter it by its full name using @kbd{M-x} notation:
1609 3 @key{RET} 4 @key{RET} M-x calc-hypot
1613 All Calculator commands begin with the word @samp{calc-}. Since it
1614 gets tiring to type this, Calc provides an @kbd{x} key which is just
1615 like the regular Emacs @kbd{M-x} key except that it types the @samp{calc-}
1624 3 @key{RET} 4 @key{RET} x hypot
1628 What happens if you take the square root of a negative number?
1632 1: 4 1: -4 1: (0, 2)
1640 The notation @expr{(a, b)} represents a complex number.
1641 Complex numbers are more traditionally written @expr{a + b i};
1642 Calc can display in this format, too, but for now we'll stick to the
1643 @expr{(a, b)} notation.
1645 If you don't know how complex numbers work, you can safely ignore this
1646 feature. Complex numbers only arise from operations that would be
1647 errors in a calculator that didn't have complex numbers. (For example,
1648 taking the square root or logarithm of a negative number produces a
1651 Complex numbers are entered in the notation shown. The @kbd{(} and
1652 @kbd{,} and @kbd{)} keys manipulate ``incomplete complex numbers.''
1656 1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3)
1664 You can perform calculations while entering parts of incomplete objects.
1665 However, an incomplete object cannot actually participate in a calculation:
1669 1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ...
1679 Adding 5 to an incomplete object makes no sense, so the last command
1680 produces an error message and leaves the stack the same.
1682 Incomplete objects can't participate in arithmetic, but they can be
1683 moved around by the regular stack commands.
1687 2: 2 3: 2 3: 3 1: ( ... 1: (2, 3)
1688 1: 3 2: 3 2: ( ... 2 .
1692 2 @key{RET} 3 @key{RET} ( M-@key{TAB} M-@key{TAB} )
1697 Note that the @kbd{,} (comma) key did not have to be used here.
1698 When you press @kbd{)} all the stack entries between the incomplete
1699 entry and the top are collected, so there's never really a reason
1700 to use the comma. It's up to you.
1702 (@bullet{}) @strong{Exercise 4.} To enter the complex number @expr{(2, 3)},
1703 your friend Joe typed @kbd{( 2 , @key{SPC} 3 )}. What happened?
1704 (Joe thought of a clever way to correct his mistake in only two
1705 keystrokes, but it didn't quite work. Try it to find out why.)
1706 @xref{RPN Answer 4, 4}. (@bullet{})
1708 Vectors are entered the same way as complex numbers, but with square
1709 brackets in place of parentheses. We'll meet vectors again later in
1712 Any Emacs command can be given a @dfn{numeric prefix argument} by
1713 typing a series of @key{META}-digits beforehand. If @key{META} is
1714 awkward for you, you can instead type @kbd{C-u} followed by the
1715 necessary digits. Numeric prefix arguments can be negative, as in
1716 @kbd{M-- M-3 M-5} or @w{@kbd{C-u - 3 5}}. Calc commands use numeric
1717 prefix arguments in a variety of ways. For example, a numeric prefix
1718 on the @kbd{+} operator adds any number of stack entries at once:
1722 1: 10 2: 10 3: 10 3: 10 1: 60
1723 . 1: 20 2: 20 2: 20 .
1727 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u 3 +
1731 For stack manipulation commands like @key{RET}, a positive numeric
1732 prefix argument operates on the top @var{n} stack entries at once. A
1733 negative argument operates on the entry in level @var{n} only. An
1734 argument of zero operates on the entire stack. In this example, we copy
1735 the second-to-top element of the stack:
1739 1: 10 2: 10 3: 10 3: 10 4: 10
1740 . 1: 20 2: 20 2: 20 3: 20
1745 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u -2 @key{RET}
1749 @cindex Clearing the stack
1750 @cindex Emptying the stack
1751 Another common idiom is @kbd{M-0 @key{DEL}}, which clears the stack.
1752 (The @kbd{M-0} numeric prefix tells @key{DEL} to operate on the
1755 @node Algebraic Tutorial, Undo Tutorial, RPN Tutorial, Basic Tutorial
1756 @subsection Algebraic-Style Calculations
1759 If you are not used to RPN notation, you may prefer to operate the
1760 Calculator in Algebraic mode, which is closer to the way
1761 non-RPN calculators work. In Algebraic mode, you enter formulas
1762 in traditional @expr{2+3} notation.
1764 @strong{Notice:} Calc gives @samp{/} lower precedence than @samp{*}, so
1765 that @samp{a/b*c} is interpreted as @samp{a/(b*c)}; this is not
1766 standard across all computer languages. See below for details.
1768 You don't really need any special ``mode'' to enter algebraic formulas.
1769 You can enter a formula at any time by pressing the apostrophe (@kbd{'})
1770 key. Answer the prompt with the desired formula, then press @key{RET}.
1771 The formula is evaluated and the result is pushed onto the RPN stack.
1772 If you don't want to think in RPN at all, you can enter your whole
1773 computation as a formula, read the result from the stack, then press
1774 @key{DEL} to delete it from the stack.
1776 Try pressing the apostrophe key, then @kbd{2+3+4}, then @key{RET}.
1777 The result should be the number 9.
1779 Algebraic formulas use the operators @samp{+}, @samp{-}, @samp{*},
1780 @samp{/}, and @samp{^}. You can use parentheses to make the order
1781 of evaluation clear. In the absence of parentheses, @samp{^} is
1782 evaluated first, then @samp{*}, then @samp{/}, then finally
1783 @samp{+} and @samp{-}. For example, the expression
1786 2 + 3*4*5 / 6*7^8 - 9
1793 2 + ((3*4*5) / (6*(7^8)) - 9
1797 or, in large mathematical notation,
1811 $$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
1816 The result of this expression will be the number @mathit{-6.99999826533}.
1818 Calc's order of evaluation is the same as for most computer languages,
1819 except that @samp{*} binds more strongly than @samp{/}, as the above
1820 example shows. As in normal mathematical notation, the @samp{*} symbol
1821 can often be omitted: @samp{2 a} is the same as @samp{2*a}.
1823 Operators at the same level are evaluated from left to right, except
1824 that @samp{^} is evaluated from right to left. Thus, @samp{2-3-4} is
1825 equivalent to @samp{(2-3)-4} or @mathit{-5}, whereas @samp{2^3^4} is equivalent
1826 to @samp{2^(3^4)} (a very large integer; try it!).
1828 If you tire of typing the apostrophe all the time, there is
1829 Algebraic mode, where Calc automatically senses
1830 when you are about to type an algebraic expression. To enter this
1831 mode, press the two letters @w{@kbd{m a}}. (An @samp{Alg} indicator
1832 should appear in the Calc window's mode line.)
1834 Press @kbd{m a}, then @kbd{2+3+4} with no apostrophe, then @key{RET}.
1836 In Algebraic mode, when you press any key that would normally begin
1837 entering a number (such as a digit, a decimal point, or the @kbd{_}
1838 key), or if you press @kbd{(} or @kbd{[}, Calc automatically begins
1841 Functions which do not have operator symbols like @samp{+} and @samp{*}
1842 must be entered in formulas using function-call notation. For example,
1843 the function name corresponding to the square-root key @kbd{Q} is
1844 @code{sqrt}. To compute a square root in a formula, you would use
1845 the notation @samp{sqrt(@var{x})}.
1847 Press the apostrophe, then type @kbd{sqrt(5*2) - 3}. The result should
1848 be @expr{0.16227766017}.
1850 Note that if the formula begins with a function name, you need to use
1851 the apostrophe even if you are in Algebraic mode. If you type @kbd{arcsin}
1852 out of the blue, the @kbd{a r} will be taken as an Algebraic Rewrite
1853 command, and the @kbd{csin} will be taken as the name of the rewrite
1856 Some people prefer to enter complex numbers and vectors in algebraic
1857 form because they find RPN entry with incomplete objects to be too
1858 distracting, even though they otherwise use Calc as an RPN calculator.
1860 Still in Algebraic mode, type:
1864 1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1)
1865 . 1: (1, -2) . 1: 1 .
1868 (2,3) @key{RET} (1,-2) @key{RET} * 1 @key{RET} +
1872 Algebraic mode allows us to enter complex numbers without pressing
1873 an apostrophe first, but it also means we need to press @key{RET}
1874 after every entry, even for a simple number like @expr{1}.
1876 (You can type @kbd{C-u m a} to enable a special Incomplete Algebraic
1877 mode in which the @kbd{(} and @kbd{[} keys use algebraic entry even
1878 though regular numeric keys still use RPN numeric entry. There is also
1879 Total Algebraic mode, started by typing @kbd{m t}, in which all
1880 normal keys begin algebraic entry. You must then use the @key{META} key
1881 to type Calc commands: @kbd{M-m t} to get back out of Total Algebraic
1882 mode, @kbd{M-q} to quit, etc.)
1884 If you're still in Algebraic mode, press @kbd{m a} again to turn it off.
1886 Actual non-RPN calculators use a mixture of algebraic and RPN styles.
1887 In general, operators of two numbers (like @kbd{+} and @kbd{*})
1888 use algebraic form, but operators of one number (like @kbd{n} and @kbd{Q})
1889 use RPN form. Also, a non-RPN calculator allows you to see the
1890 intermediate results of a calculation as you go along. You can
1891 accomplish this in Calc by performing your calculation as a series
1892 of algebraic entries, using the @kbd{$} sign to tie them together.
1893 In an algebraic formula, @kbd{$} represents the number on the top
1894 of the stack. Here, we perform the calculation
1895 @texline @math{\sqrt{2\times4+1}},
1896 @infoline @expr{sqrt(2*4+1)},
1897 which on a traditional calculator would be done by pressing
1898 @kbd{2 * 4 + 1 =} and then the square-root key.
1905 ' 2*4 @key{RET} $+1 @key{RET} Q
1910 Notice that we didn't need to press an apostrophe for the @kbd{$+1},
1911 because the dollar sign always begins an algebraic entry.
1913 (@bullet{}) @strong{Exercise 1.} How could you get the same effect as
1914 pressing @kbd{Q} but using an algebraic entry instead? How about
1915 if the @kbd{Q} key on your keyboard were broken?
1916 @xref{Algebraic Answer 1, 1}. (@bullet{})
1918 The notations @kbd{$$}, @kbd{$$$}, and so on stand for higher stack
1919 entries. For example, @kbd{' $$+$ @key{RET}} is just like typing @kbd{+}.
1921 Algebraic formulas can include @dfn{variables}. To store in a
1922 variable, press @kbd{s s}, then type the variable name, then press
1923 @key{RET}. (There are actually two flavors of store command:
1924 @kbd{s s} stores a number in a variable but also leaves the number
1925 on the stack, while @w{@kbd{s t}} removes a number from the stack and
1926 stores it in the variable.) A variable name should consist of one
1927 or more letters or digits, beginning with a letter.
1931 1: 17 . 1: a + a^2 1: 306
1934 17 s t a @key{RET} ' a+a^2 @key{RET} =
1939 The @kbd{=} key @dfn{evaluates} a formula by replacing all its
1940 variables by the values that were stored in them.
1942 For RPN calculations, you can recall a variable's value on the
1943 stack either by entering its name as a formula and pressing @kbd{=},
1944 or by using the @kbd{s r} command.
1948 1: 17 2: 17 3: 17 2: 17 1: 306
1949 . 1: 17 2: 17 1: 289 .
1953 s r a @key{RET} ' a @key{RET} = 2 ^ +
1957 If you press a single digit for a variable name (as in @kbd{s t 3}, you
1958 get one of ten @dfn{quick variables} @code{q0} through @code{q9}.
1959 They are ``quick'' simply because you don't have to type the letter
1960 @code{q} or the @key{RET} after their names. In fact, you can type
1961 simply @kbd{s 3} as a shorthand for @kbd{s s 3}, and likewise for
1962 @kbd{t 3} and @w{@kbd{r 3}}.
1964 Any variables in an algebraic formula for which you have not stored
1965 values are left alone, even when you evaluate the formula.
1969 1: 2 a + 2 b 1: 2 b + 34
1976 Calls to function names which are undefined in Calc are also left
1977 alone, as are calls for which the value is undefined.
1981 1: log10(0) + log10(x) + log10(5, 6) + foo(3) + 2
1984 ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) @key{RET}
1989 In this example, the first call to @code{log10} works, but the other
1990 calls are not evaluated. In the second call, the logarithm is
1991 undefined for that value of the argument; in the third, the argument
1992 is symbolic, and in the fourth, there are too many arguments. In the
1993 fifth case, there is no function called @code{foo}. You will see a
1994 ``Wrong number of arguments'' message referring to @samp{log10(5,6)}.
1995 Press the @kbd{w} (``why'') key to see any other messages that may
1996 have arisen from the last calculation. In this case you will get
1997 ``logarithm of zero,'' then ``number expected: @code{x}''. Calc
1998 automatically displays the first message only if the message is
1999 sufficiently important; for example, Calc considers ``wrong number
2000 of arguments'' and ``logarithm of zero'' to be important enough to
2001 report automatically, while a message like ``number expected: @code{x}''
2002 will only show up if you explicitly press the @kbd{w} key.
2004 (@bullet{}) @strong{Exercise 2.} Joe entered the formula @samp{2 x y},
2005 stored 5 in @code{x}, pressed @kbd{=}, and got the expected result,
2006 @samp{10 y}. He then tried the same for the formula @samp{2 x (1+y)},
2007 expecting @samp{10 (1+y)}, but it didn't work. Why not?
2008 @xref{Algebraic Answer 2, 2}. (@bullet{})
2010 (@bullet{}) @strong{Exercise 3.} What result would you expect
2011 @kbd{1 @key{RET} 0 /} to give? What if you then type @kbd{0 *}?
2012 @xref{Algebraic Answer 3, 3}. (@bullet{})
2014 One interesting way to work with variables is to use the
2015 @dfn{evaluates-to} (@samp{=>}) operator. It works like this:
2016 Enter a formula algebraically in the usual way, but follow
2017 the formula with an @samp{=>} symbol. (There is also an @kbd{s =}
2018 command which builds an @samp{=>} formula using the stack.) On
2019 the stack, you will see two copies of the formula with an @samp{=>}
2020 between them. The lefthand formula is exactly like you typed it;
2021 the righthand formula has been evaluated as if by typing @kbd{=}.
2025 2: 2 + 3 => 5 2: 2 + 3 => 5
2026 1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b
2029 ' 2+3 => @key{RET} ' 2a+2b @key{RET} s = 10 s t a @key{RET}
2034 Notice that the instant we stored a new value in @code{a}, all
2035 @samp{=>} operators already on the stack that referred to @expr{a}
2036 were updated to use the new value. With @samp{=>}, you can push a
2037 set of formulas on the stack, then change the variables experimentally
2038 to see the effects on the formulas' values.
2040 You can also ``unstore'' a variable when you are through with it:
2045 1: 2 a + 2 b => 2 a + 2 b
2052 We will encounter formulas involving variables and functions again
2053 when we discuss the algebra and calculus features of the Calculator.
2055 @node Undo Tutorial, Modes Tutorial, Algebraic Tutorial, Basic Tutorial
2056 @subsection Undo and Redo
2059 If you make a mistake, you can usually correct it by pressing shift-@kbd{U},
2060 the ``undo'' command. First, clear the stack (@kbd{M-0 @key{DEL}}) and exit
2061 and restart Calc (@kbd{C-x * * C-x * *}) to make sure things start off
2062 with a clean slate. Now:
2066 1: 2 2: 2 1: 8 2: 2 1: 6
2074 You can undo any number of times. Calc keeps a complete record of
2075 all you have done since you last opened the Calc window. After the
2076 above example, you could type:
2088 You can also type @kbd{D} to ``redo'' a command that you have undone
2093 . 1: 2 2: 2 1: 6 1: 6
2102 It was not possible to redo past the @expr{6}, since that was placed there
2103 by something other than an undo command.
2106 You can think of undo and redo as a sort of ``time machine.'' Press
2107 @kbd{U} to go backward in time, @kbd{D} to go forward. If you go
2108 backward and do something (like @kbd{*}) then, as any science fiction
2109 reader knows, you have changed your future and you cannot go forward
2110 again. Thus, the inability to redo past the @expr{6} even though there
2111 was an earlier undo command.
2113 You can always recall an earlier result using the Trail. We've ignored
2114 the trail so far, but it has been faithfully recording everything we
2115 did since we loaded the Calculator. If the Trail is not displayed,
2116 press @kbd{t d} now to turn it on.
2118 Let's try grabbing an earlier result. The @expr{8} we computed was
2119 undone by a @kbd{U} command, and was lost even to Redo when we pressed
2120 @kbd{*}, but it's still there in the trail. There should be a little
2121 @samp{>} arrow (the @dfn{trail pointer}) resting on the last trail
2122 entry. If there isn't, press @kbd{t ]} to reset the trail pointer.
2123 Now, press @w{@kbd{t p}} to move the arrow onto the line containing
2124 @expr{8}, and press @w{@kbd{t y}} to ``yank'' that number back onto the
2127 If you press @kbd{t ]} again, you will see that even our Yank command
2128 went into the trail.
2130 Let's go further back in time. Earlier in the tutorial we computed
2131 a huge integer using the formula @samp{2^3^4}. We don't remember
2132 what it was, but the first digits were ``241''. Press @kbd{t r}
2133 (which stands for trail-search-reverse), then type @kbd{241}.
2134 The trail cursor will jump back to the next previous occurrence of
2135 the string ``241'' in the trail. This is just a regular Emacs
2136 incremental search; you can now press @kbd{C-s} or @kbd{C-r} to
2137 continue the search forwards or backwards as you like.
2139 To finish the search, press @key{RET}. This halts the incremental
2140 search and leaves the trail pointer at the thing we found. Now we
2141 can type @kbd{t y} to yank that number onto the stack. If we hadn't
2142 remembered the ``241'', we could simply have searched for @kbd{2^3^4},
2143 then pressed @kbd{@key{RET} t n} to halt and then move to the next item.
2145 You may have noticed that all the trail-related commands begin with
2146 the letter @kbd{t}. (The store-and-recall commands, on the other hand,
2147 all began with @kbd{s}.) Calc has so many commands that there aren't
2148 enough keys for all of them, so various commands are grouped into
2149 two-letter sequences where the first letter is called the @dfn{prefix}
2150 key. If you type a prefix key by accident, you can press @kbd{C-g}
2151 to cancel it. (In fact, you can press @kbd{C-g} to cancel almost
2152 anything in Emacs.) To get help on a prefix key, press that key
2153 followed by @kbd{?}. Some prefixes have several lines of help,
2154 so you need to press @kbd{?} repeatedly to see them all.
2155 You can also type @kbd{h h} to see all the help at once.
2157 Try pressing @kbd{t ?} now. You will see a line of the form,
2160 trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t-
2164 The word ``trail'' indicates that the @kbd{t} prefix key contains
2165 trail-related commands. Each entry on the line shows one command,
2166 with a single capital letter showing which letter you press to get
2167 that command. We have used @kbd{t n}, @kbd{t p}, @kbd{t ]}, and
2168 @kbd{t y} so far. The @samp{[MORE]} means you can press @kbd{?}
2169 again to see more @kbd{t}-prefix commands. Notice that the commands
2170 are roughly divided (by semicolons) into related groups.
2172 When you are in the help display for a prefix key, the prefix is
2173 still active. If you press another key, like @kbd{y} for example,
2174 it will be interpreted as a @kbd{t y} command. If all you wanted
2175 was to look at the help messages, press @kbd{C-g} afterwards to cancel
2178 One more way to correct an error is by editing the stack entries.
2179 The actual Stack buffer is marked read-only and must not be edited
2180 directly, but you can press @kbd{`} (the backquote or accent grave)
2181 to edit a stack entry.
2183 Try entering @samp{3.141439} now. If this is supposed to represent
2184 @cpi{}, it's got several errors. Press @kbd{`} to edit this number.
2185 Now use the normal Emacs cursor motion and editing keys to change
2186 the second 4 to a 5, and to transpose the 3 and the 9. When you
2187 press @key{RET}, the number on the stack will be replaced by your
2188 new number. This works for formulas, vectors, and all other types
2189 of values you can put on the stack. The @kbd{`} key also works
2190 during entry of a number or algebraic formula.
2192 @node Modes Tutorial, , Undo Tutorial, Basic Tutorial
2193 @subsection Mode-Setting Commands
2196 Calc has many types of @dfn{modes} that affect the way it interprets
2197 your commands or the way it displays data. We have already seen one
2198 mode, namely Algebraic mode. There are many others, too; we'll
2199 try some of the most common ones here.
2201 Perhaps the most fundamental mode in Calc is the current @dfn{precision}.
2202 Notice the @samp{12} on the Calc window's mode line:
2205 --%*-Calc: 12 Deg (Calculator)----All------
2209 Most of the symbols there are Emacs things you don't need to worry
2210 about, but the @samp{12} and the @samp{Deg} are mode indicators.
2211 The @samp{12} means that calculations should always be carried to
2212 12 significant figures. That is why, when we type @kbd{1 @key{RET} 7 /},
2213 we get @expr{0.142857142857} with exactly 12 digits, not counting
2214 leading and trailing zeros.
2216 You can set the precision to anything you like by pressing @kbd{p},
2217 then entering a suitable number. Try pressing @kbd{p 30 @key{RET}},
2218 then doing @kbd{1 @key{RET} 7 /} again:
2223 2: 0.142857142857142857142857142857
2228 Although the precision can be set arbitrarily high, Calc always
2229 has to have @emph{some} value for the current precision. After
2230 all, the true value @expr{1/7} is an infinitely repeating decimal;
2231 Calc has to stop somewhere.
2233 Of course, calculations are slower the more digits you request.
2234 Press @w{@kbd{p 12}} now to set the precision back down to the default.
2236 Calculations always use the current precision. For example, even
2237 though we have a 30-digit value for @expr{1/7} on the stack, if
2238 we use it in a calculation in 12-digit mode it will be rounded
2239 down to 12 digits before it is used. Try it; press @key{RET} to
2240 duplicate the number, then @w{@kbd{1 +}}. Notice that the @key{RET}
2241 key didn't round the number, because it doesn't do any calculation.
2242 But the instant we pressed @kbd{+}, the number was rounded down.
2247 2: 0.142857142857142857142857142857
2254 In fact, since we added a digit on the left, we had to lose one
2255 digit on the right from even the 12-digit value of @expr{1/7}.
2257 How did we get more than 12 digits when we computed @samp{2^3^4}? The
2258 answer is that Calc makes a distinction between @dfn{integers} and
2259 @dfn{floating-point} numbers, or @dfn{floats}. An integer is a number
2260 that does not contain a decimal point. There is no such thing as an
2261 ``infinitely repeating fraction integer,'' so Calc doesn't have to limit
2262 itself. If you asked for @samp{2^10000} (don't try this!), you would
2263 have to wait a long time but you would eventually get an exact answer.
2264 If you ask for @samp{2.^10000}, you will quickly get an answer which is
2265 correct only to 12 places. The decimal point tells Calc that it should
2266 use floating-point arithmetic to get the answer, not exact integer
2269 You can use the @kbd{F} (@code{calc-floor}) command to convert a
2270 floating-point value to an integer, and @kbd{c f} (@code{calc-float})
2271 to convert an integer to floating-point form.
2273 Let's try entering that last calculation:
2277 1: 2. 2: 2. 1: 1.99506311689e3010
2281 2.0 @key{RET} 10000 @key{RET} ^
2286 @cindex Scientific notation, entry of
2287 Notice the letter @samp{e} in there. It represents ``times ten to the
2288 power of,'' and is used by Calc automatically whenever writing the
2289 number out fully would introduce more extra zeros than you probably
2290 want to see. You can enter numbers in this notation, too.
2294 1: 2. 2: 2. 1: 1.99506311678e3010
2298 2.0 @key{RET} 1e4 @key{RET} ^
2302 @cindex Round-off errors
2304 Hey, the answer is different! Look closely at the middle columns
2305 of the two examples. In the first, the stack contained the
2306 exact integer @expr{10000}, but in the second it contained
2307 a floating-point value with a decimal point. When you raise a
2308 number to an integer power, Calc uses repeated squaring and
2309 multiplication to get the answer. When you use a floating-point
2310 power, Calc uses logarithms and exponentials. As you can see,
2311 a slight error crept in during one of these methods. Which
2312 one should we trust? Let's raise the precision a bit and find
2317 . 1: 2. 2: 2. 1: 1.995063116880828e3010
2321 p 16 @key{RET} 2. @key{RET} 1e4 ^ p 12 @key{RET}
2326 @cindex Guard digits
2327 Presumably, it doesn't matter whether we do this higher-precision
2328 calculation using an integer or floating-point power, since we
2329 have added enough ``guard digits'' to trust the first 12 digits
2330 no matter what. And the verdict is@dots{} Integer powers were more
2331 accurate; in fact, the result was only off by one unit in the
2334 @cindex Guard digits
2335 Calc does many of its internal calculations to a slightly higher
2336 precision, but it doesn't always bump the precision up enough.
2337 In each case, Calc added about two digits of precision during
2338 its calculation and then rounded back down to 12 digits
2339 afterward. In one case, it was enough; in the other, it
2340 wasn't. If you really need @var{x} digits of precision, it
2341 never hurts to do the calculation with a few extra guard digits.
2343 What if we want guard digits but don't want to look at them?
2344 We can set the @dfn{float format}. Calc supports four major
2345 formats for floating-point numbers, called @dfn{normal},
2346 @dfn{fixed-point}, @dfn{scientific notation}, and @dfn{engineering
2347 notation}. You get them by pressing @w{@kbd{d n}}, @kbd{d f},
2348 @kbd{d s}, and @kbd{d e}, respectively. In each case, you can
2349 supply a numeric prefix argument which says how many digits
2350 should be displayed. As an example, let's put a few numbers
2351 onto the stack and try some different display modes. First,
2352 use @kbd{M-0 @key{DEL}} to clear the stack, then enter the four
2357 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2358 3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000
2359 2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450
2360 1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345
2363 d n M-3 d n d s M-3 d s M-3 d f
2368 Notice that when we typed @kbd{M-3 d n}, the numbers were rounded down
2369 to three significant digits, but then when we typed @kbd{d s} all
2370 five significant figures reappeared. The float format does not
2371 affect how numbers are stored, it only affects how they are
2372 displayed. Only the current precision governs the actual rounding
2373 of numbers in the Calculator's memory.
2375 Engineering notation, not shown here, is like scientific notation
2376 except the exponent (the power-of-ten part) is always adjusted to be
2377 a multiple of three (as in ``kilo,'' ``micro,'' etc.). As a result
2378 there will be one, two, or three digits before the decimal point.
2380 Whenever you change a display-related mode, Calc redraws everything
2381 in the stack. This may be slow if there are many things on the stack,
2382 so Calc allows you to type shift-@kbd{H} before any mode command to
2383 prevent it from updating the stack. Anything Calc displays after the
2384 mode-changing command will appear in the new format.
2388 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2389 3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345.
2390 2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45
2391 1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345
2394 H d s @key{DEL} U @key{TAB} d @key{SPC} d n
2399 Here the @kbd{H d s} command changes to scientific notation but without
2400 updating the screen. Deleting the top stack entry and undoing it back
2401 causes it to show up in the new format; swapping the top two stack
2402 entries reformats both entries. The @kbd{d @key{SPC}} command refreshes the
2403 whole stack. The @kbd{d n} command changes back to the normal float
2404 format; since it doesn't have an @kbd{H} prefix, it also updates all
2405 the stack entries to be in @kbd{d n} format.
2407 Notice that the integer @expr{12345} was not affected by any
2408 of the float formats. Integers are integers, and are always
2411 @cindex Large numbers, readability
2412 Large integers have their own problems. Let's look back at
2413 the result of @kbd{2^3^4}.
2416 2417851639229258349412352
2420 Quick---how many digits does this have? Try typing @kbd{d g}:
2423 2,417,851,639,229,258,349,412,352
2427 Now how many digits does this have? It's much easier to tell!
2428 We can actually group digits into clumps of any size. Some
2429 people prefer @kbd{M-5 d g}:
2432 24178,51639,22925,83494,12352
2435 Let's see what happens to floating-point numbers when they are grouped.
2436 First, type @kbd{p 25 @key{RET}} to make sure we have enough precision
2437 to get ourselves into trouble. Now, type @kbd{1e13 /}:
2440 24,17851,63922.9258349412352
2444 The integer part is grouped but the fractional part isn't. Now try
2445 @kbd{M-- M-5 d g} (that's meta-minus-sign, meta-five):
2448 24,17851,63922.92583,49412,352
2451 If you find it hard to tell the decimal point from the commas, try
2452 changing the grouping character to a space with @kbd{d , @key{SPC}}:
2455 24 17851 63922.92583 49412 352
2458 Type @kbd{d , ,} to restore the normal grouping character, then
2459 @kbd{d g} again to turn grouping off. Also, press @kbd{p 12} to
2460 restore the default precision.
2462 Press @kbd{U} enough times to get the original big integer back.
2463 (Notice that @kbd{U} does not undo each mode-setting command; if
2464 you want to undo a mode-setting command, you have to do it yourself.)
2465 Now, type @kbd{d r 16 @key{RET}}:
2468 16#200000000000000000000
2472 The number is now displayed in @dfn{hexadecimal}, or ``base-16'' form.
2473 Suddenly it looks pretty simple; this should be no surprise, since we
2474 got this number by computing a power of two, and 16 is a power of 2.
2475 In fact, we can use @w{@kbd{d r 2 @key{RET}}} to see it in actual binary
2479 2#1000000000000000000000000000000000000000000000000000000 @dots{}
2483 We don't have enough space here to show all the zeros! They won't
2484 fit on a typical screen, either, so you will have to use horizontal
2485 scrolling to see them all. Press @kbd{<} and @kbd{>} to scroll the
2486 stack window left and right by half its width. Another way to view
2487 something large is to press @kbd{`} (back-quote) to edit the top of
2488 stack in a separate window. (Press @kbd{C-c C-c} when you are done.)
2490 You can enter non-decimal numbers using the @kbd{#} symbol, too.
2491 Let's see what the hexadecimal number @samp{5FE} looks like in
2492 binary. Type @kbd{16#5FE} (the letters can be typed in upper or
2493 lower case; they will always appear in upper case). It will also
2494 help to turn grouping on with @kbd{d g}:
2500 Notice that @kbd{d g} groups by fours by default if the display radix
2501 is binary or hexadecimal, but by threes if it is decimal, octal, or any
2504 Now let's see that number in decimal; type @kbd{d r 10}:
2510 Numbers are not @emph{stored} with any particular radix attached. They're
2511 just numbers; they can be entered in any radix, and are always displayed
2512 in whatever radix you've chosen with @kbd{d r}. The current radix applies
2513 to integers, fractions, and floats.
2515 @cindex Roundoff errors, in non-decimal numbers
2516 (@bullet{}) @strong{Exercise 1.} Your friend Joe tried to enter one-third
2517 as @samp{3#0.1} in @kbd{d r 3} mode with a precision of 12. He got
2518 @samp{3#0.0222222...} (with 25 2's) in the display. When he multiplied
2519 that by three, he got @samp{3#0.222222...} instead of the expected
2520 @samp{3#1}. Next, Joe entered @samp{3#0.2} and, to his great relief,
2521 saw @samp{3#0.2} on the screen. But when he typed @kbd{2 /}, he got
2522 @samp{3#0.10000001} (some zeros omitted). What's going on here?
2523 @xref{Modes Answer 1, 1}. (@bullet{})
2525 @cindex Scientific notation, in non-decimal numbers
2526 (@bullet{}) @strong{Exercise 2.} Scientific notation works in non-decimal
2527 modes in the natural way (the exponent is a power of the radix instead of
2528 a power of ten, although the exponent itself is always written in decimal).
2529 Thus @samp{8#1.23e3 = 8#1230.0}. Suppose we have the hexadecimal number
2530 @samp{f.e8f} times 16 to the 15th power: We write @samp{16#f.e8fe15}.
2531 What is wrong with this picture? What could we write instead that would
2532 work better? @xref{Modes Answer 2, 2}. (@bullet{})
2534 The @kbd{m} prefix key has another set of modes, relating to the way
2535 Calc interprets your inputs and does computations. Whereas @kbd{d}-prefix
2536 modes generally affect the way things look, @kbd{m}-prefix modes affect
2537 the way they are actually computed.
2539 The most popular @kbd{m}-prefix mode is the @dfn{angular mode}. Notice
2540 the @samp{Deg} indicator in the mode line. This means that if you use
2541 a command that interprets a number as an angle, it will assume the
2542 angle is measured in degrees. For example,
2546 1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5
2554 The shift-@kbd{S} command computes the sine of an angle. The sine
2556 @texline @math{\sqrt{2}/2};
2557 @infoline @expr{sqrt(2)/2};
2558 squaring this yields @expr{2/4 = 0.5}. However, there has been a slight
2559 roundoff error because the representation of
2560 @texline @math{\sqrt{2}/2}
2561 @infoline @expr{sqrt(2)/2}
2562 wasn't exact. The @kbd{c 1} command is a handy way to clean up numbers
2563 in this case; it temporarily reduces the precision by one digit while it
2564 re-rounds the number on the top of the stack.
2566 @cindex Roundoff errors, examples
2567 (@bullet{}) @strong{Exercise 3.} Your friend Joe computed the sine
2568 of 45 degrees as shown above, then, hoping to avoid an inexact
2569 result, he increased the precision to 16 digits before squaring.
2570 What happened? @xref{Modes Answer 3, 3}. (@bullet{})
2572 To do this calculation in radians, we would type @kbd{m r} first.
2573 (The indicator changes to @samp{Rad}.) 45 degrees corresponds to
2574 @cpiover{4} radians. To get @cpi{}, press the @kbd{P} key. (Once
2575 again, this is a shifted capital @kbd{P}. Remember, unshifted
2576 @kbd{p} sets the precision.)
2580 1: 3.14159265359 1: 0.785398163398 1: 0.707106781187
2587 Likewise, inverse trigonometric functions generate results in
2588 either radians or degrees, depending on the current angular mode.
2592 1: 0.707106781187 1: 0.785398163398 1: 45.
2595 .5 Q m r I S m d U I S
2600 Here we compute the Inverse Sine of
2601 @texline @math{\sqrt{0.5}},
2602 @infoline @expr{sqrt(0.5)},
2603 first in radians, then in degrees.
2605 Use @kbd{c d} and @kbd{c r} to convert a number from radians to degrees
2610 1: 45 1: 0.785398163397 1: 45.
2617 Another interesting mode is @dfn{Fraction mode}. Normally,
2618 dividing two integers produces a floating-point result if the
2619 quotient can't be expressed as an exact integer. Fraction mode
2620 causes integer division to produce a fraction, i.e., a rational
2625 2: 12 1: 1.33333333333 1: 4:3
2629 12 @key{RET} 9 / m f U / m f
2634 In the first case, we get an approximate floating-point result.
2635 In the second case, we get an exact fractional result (four-thirds).
2637 You can enter a fraction at any time using @kbd{:} notation.
2638 (Calc uses @kbd{:} instead of @kbd{/} as the fraction separator
2639 because @kbd{/} is already used to divide the top two stack
2640 elements.) Calculations involving fractions will always
2641 produce exact fractional results; Fraction mode only says
2642 what to do when dividing two integers.
2644 @cindex Fractions vs. floats
2645 @cindex Floats vs. fractions
2646 (@bullet{}) @strong{Exercise 4.} If fractional arithmetic is exact,
2647 why would you ever use floating-point numbers instead?
2648 @xref{Modes Answer 4, 4}. (@bullet{})
2650 Typing @kbd{m f} doesn't change any existing values in the stack.
2651 In the above example, we had to Undo the division and do it over
2652 again when we changed to Fraction mode. But if you use the
2653 evaluates-to operator you can get commands like @kbd{m f} to
2658 1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3
2661 ' 12/9 => @key{RET} p 4 @key{RET} m f
2666 In this example, the righthand side of the @samp{=>} operator
2667 on the stack is recomputed when we change the precision, then
2668 again when we change to Fraction mode. All @samp{=>} expressions
2669 on the stack are recomputed every time you change any mode that
2670 might affect their values.
2672 @node Arithmetic Tutorial, Vector/Matrix Tutorial, Basic Tutorial, Tutorial
2673 @section Arithmetic Tutorial
2676 In this section, we explore the arithmetic and scientific functions
2677 available in the Calculator.
2679 The standard arithmetic commands are @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/},
2680 and @kbd{^}. Each normally takes two numbers from the top of the stack
2681 and pushes back a result. The @kbd{n} and @kbd{&} keys perform
2682 change-sign and reciprocal operations, respectively.
2686 1: 5 1: 0.2 1: 5. 1: -5. 1: 5.
2693 @cindex Binary operators
2694 You can apply a ``binary operator'' like @kbd{+} across any number of
2695 stack entries by giving it a numeric prefix. You can also apply it
2696 pairwise to several stack elements along with the top one if you use
2701 3: 2 1: 9 3: 2 4: 2 3: 12
2702 2: 3 . 2: 3 3: 3 2: 13
2703 1: 4 1: 4 2: 4 1: 14
2707 2 @key{RET} 3 @key{RET} 4 M-3 + U 10 M-- M-3 +
2711 @cindex Unary operators
2712 You can apply a ``unary operator'' like @kbd{&} to the top @var{n}
2713 stack entries with a numeric prefix, too.
2718 2: 3 2: 0.333333333333 2: 3.
2722 2 @key{RET} 3 @key{RET} 4 M-3 & M-2 &
2726 Notice that the results here are left in floating-point form.
2727 We can convert them back to integers by pressing @kbd{F}, the
2728 ``floor'' function. This function rounds down to the next lower
2729 integer. There is also @kbd{R}, which rounds to the nearest
2747 Since dividing-and-flooring (i.e., ``integer quotient'') is such a
2748 common operation, Calc provides a special command for that purpose, the
2749 backslash @kbd{\}. Another common arithmetic operator is @kbd{%}, which
2750 computes the remainder that would arise from a @kbd{\} operation, i.e.,
2751 the ``modulo'' of two numbers. For example,
2755 2: 1234 1: 12 2: 1234 1: 34
2759 1234 @key{RET} 100 \ U %
2763 These commands actually work for any real numbers, not just integers.
2767 2: 3.1415 1: 3 2: 3.1415 1: 0.1415
2771 3.1415 @key{RET} 1 \ U %
2775 (@bullet{}) @strong{Exercise 1.} The @kbd{\} command would appear to be a
2776 frill, since you could always do the same thing with @kbd{/ F}. Think
2777 of a situation where this is not true---@kbd{/ F} would be inadequate.
2778 Now think of a way you could get around the problem if Calc didn't
2779 provide a @kbd{\} command. @xref{Arithmetic Answer 1, 1}. (@bullet{})
2781 We've already seen the @kbd{Q} (square root) and @kbd{S} (sine)
2782 commands. Other commands along those lines are @kbd{C} (cosine),
2783 @kbd{T} (tangent), @kbd{E} (@expr{e^x}) and @kbd{L} (natural
2784 logarithm). These can be modified by the @kbd{I} (inverse) and
2785 @kbd{H} (hyperbolic) prefix keys.
2787 Let's compute the sine and cosine of an angle, and verify the
2789 @texline @math{\sin^2x + \cos^2x = 1}.
2790 @infoline @expr{sin(x)^2 + cos(x)^2 = 1}.
2791 We'll arbitrarily pick @mathit{-64} degrees as a good value for @expr{x}.
2792 With the angular mode set to degrees (type @w{@kbd{m d}}), do:
2796 2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1.
2797 1: -64 1: -0.89879 1: -64 1: 0.43837 .
2800 64 n @key{RET} @key{RET} S @key{TAB} C f h
2805 (For brevity, we're showing only five digits of the results here.
2806 You can of course do these calculations to any precision you like.)
2808 Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum
2809 of squares, command.
2812 @texline @math{\displaystyle\tan x = {\sin x \over \cos x}}.
2813 @infoline @expr{tan(x) = sin(x) / cos(x)}.
2817 2: -0.89879 1: -2.0503 1: -64.
2825 A physical interpretation of this calculation is that if you move
2826 @expr{0.89879} units downward and @expr{0.43837} units to the right,
2827 your direction of motion is @mathit{-64} degrees from horizontal. Suppose
2828 we move in the opposite direction, up and to the left:
2832 2: -0.89879 2: 0.89879 1: -2.0503 1: -64.
2833 1: 0.43837 1: -0.43837 . .
2841 How can the angle be the same? The answer is that the @kbd{/} operation
2842 loses information about the signs of its inputs. Because the quotient
2843 is negative, we know exactly one of the inputs was negative, but we
2844 can't tell which one. There is an @kbd{f T} [@code{arctan2}] function which
2845 computes the inverse tangent of the quotient of a pair of numbers.
2846 Since you feed it the two original numbers, it has enough information
2847 to give you a full 360-degree answer.
2851 2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180.
2852 1: -0.43837 . 2: -0.89879 1: -64. .
2856 U U f T M-@key{RET} M-2 n f T -
2861 The resulting angles differ by 180 degrees; in other words, they
2862 point in opposite directions, just as we would expect.
2864 The @key{META}-@key{RET} we used in the third step is the
2865 ``last-arguments'' command. It is sort of like Undo, except that it
2866 restores the arguments of the last command to the stack without removing
2867 the command's result. It is useful in situations like this one,
2868 where we need to do several operations on the same inputs. We could
2869 have accomplished the same thing by using @kbd{M-2 @key{RET}} to duplicate
2870 the top two stack elements right after the @kbd{U U}, then a pair of
2871 @kbd{M-@key{TAB}} commands to cycle the 116 up around the duplicates.
2873 A similar identity is supposed to hold for hyperbolic sines and cosines,
2874 except that it is the @emph{difference}
2875 @texline @math{\cosh^2x - \sinh^2x}
2876 @infoline @expr{cosh(x)^2 - sinh(x)^2}
2877 that always equals one. Let's try to verify this identity.
2881 2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54
2882 1: -64 1: -3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54
2885 64 n @key{RET} @key{RET} H C 2 ^ @key{TAB} H S 2 ^
2890 @cindex Roundoff errors, examples
2891 Something's obviously wrong, because when we subtract these numbers
2892 the answer will clearly be zero! But if you think about it, if these
2893 numbers @emph{did} differ by one, it would be in the 55th decimal
2894 place. The difference we seek has been lost entirely to roundoff
2897 We could verify this hypothesis by doing the actual calculation with,
2898 say, 60 decimal places of precision. This will be slow, but not
2899 enormously so. Try it if you wish; sure enough, the answer is
2900 0.99999, reasonably close to 1.
2902 Of course, a more reasonable way to verify the identity is to use
2903 a more reasonable value for @expr{x}!
2905 @cindex Common logarithm
2906 Some Calculator commands use the Hyperbolic prefix for other purposes.
2907 The logarithm and exponential functions, for example, work to the base
2908 @expr{e} normally but use base-10 instead if you use the Hyperbolic
2913 1: 1000 1: 6.9077 1: 1000 1: 3
2921 First, we mistakenly compute a natural logarithm. Then we undo
2922 and compute a common logarithm instead.
2924 The @kbd{B} key computes a general base-@var{b} logarithm for any
2929 2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077
2930 1: 10 . . 1: 2.71828 .
2933 1000 @key{RET} 10 B H E H P B
2938 Here we first use @kbd{B} to compute the base-10 logarithm, then use
2939 the ``hyperbolic'' exponential as a cheap hack to recover the number
2940 1000, then use @kbd{B} again to compute the natural logarithm. Note
2941 that @kbd{P} with the hyperbolic prefix pushes the constant @expr{e}
2944 You may have noticed that both times we took the base-10 logarithm
2945 of 1000, we got an exact integer result. Calc always tries to give
2946 an exact rational result for calculations involving rational numbers
2947 where possible. But when we used @kbd{H E}, the result was a
2948 floating-point number for no apparent reason. In fact, if we had
2949 computed @kbd{10 @key{RET} 3 ^} we @emph{would} have gotten an
2950 exact integer 1000. But the @kbd{H E} command is rigged to generate
2951 a floating-point result all of the time so that @kbd{1000 H E} will
2952 not waste time computing a thousand-digit integer when all you
2953 probably wanted was @samp{1e1000}.
2955 (@bullet{}) @strong{Exercise 2.} Find a pair of integer inputs to
2956 the @kbd{B} command for which Calc could find an exact rational
2957 result but doesn't. @xref{Arithmetic Answer 2, 2}. (@bullet{})
2959 The Calculator also has a set of functions relating to combinatorics
2960 and statistics. You may be familiar with the @dfn{factorial} function,
2961 which computes the product of all the integers up to a given number.
2965 1: 100 1: 93326215443... 1: 100. 1: 9.3326e157
2973 Recall, the @kbd{c f} command converts the integer or fraction at the
2974 top of the stack to floating-point format. If you take the factorial
2975 of a floating-point number, you get a floating-point result
2976 accurate to the current precision. But if you give @kbd{!} an
2977 exact integer, you get an exact integer result (158 digits long
2980 If you take the factorial of a non-integer, Calc uses a generalized
2981 factorial function defined in terms of Euler's Gamma function
2982 @texline @math{\Gamma(n)}
2983 @infoline @expr{gamma(n)}
2984 (which is itself available as the @kbd{f g} command).
2988 3: 4. 3: 24. 1: 5.5 1: 52.342777847
2989 2: 4.5 2: 52.3427777847 . .
2993 M-3 ! M-0 @key{DEL} 5.5 f g
2998 Here we verify the identity
2999 @texline @math{n! = \Gamma(n+1)}.
3000 @infoline @expr{@var{n}!@: = gamma(@var{n}+1)}.
3002 The binomial coefficient @var{n}-choose-@var{m}
3003 @texline or @math{\displaystyle {n \choose m}}
3005 @texline @math{\displaystyle {n! \over m! \, (n-m)!}}
3006 @infoline @expr{n!@: / m!@: (n-m)!}
3007 for all reals @expr{n} and @expr{m}. The intermediate results in this
3008 formula can become quite large even if the final result is small; the
3009 @kbd{k c} command computes a binomial coefficient in a way that avoids
3010 large intermediate values.
3012 The @kbd{k} prefix key defines several common functions out of
3013 combinatorics and number theory. Here we compute the binomial
3014 coefficient 30-choose-20, then determine its prime factorization.
3018 2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29]
3022 30 @key{RET} 20 k c k f
3027 You can verify these prime factors by using @kbd{V R *} to multiply
3028 together the elements of this vector. The result is the original
3032 Suppose a program you are writing needs a hash table with at least
3033 10000 entries. It's best to use a prime number as the actual size
3034 of a hash table. Calc can compute the next prime number after 10000:
3038 1: 10000 1: 10007 1: 9973
3046 Just for kicks we've also computed the next prime @emph{less} than
3049 @c [fix-ref Financial Functions]
3050 @xref{Financial Functions}, for a description of the Calculator
3051 commands that deal with business and financial calculations (functions
3052 like @code{pv}, @code{rate}, and @code{sln}).
3054 @c [fix-ref Binary Number Functions]
3055 @xref{Binary Functions}, to read about the commands for operating
3056 on binary numbers (like @code{and}, @code{xor}, and @code{lsh}).
3058 @node Vector/Matrix Tutorial, Types Tutorial, Arithmetic Tutorial, Tutorial
3059 @section Vector/Matrix Tutorial
3062 A @dfn{vector} is a list of numbers or other Calc data objects.
3063 Calc provides a large set of commands that operate on vectors. Some
3064 are familiar operations from vector analysis. Others simply treat
3065 a vector as a list of objects.
3068 * Vector Analysis Tutorial::
3073 @node Vector Analysis Tutorial, Matrix Tutorial, Vector/Matrix Tutorial, Vector/Matrix Tutorial
3074 @subsection Vector Analysis
3077 If you add two vectors, the result is a vector of the sums of the
3078 elements, taken pairwise.
3082 1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3]
3086 [1,2,3] s 1 [7 6 0] s 2 +
3091 Note that we can separate the vector elements with either commas or
3092 spaces. This is true whether we are using incomplete vectors or
3093 algebraic entry. The @kbd{s 1} and @kbd{s 2} commands save these
3094 vectors so we can easily reuse them later.
3096 If you multiply two vectors, the result is the sum of the products
3097 of the elements taken pairwise. This is called the @dfn{dot product}
3111 The dot product of two vectors is equal to the product of their
3112 lengths times the cosine of the angle between them. (Here the vector
3113 is interpreted as a line from the origin @expr{(0,0,0)} to the
3114 specified point in three-dimensional space.) The @kbd{A}
3115 (absolute value) command can be used to compute the length of a
3120 3: 19 3: 19 1: 0.550782 1: 56.579
3121 2: [1, 2, 3] 2: 3.741657 . .
3122 1: [7, 6, 0] 1: 9.219544
3125 M-@key{RET} M-2 A * / I C
3130 First we recall the arguments to the dot product command, then
3131 we compute the absolute values of the top two stack entries to
3132 obtain the lengths of the vectors, then we divide the dot product
3133 by the product of the lengths to get the cosine of the angle.
3134 The inverse cosine finds that the angle between the vectors
3135 is about 56 degrees.
3137 @cindex Cross product
3138 @cindex Perpendicular vectors
3139 The @dfn{cross product} of two vectors is a vector whose length
3140 is the product of the lengths of the inputs times the sine of the
3141 angle between them, and whose direction is perpendicular to both
3142 input vectors. Unlike the dot product, the cross product is
3143 defined only for three-dimensional vectors. Let's double-check
3144 our computation of the angle using the cross product.
3148 2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579
3149 1: [7, 6, 0] 2: [1, 2, 3] . .
3153 r 1 r 2 V C s 3 M-@key{RET} M-2 A * / A I S
3158 First we recall the original vectors and compute their cross product,
3159 which we also store for later reference. Now we divide the vector
3160 by the product of the lengths of the original vectors. The length of
3161 this vector should be the sine of the angle; sure enough, it is!
3163 @c [fix-ref General Mode Commands]
3164 Vector-related commands generally begin with the @kbd{v} prefix key.
3165 Some are uppercase letters and some are lowercase. To make it easier
3166 to type these commands, the shift-@kbd{V} prefix key acts the same as
3167 the @kbd{v} key. (@xref{General Mode Commands}, for a way to make all
3168 prefix keys have this property.)
3170 If we take the dot product of two perpendicular vectors we expect
3171 to get zero, since the cosine of 90 degrees is zero. Let's check
3172 that the cross product is indeed perpendicular to both inputs:
3176 2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0
3177 1: [-18, 21, -8] . 1: [-18, 21, -8] .
3180 r 1 r 3 * @key{DEL} r 2 r 3 *
3184 @cindex Normalizing a vector
3185 @cindex Unit vectors
3186 (@bullet{}) @strong{Exercise 1.} Given a vector on the top of the
3187 stack, what keystrokes would you use to @dfn{normalize} the
3188 vector, i.e., to reduce its length to one without changing its
3189 direction? @xref{Vector Answer 1, 1}. (@bullet{})
3191 (@bullet{}) @strong{Exercise 2.} Suppose a certain particle can be
3192 at any of several positions along a ruler. You have a list of
3193 those positions in the form of a vector, and another list of the
3194 probabilities for the particle to be at the corresponding positions.
3195 Find the average position of the particle.
3196 @xref{Vector Answer 2, 2}. (@bullet{})
3198 @node Matrix Tutorial, List Tutorial, Vector Analysis Tutorial, Vector/Matrix Tutorial
3199 @subsection Matrices
3202 A @dfn{matrix} is just a vector of vectors, all the same length.
3203 This means you can enter a matrix using nested brackets. You can
3204 also use the semicolon character to enter a matrix. We'll show
3209 1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3210 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3213 [[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6] @key{RET}
3218 We'll be using this matrix again, so type @kbd{s 4} to save it now.
3220 Note that semicolons work with incomplete vectors, but they work
3221 better in algebraic entry. That's why we use the apostrophe in
3224 When two matrices are multiplied, the lefthand matrix must have
3225 the same number of columns as the righthand matrix has rows.
3226 Row @expr{i}, column @expr{j} of the result is effectively the
3227 dot product of row @expr{i} of the left matrix by column @expr{j}
3228 of the right matrix.
3230 If we try to duplicate this matrix and multiply it by itself,
3231 the dimensions are wrong and the multiplication cannot take place:
3235 1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ]
3236 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3244 Though rather hard to read, this is a formula which shows the product
3245 of two matrices. The @samp{*} function, having invalid arguments, has
3246 been left in symbolic form.
3248 We can multiply the matrices if we @dfn{transpose} one of them first.
3252 2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ]
3253 [ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ]
3254 1: [ [ 1, 4 ] . [ 27, 36, 45 ] ]
3259 U v t * U @key{TAB} *
3263 Matrix multiplication is not commutative; indeed, switching the
3264 order of the operands can even change the dimensions of the result
3265 matrix, as happened here!
3267 If you multiply a plain vector by a matrix, it is treated as a
3268 single row or column depending on which side of the matrix it is
3269 on. The result is a plain vector which should also be interpreted
3270 as a row or column as appropriate.
3274 2: [ [ 1, 2, 3 ] 1: [14, 32]
3283 Multiplying in the other order wouldn't work because the number of
3284 rows in the matrix is different from the number of elements in the
3287 (@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows
3289 @texline @math{2\times3}
3291 matrix to get @expr{[6, 15]}. Now use @samp{*} to sum along the columns
3292 to get @expr{[5, 7, 9]}.
3293 @xref{Matrix Answer 1, 1}. (@bullet{})
3295 @cindex Identity matrix
3296 An @dfn{identity matrix} is a square matrix with ones along the
3297 diagonal and zeros elsewhere. It has the property that multiplication
3298 by an identity matrix, on the left or on the right, always produces
3299 the original matrix.
3303 1: [ [ 1, 2, 3 ] 2: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3304 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3305 . 1: [ [ 1, 0, 0 ] .
3310 r 4 v i 3 @key{RET} *
3314 If a matrix is square, it is often possible to find its @dfn{inverse},
3315 that is, a matrix which, when multiplied by the original matrix, yields
3316 an identity matrix. The @kbd{&} (reciprocal) key also computes the
3317 inverse of a matrix.
3321 1: [ [ 1, 2, 3 ] 1: [ [ -2.4, 1.2, -0.2 ]
3322 [ 4, 5, 6 ] [ 2.8, -1.4, 0.4 ]
3323 [ 7, 6, 0 ] ] [ -0.73333, 0.53333, -0.2 ] ]
3331 The vertical bar @kbd{|} @dfn{concatenates} numbers, vectors, and
3332 matrices together. Here we have used it to add a new row onto
3333 our matrix to make it square.
3335 We can multiply these two matrices in either order to get an identity.
3339 1: [ [ 1., 0., 0. ] 1: [ [ 1., 0., 0. ]
3340 [ 0., 1., 0. ] [ 0., 1., 0. ]
3341 [ 0., 0., 1. ] ] [ 0., 0., 1. ] ]
3344 M-@key{RET} * U @key{TAB} *
3348 @cindex Systems of linear equations
3349 @cindex Linear equations, systems of
3350 Matrix inverses are related to systems of linear equations in algebra.
3351 Suppose we had the following set of equations:
3364 $$ \openup1\jot \tabskip=0pt plus1fil
3365 \halign to\displaywidth{\tabskip=0pt
3366 $\hfil#$&$\hfil{}#{}$&
3367 $\hfil#$&$\hfil{}#{}$&
3368 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3377 This can be cast into the matrix equation,
3382 [ [ 1, 2, 3 ] [ [ a ] [ [ 6 ]
3383 [ 4, 5, 6 ] * [ b ] = [ 2 ]
3384 [ 7, 6, 0 ] ] [ c ] ] [ 3 ] ]
3390 $$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
3392 \pmatrix{ a \cr b \cr c } = \pmatrix{ 6 \cr 2 \cr 3 }
3397 We can solve this system of equations by multiplying both sides by the
3398 inverse of the matrix. Calc can do this all in one step:
3402 2: [6, 2, 3] 1: [-12.6, 15.2, -3.93333]
3413 The result is the @expr{[a, b, c]} vector that solves the equations.
3414 (Dividing by a square matrix is equivalent to multiplying by its
3417 Let's verify this solution:
3421 2: [ [ 1, 2, 3 ] 1: [6., 2., 3.]
3424 1: [-12.6, 15.2, -3.93333]
3432 Note that we had to be careful about the order in which we multiplied
3433 the matrix and vector. If we multiplied in the other order, Calc would
3434 assume the vector was a row vector in order to make the dimensions
3435 come out right, and the answer would be incorrect. If you
3436 don't feel safe letting Calc take either interpretation of your
3437 vectors, use explicit
3438 @texline @math{N\times1}
3441 @texline @math{1\times N}
3443 matrices instead. In this case, you would enter the original column
3444 vector as @samp{[[6], [2], [3]]} or @samp{[6; 2; 3]}.
3446 (@bullet{}) @strong{Exercise 2.} Algebraic entry allows you to make
3447 vectors and matrices that include variables. Solve the following
3448 system of equations to get expressions for @expr{x} and @expr{y}
3449 in terms of @expr{a} and @expr{b}.
3461 $$ \eqalign{ x &+ a y = 6 \cr
3468 @xref{Matrix Answer 2, 2}. (@bullet{})
3470 @cindex Least-squares for over-determined systems
3471 @cindex Over-determined systems of equations
3472 (@bullet{}) @strong{Exercise 3.} A system of equations is ``over-determined''
3473 if it has more equations than variables. It is often the case that
3474 there are no values for the variables that will satisfy all the
3475 equations at once, but it is still useful to find a set of values
3476 which ``nearly'' satisfy all the equations. In terms of matrix equations,
3477 you can't solve @expr{A X = B} directly because the matrix @expr{A}
3478 is not square for an over-determined system. Matrix inversion works
3479 only for square matrices. One common trick is to multiply both sides
3480 on the left by the transpose of @expr{A}:
3482 @samp{trn(A)*A*X = trn(A)*B}.
3485 $A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
3488 @texline @math{A^T A}
3489 @infoline @expr{trn(A)*A}
3490 is a square matrix so a solution is possible. It turns out that the
3491 @expr{X} vector you compute in this way will be a ``least-squares''
3492 solution, which can be regarded as the ``closest'' solution to the set
3493 of equations. Use Calc to solve the following over-determined
3508 $$ \openup1\jot \tabskip=0pt plus1fil
3509 \halign to\displaywidth{\tabskip=0pt
3510 $\hfil#$&$\hfil{}#{}$&
3511 $\hfil#$&$\hfil{}#{}$&
3512 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3516 2a&+&4b&+&6c&=11 \cr}
3522 @xref{Matrix Answer 3, 3}. (@bullet{})
3524 @node List Tutorial, , Matrix Tutorial, Vector/Matrix Tutorial
3525 @subsection Vectors as Lists
3529 Although Calc has a number of features for manipulating vectors and
3530 matrices as mathematical objects, you can also treat vectors as
3531 simple lists of values. For example, we saw that the @kbd{k f}
3532 command returns a vector which is a list of the prime factors of a
3535 You can pack and unpack stack entries into vectors:
3539 3: 10 1: [10, 20, 30] 3: 10
3548 You can also build vectors out of consecutive integers, or out
3549 of many copies of a given value:
3553 1: [1, 2, 3, 4] 2: [1, 2, 3, 4] 2: [1, 2, 3, 4]
3554 . 1: 17 1: [17, 17, 17, 17]
3557 v x 4 @key{RET} 17 v b 4 @key{RET}
3561 You can apply an operator to every element of a vector using the
3566 1: [17, 34, 51, 68] 1: [289, 1156, 2601, 4624] 1: [17, 34, 51, 68]
3574 In the first step, we multiply the vector of integers by the vector
3575 of 17's elementwise. In the second step, we raise each element to
3576 the power two. (The general rule is that both operands must be
3577 vectors of the same length, or else one must be a vector and the
3578 other a plain number.) In the final step, we take the square root
3581 (@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two
3583 @texline @math{2^{-4}}
3584 @infoline @expr{2^-4}
3585 to @expr{2^4}. @xref{List Answer 1, 1}. (@bullet{})
3587 You can also @dfn{reduce} a binary operator across a vector.
3588 For example, reducing @samp{*} computes the product of all the
3589 elements in the vector:
3593 1: 123123 1: [3, 7, 11, 13, 41] 1: 123123
3601 In this example, we decompose 123123 into its prime factors, then
3602 multiply those factors together again to yield the original number.
3604 We could compute a dot product ``by hand'' using mapping and
3609 2: [1, 2, 3] 1: [7, 12, 0] 1: 19
3618 Recalling two vectors from the previous section, we compute the
3619 sum of pairwise products of the elements to get the same answer
3620 for the dot product as before.
3622 A slight variant of vector reduction is the @dfn{accumulate} operation,
3623 @kbd{V U}. This produces a vector of the intermediate results from
3624 a corresponding reduction. Here we compute a table of factorials:
3628 1: [1, 2, 3, 4, 5, 6] 1: [1, 2, 6, 24, 120, 720]
3631 v x 6 @key{RET} V U *
3635 Calc allows vectors to grow as large as you like, although it gets
3636 rather slow if vectors have more than about a hundred elements.
3637 Actually, most of the time is spent formatting these large vectors
3638 for display, not calculating on them. Try the following experiment
3639 (if your computer is very fast you may need to substitute a larger
3644 1: [1, 2, 3, 4, ... 1: [2, 3, 4, 5, ...
3647 v x 500 @key{RET} 1 V M +
3651 Now press @kbd{v .} (the letter @kbd{v}, then a period) and try the
3652 experiment again. In @kbd{v .} mode, long vectors are displayed
3653 ``abbreviated'' like this:
3657 1: [1, 2, 3, ..., 500] 1: [2, 3, 4, ..., 501]
3660 v x 500 @key{RET} 1 V M +
3665 (where now the @samp{...} is actually part of the Calc display).
3666 You will find both operations are now much faster. But notice that
3667 even in @w{@kbd{v .}} mode, the full vectors are still shown in the Trail.
3668 Type @w{@kbd{t .}} to cause the trail to abbreviate as well, and try the
3669 experiment one more time. Operations on long vectors are now quite
3670 fast! (But of course if you use @kbd{t .} you will lose the ability
3671 to get old vectors back using the @kbd{t y} command.)
3673 An easy way to view a full vector when @kbd{v .} mode is active is
3674 to press @kbd{`} (back-quote) to edit the vector; editing always works
3675 with the full, unabbreviated value.
3677 @cindex Least-squares for fitting a straight line
3678 @cindex Fitting data to a line
3679 @cindex Line, fitting data to
3680 @cindex Data, extracting from buffers
3681 @cindex Columns of data, extracting
3682 As a larger example, let's try to fit a straight line to some data,
3683 using the method of least squares. (Calc has a built-in command for
3684 least-squares curve fitting, but we'll do it by hand here just to
3685 practice working with vectors.) Suppose we have the following list
3686 of values in a file we have loaded into Emacs:
3713 If you are reading this tutorial in printed form, you will find it
3714 easiest to press @kbd{C-x * i} to enter the on-line Info version of
3715 the manual and find this table there. (Press @kbd{g}, then type
3716 @kbd{List Tutorial}, to jump straight to this section.)
3718 Position the cursor at the upper-left corner of this table, just
3719 to the left of the @expr{1.34}. Press @kbd{C-@@} to set the mark.
3720 (On your system this may be @kbd{C-2}, @kbd{C-@key{SPC}}, or @kbd{NUL}.)
3721 Now position the cursor to the lower-right, just after the @expr{1.354}.
3722 You have now defined this region as an Emacs ``rectangle.'' Still
3723 in the Info buffer, type @kbd{C-x * r}. This command
3724 (@code{calc-grab-rectangle}) will pop you back into the Calculator, with
3725 the contents of the rectangle you specified in the form of a matrix.
3729 1: [ [ 1.34, 0.234 ]
3736 (You may wish to use @kbd{v .} mode to abbreviate the display of this
3739 We want to treat this as a pair of lists. The first step is to
3740 transpose this matrix into a pair of rows. Remember, a matrix is
3741 just a vector of vectors. So we can unpack the matrix into a pair
3742 of row vectors on the stack.
3746 1: [ [ 1.34, 1.41, 1.49, ... ] 2: [1.34, 1.41, 1.49, ... ]
3747 [ 0.234, 0.298, 0.402, ... ] ] 1: [0.234, 0.298, 0.402, ... ]
3755 Let's store these in quick variables 1 and 2, respectively.
3759 1: [1.34, 1.41, 1.49, ... ] .
3767 (Recall that @kbd{t 2} is a variant of @kbd{s 2} that removes the
3768 stored value from the stack.)
3770 In a least squares fit, the slope @expr{m} is given by the formula
3774 m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
3779 $$ m = {N \sum x y - \sum x \sum y \over
3780 N \sum x^2 - \left( \sum x \right)^2} $$
3786 @texline @math{\sum x}
3787 @infoline @expr{sum(x)}
3788 represents the sum of all the values of @expr{x}. While there is an
3789 actual @code{sum} function in Calc, it's easier to sum a vector using a
3790 simple reduction. First, let's compute the four different sums that
3798 r 1 V R + t 3 r 1 2 V M ^ V R + t 4
3805 1: 13.613 1: 33.36554
3808 r 2 V R + t 5 r 1 r 2 V M * V R + t 6
3814 These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
3815 respectively. (We could have used @kbd{*} to compute @samp{sum(x^2)} and
3819 These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
3820 respectively. (We could have used \kbd{*} to compute $\sum x^2$ and
3824 Finally, we also need @expr{N}, the number of data points. This is just
3825 the length of either of our lists.
3837 (That's @kbd{v} followed by a lower-case @kbd{l}.)
3839 Now we grind through the formula:
3843 1: 633.94526 2: 633.94526 1: 67.23607
3847 r 7 r 6 * r 3 r 5 * -
3854 2: 67.23607 3: 67.23607 2: 67.23607 1: 0.52141679
3855 1: 1862.0057 2: 1862.0057 1: 128.9488 .
3859 r 7 r 4 * r 3 2 ^ - / t 8
3863 That gives us the slope @expr{m}. The y-intercept @expr{b} can now
3864 be found with the simple formula,
3868 b = (sum(y) - m sum(x)) / N
3873 $$ b = {\sum y - m \sum x \over N} $$
3880 1: 13.613 2: 13.613 1: -8.09358 1: -0.425978
3884 r 5 r 8 r 3 * - r 7 / t 9
3888 Let's ``plot'' this straight line approximation,
3889 @texline @math{y \approx m x + b},
3890 @infoline @expr{m x + b},
3891 and compare it with the original data.
3895 1: [0.699, 0.735, ... ] 1: [0.273, 0.309, ... ]
3903 Notice that multiplying a vector by a constant, and adding a constant
3904 to a vector, can be done without mapping commands since these are
3905 common operations from vector algebra. As far as Calc is concerned,
3906 we've just been doing geometry in 19-dimensional space!
3908 We can subtract this vector from our original @expr{y} vector to get
3909 a feel for the error of our fit. Let's find the maximum error:
3913 1: [0.0387, 0.0112, ... ] 1: [0.0387, 0.0112, ... ] 1: 0.0897
3921 First we compute a vector of differences, then we take the absolute
3922 values of these differences, then we reduce the @code{max} function
3923 across the vector. (The @code{max} function is on the two-key sequence
3924 @kbd{f x}; because it is so common to use @code{max} in a vector
3925 operation, the letters @kbd{X} and @kbd{N} are also accepted for
3926 @code{max} and @code{min} in this context. In general, you answer
3927 the @kbd{V M} or @kbd{V R} prompt with the actual key sequence that
3928 invokes the function you want. You could have typed @kbd{V R f x} or
3929 even @kbd{V R x max @key{RET}} if you had preferred.)
3931 If your system has the GNUPLOT program, you can see graphs of your
3932 data and your straight line to see how well they match. (If you have
3933 GNUPLOT 3.0 or higher, the following instructions will work regardless
3934 of the kind of display you have. Some GNUPLOT 2.0, non-X-windows systems
3935 may require additional steps to view the graphs.)
3937 Let's start by plotting the original data. Recall the ``@var{x}'' and ``@var{y}''
3938 vectors onto the stack and press @kbd{g f}. This ``fast'' graphing
3939 command does everything you need to do for simple, straightforward
3944 2: [1.34, 1.41, 1.49, ... ]
3945 1: [0.234, 0.298, 0.402, ... ]
3952 If all goes well, you will shortly get a new window containing a graph
3953 of the data. (If not, contact your GNUPLOT or Calc installer to find
3954 out what went wrong.) In the X window system, this will be a separate
3955 graphics window. For other kinds of displays, the default is to
3956 display the graph in Emacs itself using rough character graphics.
3957 Press @kbd{q} when you are done viewing the character graphics.
3959 Next, let's add the line we got from our least-squares fit.
3961 (If you are reading this tutorial on-line while running Calc, typing
3962 @kbd{g a} may cause the tutorial to disappear from its window and be
3963 replaced by a buffer named @samp{*Gnuplot Commands*}. The tutorial
3964 will reappear when you terminate GNUPLOT by typing @kbd{g q}.)
3969 2: [1.34, 1.41, 1.49, ... ]
3970 1: [0.273, 0.309, 0.351, ... ]
3973 @key{DEL} r 0 g a g p
3977 It's not very useful to get symbols to mark the data points on this
3978 second curve; you can type @kbd{g S g p} to remove them. Type @kbd{g q}
3979 when you are done to remove the X graphics window and terminate GNUPLOT.
3981 (@bullet{}) @strong{Exercise 2.} An earlier exercise showed how to do
3982 least squares fitting to a general system of equations. Our 19 data
3983 points are really 19 equations of the form @expr{y_i = m x_i + b} for
3984 different pairs of @expr{(x_i,y_i)}. Use the matrix-transpose method
3985 to solve for @expr{m} and @expr{b}, duplicating the above result.
3986 @xref{List Answer 2, 2}. (@bullet{})
3988 @cindex Geometric mean
3989 (@bullet{}) @strong{Exercise 3.} If the input data do not form a
3990 rectangle, you can use @w{@kbd{C-x * g}} (@code{calc-grab-region})
3991 to grab the data the way Emacs normally works with regions---it reads
3992 left-to-right, top-to-bottom, treating line breaks the same as spaces.
3993 Use this command to find the geometric mean of the following numbers.
3994 (The geometric mean is the @var{n}th root of the product of @var{n} numbers.)
4003 The @kbd{C-x * g} command accepts numbers separated by spaces or commas,
4004 with or without surrounding vector brackets.
4005 @xref{List Answer 3, 3}. (@bullet{})
4008 As another example, a theorem about binomial coefficients tells
4009 us that the alternating sum of binomial coefficients
4010 @var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so
4011 on up to @var{n}-choose-@var{n},
4012 always comes out to zero. Let's verify this
4016 As another example, a theorem about binomial coefficients tells
4017 us that the alternating sum of binomial coefficients
4018 ${n \choose 0} - {n \choose 1} + {n \choose 2} - \cdots \pm {n \choose n}$
4019 always comes out to zero. Let's verify this
4025 1: [1, 2, 3, 4, 5, 6, 7] 1: [0, 1, 2, 3, 4, 5, 6]
4035 1: [1, -6, 15, -20, 15, -6, 1] 1: 0
4038 V M ' (-1)^$ choose(6,$) @key{RET} V R +
4042 The @kbd{V M '} command prompts you to enter any algebraic expression
4043 to define the function to map over the vector. The symbol @samp{$}
4044 inside this expression represents the argument to the function.
4045 The Calculator applies this formula to each element of the vector,
4046 substituting each element's value for the @samp{$} sign(s) in turn.
4048 To define a two-argument function, use @samp{$$} for the first
4049 argument and @samp{$} for the second: @kbd{V M ' $$-$ @key{RET}} is
4050 equivalent to @kbd{V M -}. This is analogous to regular algebraic
4051 entry, where @samp{$$} would refer to the next-to-top stack entry
4052 and @samp{$} would refer to the top stack entry, and @kbd{' $$-$ @key{RET}}
4053 would act exactly like @kbd{-}.
4055 Notice that the @kbd{V M '} command has recorded two things in the
4056 trail: The result, as usual, and also a funny-looking thing marked
4057 @samp{oper} that represents the operator function you typed in.
4058 The function is enclosed in @samp{< >} brackets, and the argument is
4059 denoted by a @samp{#} sign. If there were several arguments, they
4060 would be shown as @samp{#1}, @samp{#2}, and so on. (For example,
4061 @kbd{V M ' $$-$} will put the function @samp{<#1 - #2>} on the
4062 trail.) This object is a ``nameless function''; you can use nameless
4063 @w{@samp{< >}} notation to answer the @kbd{V M '} prompt if you like.
4064 Nameless function notation has the interesting, occasionally useful
4065 property that a nameless function is not actually evaluated until
4066 it is used. For example, @kbd{V M ' $+random(2.0)} evaluates
4067 @samp{random(2.0)} once and adds that random number to all elements
4068 of the vector, but @kbd{V M ' <#+random(2.0)>} evaluates the
4069 @samp{random(2.0)} separately for each vector element.
4071 Another group of operators that are often useful with @kbd{V M} are
4072 the relational operators: @kbd{a =}, for example, compares two numbers
4073 and gives the result 1 if they are equal, or 0 if not. Similarly,
4074 @w{@kbd{a <}} checks for one number being less than another.
4076 Other useful vector operations include @kbd{v v}, to reverse a
4077 vector end-for-end; @kbd{V S}, to sort the elements of a vector
4078 into increasing order; and @kbd{v r} and @w{@kbd{v c}}, to extract
4079 one row or column of a matrix, or (in both cases) to extract one
4080 element of a plain vector. With a negative argument, @kbd{v r}
4081 and @kbd{v c} instead delete one row, column, or vector element.
4083 @cindex Divisor functions
4084 (@bullet{}) @strong{Exercise 4.} The @expr{k}th @dfn{divisor function}
4088 is the sum of the @expr{k}th powers of all the divisors of an
4089 integer @expr{n}. Figure out a method for computing the divisor
4090 function for reasonably small values of @expr{n}. As a test,
4091 the 0th and 1st divisor functions of 30 are 8 and 72, respectively.
4092 @xref{List Answer 4, 4}. (@bullet{})
4094 @cindex Square-free numbers
4095 @cindex Duplicate values in a list
4096 (@bullet{}) @strong{Exercise 5.} The @kbd{k f} command produces a
4097 list of prime factors for a number. Sometimes it is important to
4098 know that a number is @dfn{square-free}, i.e., that no prime occurs
4099 more than once in its list of prime factors. Find a sequence of
4100 keystrokes to tell if a number is square-free; your method should
4101 leave 1 on the stack if it is, or 0 if it isn't.
4102 @xref{List Answer 5, 5}. (@bullet{})
4104 @cindex Triangular lists
4105 (@bullet{}) @strong{Exercise 6.} Build a list of lists that looks
4106 like the following diagram. (You may wish to use the @kbd{v /}
4107 command to enable multi-line display of vectors.)
4116 [1, 2, 3, 4, 5, 6] ]
4121 @xref{List Answer 6, 6}. (@bullet{})
4123 (@bullet{}) @strong{Exercise 7.} Build the following list of lists.
4131 [10, 11, 12, 13, 14],
4132 [15, 16, 17, 18, 19, 20] ]
4137 @xref{List Answer 7, 7}. (@bullet{})
4139 @cindex Maximizing a function over a list of values
4140 @c [fix-ref Numerical Solutions]
4141 (@bullet{}) @strong{Exercise 8.} Compute a list of values of Bessel's
4142 @texline @math{J_1(x)}
4144 function @samp{besJ(1,x)} for @expr{x} from 0 to 5 in steps of 0.25.
4145 Find the value of @expr{x} (from among the above set of values) for
4146 which @samp{besJ(1,x)} is a maximum. Use an ``automatic'' method,
4147 i.e., just reading along the list by hand to find the largest value
4148 is not allowed! (There is an @kbd{a X} command which does this kind
4149 of thing automatically; @pxref{Numerical Solutions}.)
4150 @xref{List Answer 8, 8}. (@bullet{})
4152 @cindex Digits, vectors of
4153 (@bullet{}) @strong{Exercise 9.} You are given an integer in the range
4154 @texline @math{0 \le N < 10^m}
4155 @infoline @expr{0 <= N < 10^m}
4156 for @expr{m=12} (i.e., an integer of less than
4157 twelve digits). Convert this integer into a vector of @expr{m}
4158 digits, each in the range from 0 to 9. In vector-of-digits notation,
4159 add one to this integer to produce a vector of @expr{m+1} digits
4160 (since there could be a carry out of the most significant digit).
4161 Convert this vector back into a regular integer. A good integer
4162 to try is 25129925999. @xref{List Answer 9, 9}. (@bullet{})
4164 (@bullet{}) @strong{Exercise 10.} Your friend Joe tried to use
4165 @kbd{V R a =} to test if all numbers in a list were equal. What
4166 happened? How would you do this test? @xref{List Answer 10, 10}. (@bullet{})
4168 (@bullet{}) @strong{Exercise 11.} The area of a circle of radius one
4169 is @cpi{}. The area of the
4170 @texline @math{2\times2}
4172 square that encloses that circle is 4. So if we throw @var{n} darts at
4173 random points in the square, about @cpiover{4} of them will land inside
4174 the circle. This gives us an entertaining way to estimate the value of
4175 @cpi{}. The @w{@kbd{k r}}
4176 command picks a random number between zero and the value on the stack.
4177 We could get a random floating-point number between @mathit{-1} and 1 by typing
4178 @w{@kbd{2.0 k r 1 -}}. Build a vector of 100 random @expr{(x,y)} points in
4179 this square, then use vector mapping and reduction to count how many
4180 points lie inside the unit circle. Hint: Use the @kbd{v b} command.
4181 @xref{List Answer 11, 11}. (@bullet{})
4183 @cindex Matchstick problem
4184 (@bullet{}) @strong{Exercise 12.} The @dfn{matchstick problem} provides
4185 another way to calculate @cpi{}. Say you have an infinite field
4186 of vertical lines with a spacing of one inch. Toss a one-inch matchstick
4187 onto the field. The probability that the matchstick will land crossing
4188 a line turns out to be
4189 @texline @math{2/\pi}.
4190 @infoline @expr{2/pi}.
4191 Toss 100 matchsticks to estimate @cpi{}. (If you want still more fun,
4192 the probability that the GCD (@w{@kbd{k g}}) of two large integers is
4194 @texline @math{6/\pi^2}.
4195 @infoline @expr{6/pi^2}.
4196 That provides yet another way to estimate @cpi{}.)
4197 @xref{List Answer 12, 12}. (@bullet{})
4199 (@bullet{}) @strong{Exercise 13.} An algebraic entry of a string in
4200 double-quote marks, @samp{"hello"}, creates a vector of the numerical
4201 (ASCII) codes of the characters (here, @expr{[104, 101, 108, 108, 111]}).
4202 Sometimes it is convenient to compute a @dfn{hash code} of a string,
4203 which is just an integer that represents the value of that string.
4204 Two equal strings have the same hash code; two different strings
4205 @dfn{probably} have different hash codes. (For example, Calc has
4206 over 400 function names, but Emacs can quickly find the definition for
4207 any given name because it has sorted the functions into ``buckets'' by
4208 their hash codes. Sometimes a few names will hash into the same bucket,
4209 but it is easier to search among a few names than among all the names.)
4210 One popular hash function is computed as follows: First set @expr{h = 0}.
4211 Then, for each character from the string in turn, set @expr{h = 3h + c_i}
4212 where @expr{c_i} is the character's ASCII code. If we have 511 buckets,
4213 we then take the hash code modulo 511 to get the bucket number. Develop a
4214 simple command or commands for converting string vectors into hash codes.
4215 The hash code for @samp{"Testing, 1, 2, 3"} is 1960915098, which modulo
4216 511 is 121. @xref{List Answer 13, 13}. (@bullet{})
4218 (@bullet{}) @strong{Exercise 14.} The @kbd{H V R} and @kbd{H V U}
4219 commands do nested function evaluations. @kbd{H V U} takes a starting
4220 value and a number of steps @var{n} from the stack; it then applies the
4221 function you give to the starting value 0, 1, 2, up to @var{n} times
4222 and returns a vector of the results. Use this command to create a
4223 ``random walk'' of 50 steps. Start with the two-dimensional point
4224 @expr{(0,0)}; then take one step a random distance between @mathit{-1} and 1
4225 in both @expr{x} and @expr{y}; then take another step, and so on. Use the
4226 @kbd{g f} command to display this random walk. Now modify your random
4227 walk to walk a unit distance, but in a random direction, at each step.
4228 (Hint: The @code{sincos} function returns a vector of the cosine and
4229 sine of an angle.) @xref{List Answer 14, 14}. (@bullet{})
4231 @node Types Tutorial, Algebra Tutorial, Vector/Matrix Tutorial, Tutorial
4232 @section Types Tutorial
4235 Calc understands a variety of data types as well as simple numbers.
4236 In this section, we'll experiment with each of these types in turn.
4238 The numbers we've been using so far have mainly been either @dfn{integers}
4239 or @dfn{floats}. We saw that floats are usually a good approximation to
4240 the mathematical concept of real numbers, but they are only approximations
4241 and are susceptible to roundoff error. Calc also supports @dfn{fractions},
4242 which can exactly represent any rational number.
4246 1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414
4250 10 ! 49 @key{RET} : 2 + &
4255 The @kbd{:} command divides two integers to get a fraction; @kbd{/}
4256 would normally divide integers to get a floating-point result.
4257 Notice we had to type @key{RET} between the @kbd{49} and the @kbd{:}
4258 since the @kbd{:} would otherwise be interpreted as part of a
4259 fraction beginning with 49.
4261 You can convert between floating-point and fractional format using
4262 @kbd{c f} and @kbd{c F}:
4266 1: 1.35027217629e-5 1: 7:518414
4273 The @kbd{c F} command replaces a floating-point number with the
4274 ``simplest'' fraction whose floating-point representation is the
4275 same, to within the current precision.
4279 1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113
4282 P c F @key{DEL} p 5 @key{RET} P c F
4286 (@bullet{}) @strong{Exercise 1.} A calculation has produced the
4287 result 1.26508260337. You suspect it is the square root of the
4288 product of @cpi{} and some rational number. Is it? (Be sure
4289 to allow for roundoff error!) @xref{Types Answer 1, 1}. (@bullet{})
4291 @dfn{Complex numbers} can be stored in both rectangular and polar form.
4295 1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.)
4303 The square root of @mathit{-9} is by default rendered in rectangular form
4304 (@w{@expr{0 + 3i}}), but we can convert it to polar form (3 with a
4305 phase angle of 90 degrees). All the usual arithmetic and scientific
4306 operations are defined on both types of complex numbers.
4308 Another generalized kind of number is @dfn{infinity}. Infinity
4309 isn't really a number, but it can sometimes be treated like one.
4310 Calc uses the symbol @code{inf} to represent positive infinity,
4311 i.e., a value greater than any real number. Naturally, you can
4312 also write @samp{-inf} for minus infinity, a value less than any
4313 real number. The word @code{inf} can only be input using
4318 2: inf 2: -inf 2: -inf 2: -inf 1: nan
4319 1: -17 1: -inf 1: -inf 1: inf .
4322 ' inf @key{RET} 17 n * @key{RET} 72 + A +
4327 Since infinity is infinitely large, multiplying it by any finite
4328 number (like @mathit{-17}) has no effect, except that since @mathit{-17}
4329 is negative, it changes a plus infinity to a minus infinity.
4330 (``A huge positive number, multiplied by @mathit{-17}, yields a huge
4331 negative number.'') Adding any finite number to infinity also
4332 leaves it unchanged. Taking an absolute value gives us plus
4333 infinity again. Finally, we add this plus infinity to the minus
4334 infinity we had earlier. If you work it out, you might expect
4335 the answer to be @mathit{-72} for this. But the 72 has been completely
4336 lost next to the infinities; by the time we compute @w{@samp{inf - inf}}
4337 the finite difference between them, if any, is undetectable.
4338 So we say the result is @dfn{indeterminate}, which Calc writes
4339 with the symbol @code{nan} (for Not A Number).
4341 Dividing by zero is normally treated as an error, but you can get
4342 Calc to write an answer in terms of infinity by pressing @kbd{m i}
4343 to turn on Infinite mode.
4347 3: nan 2: nan 2: nan 2: nan 1: nan
4348 2: 1 1: 1 / 0 1: uinf 1: uinf .
4352 1 @key{RET} 0 / m i U / 17 n * +
4357 Dividing by zero normally is left unevaluated, but after @kbd{m i}
4358 it instead gives an infinite result. The answer is actually
4359 @code{uinf}, ``undirected infinity.'' If you look at a graph of
4360 @expr{1 / x} around @w{@expr{x = 0}}, you'll see that it goes toward
4361 plus infinity as you approach zero from above, but toward minus
4362 infinity as you approach from below. Since we said only @expr{1 / 0},
4363 Calc knows that the answer is infinite but not in which direction.
4364 That's what @code{uinf} means. Notice that multiplying @code{uinf}
4365 by a negative number still leaves plain @code{uinf}; there's no
4366 point in saying @samp{-uinf} because the sign of @code{uinf} is
4367 unknown anyway. Finally, we add @code{uinf} to our @code{nan},
4368 yielding @code{nan} again. It's easy to see that, because
4369 @code{nan} means ``totally unknown'' while @code{uinf} means
4370 ``unknown sign but known to be infinite,'' the more mysterious
4371 @code{nan} wins out when it is combined with @code{uinf}, or, for
4372 that matter, with anything else.
4374 (@bullet{}) @strong{Exercise 2.} Predict what Calc will answer
4375 for each of these formulas: @samp{inf / inf}, @samp{exp(inf)},
4376 @samp{exp(-inf)}, @samp{sqrt(-inf)}, @samp{sqrt(uinf)},
4377 @samp{abs(uinf)}, @samp{ln(0)}.
4378 @xref{Types Answer 2, 2}. (@bullet{})
4380 (@bullet{}) @strong{Exercise 3.} We saw that @samp{inf - inf = nan},
4381 which stands for an unknown value. Can @code{nan} stand for
4382 a complex number? Can it stand for infinity?
4383 @xref{Types Answer 3, 3}. (@bullet{})
4385 @dfn{HMS forms} represent a value in terms of hours, minutes, and
4390 1: 2@@ 30' 0" 1: 3@@ 30' 0" 2: 3@@ 30' 0" 1: 2.
4391 . . 1: 1@@ 45' 0." .
4394 2@@ 30' @key{RET} 1 + @key{RET} 2 / /
4398 HMS forms can also be used to hold angles in degrees, minutes, and
4403 1: 0.5 1: 26.56505 1: 26@@ 33' 54.18" 1: 0.44721
4411 First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
4412 form, then we take the sine of that angle. Note that the trigonometric
4413 functions will accept HMS forms directly as input.
4416 (@bullet{}) @strong{Exercise 4.} The Beatles' @emph{Abbey Road} is
4417 47 minutes and 26 seconds long, and contains 17 songs. What is the
4418 average length of a song on @emph{Abbey Road}? If the Extended Disco
4419 Version of @emph{Abbey Road} added 20 seconds to the length of each
4420 song, how long would the album be? @xref{Types Answer 4, 4}. (@bullet{})
4422 A @dfn{date form} represents a date, or a date and time. Dates must
4423 be entered using algebraic entry. Date forms are surrounded by
4424 @samp{< >} symbols; most standard formats for dates are recognized.
4428 2: <Sun Jan 13, 1991> 1: 2.25
4429 1: <6:00pm Thu Jan 10, 1991> .
4432 ' <13 Jan 1991>, <1/10/91, 6pm> @key{RET} -
4437 In this example, we enter two dates, then subtract to find the
4438 number of days between them. It is also possible to add an
4439 HMS form or a number (of days) to a date form to get another
4444 1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991>
4451 @c [fix-ref Date Arithmetic]
4453 The @kbd{t N} (``now'') command pushes the current date and time on the
4454 stack; then we add two days, ten hours and five minutes to the date and
4455 time. Other date-and-time related commands include @kbd{t J}, which
4456 does Julian day conversions, @kbd{t W}, which finds the beginning of
4457 the week in which a date form lies, and @kbd{t I}, which increments a
4458 date by one or several months. @xref{Date Arithmetic}, for more.
4460 (@bullet{}) @strong{Exercise 5.} How many days until the next
4461 Friday the 13th? @xref{Types Answer 5, 5}. (@bullet{})
4463 (@bullet{}) @strong{Exercise 6.} How many leap years will there be
4464 between now and the year 10001 AD@? @xref{Types Answer 6, 6}. (@bullet{})
4466 @cindex Slope and angle of a line
4467 @cindex Angle and slope of a line
4468 An @dfn{error form} represents a mean value with an attached standard
4469 deviation, or error estimate. Suppose our measurements indicate that
4470 a certain telephone pole is about 30 meters away, with an estimated
4471 error of 1 meter, and 8 meters tall, with an estimated error of 0.2
4472 meters. What is the slope of a line from here to the top of the
4473 pole, and what is the equivalent angle in degrees?
4477 1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594
4481 8 p .2 @key{RET} 30 p 1 / I T
4486 This means that the angle is about 15 degrees, and, assuming our
4487 original error estimates were valid standard deviations, there is about
4488 a 60% chance that the result is correct within 0.59 degrees.
4490 @cindex Torus, volume of
4491 (@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is
4492 @texline @math{2 \pi^2 R r^2}
4493 @infoline @w{@expr{2 pi^2 R r^2}}
4494 where @expr{R} is the radius of the circle that
4495 defines the center of the tube and @expr{r} is the radius of the tube
4496 itself. Suppose @expr{R} is 20 cm and @expr{r} is 4 cm, each known to
4497 within 5 percent. What is the volume and the relative uncertainty of
4498 the volume? @xref{Types Answer 7, 7}. (@bullet{})
4500 An @dfn{interval form} represents a range of values. While an
4501 error form is best for making statistical estimates, intervals give
4502 you exact bounds on an answer. Suppose we additionally know that
4503 our telephone pole is definitely between 28 and 31 meters away,
4504 and that it is between 7.7 and 8.1 meters tall.
4508 1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1]
4512 [ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T
4517 If our bounds were correct, then the angle to the top of the pole
4518 is sure to lie in the range shown.
4520 The square brackets around these intervals indicate that the endpoints
4521 themselves are allowable values. In other words, the distance to the
4522 telephone pole is between 28 and 31, @emph{inclusive}. You can also
4523 make an interval that is exclusive of its endpoints by writing
4524 parentheses instead of square brackets. You can even make an interval
4525 which is inclusive (``closed'') on one end and exclusive (``open'') on
4530 1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3)
4534 [ 1 .. 10 ) & [ 2 .. 3 ) *
4539 The Calculator automatically keeps track of which end values should
4540 be open and which should be closed. You can also make infinite or
4541 semi-infinite intervals by using @samp{-inf} or @samp{inf} for one
4544 (@bullet{}) @strong{Exercise 8.} What answer would you expect from
4545 @samp{@w{1 /} @w{(0 .. 10)}}? What about @samp{@w{1 /} @w{(-10 .. 0)}}? What
4546 about @samp{@w{1 /} @w{[0 .. 10]}} (where the interval actually includes
4547 zero)? What about @samp{@w{1 /} @w{(-10 .. 10)}}?
4548 @xref{Types Answer 8, 8}. (@bullet{})
4550 (@bullet{}) @strong{Exercise 9.} Two easy ways of squaring a number
4551 are @kbd{@key{RET} *} and @w{@kbd{2 ^}}. Normally these produce the same
4552 answer. Would you expect this still to hold true for interval forms?
4553 If not, which of these will result in a larger interval?
4554 @xref{Types Answer 9, 9}. (@bullet{})
4556 A @dfn{modulo form} is used for performing arithmetic modulo @var{m}.
4557 For example, arithmetic involving time is generally done modulo 12
4562 1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24
4565 17 M 24 @key{RET} 10 + n 5 /
4570 In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
4571 new number which, when multiplied by 5 modulo 24, produces the original
4572 number, 21. If @var{m} is prime and the divisor is not a multiple of
4573 @var{m}, it is always possible to find such a number. For non-prime
4574 @var{m} like 24, it is only sometimes possible.
4578 1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16
4581 10 M 24 @key{RET} 100 ^ 10 @key{RET} 100 ^ 24 %
4586 These two calculations get the same answer, but the first one is
4587 much more efficient because it avoids the huge intermediate value
4588 that arises in the second one.
4590 @cindex Fermat, primality test of
4591 (@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat
4593 @texline @math{x^{n-1} \bmod n = 1}
4594 @infoline @expr{x^(n-1) mod n = 1}
4595 if @expr{n} is a prime number and @expr{x} is an integer less than
4596 @expr{n}. If @expr{n} is @emph{not} a prime number, this will
4597 @emph{not} be true for most values of @expr{x}. Thus we can test
4598 informally if a number is prime by trying this formula for several
4599 values of @expr{x}. Use this test to tell whether the following numbers
4600 are prime: 811749613, 15485863. @xref{Types Answer 10, 10}. (@bullet{})
4602 It is possible to use HMS forms as parts of error forms, intervals,
4603 modulo forms, or as the phase part of a polar complex number.
4604 For example, the @code{calc-time} command pushes the current time
4605 of day on the stack as an HMS/modulo form.
4609 1: 17@@ 34' 45" mod 24@@ 0' 0" 1: 6@@ 22' 15" mod 24@@ 0' 0"
4617 This calculation tells me it is six hours and 22 minutes until midnight.
4619 (@bullet{}) @strong{Exercise 11.} A rule of thumb is that one year
4621 @texline @math{\pi \times 10^7}
4622 @infoline @w{@expr{pi * 10^7}}
4623 seconds. What time will it be that many seconds from right now?
4624 @xref{Types Answer 11, 11}. (@bullet{})
4626 (@bullet{}) @strong{Exercise 12.} You are preparing to order packaging
4627 for the CD release of the Extended Disco Version of @emph{Abbey Road}.
4628 You are told that the songs will actually be anywhere from 20 to 60
4629 seconds longer than the originals. One CD can hold about 75 minutes
4630 of music. Should you order single or double packages?
4631 @xref{Types Answer 12, 12}. (@bullet{})
4633 Another kind of data the Calculator can manipulate is numbers with
4634 @dfn{units}. This isn't strictly a new data type; it's simply an
4635 application of algebraic expressions, where we use variables with
4636 suggestive names like @samp{cm} and @samp{in} to represent units
4637 like centimeters and inches.
4641 1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m
4644 ' 2in @key{RET} u c cm @key{RET} u c fath @key{RET} u b
4649 We enter the quantity ``2 inches'' (actually an algebraic expression
4650 which means two times the variable @samp{in}), then we convert it
4651 first to centimeters, then to fathoms, then finally to ``base'' units,
4652 which in this case means meters.
4656 1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm
4659 ' 9 acre @key{RET} Q u s ' $+30 cm @key{RET}
4666 1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2
4674 Since units expressions are really just formulas, taking the square
4675 root of @samp{acre} is undefined. After all, @code{acre} might be an
4676 algebraic variable that you will someday assign a value. We use the
4677 ``units-simplify'' command to simplify the expression with variables
4678 being interpreted as unit names.
4680 In the final step, we have converted not to a particular unit, but to a
4681 units system. The ``cgs'' system uses centimeters instead of meters
4682 as its standard unit of length.
4684 There is a wide variety of units defined in the Calculator.
4688 1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c
4691 ' 55 mph @key{RET} u c kph @key{RET} u c km/hr @key{RET} u c c @key{RET}
4696 We express a speed first in miles per hour, then in kilometers per
4697 hour, then again using a slightly more explicit notation, then
4698 finally in terms of fractions of the speed of light.
4700 Temperature conversions are a bit more tricky. There are two ways to
4701 interpret ``20 degrees Fahrenheit''---it could mean an actual
4702 temperature, or it could mean a change in temperature. For normal
4703 units there is no difference, but temperature units have an offset
4704 as well as a scale factor and so there must be two explicit commands
4709 1: 20 degF 1: 11.1111 degC 1: -6.666 degC
4712 ' 20 degF @key{RET} u c degC @key{RET} U u t degC @key{RET}
4717 First we convert a change of 20 degrees Fahrenheit into an equivalent
4718 change in degrees Celsius (or Centigrade). Then, we convert the
4719 absolute temperature 20 degrees Fahrenheit into Celsius.
4721 For simple unit conversions, you can put a plain number on the stack.
4722 Then @kbd{u c} and @kbd{u t} will prompt for both old and new units.
4723 When you use this method, you're responsible for remembering which
4724 numbers are in which units:
4728 1: 55 1: 88.5139 1: 8.201407e-8
4731 55 u c mph @key{RET} kph @key{RET} u c km/hr @key{RET} c @key{RET}
4735 To see a complete list of built-in units, type @kbd{u v}. Press
4736 @w{@kbd{C-x * c}} again to re-enter the Calculator when you're done looking
4739 (@bullet{}) @strong{Exercise 13.} How many seconds are there really
4740 in a year? @xref{Types Answer 13, 13}. (@bullet{})
4742 @cindex Speed of light
4743 (@bullet{}) @strong{Exercise 14.} Supercomputer designs are limited by
4744 the speed of light (and of electricity, which is nearly as fast).
4745 Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its
4746 cabinet is one meter across. Is speed of light going to be a
4747 significant factor in its design? @xref{Types Answer 14, 14}. (@bullet{})
4749 (@bullet{}) @strong{Exercise 15.} Sam the Slug normally travels about
4750 five yards in an hour. He has obtained a supply of Power Pills; each
4751 Power Pill he eats doubles his speed. How many Power Pills can he
4752 swallow and still travel legally on most US highways?
4753 @xref{Types Answer 15, 15}. (@bullet{})
4755 @node Algebra Tutorial, Programming Tutorial, Types Tutorial, Tutorial
4756 @section Algebra and Calculus Tutorial
4759 This section shows how to use Calc's algebra facilities to solve
4760 equations, do simple calculus problems, and manipulate algebraic
4764 * Basic Algebra Tutorial::
4765 * Rewrites Tutorial::
4768 @node Basic Algebra Tutorial, Rewrites Tutorial, Algebra Tutorial, Algebra Tutorial
4769 @subsection Basic Algebra
4772 If you enter a formula in Algebraic mode that refers to variables,
4773 the formula itself is pushed onto the stack. You can manipulate
4774 formulas as regular data objects.
4778 1: 2 x^2 - 6 1: 6 - 2 x^2 1: (3 x^2 + y) (6 - 2 x^2)
4781 ' 2x^2-6 @key{RET} n ' 3x^2+y @key{RET} *
4785 (@bullet{}) @strong{Exercise 1.} Do @kbd{' x @key{RET} Q 2 ^} and
4786 @kbd{' x @key{RET} 2 ^ Q} both wind up with the same result (@samp{x})?
4787 Why or why not? @xref{Algebra Answer 1, 1}. (@bullet{})
4789 There are also commands for doing common algebraic operations on
4790 formulas. Continuing with the formula from the last example,
4794 1: 18 x^2 - 6 x^4 + 6 y - 2 y x^2 1: (18 - 2 y) x^2 - 6 x^4 + 6 y
4802 First we ``expand'' using the distributive law, then we ``collect''
4803 terms involving like powers of @expr{x}.
4805 Let's find the value of this expression when @expr{x} is 2 and @expr{y}
4810 1: 17 x^2 - 6 x^4 + 3 1: -25
4813 1:2 s l y @key{RET} 2 s l x @key{RET}
4818 The @kbd{s l} command means ``let''; it takes a number from the top of
4819 the stack and temporarily assigns it as the value of the variable
4820 you specify. It then evaluates (as if by the @kbd{=} key) the
4821 next expression on the stack. After this command, the variable goes
4822 back to its original value, if any.
4824 (An earlier exercise in this tutorial involved storing a value in the
4825 variable @code{x}; if this value is still there, you will have to
4826 unstore it with @kbd{s u x @key{RET}} before the above example will work
4829 @cindex Maximum of a function using Calculus
4830 Let's find the maximum value of our original expression when @expr{y}
4831 is one-half and @expr{x} ranges over all possible values. We can
4832 do this by taking the derivative with respect to @expr{x} and examining
4833 values of @expr{x} for which the derivative is zero. If the second
4834 derivative of the function at that value of @expr{x} is negative,
4835 the function has a local maximum there.
4839 1: 17 x^2 - 6 x^4 + 3 1: 34 x - 24 x^3
4842 U @key{DEL} s 1 a d x @key{RET} s 2
4847 Well, the derivative is clearly zero when @expr{x} is zero. To find
4848 the other root(s), let's divide through by @expr{x} and then solve:
4852 1: (34 x - 24 x^3) / x 1: 34 - 24 x^2
4862 1: 0.70588 x^2 = 1 1: x = 1.19023
4865 0 a = s 3 a S x @key{RET}
4870 Now we compute the second derivative and plug in our values of @expr{x}:
4874 1: 1.19023 2: 1.19023 2: 1.19023
4875 . 1: 34 x - 24 x^3 1: 34 - 72 x^2
4878 a . r 2 a d x @key{RET} s 4
4883 (The @kbd{a .} command extracts just the righthand side of an equation.
4884 Another method would have been to use @kbd{v u} to unpack the equation
4885 @w{@samp{x = 1.19}} to @samp{x} and @samp{1.19}, then use @kbd{M-- M-2 @key{DEL}}
4886 to delete the @samp{x}.)
4890 2: 34 - 72 x^2 1: -68. 2: 34 - 72 x^2 1: 34
4894 @key{TAB} s l x @key{RET} U @key{DEL} 0 s l x @key{RET}
4899 The first of these second derivatives is negative, so we know the function
4900 has a maximum value at @expr{x = 1.19023}. (The function also has a
4901 local @emph{minimum} at @expr{x = 0}.)
4903 When we solved for @expr{x}, we got only one value even though
4904 @expr{0.70588 x^2 = 1} is a quadratic equation that ought to have
4905 two solutions. The reason is that @w{@kbd{a S}} normally returns a
4906 single ``principal'' solution. If it needs to come up with an
4907 arbitrary sign (as occurs in the quadratic formula) it picks @expr{+}.
4908 If it needs an arbitrary integer, it picks zero. We can get a full
4909 solution by pressing @kbd{H} (the Hyperbolic flag) before @kbd{a S}.
4913 1: 0.70588 x^2 = 1 1: x = 1.19023 s1 1: x = -1.19023
4916 r 3 H a S x @key{RET} s 5 1 n s l s1 @key{RET}
4921 Calc has invented the variable @samp{s1} to represent an unknown sign;
4922 it is supposed to be either @mathit{+1} or @mathit{-1}. Here we have used
4923 the ``let'' command to evaluate the expression when the sign is negative.
4924 If we plugged this into our second derivative we would get the same,
4925 negative, answer, so @expr{x = -1.19023} is also a maximum.
4927 To find the actual maximum value, we must plug our two values of @expr{x}
4928 into the original formula.
4932 2: 17 x^2 - 6 x^4 + 3 1: 24.08333 s1^2 - 12.04166 s1^4 + 3
4936 r 1 r 5 s l @key{RET}
4941 (Here we see another way to use @kbd{s l}; if its input is an equation
4942 with a variable on the lefthand side, then @kbd{s l} treats the equation
4943 like an assignment to that variable if you don't give a variable name.)
4945 It's clear that this will have the same value for either sign of
4946 @code{s1}, but let's work it out anyway, just for the exercise:
4950 2: [-1, 1] 1: [15.04166, 15.04166]
4951 1: 24.08333 s1^2 ... .
4954 [ 1 n , 1 ] @key{TAB} V M $ @key{RET}
4959 Here we have used a vector mapping operation to evaluate the function
4960 at several values of @samp{s1} at once. @kbd{V M $} is like @kbd{V M '}
4961 except that it takes the formula from the top of the stack. The
4962 formula is interpreted as a function to apply across the vector at the
4963 next-to-top stack level. Since a formula on the stack can't contain
4964 @samp{$} signs, Calc assumes the variables in the formula stand for
4965 different arguments. It prompts you for an @dfn{argument list}, giving
4966 the list of all variables in the formula in alphabetical order as the
4967 default list. In this case the default is @samp{(s1)}, which is just
4968 what we want so we simply press @key{RET} at the prompt.
4970 If there had been several different values, we could have used
4971 @w{@kbd{V R X}} to find the global maximum.
4973 Calc has a built-in @kbd{a P} command that solves an equation using
4974 @w{@kbd{H a S}} and returns a vector of all the solutions. It simply
4975 automates the job we just did by hand. Applied to our original
4976 cubic polynomial, it would produce the vector of solutions
4977 @expr{[1.19023, -1.19023, 0]}. (There is also an @kbd{a X} command
4978 which finds a local maximum of a function. It uses a numerical search
4979 method rather than examining the derivatives, and thus requires you
4980 to provide some kind of initial guess to show it where to look.)
4982 (@bullet{}) @strong{Exercise 2.} Given a vector of the roots of a
4983 polynomial (such as the output of an @kbd{a P} command), what
4984 sequence of commands would you use to reconstruct the original
4985 polynomial? (The answer will be unique to within a constant
4986 multiple; choose the solution where the leading coefficient is one.)
4987 @xref{Algebra Answer 2, 2}. (@bullet{})
4989 The @kbd{m s} command enables Symbolic mode, in which formulas
4990 like @samp{sqrt(5)} that can't be evaluated exactly are left in
4991 symbolic form rather than giving a floating-point approximate answer.
4992 Fraction mode (@kbd{m f}) is also useful when doing algebra.
4996 2: 34 x - 24 x^3 2: 34 x - 24 x^3
4997 1: 34 x - 24 x^3 1: [sqrt(51) / 6, sqrt(51) / -6, 0]
5000 r 2 @key{RET} m s m f a P x @key{RET}
5004 One more mode that makes reading formulas easier is Big mode.
5013 1: [-----, -----, 0]
5022 Here things like powers, square roots, and quotients and fractions
5023 are displayed in a two-dimensional pictorial form. Calc has other
5024 language modes as well, such as C mode, FORTRAN mode, @TeX{} mode
5029 2: 34*x - 24*pow(x, 3) 2: 34*x - 24*x**3
5030 1: @{sqrt(51) / 6, sqrt(51) / -6, 0@} 1: /sqrt(51) / 6, sqrt(51) / -6, 0/
5041 2: [@{\sqrt@{51@} \over 6@}, @{\sqrt@{51@} \over -6@}, 0]
5042 1: @{2 \over 3@} \sqrt@{5@}
5045 d T ' 2 \sqrt@{5@} \over 3 @key{RET}
5050 As you can see, language modes affect both entry and display of
5051 formulas. They affect such things as the names used for built-in
5052 functions, the set of arithmetic operators and their precedences,
5053 and notations for vectors and matrices.
5055 Notice that @samp{sqrt(51)} may cause problems with older
5056 implementations of C and FORTRAN, which would require something more
5057 like @samp{sqrt(51.0)}. It is always wise to check over the formulas
5058 produced by the various language modes to make sure they are fully
5061 Type @kbd{m s}, @kbd{m f}, and @kbd{d N} to reset these modes. (You
5062 may prefer to remain in Big mode, but all the examples in the tutorial
5063 are shown in normal mode.)
5065 @cindex Area under a curve
5066 What is the area under the portion of this curve from @expr{x = 1} to @expr{2}?
5067 This is simply the integral of the function:
5071 1: 17 x^2 - 6 x^4 + 3 1: 5.6666 x^3 - 1.2 x^5 + 3 x
5079 We want to evaluate this at our two values for @expr{x} and subtract.
5080 One way to do it is again with vector mapping and reduction:
5084 2: [2, 1] 1: [12.93333, 7.46666] 1: 5.46666
5085 1: 5.6666 x^3 ... . .
5087 [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5091 (@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @expr{y}
5093 @texline @math{x \sin \pi x}
5094 @infoline @w{@expr{x sin(pi x)}}
5095 (where the sine is calculated in radians). Find the values of the
5096 integral for integers @expr{y} from 1 to 5. @xref{Algebra Answer 3,
5099 Calc's integrator can do many simple integrals symbolically, but many
5100 others are beyond its capabilities. Suppose we wish to find the area
5102 @texline @math{\sin x \ln x}
5103 @infoline @expr{sin(x) ln(x)}
5104 over the same range of @expr{x}. If you entered this formula and typed
5105 @kbd{a i x @key{RET}} (don't bother to try this), Calc would work for a
5106 long time but would be unable to find a solution. In fact, there is no
5107 closed-form solution to this integral. Now what do we do?
5109 @cindex Integration, numerical
5110 @cindex Numerical integration
5111 One approach would be to do the integral numerically. It is not hard
5112 to do this by hand using vector mapping and reduction. It is rather
5113 slow, though, since the sine and logarithm functions take a long time.
5114 We can save some time by reducing the working precision.
5118 3: 10 1: [1, 1.1, 1.2, ... , 1.8, 1.9]
5123 10 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
5128 (Note that we have used the extended version of @kbd{v x}; we could
5129 also have used plain @kbd{v x} as follows: @kbd{v x 10 @key{RET} 9 + .1 *}.)
5133 2: [1, 1.1, ... ] 1: [0., 0.084941, 0.16993, ... ]
5137 ' sin(x) ln(x) @key{RET} s 1 m r p 5 @key{RET} V M $ @key{RET}
5152 (If you got wildly different results, did you remember to switch
5155 Here we have divided the curve into ten segments of equal width;
5156 approximating these segments as rectangular boxes (i.e., assuming
5157 the curve is nearly flat at that resolution), we compute the areas
5158 of the boxes (height times width), then sum the areas. (It is
5159 faster to sum first, then multiply by the width, since the width
5160 is the same for every box.)
5162 The true value of this integral turns out to be about 0.374, so
5163 we're not doing too well. Let's try another approach.
5167 1: ln(x) sin(x) 1: 0.84147 x + 0.11957 (x - 1)^2 - ...
5170 r 1 a t x=1 @key{RET} 4 @key{RET}
5175 Here we have computed the Taylor series expansion of the function
5176 about the point @expr{x=1}. We can now integrate this polynomial
5177 approximation, since polynomials are easy to integrate.
5181 1: 0.42074 x^2 + ... 1: [-0.0446, -0.42073] 1: 0.3761
5184 a i x @key{RET} [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5189 Better! By increasing the precision and/or asking for more terms
5190 in the Taylor series, we can get a result as accurate as we like.
5191 (Taylor series converge better away from singularities in the
5192 function such as the one at @code{ln(0)}, so it would also help to
5193 expand the series about the points @expr{x=2} or @expr{x=1.5} instead
5196 @cindex Simpson's rule
5197 @cindex Integration by Simpson's rule
5198 (@bullet{}) @strong{Exercise 4.} Our first method approximated the
5199 curve by stairsteps of width 0.1; the total area was then the sum
5200 of the areas of the rectangles under these stairsteps. Our second
5201 method approximated the function by a polynomial, which turned out
5202 to be a better approximation than stairsteps. A third method is
5203 @dfn{Simpson's rule}, which is like the stairstep method except
5204 that the steps are not required to be flat. Simpson's rule boils
5205 down to the formula,
5209 (h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
5210 + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
5216 \qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
5217 \hfill \cr \hfill {} + 2 f(a+(n-2)h) + 4 f(a+(n-1)h) + f(a+n h)) \qquad
5223 where @expr{n} (which must be even) is the number of slices and @expr{h}
5224 is the width of each slice. These are 10 and 0.1 in our example.
5225 For reference, here is the corresponding formula for the stairstep
5230 h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
5231 + f(a+(n-2)*h) + f(a+(n-1)*h))
5236 $$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
5237 + f(a+(n-2)h) + f(a+(n-1)h)) $$
5241 Compute the integral from 1 to 2 of
5242 @texline @math{\sin x \ln x}
5243 @infoline @expr{sin(x) ln(x)}
5244 using Simpson's rule with 10 slices.
5245 @xref{Algebra Answer 4, 4}. (@bullet{})
5247 Calc has a built-in @kbd{a I} command for doing numerical integration.
5248 It uses @dfn{Romberg's method}, which is a more sophisticated cousin
5249 of Simpson's rule. In particular, it knows how to keep refining the
5250 result until the current precision is satisfied.
5252 @c [fix-ref Selecting Sub-Formulas]
5253 Aside from the commands we've seen so far, Calc also provides a
5254 large set of commands for operating on parts of formulas. You
5255 indicate the desired sub-formula by placing the cursor on any part
5256 of the formula before giving a @dfn{selection} command. Selections won't
5257 be covered in the tutorial; @pxref{Selecting Subformulas}, for
5258 details and examples.
5260 @c hard exercise: simplify (2^(n r) - 2^(r*(n - 1))) / (2^r - 1) 2^(n - 1)
5261 @c to 2^((n-1)*(r-1)).
5263 @node Rewrites Tutorial, , Basic Algebra Tutorial, Algebra Tutorial
5264 @subsection Rewrite Rules
5267 No matter how many built-in commands Calc provided for doing algebra,
5268 there would always be something you wanted to do that Calc didn't have
5269 in its repertoire. So Calc also provides a @dfn{rewrite rule} system
5270 that you can use to define your own algebraic manipulations.
5272 Suppose we want to simplify this trigonometric formula:
5276 1: 2 sec(x)^2 / tan(x)^2 - 2 / tan(x)^2
5279 ' 2sec(x)^2/tan(x)^2 - 2/tan(x)^2 @key{RET} s 1
5284 If we were simplifying this by hand, we'd probably combine over the common
5285 denominator. The @kbd{a n} algebra command will do this, but we'll do
5286 it with a rewrite rule just for practice.
5288 Rewrite rules are written with the @samp{:=} symbol.
5292 1: (2 sec(x)^2 - 2) / tan(x)^2
5295 a r a/x + b/x := (a+b)/x @key{RET}
5300 (The ``assignment operator'' @samp{:=} has several uses in Calc. All
5301 by itself the formula @samp{a/x + b/x := (a+b)/x} doesn't do anything,
5302 but when it is given to the @kbd{a r} command, that command interprets
5303 it as a rewrite rule.)
5305 The lefthand side, @samp{a/x + b/x}, is called the @dfn{pattern} of the
5306 rewrite rule. Calc searches the formula on the stack for parts that
5307 match the pattern. Variables in a rewrite pattern are called
5308 @dfn{meta-variables}, and when matching the pattern each meta-variable
5309 can match any sub-formula. Here, the meta-variable @samp{a} matched
5310 the expression @samp{2 sec(x)^2}, the meta-variable @samp{b} matched
5311 the constant @samp{-2} and the meta-variable @samp{x} matched
5312 the expression @samp{tan(x)^2}.
5314 This rule points out several interesting features of rewrite patterns.
5315 First, if a meta-variable appears several times in a pattern, it must
5316 match the same thing everywhere. This rule detects common denominators
5317 because the same meta-variable @samp{x} is used in both of the
5320 Second, meta-variable names are independent from variables in the
5321 target formula. Notice that the meta-variable @samp{x} here matches
5322 the subformula @samp{tan(x)^2}; Calc never confuses the two meanings of
5325 And third, rewrite patterns know a little bit about the algebraic
5326 properties of formulas. The pattern called for a sum of two quotients;
5327 Calc was able to match a difference of two quotients by matching
5328 @samp{a = 2 sec(x)^2}, @samp{b = -2}, and @samp{x = tan(x)^2}.
5330 When the pattern part of a rewrite rule matches a part of the formula,
5331 that part is replaced by the righthand side with all the meta-variables
5332 substituted with the things they matched. So the result is
5333 @samp{(2 sec(x)^2 - 2) / tan(x)^2}.
5335 @c [fix-ref Algebraic Properties of Rewrite Rules]
5336 We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
5337 the rule. It would have worked just the same in all cases. (If we
5338 really wanted the rule to apply only to @samp{+} or only to @samp{-},
5339 we could have used the @code{plain} symbol. @xref{Algebraic Properties
5340 of Rewrite Rules}, for some examples of this.)
5342 One more rewrite will complete the job. We want to use the identity
5343 @samp{tan(x)^2 + 1 = sec(x)^2}, but of course we must first rearrange
5344 the identity in a way that matches our formula. The obvious rule
5345 would be @samp{@w{2 sec(x)^2 - 2} := 2 tan(x)^2}, but a little thought shows
5346 that the rule @samp{sec(x)^2 := 1 + tan(x)^2} will also work. The
5347 latter rule has a more general pattern so it will work in many other
5355 a r sec(x)^2 := 1 + tan(x)^2 @key{RET}
5359 You may ask, what's the point of using the most general rule if you
5360 have to type it in every time anyway? The answer is that Calc allows
5361 you to store a rewrite rule in a variable, then give the variable
5362 name in the @kbd{a r} command. In fact, this is the preferred way to
5363 use rewrites. For one, if you need a rule once you'll most likely
5364 need it again later. Also, if the rule doesn't work quite right you
5365 can simply Undo, edit the variable, and run the rule again without
5366 having to retype it.
5370 ' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET}
5371 ' sec(x)^2 := 1 + tan(x)^2 @key{RET} s t secsqr @key{RET}
5373 1: 2 sec(x)^2 / tan(x)^2 - 2 / tan(x)^2 1: 2
5376 r 1 a r merge @key{RET} a r secsqr @key{RET}
5380 To edit a variable, type @kbd{s e} and the variable name, use regular
5381 Emacs editing commands as necessary, then type @kbd{C-c C-c} to store
5382 the edited value back into the variable.
5383 You can also use @w{@kbd{s e}} to create a new variable if you wish.
5385 Notice that the first time you use each rule, Calc puts up a ``compiling''
5386 message briefly. The pattern matcher converts rules into a special
5387 optimized pattern-matching language rather than using them directly.
5388 This allows @kbd{a r} to apply even rather complicated rules very
5389 efficiently. If the rule is stored in a variable, Calc compiles it
5390 only once and stores the compiled form along with the variable. That's
5391 another good reason to store your rules in variables rather than
5392 entering them on the fly.
5394 (@bullet{}) @strong{Exercise 1.} Type @kbd{m s} to get Symbolic
5395 mode, then enter the formula @samp{@w{(2 + sqrt(2))} / @w{(1 + sqrt(2))}}.
5396 Using a rewrite rule, simplify this formula by multiplying the top and
5397 bottom by the conjugate @w{@samp{1 - sqrt(2)}}. The result will have
5398 to be expanded by the distributive law; do this with another
5399 rewrite. @xref{Rewrites Answer 1, 1}. (@bullet{})
5401 The @kbd{a r} command can also accept a vector of rewrite rules, or
5402 a variable containing a vector of rules.
5406 1: [merge, secsqr] 1: [a/x + b/x := (a + b)/x, ... ]
5409 ' [merge,sinsqr] @key{RET} =
5416 1: 2 sec(x)^2 / tan(x)^2 - 2 / tan(x)^2 1: 2
5419 s t trig @key{RET} r 1 a r trig @key{RET}
5423 @c [fix-ref Nested Formulas with Rewrite Rules]
5424 Calc tries all the rules you give against all parts of the formula,
5425 repeating until no further change is possible. (The exact order in
5426 which things are tried is rather complex, but for simple rules like
5427 the ones we've used here the order doesn't really matter.
5428 @xref{Nested Formulas with Rewrite Rules}.)
5430 Calc actually repeats only up to 100 times, just in case your rule set
5431 has gotten into an infinite loop. You can give a numeric prefix argument
5432 to @kbd{a r} to specify any limit. In particular, @kbd{M-1 a r} does
5433 only one rewrite at a time.
5437 1: (2 sec(x)^2 - 2) / tan(x)^2 1: 2
5440 r 1 M-1 a r trig @key{RET} M-1 a r trig @key{RET}
5444 You can type @kbd{M-0 a r} if you want no limit at all on the number
5445 of rewrites that occur.
5447 Rewrite rules can also be @dfn{conditional}. Simply follow the rule
5448 with a @samp{::} symbol and the desired condition. For example,
5452 1: sin(x + 2 pi) + sin(x + 3 pi) + sin(x + 4 pi)
5455 ' sin(x+2pi) + sin(x+3pi) + sin(x+4pi) @key{RET}
5462 1: sin(x + 3 pi) + 2 sin(x)
5465 a r sin(a + k pi) := sin(a) :: k % 2 = 0 @key{RET}
5470 (Recall, @samp{k % 2} is the remainder from dividing @samp{k} by 2,
5471 which will be zero only when @samp{k} is an even integer.)
5473 An interesting point is that the variable @samp{pi} was matched
5474 literally rather than acting as a meta-variable.
5475 This is because it is a special-constant variable. The special
5476 constants @samp{e}, @samp{i}, @samp{phi}, and so on also match literally.
5477 A common error with rewrite
5478 rules is to write, say, @samp{f(a,b,c,d,e) := g(a+b+c+d+e)}, expecting
5479 to match any @samp{f} with five arguments but in fact matching
5480 only when the fifth argument is literally @samp{e}!
5482 @cindex Fibonacci numbers
5487 Rewrite rules provide an interesting way to define your own functions.
5488 Suppose we want to define @samp{fib(n)} to produce the @var{n}th
5489 Fibonacci number. The first two Fibonacci numbers are each 1;
5490 later numbers are formed by summing the two preceding numbers in
5491 the sequence. This is easy to express in a set of three rules:
5495 ' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] @key{RET} s t fib
5500 ' fib(7) @key{RET} a r fib @key{RET}
5504 One thing that is guaranteed about the order that rewrites are tried
5505 is that, for any given subformula, earlier rules in the rule set will
5506 be tried for that subformula before later ones. So even though the
5507 first and third rules both match @samp{fib(1)}, we know the first will
5508 be used preferentially.
5510 This rule set has one dangerous bug: Suppose we apply it to the
5511 formula @samp{fib(x)}? (Don't actually try this.) The third rule
5512 will match @samp{fib(x)} and replace it with @w{@samp{fib(x-1) + fib(x-2)}}.
5513 Each of these will then be replaced to get @samp{fib(x-2) + 2 fib(x-3) +
5514 fib(x-4)}, and so on, expanding forever. What we really want is to apply
5515 the third rule only when @samp{n} is an integer greater than two. Type
5516 @w{@kbd{s e fib @key{RET}}}, then edit the third rule to:
5519 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2
5527 1: fib(6) + fib(x) + fib(0) 1: fib(x) + fib(0) + 8
5530 ' fib(6)+fib(x)+fib(0) @key{RET} a r fib @key{RET}
5535 We've created a new function, @code{fib}, and a new command,
5536 @w{@kbd{a r fib @key{RET}}}, which means ``evaluate all @code{fib} calls in
5537 this formula.'' To make things easier still, we can tell Calc to
5538 apply these rules automatically by storing them in the special
5539 variable @code{EvalRules}.
5543 1: [fib(1) := ...] . 1: [8, 13]
5546 s r fib @key{RET} s t EvalRules @key{RET} ' [fib(6), fib(7)] @key{RET}
5550 It turns out that this rule set has the problem that it does far
5551 more work than it needs to when @samp{n} is large. Consider the
5552 first few steps of the computation of @samp{fib(6)}:
5558 fib(4) + fib(3) + fib(3) + fib(2) =
5559 fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...
5564 Note that @samp{fib(3)} appears three times here. Unless Calc's
5565 algebraic simplifier notices the multiple @samp{fib(3)}s and combines
5566 them (and, as it happens, it doesn't), this rule set does lots of
5567 needless recomputation. To cure the problem, type @code{s e EvalRules}
5568 to edit the rules (or just @kbd{s E}, a shorthand command for editing
5569 @code{EvalRules}) and add another condition:
5572 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember
5576 If a @samp{:: remember} condition appears anywhere in a rule, then if
5577 that rule succeeds Calc will add another rule that describes that match
5578 to the front of the rule set. (Remembering works in any rule set, but
5579 for technical reasons it is most effective in @code{EvalRules}.) For
5580 example, if the rule rewrites @samp{fib(7)} to something that evaluates
5581 to 13, then the rule @samp{fib(7) := 13} will be added to the rule set.
5583 Type @kbd{' fib(8) @key{RET}} to compute the eighth Fibonacci number, then
5584 type @kbd{s E} again to see what has happened to the rule set.
5586 With the @code{remember} feature, our rule set can now compute
5587 @samp{fib(@var{n})} in just @var{n} steps. In the process it builds
5588 up a table of all Fibonacci numbers up to @var{n}. After we have
5589 computed the result for a particular @var{n}, we can get it back
5590 (and the results for all smaller @var{n}) later in just one step.
5592 All Calc operations will run somewhat slower whenever @code{EvalRules}
5593 contains any rules. You should type @kbd{s u EvalRules @key{RET}} now to
5594 un-store the variable.
5596 (@bullet{}) @strong{Exercise 2.} Sometimes it is possible to reformulate
5597 a problem to reduce the amount of recursion necessary to solve it.
5598 Create a rule that, in about @var{n} simple steps and without recourse
5599 to the @code{remember} option, replaces @samp{fib(@var{n}, 1, 1)} with
5600 @samp{fib(1, @var{x}, @var{y})} where @var{x} and @var{y} are the
5601 @var{n}th and @var{n+1}st Fibonacci numbers, respectively. This rule is
5602 rather clunky to use, so add a couple more rules to make the ``user
5603 interface'' the same as for our first version: enter @samp{fib(@var{n})},
5604 get back a plain number. @xref{Rewrites Answer 2, 2}. (@bullet{})
5606 There are many more things that rewrites can do. For example, there
5607 are @samp{&&&} and @samp{|||} pattern operators that create ``and''
5608 and ``or'' combinations of rules. As one really simple example, we
5609 could combine our first two Fibonacci rules thusly:
5612 [fib(1 ||| 2) := 1, fib(n) := ... ]
5616 That means ``@code{fib} of something matching either 1 or 2 rewrites
5619 You can also make meta-variables optional by enclosing them in @code{opt}.
5620 For example, the pattern @samp{a + b x} matches @samp{2 + 3 x} but not
5621 @samp{2 + x} or @samp{3 x} or @samp{x}. The pattern @samp{opt(a) + opt(b) x}
5622 matches all of these forms, filling in a default of zero for @samp{a}
5623 and one for @samp{b}.
5625 (@bullet{}) @strong{Exercise 3.} Your friend Joe had @samp{2 + 3 x}
5626 on the stack and tried to use the rule
5627 @samp{opt(a) + opt(b) x := f(a, b, x)}. What happened?
5628 @xref{Rewrites Answer 3, 3}. (@bullet{})
5630 (@bullet{}) @strong{Exercise 4.} Starting with a positive integer @expr{a},
5631 divide @expr{a} by two if it is even, otherwise compute @expr{3 a + 1}.
5632 Now repeat this step over and over. A famous unproved conjecture
5633 is that for any starting @expr{a}, the sequence always eventually
5634 reaches 1. Given the formula @samp{seq(@var{a}, 0)}, write a set of
5635 rules that convert this into @samp{seq(1, @var{n})} where @var{n}
5636 is the number of steps it took the sequence to reach the value 1.
5637 Now enhance the rules to accept @samp{seq(@var{a})} as a starting
5638 configuration, and to stop with just the number @var{n} by itself.
5639 Now make the result be a vector of values in the sequence, from @var{a}
5640 to 1. (The formula @samp{@var{x}|@var{y}} appends the vectors @var{x}
5641 and @var{y}.) For example, rewriting @samp{seq(6)} should yield the
5642 vector @expr{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
5643 @xref{Rewrites Answer 4, 4}. (@bullet{})
5645 (@bullet{}) @strong{Exercise 5.} Define, using rewrite rules, a function
5646 @samp{nterms(@var{x})} that returns the number of terms in the sum
5647 @var{x}, or 1 if @var{x} is not a sum. (A @dfn{sum} for our purposes
5648 is one or more non-sum terms separated by @samp{+} or @samp{-} signs,
5649 so that @expr{2 - 3 (x + y) + x y} is a sum of three terms.)
5650 @xref{Rewrites Answer 5, 5}. (@bullet{})
5652 (@bullet{}) @strong{Exercise 6.} A Taylor series for a function is an
5653 infinite series that exactly equals the value of that function at
5654 values of @expr{x} near zero.
5658 cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
5663 $$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
5667 The @kbd{a t} command produces a @dfn{truncated Taylor series} which
5668 is obtained by dropping all the terms higher than, say, @expr{x^2}.
5669 Calc represents the truncated Taylor series as a polynomial in @expr{x}.
5670 Mathematicians often write a truncated series using a ``big-O'' notation
5671 that records what was the lowest term that was truncated.
5675 cos(x) = 1 - x^2 / 2! + O(x^3)
5680 $$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
5685 The meaning of @expr{O(x^3)} is ``a quantity which is negligibly small
5686 if @expr{x^3} is considered negligibly small as @expr{x} goes to zero.''
5688 The exercise is to create rewrite rules that simplify sums and products of
5689 power series represented as @samp{@var{polynomial} + O(@var{var}^@var{n})}.
5690 For example, given @samp{1 - x^2 / 2 + O(x^3)} and @samp{x - x^3 / 6 + O(x^4)}
5691 on the stack, we want to be able to type @kbd{*} and get the result
5692 @samp{x - 2:3 x^3 + O(x^4)}. Don't worry if the terms of the sum are
5693 rearranged. (This one is rather tricky; the solution at the end of
5694 this chapter uses 6 rewrite rules. Hint: The @samp{constant(x)}
5695 condition tests whether @samp{x} is a number.) @xref{Rewrites Answer
5698 Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}.
5699 What happens? (Be sure to remove this rule afterward, or you might get
5700 a nasty surprise when you use Calc to balance your checkbook!)
5702 @xref{Rewrite Rules}, for the whole story on rewrite rules.
5704 @node Programming Tutorial, Answers to Exercises, Algebra Tutorial, Tutorial
5705 @section Programming Tutorial
5708 The Calculator is written entirely in Emacs Lisp, a highly extensible
5709 language. If you know Lisp, you can program the Calculator to do
5710 anything you like. Rewrite rules also work as a powerful programming
5711 system. But Lisp and rewrite rules take a while to master, and often
5712 all you want to do is define a new function or repeat a command a few
5713 times. Calc has features that allow you to do these things easily.
5715 One very limited form of programming is defining your own functions.
5716 Calc's @kbd{Z F} command allows you to define a function name and
5717 key sequence to correspond to any formula. Programming commands use
5718 the shift-@kbd{Z} prefix; the user commands they create use the lower
5719 case @kbd{z} prefix.
5723 1: x + x^2 / 2 + x^3 / 6 + 1 1: x + x^2 / 2 + x^3 / 6 + 1
5726 ' 1 + x + x^2/2! + x^3/3! @key{RET} Z F e myexp @key{RET} @key{RET} @key{RET} y
5730 This polynomial is a Taylor series approximation to @samp{exp(x)}.
5731 The @kbd{Z F} command asks a number of questions. The above answers
5732 say that the key sequence for our function should be @kbd{z e}; the
5733 @kbd{M-x} equivalent should be @code{calc-myexp}; the name of the
5734 function in algebraic formulas should also be @code{myexp}; the
5735 default argument list @samp{(x)} is acceptable; and finally @kbd{y}
5736 answers the question ``leave it in symbolic form for non-constant
5741 1: 1.3495 2: 1.3495 3: 1.3495
5742 . 1: 1.34986 2: 1.34986
5746 .3 z e .3 E ' a+1 @key{RET} z e
5751 First we call our new @code{exp} approximation with 0.3 as an
5752 argument, and compare it with the true @code{exp} function. Then
5753 we note that, as requested, if we try to give @kbd{z e} an
5754 argument that isn't a plain number, it leaves the @code{myexp}
5755 function call in symbolic form. If we had answered @kbd{n} to the
5756 final question, @samp{myexp(a + 1)} would have evaluated by plugging
5757 in @samp{a + 1} for @samp{x} in the defining formula.
5759 @cindex Sine integral Si(x)
5764 (@bullet{}) @strong{Exercise 1.} The ``sine integral'' function
5765 @texline @math{{\rm Si}(x)}
5766 @infoline @expr{Si(x)}
5767 is defined as the integral of @samp{sin(t)/t} for
5768 @expr{t = 0} to @expr{x} in radians. (It was invented because this
5769 integral has no solution in terms of basic functions; if you give it
5770 to Calc's @kbd{a i} command, it will ponder it for a long time and then
5771 give up.) We can use the numerical integration command, however,
5772 which in algebraic notation is written like @samp{ninteg(f(t), t, 0, x)}
5773 with any integrand @samp{f(t)}. Define a @kbd{z s} command and
5774 @code{Si} function that implement this. You will need to edit the
5775 default argument list a bit. As a test, @samp{Si(1)} should return
5776 0.946083. (If you don't get this answer, you might want to check that
5777 Calc is in Radians mode. Also, @code{ninteg} will run a lot faster if
5778 you reduce the precision to, say, six digits beforehand.)
5779 @xref{Programming Answer 1, 1}. (@bullet{})
5781 The simplest way to do real ``programming'' of Emacs is to define a
5782 @dfn{keyboard macro}. A keyboard macro is simply a sequence of
5783 keystrokes which Emacs has stored away and can play back on demand.
5784 For example, if you find yourself typing @kbd{H a S x @key{RET}} often,
5785 you may wish to program a keyboard macro to type this for you.
5789 1: y = sqrt(x) 1: x = y^2
5792 ' y=sqrt(x) @key{RET} C-x ( H a S x @key{RET} C-x )
5794 1: y = cos(x) 1: x = s1 arccos(y) + 2 n1 pi
5797 ' y=cos(x) @key{RET} X
5802 When you type @kbd{C-x (}, Emacs begins recording. But it is also
5803 still ready to execute your keystrokes, so you're really ``training''
5804 Emacs by walking it through the procedure once. When you type
5805 @w{@kbd{C-x )}}, the macro is recorded. You can now type @kbd{X} to
5806 re-execute the same keystrokes.
5808 You can give a name to your macro by typing @kbd{Z K}.
5812 1: . 1: y = x^4 1: x = s2 sqrt(s1 sqrt(y))
5815 Z K x @key{RET} ' y=x^4 @key{RET} z x
5820 Notice that we use shift-@kbd{Z} to define the command, and lower-case
5821 @kbd{z} to call it up.
5823 Keyboard macros can call other macros.
5827 1: abs(x) 1: x = s1 y 1: 2 / x 1: x = 2 / y
5830 ' abs(x) @key{RET} C-x ( ' y @key{RET} a = z x C-x ) ' 2/x @key{RET} X
5834 (@bullet{}) @strong{Exercise 2.} Define a keyboard macro to negate
5835 the item in level 3 of the stack, without disturbing the rest of
5836 the stack. @xref{Programming Answer 2, 2}. (@bullet{})
5838 (@bullet{}) @strong{Exercise 3.} Define keyboard macros to compute
5839 the following functions:
5844 @texline @math{\displaystyle{\sin x \over x}},
5845 @infoline @expr{sin(x) / x},
5846 where @expr{x} is the number on the top of the stack.
5849 Compute the base-@expr{b} logarithm, just like the @kbd{B} key except
5850 the arguments are taken in the opposite order.
5853 Produce a vector of integers from 1 to the integer on the top of
5857 @xref{Programming Answer 3, 3}. (@bullet{})
5859 (@bullet{}) @strong{Exercise 4.} Define a keyboard macro to compute
5860 the average (mean) value of a list of numbers.
5861 @xref{Programming Answer 4, 4}. (@bullet{})
5863 In many programs, some of the steps must execute several times.
5864 Calc has @dfn{looping} commands that allow this. Loops are useful
5865 inside keyboard macros, but actually work at any time.
5869 1: x^6 2: x^6 1: 360 x^2
5873 ' x^6 @key{RET} 4 Z < a d x @key{RET} Z >
5878 Here we have computed the fourth derivative of @expr{x^6} by
5879 enclosing a derivative command in a ``repeat loop'' structure.
5880 This structure pops a repeat count from the stack, then
5881 executes the body of the loop that many times.
5883 If you make a mistake while entering the body of the loop,
5884 type @w{@kbd{Z C-g}} to cancel the loop command.
5886 @cindex Fibonacci numbers
5887 Here's another example:
5896 1 @key{RET} @key{RET} 20 Z < @key{TAB} C-j + Z >
5901 The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
5902 numbers, respectively. (To see what's going on, try a few repetitions
5903 of the loop body by hand; @kbd{C-j}, also on the Line-Feed or @key{LFD}
5904 key if you have one, makes a copy of the number in level 2.)
5906 @cindex Golden ratio
5907 @cindex Phi, golden ratio
5908 A fascinating property of the Fibonacci numbers is that the @expr{n}th
5909 Fibonacci number can be found directly by computing
5910 @texline @math{\phi^n / \sqrt{5}}
5911 @infoline @expr{phi^n / sqrt(5)}
5912 and then rounding to the nearest integer, where
5913 @texline @math{\phi} (``phi''),
5914 @infoline @expr{phi},
5915 the ``golden ratio,'' is
5916 @texline @math{(1 + \sqrt{5}) / 2}.
5917 @infoline @expr{(1 + sqrt(5)) / 2}.
5918 (For convenience, this constant is available from the @code{phi}
5919 variable, or the @kbd{I H P} command.)
5923 1: 1.61803 1: 24476.0000409 1: 10945.9999817 1: 10946
5930 @cindex Continued fractions
5931 (@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction}
5933 @texline @math{\phi}
5934 @infoline @expr{phi}
5936 @texline @math{1 + 1/(1 + 1/(1 + 1/( \ldots )))}.
5937 @infoline @expr{1 + 1/(1 + 1/(1 + 1/( ...@: )))}.
5938 We can compute an approximate value by carrying this however far
5939 and then replacing the innermost
5940 @texline @math{1/( \ldots )}
5941 @infoline @expr{1/( ...@: )}
5943 @texline @math{\phi}
5944 @infoline @expr{phi}
5945 using a twenty-term continued fraction.
5946 @xref{Programming Answer 5, 5}. (@bullet{})
5948 (@bullet{}) @strong{Exercise 6.} Linear recurrences like the one for
5949 Fibonacci numbers can be expressed in terms of matrices. Given a
5950 vector @w{@expr{[a, b]}} determine a matrix which, when multiplied by this
5951 vector, produces the vector @expr{[b, c]}, where @expr{a}, @expr{b} and
5952 @expr{c} are three successive Fibonacci numbers. Now write a program
5953 that, given an integer @expr{n}, computes the @expr{n}th Fibonacci number
5954 using matrix arithmetic. @xref{Programming Answer 6, 6}. (@bullet{})
5956 @cindex Harmonic numbers
5957 A more sophisticated kind of loop is the @dfn{for} loop. Suppose
5958 we wish to compute the 20th ``harmonic'' number, which is equal to
5959 the sum of the reciprocals of the integers from 1 to 20.
5968 0 @key{RET} 1 @key{RET} 20 Z ( & + 1 Z )
5973 The ``for'' loop pops two numbers, the lower and upper limits, then
5974 repeats the body of the loop as an internal counter increases from
5975 the lower limit to the upper one. Just before executing the loop
5976 body, it pushes the current loop counter. When the loop body
5977 finishes, it pops the ``step,'' i.e., the amount by which to
5978 increment the loop counter. As you can see, our loop always
5981 This harmonic number function uses the stack to hold the running
5982 total as well as for the various loop housekeeping functions. If
5983 you find this disorienting, you can sum in a variable instead:
5987 1: 0 2: 1 . 1: 3.597739
5991 0 t 7 1 @key{RET} 20 Z ( & s + 7 1 Z ) r 7
5996 The @kbd{s +} command adds the top-of-stack into the value in a
5997 variable (and removes that value from the stack).
5999 It's worth noting that many jobs that call for a ``for'' loop can
6000 also be done more easily by Calc's high-level operations. Two
6001 other ways to compute harmonic numbers are to use vector mapping
6002 and reduction (@kbd{v x 20}, then @w{@kbd{V M &}}, then @kbd{V R +}),
6003 or to use the summation command @kbd{a +}. Both of these are
6004 probably easier than using loops. However, there are some
6005 situations where loops really are the way to go:
6007 (@bullet{}) @strong{Exercise 7.} Use a ``for'' loop to find the first
6008 harmonic number which is greater than 4.0.
6009 @xref{Programming Answer 7, 7}. (@bullet{})
6011 Of course, if we're going to be using variables in our programs,
6012 we have to worry about the programs clobbering values that the
6013 caller was keeping in those same variables. This is easy to
6018 . 1: 0.6667 1: 0.6667 3: 0.6667
6023 Z ` p 4 @key{RET} 2 @key{RET} 3 / s 7 s s a @key{RET} Z ' r 7 s r a @key{RET}
6028 When we type @kbd{Z `} (that's a back-quote character), Calc saves
6029 its mode settings and the contents of the ten ``quick variables''
6030 for later reference. When we type @kbd{Z '} (that's an apostrophe
6031 now), Calc restores those saved values. Thus the @kbd{p 4} and
6032 @kbd{s 7} commands have no effect outside this sequence. Wrapping
6033 this around the body of a keyboard macro ensures that it doesn't
6034 interfere with what the user of the macro was doing. Notice that
6035 the contents of the stack, and the values of named variables,
6036 survive past the @kbd{Z '} command.
6038 @cindex Bernoulli numbers, approximate
6039 The @dfn{Bernoulli numbers} are a sequence with the interesting
6040 property that all of the odd Bernoulli numbers are zero, and the
6041 even ones, while difficult to compute, can be roughly approximated
6043 @texline @math{\displaystyle{2 n! \over (2 \pi)^n}}.
6044 @infoline @expr{2 n!@: / (2 pi)^n}.
6045 Let's write a keyboard macro to compute (approximate) Bernoulli numbers.
6046 (Calc has a command, @kbd{k b}, to compute exact Bernoulli numbers, but
6047 this command is very slow for large @expr{n} since the higher Bernoulli
6048 numbers are very large fractions.)
6055 10 C-x ( @key{RET} 2 % Z [ @key{DEL} 0 Z : ' 2 $! / (2 pi)^$ @key{RET} = Z ] C-x )
6060 You can read @kbd{Z [} as ``then,'' @kbd{Z :} as ``else,'' and
6061 @kbd{Z ]} as ``end-if.'' There is no need for an explicit ``if''
6062 command. For the purposes of @w{@kbd{Z [}}, the condition is ``true''
6063 if the value it pops from the stack is a nonzero number, or ``false''
6064 if it pops zero or something that is not a number (like a formula).
6065 Here we take our integer argument modulo 2; this will be nonzero
6066 if we're asking for an odd Bernoulli number.
6068 The actual tenth Bernoulli number is @expr{5/66}.
6072 3: 0.0756823 1: 0 1: 0.25305 1: 0 1: 1.16659
6077 10 k b @key{RET} c f M-0 @key{DEL} 11 X @key{DEL} 12 X @key{DEL} 13 X @key{DEL} 14 X
6081 Just to exercise loops a bit more, let's compute a table of even
6086 3: [] 1: [0.10132, 0.03079, 0.02340, 0.033197, ...]
6091 [ ] 2 @key{RET} 30 Z ( X | 2 Z )
6096 The vertical-bar @kbd{|} is the vector-concatenation command. When
6097 we execute it, the list we are building will be in stack level 2
6098 (initially this is an empty list), and the next Bernoulli number
6099 will be in level 1. The effect is to append the Bernoulli number
6100 onto the end of the list. (To create a table of exact fractional
6101 Bernoulli numbers, just replace @kbd{X} with @kbd{k b} in the above
6102 sequence of keystrokes.)
6104 With loops and conditionals, you can program essentially anything
6105 in Calc. One other command that makes looping easier is @kbd{Z /},
6106 which takes a condition from the stack and breaks out of the enclosing
6107 loop if the condition is true (non-zero). You can use this to make
6108 ``while'' and ``until'' style loops.
6110 If you make a mistake when entering a keyboard macro, you can edit
6111 it using @kbd{Z E}. First, you must attach it to a key with @kbd{Z K}.
6112 One technique is to enter a throwaway dummy definition for the macro,
6113 then enter the real one in the edit command.
6117 1: 3 1: 3 Calc Macro Edit Mode.
6118 . . Original keys: 1 <return> 2 +
6125 C-x ( 1 @key{RET} 2 + C-x ) Z K h @key{RET} Z E h
6130 A keyboard macro is stored as a pure keystroke sequence. The
6131 @file{edmacro} package (invoked by @kbd{Z E}) scans along the
6132 macro and tries to decode it back into human-readable steps.
6133 Descriptions of the keystrokes are given as comments, which begin with
6134 @samp{;;}, and which are ignored when the edited macro is saved.
6135 Spaces and line breaks are also ignored when the edited macro is saved.
6136 To enter a space into the macro, type @code{SPC}. All the special
6137 characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC}, @code{DEL},
6138 and @code{NUL} must be written in all uppercase, as must the prefixes
6139 @code{C-} and @code{M-}.
6141 Let's edit in a new definition, for computing harmonic numbers.
6142 First, erase the four lines of the old definition. Then, type
6143 in the new definition (or use Emacs @kbd{M-w} and @kbd{C-y} commands
6144 to copy it from this page of the Info file; you can of course skip
6145 typing the comments, which begin with @samp{;;}).
6148 Z` ;; calc-kbd-push (Save local values)
6149 0 ;; calc digits (Push a zero onto the stack)
6150 st ;; calc-store-into (Store it in the following variable)
6151 1 ;; calc quick variable (Quick variable q1)
6152 1 ;; calc digits (Initial value for the loop)
6153 TAB ;; calc-roll-down (Swap initial and final)
6154 Z( ;; calc-kbd-for (Begin the "for" loop)
6155 & ;; calc-inv (Take the reciprocal)
6156 s+ ;; calc-store-plus (Add to the following variable)
6157 1 ;; calc quick variable (Quick variable q1)
6158 1 ;; calc digits (The loop step is 1)
6159 Z) ;; calc-kbd-end-for (End the "for" loop)
6160 sr ;; calc-recall (Recall the final accumulated value)
6161 1 ;; calc quick variable (Quick variable q1)
6162 Z' ;; calc-kbd-pop (Restore values)
6166 Press @kbd{C-c C-c} to finish editing and return to the Calculator.
6177 The @file{edmacro} package defines a handy @code{read-kbd-macro} command
6178 which reads the current region of the current buffer as a sequence of
6179 keystroke names, and defines that sequence on the @kbd{X}
6180 (and @kbd{C-x e}) key. Because this is so useful, Calc puts this
6181 command on the @kbd{C-x * m} key. Try reading in this macro in the
6182 following form: Press @kbd{C-@@} (or @kbd{C-@key{SPC}}) at
6183 one end of the text below, then type @kbd{C-x * m} at the other.
6195 (@bullet{}) @strong{Exercise 8.} A general algorithm for solving
6196 equations numerically is @dfn{Newton's Method}. Given the equation
6197 @expr{f(x) = 0} for any function @expr{f}, and an initial guess
6198 @expr{x_0} which is reasonably close to the desired solution, apply
6199 this formula over and over:
6203 new_x = x - f(x)/f'(x)
6208 $$ x_{\rm new} = x - {f(x) \over f^{\prime}(x)} $$
6213 where @expr{f'(x)} is the derivative of @expr{f}. The @expr{x}
6214 values will quickly converge to a solution, i.e., eventually
6215 @texline @math{x_{\rm new}}
6216 @infoline @expr{new_x}
6217 and @expr{x} will be equal to within the limits
6218 of the current precision. Write a program which takes a formula
6219 involving the variable @expr{x}, and an initial guess @expr{x_0},
6220 on the stack, and produces a value of @expr{x} for which the formula
6221 is zero. Use it to find a solution of
6222 @texline @math{\sin(\cos x) = 0.5}
6223 @infoline @expr{sin(cos(x)) = 0.5}
6224 near @expr{x = 4.5}. (Use angles measured in radians.) Note that
6225 the built-in @w{@kbd{a R}} (@code{calc-find-root}) command uses Newton's
6226 method when it is able. @xref{Programming Answer 8, 8}. (@bullet{})
6228 @cindex Digamma function
6229 @cindex Gamma constant, Euler's
6230 @cindex Euler's gamma constant
6231 (@bullet{}) @strong{Exercise 9.} The @dfn{digamma} function
6232 @texline @math{\psi(z) (``psi'')}
6233 @infoline @expr{psi(z)}
6234 is defined as the derivative of
6235 @texline @math{\ln \Gamma(z)}.
6236 @infoline @expr{ln(gamma(z))}.
6237 For large values of @expr{z}, it can be approximated by the infinite sum
6241 psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
6246 $$ \psi(z) \approx \ln z - {1\over2z} -
6247 \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}}
6254 @texline @math{\sum}
6255 @infoline @expr{sum}
6256 represents the sum over @expr{n} from 1 to infinity
6257 (or to some limit high enough to give the desired accuracy), and
6258 the @code{bern} function produces (exact) Bernoulli numbers.
6259 While this sum is not guaranteed to converge, in practice it is safe.
6260 An interesting mathematical constant is Euler's gamma, which is equal
6261 to about 0.5772. One way to compute it is by the formula,
6262 @texline @math{\gamma = -\psi(1)}.
6263 @infoline @expr{gamma = -psi(1)}.
6264 Unfortunately, 1 isn't a large enough argument
6265 for the above formula to work (5 is a much safer value for @expr{z}).
6266 Fortunately, we can compute
6267 @texline @math{\psi(1)}
6268 @infoline @expr{psi(1)}
6270 @texline @math{\psi(5)}
6271 @infoline @expr{psi(5)}
6272 using the recurrence
6273 @texline @math{\psi(z+1) = \psi(z) + {1 \over z}}.
6274 @infoline @expr{psi(z+1) = psi(z) + 1/z}.
6275 Your task: Develop a program to compute
6276 @texline @math{\psi(z)};
6277 @infoline @expr{psi(z)};
6278 it should ``pump up'' @expr{z}
6279 if necessary to be greater than 5, then use the above summation
6280 formula. Use looping commands to compute the sum. Use your function
6282 @texline @math{\gamma}
6283 @infoline @expr{gamma}
6284 to twelve decimal places. (Calc has a built-in command
6285 for Euler's constant, @kbd{I P}, which you can use to check your answer.)
6286 @xref{Programming Answer 9, 9}. (@bullet{})
6288 @cindex Polynomial, list of coefficients
6289 (@bullet{}) @strong{Exercise 10.} Given a polynomial in @expr{x} and
6290 a number @expr{m} on the stack, where the polynomial is of degree
6291 @expr{m} or less (i.e., does not have any terms higher than @expr{x^m}),
6292 write a program to convert the polynomial into a list-of-coefficients
6293 notation. For example, @expr{5 x^4 + (x + 1)^2} with @expr{m = 6}
6294 should produce the list @expr{[1, 2, 1, 0, 5, 0, 0]}. Also develop
6295 a way to convert from this form back to the standard algebraic form.
6296 @xref{Programming Answer 10, 10}. (@bullet{})
6299 (@bullet{}) @strong{Exercise 11.} The @dfn{Stirling numbers of the
6300 first kind} are defined by the recurrences,
6304 s(n,n) = 1 for n >= 0,
6305 s(n,0) = 0 for n > 0,
6306 s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1.
6311 $$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr
6312 s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr
6313 s(n+1,m) &= s(n,m-1) - n \, s(n,m) \qquad
6314 \hbox{for } n \ge m \ge 1.}
6318 (These numbers are also sometimes written $\displaystyle{n \brack m}$.)
6321 This can be implemented using a @dfn{recursive} program in Calc; the
6322 program must invoke itself in order to calculate the two righthand
6323 terms in the general formula. Since it always invokes itself with
6324 ``simpler'' arguments, it's easy to see that it must eventually finish
6325 the computation. Recursion is a little difficult with Emacs keyboard
6326 macros since the macro is executed before its definition is complete.
6327 So here's the recommended strategy: Create a ``dummy macro'' and assign
6328 it to a key with, e.g., @kbd{Z K s}. Now enter the true definition,
6329 using the @kbd{z s} command to call itself recursively, then assign it
6330 to the same key with @kbd{Z K s}. Now the @kbd{z s} command will run
6331 the complete recursive program. (Another way is to use @w{@kbd{Z E}}
6332 or @kbd{C-x * m} (@code{read-kbd-macro}) to read the whole macro at once,
6333 thus avoiding the ``training'' phase.) The task: Write a program
6334 that computes Stirling numbers of the first kind, given @expr{n} and
6335 @expr{m} on the stack. Test it with @emph{small} inputs like
6336 @expr{s(4,2)}. (There is a built-in command for Stirling numbers,
6337 @kbd{k s}, which you can use to check your answers.)
6338 @xref{Programming Answer 11, 11}. (@bullet{})
6340 The programming commands we've seen in this part of the tutorial
6341 are low-level, general-purpose operations. Often you will find
6342 that a higher-level function, such as vector mapping or rewrite
6343 rules, will do the job much more easily than a detailed, step-by-step
6346 (@bullet{}) @strong{Exercise 12.} Write another program for
6347 computing Stirling numbers of the first kind, this time using
6348 rewrite rules. Once again, @expr{n} and @expr{m} should be taken
6349 from the stack. @xref{Programming Answer 12, 12}. (@bullet{})
6354 This ends the tutorial section of the Calc manual. Now you know enough
6355 about Calc to use it effectively for many kinds of calculations. But
6356 Calc has many features that were not even touched upon in this tutorial.
6358 The rest of this manual tells the whole story.
6360 @c Volume II of this manual, the @dfn{Calc Reference}, tells the whole story.
6363 @node Answers to Exercises, , Programming Tutorial, Tutorial
6364 @section Answers to Exercises
6367 This section includes answers to all the exercises in the Calc tutorial.
6370 * RPN Answer 1:: 1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -
6371 * RPN Answer 2:: 2*4 + 7*9.5 + 5/4
6372 * RPN Answer 3:: Operating on levels 2 and 3
6373 * RPN Answer 4:: Joe's complex problems
6374 * Algebraic Answer 1:: Simulating Q command
6375 * Algebraic Answer 2:: Joe's algebraic woes
6376 * Algebraic Answer 3:: 1 / 0
6377 * Modes Answer 1:: 3#0.1 = 3#0.0222222?
6378 * Modes Answer 2:: 16#f.e8fe15
6379 * Modes Answer 3:: Joe's rounding bug
6380 * Modes Answer 4:: Why floating point?
6381 * Arithmetic Answer 1:: Why the \ command?
6382 * Arithmetic Answer 2:: Tripping up the B command
6383 * Vector Answer 1:: Normalizing a vector
6384 * Vector Answer 2:: Average position
6385 * Matrix Answer 1:: Row and column sums
6386 * Matrix Answer 2:: Symbolic system of equations
6387 * Matrix Answer 3:: Over-determined system
6388 * List Answer 1:: Powers of two
6389 * List Answer 2:: Least-squares fit with matrices
6390 * List Answer 3:: Geometric mean
6391 * List Answer 4:: Divisor function
6392 * List Answer 5:: Duplicate factors
6393 * List Answer 6:: Triangular list
6394 * List Answer 7:: Another triangular list
6395 * List Answer 8:: Maximum of Bessel function
6396 * List Answer 9:: Integers the hard way
6397 * List Answer 10:: All elements equal
6398 * List Answer 11:: Estimating pi with darts
6399 * List Answer 12:: Estimating pi with matchsticks
6400 * List Answer 13:: Hash codes
6401 * List Answer 14:: Random walk
6402 * Types Answer 1:: Square root of pi times rational
6403 * Types Answer 2:: Infinities
6404 * Types Answer 3:: What can "nan" be?
6405 * Types Answer 4:: Abbey Road
6406 * Types Answer 5:: Friday the 13th
6407 * Types Answer 6:: Leap years
6408 * Types Answer 7:: Erroneous donut
6409 * Types Answer 8:: Dividing intervals
6410 * Types Answer 9:: Squaring intervals
6411 * Types Answer 10:: Fermat's primality test
6412 * Types Answer 11:: pi * 10^7 seconds
6413 * Types Answer 12:: Abbey Road on CD
6414 * Types Answer 13:: Not quite pi * 10^7 seconds
6415 * Types Answer 14:: Supercomputers and c
6416 * Types Answer 15:: Sam the Slug
6417 * Algebra Answer 1:: Squares and square roots
6418 * Algebra Answer 2:: Building polynomial from roots
6419 * Algebra Answer 3:: Integral of x sin(pi x)
6420 * Algebra Answer 4:: Simpson's rule
6421 * Rewrites Answer 1:: Multiplying by conjugate
6422 * Rewrites Answer 2:: Alternative fib rule
6423 * Rewrites Answer 3:: Rewriting opt(a) + opt(b) x
6424 * Rewrites Answer 4:: Sequence of integers
6425 * Rewrites Answer 5:: Number of terms in sum
6426 * Rewrites Answer 6:: Truncated Taylor series
6427 * Programming Answer 1:: Fresnel's C(x)
6428 * Programming Answer 2:: Negate third stack element
6429 * Programming Answer 3:: Compute sin(x) / x, etc.
6430 * Programming Answer 4:: Average value of a list
6431 * Programming Answer 5:: Continued fraction phi
6432 * Programming Answer 6:: Matrix Fibonacci numbers
6433 * Programming Answer 7:: Harmonic number greater than 4
6434 * Programming Answer 8:: Newton's method
6435 * Programming Answer 9:: Digamma function
6436 * Programming Answer 10:: Unpacking a polynomial
6437 * Programming Answer 11:: Recursive Stirling numbers
6438 * Programming Answer 12:: Stirling numbers with rewrites
6441 @c The following kludgery prevents the individual answers from
6442 @c being entered on the table of contents.
6444 \global\let\oldwrite=\write
6445 \gdef\skipwrite#1#2{\let\write=\oldwrite}
6446 \global\let\oldchapternofonts=\chapternofonts
6447 \gdef\chapternofonts{\let\write=\skipwrite\oldchapternofonts}
6450 @node RPN Answer 1, RPN Answer 2, Answers to Exercises, Answers to Exercises
6451 @subsection RPN Tutorial Exercise 1
6454 @kbd{1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -}
6457 @texline @math{1 - (2 \times (3 + 4)) = -13}.
6458 @infoline @expr{1 - (2 * (3 + 4)) = -13}.
6460 @node RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises
6461 @subsection RPN Tutorial Exercise 2
6464 @texline @math{2\times4 + 7\times9.5 + {5\over4} = 75.75}
6465 @infoline @expr{2*4 + 7*9.5 + 5/4 = 75.75}
6467 After computing the intermediate term
6468 @texline @math{2\times4 = 8},
6469 @infoline @expr{2*4 = 8},
6470 you can leave that result on the stack while you compute the second
6471 term. With both of these results waiting on the stack you can then
6472 compute the final term, then press @kbd{+ +} to add everything up.
6481 2 @key{RET} 4 * 7 @key{RET} 9.5 *
6488 4: 8 3: 8 2: 8 1: 75.75
6489 3: 66.5 2: 66.5 1: 67.75 .
6498 Alternatively, you could add the first two terms before going on
6499 with the third term.
6503 2: 8 1: 74.5 3: 74.5 2: 74.5 1: 75.75
6504 1: 66.5 . 2: 5 1: 1.25 .
6508 ... + 5 @key{RET} 4 / +
6512 On an old-style RPN calculator this second method would have the
6513 advantage of using only three stack levels. But since Calc's stack
6514 can grow arbitrarily large this isn't really an issue. Which method
6515 you choose is purely a matter of taste.
6517 @node RPN Answer 3, RPN Answer 4, RPN Answer 2, Answers to Exercises
6518 @subsection RPN Tutorial Exercise 3
6521 The @key{TAB} key provides a way to operate on the number in level 2.
6525 3: 10 3: 10 4: 10 3: 10 3: 10
6526 2: 20 2: 30 3: 30 2: 30 2: 21
6527 1: 30 1: 20 2: 20 1: 21 1: 30
6531 @key{TAB} 1 + @key{TAB}
6535 Similarly, @kbd{M-@key{TAB}} gives you access to the number in level 3.
6539 3: 10 3: 21 3: 21 3: 30 3: 11
6540 2: 21 2: 30 2: 30 2: 11 2: 21
6541 1: 30 1: 10 1: 11 1: 21 1: 30
6544 M-@key{TAB} 1 + M-@key{TAB} M-@key{TAB}
6548 @node RPN Answer 4, Algebraic Answer 1, RPN Answer 3, Answers to Exercises
6549 @subsection RPN Tutorial Exercise 4
6552 Either @kbd{( 2 , 3 )} or @kbd{( 2 @key{SPC} 3 )} would have worked,
6553 but using both the comma and the space at once yields:
6557 1: ( ... 2: ( ... 1: (2, ... 2: (2, ... 2: (2, ...
6558 . 1: 2 . 1: (2, ... 1: (2, 3)
6565 Joe probably tried to type @kbd{@key{TAB} @key{DEL}} to swap the
6566 extra incomplete object to the top of the stack and delete it.
6567 But a feature of Calc is that @key{DEL} on an incomplete object
6568 deletes just one component out of that object, so he had to press
6569 @key{DEL} twice to finish the job.
6573 2: (2, ... 2: (2, 3) 2: (2, 3) 1: (2, 3)
6574 1: (2, 3) 1: (2, ... 1: ( ... .
6577 @key{TAB} @key{DEL} @key{DEL}
6581 (As it turns out, deleting the second-to-top stack entry happens often
6582 enough that Calc provides a special key, @kbd{M-@key{DEL}}, to do just that.
6583 @kbd{M-@key{DEL}} is just like @kbd{@key{TAB} @key{DEL}}, except that it doesn't exhibit
6584 the ``feature'' that tripped poor Joe.)
6586 @node Algebraic Answer 1, Algebraic Answer 2, RPN Answer 4, Answers to Exercises
6587 @subsection Algebraic Entry Tutorial Exercise 1
6590 Type @kbd{' sqrt($) @key{RET}}.
6592 If the @kbd{Q} key is broken, you could use @kbd{' $^0.5 @key{RET}}.
6593 Or, RPN style, @kbd{0.5 ^}.
6595 (Actually, @samp{$^1:2}, using the fraction one-half as the power, is
6596 a closer equivalent, since @samp{9^0.5} yields @expr{3.0} whereas
6597 @samp{sqrt(9)} and @samp{9^1:2} yield the exact integer @expr{3}.)
6599 @node Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1, Answers to Exercises
6600 @subsection Algebraic Entry Tutorial Exercise 2
6603 In the formula @samp{2 x (1+y)}, @samp{x} was interpreted as a function
6604 name with @samp{1+y} as its argument. Assigning a value to a variable
6605 has no relation to a function by the same name. Joe needed to use an
6606 explicit @samp{*} symbol here: @samp{2 x*(1+y)}.
6608 @node Algebraic Answer 3, Modes Answer 1, Algebraic Answer 2, Answers to Exercises
6609 @subsection Algebraic Entry Tutorial Exercise 3
6612 The result from @kbd{1 @key{RET} 0 /} will be the formula @expr{1 / 0}.
6613 The ``function'' @samp{/} cannot be evaluated when its second argument
6614 is zero, so it is left in symbolic form. When you now type @kbd{0 *},
6615 the result will be zero because Calc uses the general rule that ``zero
6616 times anything is zero.''
6618 @c [fix-ref Infinities]
6619 The @kbd{m i} command enables an @dfn{Infinite mode} in which @expr{1 / 0}
6620 results in a special symbol that represents ``infinity.'' If you
6621 multiply infinity by zero, Calc uses another special new symbol to
6622 show that the answer is ``indeterminate.'' @xref{Infinities}, for
6623 further discussion of infinite and indeterminate values.
6625 @node Modes Answer 1, Modes Answer 2, Algebraic Answer 3, Answers to Exercises
6626 @subsection Modes Tutorial Exercise 1
6629 Calc always stores its numbers in decimal, so even though one-third has
6630 an exact base-3 representation (@samp{3#0.1}), it is still stored as
6631 0.3333333 (chopped off after 12 or however many decimal digits) inside
6632 the calculator's memory. When this inexact number is converted back
6633 to base 3 for display, it may still be slightly inexact. When we
6634 multiply this number by 3, we get 0.999999, also an inexact value.
6636 When Calc displays a number in base 3, it has to decide how many digits
6637 to show. If the current precision is 12 (decimal) digits, that corresponds
6638 to @samp{12 / log10(3) = 25.15} base-3 digits. Because 25.15 is not an
6639 exact integer, Calc shows only 25 digits, with the result that stored
6640 numbers carry a little bit of extra information that may not show up on
6641 the screen. When Joe entered @samp{3#0.2}, the stored number 0.666666
6642 happened to round to a pleasing value when it lost that last 0.15 of a
6643 digit, but it was still inexact in Calc's memory. When he divided by 2,
6644 he still got the dreaded inexact value 0.333333. (Actually, he divided
6645 0.666667 by 2 to get 0.333334, which is why he got something a little
6646 higher than @code{3#0.1} instead of a little lower.)
6648 If Joe didn't want to be bothered with all this, he could have typed
6649 @kbd{M-24 d n} to display with one less digit than the default. (If
6650 you give @kbd{d n} a negative argument, it uses default-minus-that,
6651 so @kbd{M-- d n} would be an easier way to get the same effect.) Those
6652 inexact results would still be lurking there, but they would now be
6653 rounded to nice, natural-looking values for display purposes. (Remember,
6654 @samp{0.022222} in base 3 is like @samp{0.099999} in base 10; rounding
6655 off one digit will round the number up to @samp{0.1}.) Depending on the
6656 nature of your work, this hiding of the inexactness may be a benefit or
6657 a danger. With the @kbd{d n} command, Calc gives you the choice.
6659 Incidentally, another consequence of all this is that if you type
6660 @kbd{M-30 d n} to display more digits than are ``really there,''
6661 you'll see garbage digits at the end of the number. (In decimal
6662 display mode, with decimally-stored numbers, these garbage digits are
6663 always zero so they vanish and you don't notice them.) Because Calc
6664 rounds off that 0.15 digit, there is the danger that two numbers could
6665 be slightly different internally but still look the same. If you feel
6666 uneasy about this, set the @kbd{d n} precision to be a little higher
6667 than normal; you'll get ugly garbage digits, but you'll always be able
6668 to tell two distinct numbers apart.
6670 An interesting side note is that most computers store their
6671 floating-point numbers in binary, and convert to decimal for display.
6672 Thus everyday programs have the same problem: Decimal 0.1 cannot be
6673 represented exactly in binary (try it: @kbd{0.1 d 2}), so @samp{0.1 * 10}
6674 comes out as an inexact approximation to 1 on some machines (though
6675 they generally arrange to hide it from you by rounding off one digit as
6676 we did above). Because Calc works in decimal instead of binary, you can
6677 be sure that numbers that look exact @emph{are} exact as long as you stay
6678 in decimal display mode.
6680 It's not hard to show that any number that can be represented exactly
6681 in binary, octal, or hexadecimal is also exact in decimal, so the kinds
6682 of problems we saw in this exercise are likely to be severe only when
6683 you use a relatively unusual radix like 3.
6685 @node Modes Answer 2, Modes Answer 3, Modes Answer 1, Answers to Exercises
6686 @subsection Modes Tutorial Exercise 2
6688 If the radix is 15 or higher, we can't use the letter @samp{e} to mark
6689 the exponent because @samp{e} is interpreted as a digit. When Calc
6690 needs to display scientific notation in a high radix, it writes
6691 @samp{16#F.E8F*16.^15}. You can enter a number like this as an
6692 algebraic entry. Also, pressing @kbd{e} without any digits before it
6693 normally types @kbd{1e}, but in a high radix it types @kbd{16.^} and
6694 puts you in algebraic entry: @kbd{16#f.e8f @key{RET} e 15 @key{RET} *} is another
6695 way to enter this number.
6697 The reason Calc puts a decimal point in the @samp{16.^} is to prevent
6698 huge integers from being generated if the exponent is large (consider
6699 @samp{16#1.23*16^1000}, where we compute @samp{16^1000} as a giant
6700 exact integer and then throw away most of the digits when we multiply
6701 it by the floating-point @samp{16#1.23}). While this wouldn't normally
6702 matter for display purposes, it could give you a nasty surprise if you
6703 copied that number into a file and later moved it back into Calc.
6705 @node Modes Answer 3, Modes Answer 4, Modes Answer 2, Answers to Exercises
6706 @subsection Modes Tutorial Exercise 3
6709 The answer he got was @expr{0.5000000000006399}.
6711 The problem is not that the square operation is inexact, but that the
6712 sine of 45 that was already on the stack was accurate to only 12 places.
6713 Arbitrary-precision calculations still only give answers as good as
6716 The real problem is that there is no 12-digit number which, when
6717 squared, comes out to 0.5 exactly. The @kbd{f [} and @kbd{f ]}
6718 commands decrease or increase a number by one unit in the last
6719 place (according to the current precision). They are useful for
6720 determining facts like this.
6724 1: 0.707106781187 1: 0.500000000001
6734 1: 0.707106781187 1: 0.707106781186 1: 0.499999999999
6741 A high-precision calculation must be carried out in high precision
6742 all the way. The only number in the original problem which was known
6743 exactly was the quantity 45 degrees, so the precision must be raised
6744 before anything is done after the number 45 has been entered in order
6745 for the higher precision to be meaningful.
6747 @node Modes Answer 4, Arithmetic Answer 1, Modes Answer 3, Answers to Exercises
6748 @subsection Modes Tutorial Exercise 4
6751 Many calculations involve real-world quantities, like the width and
6752 height of a piece of wood or the volume of a jar. Such quantities
6753 can't be measured exactly anyway, and if the data that is input to
6754 a calculation is inexact, doing exact arithmetic on it is a waste
6757 Fractions become unwieldy after too many calculations have been
6758 done with them. For example, the sum of the reciprocals of the
6759 integers from 1 to 10 is 7381:2520. The sum from 1 to 30 is
6760 9304682830147:2329089562800. After a point it will take a long
6761 time to add even one more term to this sum, but a floating-point
6762 calculation of the sum will not have this problem.
6764 Also, rational numbers cannot express the results of all calculations.
6765 There is no fractional form for the square root of two, so if you type
6766 @w{@kbd{2 Q}}, Calc has no choice but to give you a floating-point answer.
6768 @node Arithmetic Answer 1, Arithmetic Answer 2, Modes Answer 4, Answers to Exercises
6769 @subsection Arithmetic Tutorial Exercise 1
6772 Dividing two integers that are larger than the current precision may
6773 give a floating-point result that is inaccurate even when rounded
6774 down to an integer. Consider @expr{123456789 / 2} when the current
6775 precision is 6 digits. The true answer is @expr{61728394.5}, but
6776 with a precision of 6 this will be rounded to
6777 @texline @math{12345700.0/2.0 = 61728500.0}.
6778 @infoline @expr{12345700.@: / 2.@: = 61728500.}.
6779 The result, when converted to an integer, will be off by 106.
6781 Here are two solutions: Raise the precision enough that the
6782 floating-point round-off error is strictly to the right of the
6783 decimal point. Or, convert to Fraction mode so that @expr{123456789 / 2}
6784 produces the exact fraction @expr{123456789:2}, which can be rounded
6785 down by the @kbd{F} command without ever switching to floating-point
6788 @node Arithmetic Answer 2, Vector Answer 1, Arithmetic Answer 1, Answers to Exercises
6789 @subsection Arithmetic Tutorial Exercise 2
6792 @kbd{27 @key{RET} 9 B} could give the exact result @expr{3:2}, but it
6793 does a floating-point calculation instead and produces @expr{1.5}.
6795 Calc will find an exact result for a logarithm if the result is an integer
6796 or (when in Fraction mode) the reciprocal of an integer. But there is
6797 no efficient way to search the space of all possible rational numbers
6798 for an exact answer, so Calc doesn't try.
6800 @node Vector Answer 1, Vector Answer 2, Arithmetic Answer 2, Answers to Exercises
6801 @subsection Vector Tutorial Exercise 1
6804 Duplicate the vector, compute its length, then divide the vector
6805 by its length: @kbd{@key{RET} A /}.
6809 1: [1, 2, 3] 2: [1, 2, 3] 1: [0.27, 0.53, 0.80] 1: 1.
6810 . 1: 3.74165738677 . .
6817 The final @kbd{A} command shows that the normalized vector does
6818 indeed have unit length.
6820 @node Vector Answer 2, Matrix Answer 1, Vector Answer 1, Answers to Exercises
6821 @subsection Vector Tutorial Exercise 2
6824 The average position is equal to the sum of the products of the
6825 positions times their corresponding probabilities. This is the
6826 definition of the dot product operation. So all you need to do
6827 is to put the two vectors on the stack and press @kbd{*}.
6829 @node Matrix Answer 1, Matrix Answer 2, Vector Answer 2, Answers to Exercises
6830 @subsection Matrix Tutorial Exercise 1
6833 The trick is to multiply by a vector of ones. Use @kbd{r 4 [1 1 1] *} to
6834 get the row sum. Similarly, use @kbd{[1 1] r 4 *} to get the column sum.
6836 @node Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to Exercises
6837 @subsection Matrix Tutorial Exercise 2
6849 $$ \eqalign{ x &+ a y = 6 \cr
6855 Just enter the righthand side vector, then divide by the lefthand side
6860 1: [6, 10] 2: [6, 10] 1: [4 a / (a - b) + 6, 4 / (b - a) ]
6865 ' [6 10] @key{RET} ' [1 a; 1 b] @key{RET} /
6869 This can be made more readable using @kbd{d B} to enable Big display
6875 1: [----- + 6, -----]
6880 Type @kbd{d N} to return to Normal display mode afterwards.
6882 @node Matrix Answer 3, List Answer 1, Matrix Answer 2, Answers to Exercises
6883 @subsection Matrix Tutorial Exercise 3
6887 @texline @math{A^T A \, X = A^T B},
6888 @infoline @expr{trn(A) * A * X = trn(A) * B},
6890 @texline @math{A' = A^T A}
6891 @infoline @expr{A2 = trn(A) * A}
6893 @texline @math{B' = A^T B};
6894 @infoline @expr{B2 = trn(A) * B};
6895 now, we have a system
6896 @texline @math{A' X = B'}
6897 @infoline @expr{A2 * X = B2}
6898 which we can solve using Calc's @samp{/} command.
6912 $$ \openup1\jot \tabskip=0pt plus1fil
6913 \halign to\displaywidth{\tabskip=0pt
6914 $\hfil#$&$\hfil{}#{}$&
6915 $\hfil#$&$\hfil{}#{}$&
6916 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
6920 2a&+&4b&+&6c&=11 \cr}
6925 The first step is to enter the coefficient matrix. We'll store it in
6926 quick variable number 7 for later reference. Next, we compute the
6933 1: [ [ 1, 2, 3 ] 2: [ [ 1, 4, 7, 2 ] 1: [57, 84, 96]
6934 [ 4, 5, 6 ] [ 2, 5, 6, 4 ] .
6935 [ 7, 6, 0 ] [ 3, 6, 0, 6 ] ]
6936 [ 2, 4, 6 ] ] 1: [6, 2, 3, 11]
6939 ' [1 2 3; 4 5 6; 7 6 0; 2 4 6] @key{RET} s 7 v t [6 2 3 11] *
6944 Now we compute the matrix
6951 2: [57, 84, 96] 1: [-11.64, 14.08, -3.64]
6952 1: [ [ 70, 72, 39 ] .
6962 (The actual computed answer will be slightly inexact due to
6965 Notice that the answers are similar to those for the
6966 @texline @math{3\times3}
6968 system solved in the text. That's because the fourth equation that was
6969 added to the system is almost identical to the first one multiplied
6970 by two. (If it were identical, we would have gotten the exact same
6972 @texline @math{4\times3}
6974 system would be equivalent to the original
6975 @texline @math{3\times3}
6979 Since the first and fourth equations aren't quite equivalent, they
6980 can't both be satisfied at once. Let's plug our answers back into
6981 the original system of equations to see how well they match.
6985 2: [-11.64, 14.08, -3.64] 1: [5.6, 2., 3., 11.2]
6997 This is reasonably close to our original @expr{B} vector,
6998 @expr{[6, 2, 3, 11]}.
7000 @node List Answer 1, List Answer 2, Matrix Answer 3, Answers to Exercises
7001 @subsection List Tutorial Exercise 1
7004 We can use @kbd{v x} to build a vector of integers. This needs to be
7005 adjusted to get the range of integers we desire. Mapping @samp{-}
7006 across the vector will accomplish this, although it turns out the
7007 plain @samp{-} key will work just as well.
7012 1: [1, 2, 3, 4, 5, 6, 7, 8, 9] 1: [-4, -3, -2, -1, 0, 1, 2, 3, 4]
7015 2 v x 9 @key{RET} 5 V M - or 5 -
7020 Now we use @kbd{V M ^} to map the exponentiation operator across the
7025 1: [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
7032 @node List Answer 2, List Answer 3, List Answer 1, Answers to Exercises
7033 @subsection List Tutorial Exercise 2
7036 Given @expr{x} and @expr{y} vectors in quick variables 1 and 2 as before,
7037 the first job is to form the matrix that describes the problem.
7046 $$ m \times x + b \times 1 = y $$
7051 @texline @math{19\times2}
7053 matrix with our @expr{x} vector as one column and
7054 ones as the other column. So, first we build the column of ones, then
7055 we combine the two columns to form our @expr{A} matrix.
7059 2: [1.34, 1.41, 1.49, ... ] 1: [ [ 1.34, 1 ]
7060 1: [1, 1, 1, ...] [ 1.41, 1 ]
7064 r 1 1 v b 19 @key{RET} M-2 v p v t s 3
7070 @texline @math{A^T y}
7071 @infoline @expr{trn(A) * y}
7073 @texline @math{A^T A}
7074 @infoline @expr{trn(A) * A}
7079 1: [33.36554, 13.613] 2: [33.36554, 13.613]
7080 . 1: [ [ 98.0003, 41.63 ]
7084 v t r 2 * r 3 v t r 3 *
7089 (Hey, those numbers look familiar!)
7093 1: [0.52141679, -0.425978]
7100 Since we were solving equations of the form
7101 @texline @math{m \times x + b \times 1 = y},
7102 @infoline @expr{m*x + b*1 = y},
7103 these numbers should be @expr{m} and @expr{b}, respectively. Sure
7104 enough, they agree exactly with the result computed using @kbd{V M} and
7107 The moral of this story: @kbd{V M} and @kbd{V R} will probably solve
7108 your problem, but there is often an easier way using the higher-level
7109 arithmetic functions!
7111 @c [fix-ref Curve Fitting]
7112 In fact, there is a built-in @kbd{a F} command that does least-squares
7113 fits. @xref{Curve Fitting}.
7115 @node List Answer 3, List Answer 4, List Answer 2, Answers to Exercises
7116 @subsection List Tutorial Exercise 3
7119 Move to one end of the list and press @kbd{C-@@} (or @kbd{C-@key{SPC}} or
7120 whatever) to set the mark, then move to the other end of the list
7121 and type @w{@kbd{C-x * g}}.
7125 1: [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5]
7130 To make things interesting, let's assume we don't know at a glance
7131 how many numbers are in this list. Then we could type:
7135 2: [2.3, 6, 22, ... ] 2: [2.3, 6, 22, ... ]
7136 1: [2.3, 6, 22, ... ] 1: 126356422.5
7146 2: 126356422.5 2: 126356422.5 1: 7.94652913734
7147 1: [2.3, 6, 22, ... ] 1: 9 .
7155 (The @kbd{I ^} command computes the @var{n}th root of a number.
7156 You could also type @kbd{& ^} to take the reciprocal of 9 and
7157 then raise the number to that power.)
7159 @node List Answer 4, List Answer 5, List Answer 3, Answers to Exercises
7160 @subsection List Tutorial Exercise 4
7163 A number @expr{j} is a divisor of @expr{n} if
7164 @texline @math{n \mathbin{\hbox{\code{\%}}} j = 0}.
7165 @infoline @samp{n % j = 0}.
7166 The first step is to get a vector that identifies the divisors.
7170 2: 30 2: [0, 0, 0, 2, ...] 1: [1, 1, 1, 0, ...]
7171 1: [1, 2, 3, 4, ...] 1: 0 .
7174 30 @key{RET} v x 30 @key{RET} s 1 V M % 0 V M a = s 2
7179 This vector has 1's marking divisors of 30 and 0's marking non-divisors.
7181 The zeroth divisor function is just the total number of divisors.
7182 The first divisor function is the sum of the divisors.
7187 2: [1, 2, 3, 4, ...] 1: [1, 2, 3, 0, ...] 1: 72
7188 1: [1, 1, 1, 0, ...] . .
7191 V R + r 1 r 2 V M * V R +
7196 Once again, the last two steps just compute a dot product for which
7197 a simple @kbd{*} would have worked equally well.
7199 @node List Answer 5, List Answer 6, List Answer 4, Answers to Exercises
7200 @subsection List Tutorial Exercise 5
7203 The obvious first step is to obtain the list of factors with @kbd{k f}.
7204 This list will always be in sorted order, so if there are duplicates
7205 they will be right next to each other. A suitable method is to compare
7206 the list with a copy of itself shifted over by one.
7210 1: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19, 0]
7211 . 1: [3, 7, 7, 7, 19, 0] 1: [0, 3, 7, 7, 7, 19]
7214 19551 k f @key{RET} 0 | @key{TAB} 0 @key{TAB} |
7221 1: [0, 0, 1, 1, 0, 0] 1: 2 1: 0
7229 Note that we have to arrange for both vectors to have the same length
7230 so that the mapping operation works; no prime factor will ever be
7231 zero, so adding zeros on the left and right is safe. From then on
7232 the job is pretty straightforward.
7234 Incidentally, Calc provides the
7235 @texline @dfn{M@"obius} @math{\mu}
7236 @infoline @dfn{Moebius mu}
7237 function which is zero if and only if its argument is square-free. It
7238 would be a much more convenient way to do the above test in practice.
7240 @node List Answer 6, List Answer 7, List Answer 5, Answers to Exercises
7241 @subsection List Tutorial Exercise 6
7244 First use @kbd{v x 6 @key{RET}} to get a list of integers, then @kbd{V M v x}
7245 to get a list of lists of integers!
7247 @node List Answer 7, List Answer 8, List Answer 6, Answers to Exercises
7248 @subsection List Tutorial Exercise 7
7251 Here's one solution. First, compute the triangular list from the previous
7252 exercise and type @kbd{1 -} to subtract one from all the elements.
7265 The numbers down the lefthand edge of the list we desire are called
7266 the ``triangular numbers'' (now you know why!). The @expr{n}th
7267 triangular number is the sum of the integers from 1 to @expr{n}, and
7268 can be computed directly by the formula
7269 @texline @math{n (n+1) \over 2}.
7270 @infoline @expr{n * (n+1) / 2}.
7274 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7275 1: [0, 1, 2, 3, 4, 5] 1: [0, 1, 3, 6, 10, 15]
7278 v x 6 @key{RET} 1 - V M ' $ ($+1)/2 @key{RET}
7283 Adding this list to the above list of lists produces the desired
7292 [10, 11, 12, 13, 14],
7293 [15, 16, 17, 18, 19, 20] ]
7300 If we did not know the formula for triangular numbers, we could have
7301 computed them using a @kbd{V U +} command. We could also have
7302 gotten them the hard way by mapping a reduction across the original
7307 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7308 1: [ [0], [0, 1], ... ] 1: [0, 1, 3, 6, 10, 15]
7316 (This means ``map a @kbd{V R +} command across the vector,'' and
7317 since each element of the main vector is itself a small vector,
7318 @kbd{V R +} computes the sum of its elements.)
7320 @node List Answer 8, List Answer 9, List Answer 7, Answers to Exercises
7321 @subsection List Tutorial Exercise 8
7324 The first step is to build a list of values of @expr{x}.
7328 1: [1, 2, 3, ..., 21] 1: [0, 1, 2, ..., 20] 1: [0, 0.25, 0.5, ..., 5]
7331 v x 21 @key{RET} 1 - 4 / s 1
7335 Next, we compute the Bessel function values.
7339 1: [0., 0.124, 0.242, ..., -0.328]
7342 V M ' besJ(1,$) @key{RET}
7347 (Another way to do this would be @kbd{1 @key{TAB} V M f j}.)
7349 A way to isolate the maximum value is to compute the maximum using
7350 @kbd{V R X}, then compare all the Bessel values with that maximum.
7354 2: [0., 0.124, 0.242, ... ] 1: [0, 0, 0, ... ] 2: [0, 0, 0, ... ]
7358 @key{RET} V R X V M a = @key{RET} V R + @key{DEL}
7363 It's a good idea to verify, as in the last step above, that only
7364 one value is equal to the maximum. (After all, a plot of
7365 @texline @math{\sin x}
7366 @infoline @expr{sin(x)}
7367 might have many points all equal to the maximum value, 1.)
7369 The vector we have now has a single 1 in the position that indicates
7370 the maximum value of @expr{x}. Now it is a simple matter to convert
7371 this back into the corresponding value itself.
7375 2: [0, 0, 0, ... ] 1: [0, 0., 0., ... ] 1: 1.75
7376 1: [0, 0.25, 0.5, ... ] . .
7383 If @kbd{a =} had produced more than one @expr{1} value, this method
7384 would have given the sum of all maximum @expr{x} values; not very
7385 useful! In this case we could have used @kbd{v m} (@code{calc-mask-vector})
7386 instead. This command deletes all elements of a ``data'' vector that
7387 correspond to zeros in a ``mask'' vector, leaving us with, in this
7388 example, a vector of maximum @expr{x} values.
7390 The built-in @kbd{a X} command maximizes a function using more
7391 efficient methods. Just for illustration, let's use @kbd{a X}
7392 to maximize @samp{besJ(1,x)} over this same interval.
7396 2: besJ(1, x) 1: [1.84115, 0.581865]
7400 ' besJ(1,x), [0..5] @key{RET} a X x @key{RET}
7405 The output from @kbd{a X} is a vector containing the value of @expr{x}
7406 that maximizes the function, and the function's value at that maximum.
7407 As you can see, our simple search got quite close to the right answer.
7409 @node List Answer 9, List Answer 10, List Answer 8, Answers to Exercises
7410 @subsection List Tutorial Exercise 9
7413 Step one is to convert our integer into vector notation.
7417 1: 25129925999 3: 25129925999
7419 1: [11, 10, 9, ..., 1, 0]
7422 25129925999 @key{RET} 10 @key{RET} 12 @key{RET} v x 12 @key{RET} -
7429 1: 25129925999 1: [0, 2, 25, 251, 2512, ... ]
7430 2: [100000000000, ... ] .
7438 (Recall, the @kbd{\} command computes an integer quotient.)
7442 1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
7449 Next we must increment this number. This involves adding one to
7450 the last digit, plus handling carries. There is a carry to the
7451 left out of a digit if that digit is a nine and all the digits to
7452 the right of it are nines.
7456 1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ]
7466 1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
7474 Accumulating @kbd{*} across a vector of ones and zeros will preserve
7475 only the initial run of ones. These are the carries into all digits
7476 except the rightmost digit. Concatenating a one on the right takes
7477 care of aligning the carries properly, and also adding one to the
7482 2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
7483 1: [0, 0, 2, 5, ... ] .
7486 0 r 2 | V M + 10 V M %
7491 Here we have concatenated 0 to the @emph{left} of the original number;
7492 this takes care of shifting the carries by one with respect to the
7493 digits that generated them.
7495 Finally, we must convert this list back into an integer.
7499 3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ]
7500 2: 1000000000000 1: [1000000000000, 100000000000, ... ]
7501 1: [100000000000, ... ] .
7504 10 @key{RET} 12 ^ r 1 |
7511 1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000
7519 Another way to do this final step would be to reduce the formula
7520 @w{@samp{10 $$ + $}} across the vector of digits.
7524 1: [0, 0, 2, 5, ... ] 1: 25129926000
7527 V R ' 10 $$ + $ @key{RET}
7531 @node List Answer 10, List Answer 11, List Answer 9, Answers to Exercises
7532 @subsection List Tutorial Exercise 10
7535 For the list @expr{[a, b, c, d]}, the result is @expr{((a = b) = c) = d},
7536 which will compare @expr{a} and @expr{b} to produce a 1 or 0, which is
7537 then compared with @expr{c} to produce another 1 or 0, which is then
7538 compared with @expr{d}. This is not at all what Joe wanted.
7540 Here's a more correct method:
7544 1: [7, 7, 7, 8, 7] 2: [7, 7, 7, 8, 7]
7548 ' [7,7,7,8,7] @key{RET} @key{RET} v r 1 @key{RET}
7555 1: [1, 1, 1, 0, 1] 1: 0
7562 @node List Answer 11, List Answer 12, List Answer 10, Answers to Exercises
7563 @subsection List Tutorial Exercise 11
7566 The circle of unit radius consists of those points @expr{(x,y)} for which
7567 @expr{x^2 + y^2 < 1}. We start by generating a vector of @expr{x^2}
7568 and a vector of @expr{y^2}.
7570 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
7575 2: [2., 2., ..., 2.] 2: [2., 2., ..., 2.]
7576 1: [2., 2., ..., 2.] 1: [1.16, 1.98, ..., 0.81]
7579 v . t . 2. v b 100 @key{RET} @key{RET} V M k r
7586 2: [2., 2., ..., 2.] 1: [0.026, 0.96, ..., 0.036]
7587 1: [0.026, 0.96, ..., 0.036] 2: [0.53, 0.81, ..., 0.094]
7590 1 - 2 V M ^ @key{TAB} V M k r 1 - 2 V M ^
7594 Now we sum the @expr{x^2} and @expr{y^2} values, compare with 1 to
7595 get a vector of 1/0 truth values, then sum the truth values.
7599 1: [0.56, 1.78, ..., 0.13] 1: [1, 0, ..., 1] 1: 84
7607 The ratio @expr{84/100} should approximate the ratio @cpiover{4}.
7611 1: 0.84 1: 3.36 2: 3.36 1: 1.0695
7619 Our estimate, 3.36, is off by about 7%. We could get a better estimate
7620 by taking more points (say, 1000), but it's clear that this method is
7623 (Naturally, since this example uses random numbers your own answer
7624 will be slightly different from the one shown here!)
7626 If you typed @kbd{v .} and @kbd{t .} before, type them again to
7627 return to full-sized display of vectors.
7629 @node List Answer 12, List Answer 13, List Answer 11, Answers to Exercises
7630 @subsection List Tutorial Exercise 12
7633 This problem can be made a lot easier by taking advantage of some
7634 symmetries. First of all, after some thought it's clear that the
7635 @expr{y} axis can be ignored altogether. Just pick a random @expr{x}
7636 component for one end of the match, pick a random direction
7637 @texline @math{\theta},
7638 @infoline @expr{theta},
7639 and see if @expr{x} and
7640 @texline @math{x + \cos \theta}
7641 @infoline @expr{x + cos(theta)}
7642 (which is the @expr{x} coordinate of the other endpoint) cross a line.
7643 The lines are at integer coordinates, so this happens when the two
7644 numbers surround an integer.
7646 Since the two endpoints are equivalent, we may as well choose the leftmost
7647 of the two endpoints as @expr{x}. Then @expr{theta} is an angle pointing
7648 to the right, in the range -90 to 90 degrees. (We could use radians, but
7649 it would feel like cheating to refer to @cpiover{2} radians while trying
7650 to estimate @cpi{}!)
7652 In fact, since the field of lines is infinite we can choose the
7653 coordinates 0 and 1 for the lines on either side of the leftmost
7654 endpoint. The rightmost endpoint will be between 0 and 1 if the
7655 match does not cross a line, or between 1 and 2 if it does. So:
7656 Pick random @expr{x} and
7657 @texline @math{\theta},
7658 @infoline @expr{theta},
7660 @texline @math{x + \cos \theta},
7661 @infoline @expr{x + cos(theta)},
7662 and count how many of the results are greater than one. Simple!
7664 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
7669 1: [0.52, 0.71, ..., 0.72] 2: [0.52, 0.71, ..., 0.72]
7670 . 1: [78.4, 64.5, ..., -42.9]
7673 v . t . 1. v b 100 @key{RET} V M k r 180. v b 100 @key{RET} V M k r 90 -
7678 (The next step may be slow, depending on the speed of your computer.)
7682 2: [0.52, 0.71, ..., 0.72] 1: [0.72, 1.14, ..., 1.45]
7683 1: [0.20, 0.43, ..., 0.73] .
7693 1: [0, 1, ..., 1] 1: 0.64 1: 3.125
7696 1 V M a > V R + 100 / 2 @key{TAB} /
7700 Let's try the third method, too. We'll use random integers up to
7701 one million. The @kbd{k r} command with an integer argument picks
7706 2: [1000000, 1000000, ..., 1000000] 2: [78489, 527587, ..., 814975]
7707 1: [1000000, 1000000, ..., 1000000] 1: [324014, 358783, ..., 955450]
7710 1000000 v b 100 @key{RET} @key{RET} V M k r @key{TAB} V M k r
7717 1: [1, 1, ..., 25] 1: [1, 1, ..., 0] 1: 0.56
7720 V M k g 1 V M a = V R + 100 /
7734 For a proof of this property of the GCD function, see section 4.5.2,
7735 exercise 10, of Knuth's @emph{Art of Computer Programming}, volume II.
7737 If you typed @kbd{v .} and @kbd{t .} before, type them again to
7738 return to full-sized display of vectors.
7740 @node List Answer 13, List Answer 14, List Answer 12, Answers to Exercises
7741 @subsection List Tutorial Exercise 13
7744 First, we put the string on the stack as a vector of ASCII codes.
7748 1: [84, 101, 115, ..., 51]
7751 "Testing, 1, 2, 3 @key{RET}
7756 Note that the @kbd{"} key, like @kbd{$}, initiates algebraic entry so
7757 there was no need to type an apostrophe. Also, Calc didn't mind that
7758 we omitted the closing @kbd{"}. (The same goes for all closing delimiters
7759 like @kbd{)} and @kbd{]} at the end of a formula.
7761 We'll show two different approaches here. In the first, we note that
7762 if the input vector is @expr{[a, b, c, d]}, then the hash code is
7763 @expr{3 (3 (3a + b) + c) + d = 27a + 9b + 3c + d}. In other words,
7764 it's a sum of descending powers of three times the ASCII codes.
7768 2: [84, 101, 115, ..., 51] 2: [84, 101, 115, ..., 51]
7769 1: 16 1: [15, 14, 13, ..., 0]
7772 @key{RET} v l v x 16 @key{RET} -
7779 2: [84, 101, 115, ..., 51] 1: 1960915098 1: 121
7780 1: [14348907, ..., 1] . .
7783 3 @key{TAB} V M ^ * 511 %
7788 Once again, @kbd{*} elegantly summarizes most of the computation.
7789 But there's an even more elegant approach: Reduce the formula
7790 @kbd{3 $$ + $} across the vector. Recall that this represents a
7791 function of two arguments that computes its first argument times three
7792 plus its second argument.
7796 1: [84, 101, 115, ..., 51] 1: 1960915098
7799 "Testing, 1, 2, 3 @key{RET} V R ' 3$$+$ @key{RET}
7804 If you did the decimal arithmetic exercise, this will be familiar.
7805 Basically, we're turning a base-3 vector of digits into an integer,
7806 except that our ``digits'' are much larger than real digits.
7808 Instead of typing @kbd{511 %} again to reduce the result, we can be
7809 cleverer still and notice that rather than computing a huge integer
7810 and taking the modulo at the end, we can take the modulo at each step
7811 without affecting the result. While this means there are more
7812 arithmetic operations, the numbers we operate on remain small so
7813 the operations are faster.
7817 1: [84, 101, 115, ..., 51] 1: 121
7820 "Testing, 1, 2, 3 @key{RET} V R ' (3$$+$)%511 @key{RET}
7824 Why does this work? Think about a two-step computation:
7825 @w{@expr{3 (3a + b) + c}}. Taking a result modulo 511 basically means
7826 subtracting off enough 511's to put the result in the desired range.
7827 So the result when we take the modulo after every step is,
7831 3 (3 a + b - 511 m) + c - 511 n
7836 $$ 3 (3 a + b - 511 m) + c - 511 n $$
7841 for some suitable integers @expr{m} and @expr{n}. Expanding out by
7842 the distributive law yields
7846 9 a + 3 b + c - 511*3 m - 511 n
7851 $$ 9 a + 3 b + c - 511\times3 m - 511 n $$
7856 The @expr{m} term in the latter formula is redundant because any
7857 contribution it makes could just as easily be made by the @expr{n}
7858 term. So we can take it out to get an equivalent formula with
7863 9 a + 3 b + c - 511 n'
7868 $$ 9 a + 3 b + c - 511 n^{\prime} $$
7873 which is just the formula for taking the modulo only at the end of
7874 the calculation. Therefore the two methods are essentially the same.
7876 Later in the tutorial we will encounter @dfn{modulo forms}, which
7877 basically automate the idea of reducing every intermediate result
7878 modulo some value @var{m}.
7880 @node List Answer 14, Types Answer 1, List Answer 13, Answers to Exercises
7881 @subsection List Tutorial Exercise 14
7883 We want to use @kbd{H V U} to nest a function which adds a random
7884 step to an @expr{(x,y)} coordinate. The function is a bit long, but
7885 otherwise the problem is quite straightforward.
7889 2: [0, 0] 1: [ [ 0, 0 ]
7890 1: 50 [ 0.4288, -0.1695 ]
7891 . [ -0.4787, -0.9027 ]
7894 [0,0] 50 H V U ' <# + [random(2.0)-1, random(2.0)-1]> @key{RET}
7898 Just as the text recommended, we used @samp{< >} nameless function
7899 notation to keep the two @code{random} calls from being evaluated
7900 before nesting even begins.
7902 We now have a vector of @expr{[x, y]} sub-vectors, which by Calc's
7903 rules acts like a matrix. We can transpose this matrix and unpack
7904 to get a pair of vectors, @expr{x} and @expr{y}, suitable for graphing.
7908 2: [ 0, 0.4288, -0.4787, ... ]
7909 1: [ 0, -0.1696, -0.9027, ... ]
7916 Incidentally, because the @expr{x} and @expr{y} are completely
7917 independent in this case, we could have done two separate commands
7918 to create our @expr{x} and @expr{y} vectors of numbers directly.
7920 To make a random walk of unit steps, we note that @code{sincos} of
7921 a random direction exactly gives us an @expr{[x, y]} step of unit
7922 length; in fact, the new nesting function is even briefer, though
7923 we might want to lower the precision a bit for it.
7927 2: [0, 0] 1: [ [ 0, 0 ]
7928 1: 50 [ 0.1318, 0.9912 ]
7929 . [ -0.5965, 0.3061 ]
7932 [0,0] 50 m d p 6 @key{RET} H V U ' <# + sincos(random(360.0))> @key{RET}
7936 Another @kbd{v t v u g f} sequence will graph this new random walk.
7938 An interesting twist on these random walk functions would be to use
7939 complex numbers instead of 2-vectors to represent points on the plane.
7940 In the first example, we'd use something like @samp{random + random*(0,1)},
7941 and in the second we could use polar complex numbers with random phase
7942 angles. (This exercise was first suggested in this form by Randal
7945 @node Types Answer 1, Types Answer 2, List Answer 14, Answers to Exercises
7946 @subsection Types Tutorial Exercise 1
7949 If the number is the square root of @cpi{} times a rational number,
7950 then its square, divided by @cpi{}, should be a rational number.
7954 1: 1.26508260337 1: 0.509433962268 1: 2486645810:4881193627
7962 Technically speaking this is a rational number, but not one that is
7963 likely to have arisen in the original problem. More likely, it just
7964 happens to be the fraction which most closely represents some
7965 irrational number to within 12 digits.
7967 But perhaps our result was not quite exact. Let's reduce the
7968 precision slightly and try again:
7972 1: 0.509433962268 1: 27:53
7975 U p 10 @key{RET} c F
7980 Aha! It's unlikely that an irrational number would equal a fraction
7981 this simple to within ten digits, so our original number was probably
7982 @texline @math{\sqrt{27 \pi / 53}}.
7983 @infoline @expr{sqrt(27 pi / 53)}.
7985 Notice that we didn't need to re-round the number when we reduced the
7986 precision. Remember, arithmetic operations always round their inputs
7987 to the current precision before they begin.
7989 @node Types Answer 2, Types Answer 3, Types Answer 1, Answers to Exercises
7990 @subsection Types Tutorial Exercise 2
7993 @samp{inf / inf = nan}. Perhaps @samp{1} is the ``obvious'' answer.
7994 But if @w{@samp{17 inf = inf}}, then @samp{17 inf / inf = inf / inf = 17}, too.
7996 @samp{exp(inf) = inf}. It's tempting to say that the exponential
7997 of infinity must be ``bigger'' than ``regular'' infinity, but as
7998 far as Calc is concerned all infinities are the same size.
7999 In other words, as @expr{x} goes to infinity, @expr{e^x} also goes
8000 to infinity, but the fact the @expr{e^x} grows much faster than
8001 @expr{x} is not relevant here.
8003 @samp{exp(-inf) = 0}. Here we have a finite answer even though
8004 the input is infinite.
8006 @samp{sqrt(-inf) = (0, 1) inf}. Remember that @expr{(0, 1)}
8007 represents the imaginary number @expr{i}. Here's a derivation:
8008 @samp{sqrt(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}.
8009 The first part is, by definition, @expr{i}; the second is @code{inf}
8010 because, once again, all infinities are the same size.
8012 @samp{sqrt(uinf) = uinf}. In fact, we do know something about the
8013 direction because @code{sqrt} is defined to return a value in the
8014 right half of the complex plane. But Calc has no notation for this,
8015 so it settles for the conservative answer @code{uinf}.
8017 @samp{abs(uinf) = inf}. No matter which direction @expr{x} points,
8018 @samp{abs(x)} always points along the positive real axis.
8020 @samp{ln(0) = -inf}. Here we have an infinite answer to a finite
8021 input. As in the @expr{1 / 0} case, Calc will only use infinities
8022 here if you have turned on Infinite mode. Otherwise, it will
8023 treat @samp{ln(0)} as an error.
8025 @node Types Answer 3, Types Answer 4, Types Answer 2, Answers to Exercises
8026 @subsection Types Tutorial Exercise 3
8029 We can make @samp{inf - inf} be any real number we like, say,
8030 @expr{a}, just by claiming that we added @expr{a} to the first
8031 infinity but not to the second. This is just as true for complex
8032 values of @expr{a}, so @code{nan} can stand for a complex number.
8033 (And, similarly, @code{uinf} can stand for an infinity that points
8034 in any direction in the complex plane, such as @samp{(0, 1) inf}).
8036 In fact, we can multiply the first @code{inf} by two. Surely
8037 @w{@samp{2 inf - inf = inf}}, but also @samp{2 inf - inf = inf - inf = nan}.
8038 So @code{nan} can even stand for infinity. Obviously it's just
8039 as easy to make it stand for minus infinity as for plus infinity.
8041 The moral of this story is that ``infinity'' is a slippery fish
8042 indeed, and Calc tries to handle it by having a very simple model
8043 for infinities (only the direction counts, not the ``size''); but
8044 Calc is careful to write @code{nan} any time this simple model is
8045 unable to tell what the true answer is.
8047 @node Types Answer 4, Types Answer 5, Types Answer 3, Answers to Exercises
8048 @subsection Types Tutorial Exercise 4
8052 2: 0@@ 47' 26" 1: 0@@ 2' 47.411765"
8056 0@@ 47' 26" @key{RET} 17 /
8061 The average song length is two minutes and 47.4 seconds.
8065 2: 0@@ 2' 47.411765" 1: 0@@ 3' 7.411765" 1: 0@@ 53' 6.000005"
8074 The album would be 53 minutes and 6 seconds long.
8076 @node Types Answer 5, Types Answer 6, Types Answer 4, Answers to Exercises
8077 @subsection Types Tutorial Exercise 5
8080 Let's suppose it's January 14, 1991. The easiest thing to do is
8081 to keep trying 13ths of months until Calc reports a Friday.
8082 We can do this by manually entering dates, or by using @kbd{t I}:
8086 1: <Wed Feb 13, 1991> 1: <Wed Mar 13, 1991> 1: <Sat Apr 13, 1991>
8089 ' <2/13> @key{RET} @key{DEL} ' <3/13> @key{RET} t I
8094 (Calc assumes the current year if you don't say otherwise.)
8096 This is getting tedious---we can keep advancing the date by typing
8097 @kbd{t I} over and over again, but let's automate the job by using
8098 vector mapping. The @kbd{t I} command actually takes a second
8099 ``how-many-months'' argument, which defaults to one. This
8100 argument is exactly what we want to map over:
8104 2: <Sat Apr 13, 1991> 1: [<Mon May 13, 1991>, <Thu Jun 13, 1991>,
8105 1: [1, 2, 3, 4, 5, 6] <Sat Jul 13, 1991>, <Tue Aug 13, 1991>,
8106 . <Fri Sep 13, 1991>, <Sun Oct 13, 1991>]
8109 v x 6 @key{RET} V M t I
8114 Et voil@`a, September 13, 1991 is a Friday.
8121 ' <sep 13> - <jan 14> @key{RET}
8126 And the answer to our original question: 242 days to go.
8128 @node Types Answer 6, Types Answer 7, Types Answer 5, Answers to Exercises
8129 @subsection Types Tutorial Exercise 6
8132 The full rule for leap years is that they occur in every year divisible
8133 by four, except that they don't occur in years divisible by 100, except
8134 that they @emph{do} in years divisible by 400. We could work out the
8135 answer by carefully counting the years divisible by four and the
8136 exceptions, but there is a much simpler way that works even if we
8137 don't know the leap year rule.
8139 Let's assume the present year is 1991. Years have 365 days, except
8140 that leap years (whenever they occur) have 366 days. So let's count
8141 the number of days between now and then, and compare that to the
8142 number of years times 365. The number of extra days we find must be
8143 equal to the number of leap years there were.
8147 1: <Mon Jan 1, 10001> 2: <Mon Jan 1, 10001> 1: 2925593
8148 . 1: <Tue Jan 1, 1991> .
8151 ' <jan 1 10001> @key{RET} ' <jan 1 1991> @key{RET} -
8158 3: 2925593 2: 2925593 2: 2925593 1: 1943
8159 2: 10001 1: 8010 1: 2923650 .
8163 10001 @key{RET} 1991 - 365 * -
8167 @c [fix-ref Date Forms]
8169 There will be 1943 leap years before the year 10001. (Assuming,
8170 of course, that the algorithm for computing leap years remains
8171 unchanged for that long. @xref{Date Forms}, for some interesting
8172 background information in that regard.)
8174 @node Types Answer 7, Types Answer 8, Types Answer 6, Answers to Exercises
8175 @subsection Types Tutorial Exercise 7
8178 The relative errors must be converted to absolute errors so that
8179 @samp{+/-} notation may be used.
8187 20 @key{RET} .05 * 4 @key{RET} .05 *
8191 Now we simply chug through the formula.
8195 1: 19.7392088022 1: 394.78 +/- 19.739 1: 6316.5 +/- 706.21
8198 2 P 2 ^ * 20 p 1 * 4 p .2 @key{RET} 2 ^ *
8202 It turns out the @kbd{v u} command will unpack an error form as
8203 well as a vector. This saves us some retyping of numbers.
8207 3: 6316.5 +/- 706.21 2: 6316.5 +/- 706.21
8212 @key{RET} v u @key{TAB} /
8217 Thus the volume is 6316 cubic centimeters, within about 11 percent.
8219 @node Types Answer 8, Types Answer 9, Types Answer 7, Answers to Exercises
8220 @subsection Types Tutorial Exercise 8
8223 The first answer is pretty simple: @samp{1 / (0 .. 10) = (0.1 .. inf)}.
8224 Since a number in the interval @samp{(0 .. 10)} can get arbitrarily
8225 close to zero, its reciprocal can get arbitrarily large, so the answer
8226 is an interval that effectively means, ``any number greater than 0.1''
8227 but with no upper bound.
8229 The second answer, similarly, is @samp{1 / (-10 .. 0) = (-inf .. -0.1)}.
8231 Calc normally treats division by zero as an error, so that the formula
8232 @w{@samp{1 / 0}} is left unsimplified. Our third problem,
8233 @w{@samp{1 / [0 .. 10]}}, also (potentially) divides by zero because zero
8234 is now a member of the interval. So Calc leaves this one unevaluated, too.
8236 If you turn on Infinite mode by pressing @kbd{m i}, you will
8237 instead get the answer @samp{[0.1 .. inf]}, which includes infinity
8238 as a possible value.
8240 The fourth calculation, @samp{1 / (-10 .. 10)}, has the same problem.
8241 Zero is buried inside the interval, but it's still a possible value.
8242 It's not hard to see that the actual result of @samp{1 / (-10 .. 10)}
8243 will be either greater than @mathit{0.1}, or less than @mathit{-0.1}. Thus
8244 the interval goes from minus infinity to plus infinity, with a ``hole''
8245 in it from @mathit{-0.1} to @mathit{0.1}. Calc doesn't have any way to
8246 represent this, so it just reports @samp{[-inf .. inf]} as the answer.
8247 It may be disappointing to hear ``the answer lies somewhere between
8248 minus infinity and plus infinity, inclusive,'' but that's the best
8249 that interval arithmetic can do in this case.
8251 @node Types Answer 9, Types Answer 10, Types Answer 8, Answers to Exercises
8252 @subsection Types Tutorial Exercise 9
8256 1: [-3 .. 3] 2: [-3 .. 3] 2: [0 .. 9]
8257 . 1: [0 .. 9] 1: [-9 .. 9]
8260 [ 3 n .. 3 ] @key{RET} 2 ^ @key{TAB} @key{RET} *
8265 In the first case the result says, ``if a number is between @mathit{-3} and
8266 3, its square is between 0 and 9.'' The second case says, ``the product
8267 of two numbers each between @mathit{-3} and 3 is between @mathit{-9} and 9.''
8269 An interval form is not a number; it is a symbol that can stand for
8270 many different numbers. Two identical-looking interval forms can stand
8271 for different numbers.
8273 The same issue arises when you try to square an error form.
8275 @node Types Answer 10, Types Answer 11, Types Answer 9, Answers to Exercises
8276 @subsection Types Tutorial Exercise 10
8279 Testing the first number, we might arbitrarily choose 17 for @expr{x}.
8283 1: 17 mod 811749613 2: 17 mod 811749613 1: 533694123 mod 811749613
8287 17 M 811749613 @key{RET} 811749612 ^
8292 Since 533694123 is (considerably) different from 1, the number 811749613
8295 It's awkward to type the number in twice as we did above. There are
8296 various ways to avoid this, and algebraic entry is one. In fact, using
8297 a vector mapping operation we can perform several tests at once. Let's
8298 use this method to test the second number.
8302 2: [17, 42, 100000] 1: [1 mod 15485863, 1 mod ... ]
8306 [17 42 100000] 15485863 @key{RET} V M ' ($$ mod $)^($-1) @key{RET}
8311 The result is three ones (modulo @expr{n}), so it's very probable that
8312 15485863 is prime. (In fact, this number is the millionth prime.)
8314 Note that the functions @samp{($$^($-1)) mod $} or @samp{$$^($-1) % $}
8315 would have been hopelessly inefficient, since they would have calculated
8316 the power using full integer arithmetic.
8318 Calc has a @kbd{k p} command that does primality testing. For small
8319 numbers it does an exact test; for large numbers it uses a variant
8320 of the Fermat test we used here. You can use @kbd{k p} repeatedly
8321 to prove that a large integer is prime with any desired probability.
8323 @node Types Answer 11, Types Answer 12, Types Answer 10, Answers to Exercises
8324 @subsection Types Tutorial Exercise 11
8327 There are several ways to insert a calculated number into an HMS form.
8328 One way to convert a number of seconds to an HMS form is simply to
8329 multiply the number by an HMS form representing one second:
8333 1: 31415926.5359 2: 31415926.5359 1: 8726@@ 38' 46.5359"
8344 2: 8726@@ 38' 46.5359" 1: 6@@ 6' 2.5359" mod 24@@ 0' 0"
8345 1: 15@@ 27' 16" mod 24@@ 0' 0" .
8353 It will be just after six in the morning.
8355 The algebraic @code{hms} function can also be used to build an
8360 1: hms(0, 0, 10000000. pi) 1: 8726@@ 38' 46.5359"
8363 ' hms(0, 0, 1e7 pi) @key{RET} =
8368 The @kbd{=} key is necessary to evaluate the symbol @samp{pi} to
8369 the actual number 3.14159...
8371 @node Types Answer 12, Types Answer 13, Types Answer 11, Answers to Exercises
8372 @subsection Types Tutorial Exercise 12
8375 As we recall, there are 17 songs of about 2 minutes and 47 seconds
8380 2: 0@@ 2' 47" 1: [0@@ 3' 7" .. 0@@ 3' 47"]
8381 1: [0@@ 0' 20" .. 0@@ 1' 0"] .
8384 [ 0@@ 20" .. 0@@ 1' ] +
8391 1: [0@@ 52' 59." .. 1@@ 4' 19."]
8399 No matter how long it is, the album will fit nicely on one CD.
8401 @node Types Answer 13, Types Answer 14, Types Answer 12, Answers to Exercises
8402 @subsection Types Tutorial Exercise 13
8405 Type @kbd{' 1 yr @key{RET} u c s @key{RET}}. The answer is 31557600 seconds.
8407 @node Types Answer 14, Types Answer 15, Types Answer 13, Answers to Exercises
8408 @subsection Types Tutorial Exercise 14
8411 How long will it take for a signal to get from one end of the computer
8416 1: m / c 1: 3.3356 ns
8419 ' 1 m / c @key{RET} u c ns @key{RET}
8424 (Recall, @samp{c} is a ``unit'' corresponding to the speed of light.)
8428 1: 3.3356 ns 1: 0.81356
8432 ' 4.1 ns @key{RET} /
8437 Thus a signal could take up to 81 percent of a clock cycle just to
8438 go from one place to another inside the computer, assuming the signal
8439 could actually attain the full speed of light. Pretty tight!
8441 @node Types Answer 15, Algebra Answer 1, Types Answer 14, Answers to Exercises
8442 @subsection Types Tutorial Exercise 15
8445 The speed limit is 55 miles per hour on most highways. We want to
8446 find the ratio of Sam's speed to the US speed limit.
8450 1: 55 mph 2: 55 mph 3: 11 hr mph / yd
8454 ' 55 mph @key{RET} ' 5 yd/hr @key{RET} /
8458 The @kbd{u s} command cancels out these units to get a plain
8459 number. Now we take the logarithm base two to find the final
8460 answer, assuming that each successive pill doubles his speed.
8464 1: 19360. 2: 19360. 1: 14.24
8473 Thus Sam can take up to 14 pills without a worry.
8475 @node Algebra Answer 1, Algebra Answer 2, Types Answer 15, Answers to Exercises
8476 @subsection Algebra Tutorial Exercise 1
8479 @c [fix-ref Declarations]
8480 The result @samp{sqrt(x)^2} is simplified back to @expr{x} by the
8481 Calculator, but @samp{sqrt(x^2)} is not. (Consider what happens
8482 if @w{@expr{x = -4}}.) If @expr{x} is real, this formula could be
8483 simplified to @samp{abs(x)}, but for general complex arguments even
8484 that is not safe. (@xref{Declarations}, for a way to tell Calc
8485 that @expr{x} is known to be real.)
8487 @node Algebra Answer 2, Algebra Answer 3, Algebra Answer 1, Answers to Exercises
8488 @subsection Algebra Tutorial Exercise 2
8491 Suppose our roots are @expr{[a, b, c]}. We want a polynomial which
8492 is zero when @expr{x} is any of these values. The trivial polynomial
8493 @expr{x-a} is zero when @expr{x=a}, so the product @expr{(x-a)(x-b)(x-c)}
8494 will do the job. We can use @kbd{a c x} to write this in a more
8499 1: 34 x - 24 x^3 1: [1.19023, -1.19023, 0]
8509 1: [x - 1.19023, x + 1.19023, x] 1: x*(x + 1.19023) (x - 1.19023)
8512 V M ' x-$ @key{RET} V R *
8519 1: x^3 - 1.41666 x 1: 34 x - 24 x^3
8522 a c x @key{RET} 24 n * a x
8527 Sure enough, our answer (multiplied by a suitable constant) is the
8528 same as the original polynomial.
8530 @node Algebra Answer 3, Algebra Answer 4, Algebra Answer 2, Answers to Exercises
8531 @subsection Algebra Tutorial Exercise 3
8535 1: x sin(pi x) 1: sin(pi x) / pi^2 - x cos(pi x) / pi
8538 ' x sin(pi x) @key{RET} m r a i x @key{RET}
8546 2: sin(pi x) / pi^2 - x cos(pi x) / pi
8549 ' [y,1] @key{RET} @key{TAB}
8556 1: [sin(pi y) / pi^2 - y cos(pi y) / pi, 1 / pi]
8566 1: sin(pi y) / pi^2 - y cos(pi y) / pi - 1 / pi
8576 1: sin(3.14159 y) / 9.8696 - y cos(3.14159 y) / 3.14159 - 0.3183
8586 1: [0., -0.95493, 0.63662, -1.5915, 1.2732]
8589 v x 5 @key{RET} @key{TAB} V M $ @key{RET}
8593 @node Algebra Answer 4, Rewrites Answer 1, Algebra Answer 3, Answers to Exercises
8594 @subsection Algebra Tutorial Exercise 4
8597 The hard part is that @kbd{V R +} is no longer sufficient to add up all
8598 the contributions from the slices, since the slices have varying
8599 coefficients. So first we must come up with a vector of these
8600 coefficients. Here's one way:
8604 2: -1 2: 3 1: [4, 2, ..., 4]
8605 1: [1, 2, ..., 9] 1: [-1, 1, ..., -1] .
8608 1 n v x 9 @key{RET} V M ^ 3 @key{TAB} -
8615 1: [4, 2, ..., 4, 1] 1: [1, 4, 2, ..., 4, 1]
8623 Now we compute the function values. Note that for this method we need
8624 eleven values, including both endpoints of the desired interval.
8628 2: [1, 4, 2, ..., 4, 1]
8629 1: [1, 1.1, 1.2, ... , 1.8, 1.9, 2.]
8632 11 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
8639 2: [1, 4, 2, ..., 4, 1]
8640 1: [0., 0.084941, 0.16993, ... ]
8643 ' sin(x) ln(x) @key{RET} m r p 5 @key{RET} V M $ @key{RET}
8648 Once again this calls for @kbd{V M * V R +}; a simple @kbd{*} does the
8653 1: 11.22 1: 1.122 1: 0.374
8661 Wow! That's even better than the result from the Taylor series method.
8663 @node Rewrites Answer 1, Rewrites Answer 2, Algebra Answer 4, Answers to Exercises
8664 @subsection Rewrites Tutorial Exercise 1
8667 We'll use Big mode to make the formulas more readable.
8673 1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: ---------
8679 ' (2+sqrt(2)) / (1+sqrt(2)) @key{RET} d B
8684 Multiplying by the conjugate helps because @expr{(a+b) (a-b) = a^2 - b^2}.
8689 1: (2 + V 2 ) (V 2 - 1)
8692 a r a/(b+c) := a*(b-c) / (b^2-c^2) @key{RET}
8703 a r a*(b+c) := a*b + a*c
8708 (We could have used @kbd{a x} instead of a rewrite rule for the
8711 The multiply-by-conjugate rule turns out to be useful in many
8712 different circumstances, such as when the denominator involves
8713 sines and cosines or the imaginary constant @code{i}.
8715 @node Rewrites Answer 2, Rewrites Answer 3, Rewrites Answer 1, Answers to Exercises
8716 @subsection Rewrites Tutorial Exercise 2
8719 Here is the rule set:
8723 [ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1,
8725 fib(n, x, y) := fib(n-1, y, x+y) ]
8730 The first rule turns a one-argument @code{fib} that people like to write
8731 into a three-argument @code{fib} that makes computation easier. The
8732 second rule converts back from three-argument form once the computation
8733 is done. The third rule does the computation itself. It basically
8734 says that if @expr{x} and @expr{y} are two consecutive Fibonacci numbers,
8735 then @expr{y} and @expr{x+y} are the next (overlapping) pair of Fibonacci
8738 Notice that because the number @expr{n} was ``validated'' by the
8739 conditions on the first rule, there is no need to put conditions on
8740 the other rules because the rule set would never get that far unless
8741 the input were valid. That further speeds computation, since no
8742 extra conditions need to be checked at every step.
8744 Actually, a user with a nasty sense of humor could enter a bad
8745 three-argument @code{fib} call directly, say, @samp{fib(0, 1, 1)},
8746 which would get the rules into an infinite loop. One thing that would
8747 help keep this from happening by accident would be to use something like
8748 @samp{ZzFib} instead of @code{fib} as the name of the three-argument
8751 @node Rewrites Answer 3, Rewrites Answer 4, Rewrites Answer 2, Answers to Exercises
8752 @subsection Rewrites Tutorial Exercise 3
8755 He got an infinite loop. First, Calc did as expected and rewrote
8756 @w{@samp{2 + 3 x}} to @samp{f(2, 3, x)}. Then it looked for ways to
8757 apply the rule again, and found that @samp{f(2, 3, x)} looks like
8758 @samp{a + b x} with @w{@samp{a = 0}} and @samp{b = 1}, so it rewrote to
8759 @samp{f(0, 1, f(2, 3, x))}. It then wrapped another @samp{f(0, 1, ...)}
8760 around that, and so on, ad infinitum. Joe should have used @kbd{M-1 a r}
8761 to make sure the rule applied only once.
8763 (Actually, even the first step didn't work as he expected. What Calc
8764 really gives for @kbd{M-1 a r} in this situation is @samp{f(3 x, 1, 2)},
8765 treating 2 as the ``variable,'' and @samp{3 x} as a constant being added
8766 to it. While this may seem odd, it's just as valid a solution as the
8767 ``obvious'' one. One way to fix this would be to add the condition
8768 @samp{:: variable(x)} to the rule, to make sure the thing that matches
8769 @samp{x} is indeed a variable, or to change @samp{x} to @samp{quote(x)}
8770 on the lefthand side, so that the rule matches the actual variable
8771 @samp{x} rather than letting @samp{x} stand for something else.)
8773 @node Rewrites Answer 4, Rewrites Answer 5, Rewrites Answer 3, Answers to Exercises
8774 @subsection Rewrites Tutorial Exercise 4
8781 Here is a suitable set of rules to solve the first part of the problem:
8785 [ seq(n, c) := seq(n/2, c+1) :: n%2 = 0,
8786 seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]
8790 Given the initial formula @samp{seq(6, 0)}, application of these
8791 rules produces the following sequence of formulas:
8805 whereupon neither of the rules match, and rewriting stops.
8807 We can pretty this up a bit with a couple more rules:
8811 [ seq(n) := seq(n, 0),
8818 Now, given @samp{seq(6)} as the starting configuration, we get 8
8821 The change to return a vector is quite simple:
8825 [ seq(n) := seq(n, []) :: integer(n) :: n > 0,
8827 seq(n, v) := seq(n/2, v | n) :: n%2 = 0,
8828 seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]
8833 Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
8835 Notice that the @expr{n > 1} guard is no longer necessary on the last
8836 rule since the @expr{n = 1} case is now detected by another rule.
8837 But a guard has been added to the initial rule to make sure the
8838 initial value is suitable before the computation begins.
8840 While still a good idea, this guard is not as vitally important as it
8841 was for the @code{fib} function, since calling, say, @samp{seq(x, [])}
8842 will not get into an infinite loop. Calc will not be able to prove
8843 the symbol @samp{x} is either even or odd, so none of the rules will
8844 apply and the rewrites will stop right away.
8846 @node Rewrites Answer 5, Rewrites Answer 6, Rewrites Answer 4, Answers to Exercises
8847 @subsection Rewrites Tutorial Exercise 5
8854 If @expr{x} is the sum @expr{a + b}, then `@tfn{nterms(}@var{x}@tfn{)}' must
8855 be `@tfn{nterms(}@var{a}@tfn{)}' plus `@tfn{nterms(}@var{b}@tfn{)}'. If @expr{x}
8856 is not a sum, then `@tfn{nterms(}@var{x}@tfn{)}' = 1.
8860 [ nterms(a + b) := nterms(a) + nterms(b),
8866 Here we have taken advantage of the fact that earlier rules always
8867 match before later rules; @samp{nterms(x)} will only be tried if we
8868 already know that @samp{x} is not a sum.
8870 @node Rewrites Answer 6, Programming Answer 1, Rewrites Answer 5, Answers to Exercises
8871 @subsection Rewrites Tutorial Exercise 6
8874 Here is a rule set that will do the job:
8878 [ a*(b + c) := a*b + a*c,
8879 opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m
8880 :: constant(a) :: constant(b),
8881 opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m
8882 :: constant(a) :: constant(b),
8883 a O(x^n) := O(x^n) :: constant(a),
8884 x^opt(m) O(x^n) := O(x^(n+m)),
8885 O(x^n) O(x^m) := O(x^(n+m)) ]
8889 If we really want the @kbd{+} and @kbd{*} keys to operate naturally
8890 on power series, we should put these rules in @code{EvalRules}. For
8891 testing purposes, it is better to put them in a different variable,
8892 say, @code{O}, first.
8894 The first rule just expands products of sums so that the rest of the
8895 rules can assume they have an expanded-out polynomial to work with.
8896 Note that this rule does not mention @samp{O} at all, so it will
8897 apply to any product-of-sum it encounters---this rule may surprise
8898 you if you put it into @code{EvalRules}!
8900 In the second rule, the sum of two O's is changed to the smaller O@.
8901 The optional constant coefficients are there mostly so that
8902 @samp{O(x^2) - O(x^3)} and @samp{O(x^3) - O(x^2)} are handled
8903 as well as @samp{O(x^2) + O(x^3)}.
8905 The third rule absorbs higher powers of @samp{x} into O's.
8907 The fourth rule says that a constant times a negligible quantity
8908 is still negligible. (This rule will also match @samp{O(x^3) / 4},
8909 with @samp{a = 1/4}.)
8911 The fifth rule rewrites, for example, @samp{x^2 O(x^3)} to @samp{O(x^5)}.
8912 (It is easy to see that if one of these forms is negligible, the other
8913 is, too.) Notice the @samp{x^opt(m)} to pick up terms like
8914 @w{@samp{x O(x^3)}}. Optional powers will match @samp{x} as @samp{x^1}
8915 but not 1 as @samp{x^0}. This turns out to be exactly what we want here.
8917 The sixth rule is the corresponding rule for products of two O's.
8919 Another way to solve this problem would be to create a new ``data type''
8920 that represents truncated power series. We might represent these as
8921 function calls @samp{series(@var{coefs}, @var{x})} where @var{coefs} is
8922 a vector of coefficients for @expr{x^0}, @expr{x^1}, @expr{x^2}, and so
8923 on. Rules would exist for sums and products of such @code{series}
8924 objects, and as an optional convenience could also know how to combine a
8925 @code{series} object with a normal polynomial. (With this, and with a
8926 rule that rewrites @samp{O(x^n)} to the equivalent @code{series} form,
8927 you could still enter power series in exactly the same notation as
8928 before.) Operations on such objects would probably be more efficient,
8929 although the objects would be a bit harder to read.
8931 @c [fix-ref Compositions]
8932 Some other symbolic math programs provide a power series data type
8933 similar to this. Mathematica, for example, has an object that looks
8934 like @samp{PowerSeries[@var{x}, @var{x0}, @var{coefs}, @var{nmin},
8935 @var{nmax}, @var{den}]}, where @var{x0} is the point about which the
8936 power series is taken (we've been assuming this was always zero),
8937 and @var{nmin}, @var{nmax}, and @var{den} allow pseudo-power-series
8938 with fractional or negative powers. Also, the @code{PowerSeries}
8939 objects have a special display format that makes them look like
8940 @samp{2 x^2 + O(x^4)} when they are printed out. (@xref{Compositions},
8941 for a way to do this in Calc, although for something as involved as
8942 this it would probably be better to write the formatting routine
8945 @node Programming Answer 1, Programming Answer 2, Rewrites Answer 6, Answers to Exercises
8946 @subsection Programming Tutorial Exercise 1
8949 Just enter the formula @samp{ninteg(sin(t)/t, t, 0, x)}, type
8950 @kbd{Z F}, and answer the questions. Since this formula contains two
8951 variables, the default argument list will be @samp{(t x)}. We want to
8952 change this to @samp{(x)} since @expr{t} is really a dummy variable
8953 to be used within @code{ninteg}.
8955 The exact keystrokes are @kbd{Z F s Si @key{RET} @key{RET} C-b C-b @key{DEL} @key{DEL} @key{RET} y}.
8956 (The @kbd{C-b C-b @key{DEL} @key{DEL}} are what fix the argument list.)
8958 @node Programming Answer 2, Programming Answer 3, Programming Answer 1, Answers to Exercises
8959 @subsection Programming Tutorial Exercise 2
8962 One way is to move the number to the top of the stack, operate on
8963 it, then move it back: @kbd{C-x ( M-@key{TAB} n M-@key{TAB} M-@key{TAB} C-x )}.
8965 Another way is to negate the top three stack entries, then negate
8966 again the top two stack entries: @kbd{C-x ( M-3 n M-2 n C-x )}.
8968 Finally, it turns out that a negative prefix argument causes a
8969 command like @kbd{n} to operate on the specified stack entry only,
8970 which is just what we want: @kbd{C-x ( M-- 3 n C-x )}.
8972 Just for kicks, let's also do it algebraically:
8973 @w{@kbd{C-x ( ' -$$$, $$, $ @key{RET} C-x )}}.
8975 @node Programming Answer 3, Programming Answer 4, Programming Answer 2, Answers to Exercises
8976 @subsection Programming Tutorial Exercise 3
8979 Each of these functions can be computed using the stack, or using
8980 algebraic entry, whichever way you prefer:
8984 @texline @math{\displaystyle{\sin x \over x}}:
8985 @infoline @expr{sin(x) / x}:
8987 Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}.
8989 Using algebraic entry: @kbd{C-x ( ' sin($)/$ @key{RET} C-x )}.
8992 Computing the logarithm:
8994 Using the stack: @kbd{C-x ( @key{TAB} B C-x )}
8996 Using algebraic entry: @kbd{C-x ( ' log($,$$) @key{RET} C-x )}.
8999 Computing the vector of integers:
9001 Using the stack: @kbd{C-x ( 1 @key{RET} 1 C-u v x C-x )}. (Recall that
9002 @kbd{C-u v x} takes the vector size, starting value, and increment
9005 Alternatively: @kbd{C-x ( ~ v x C-x )}. (The @kbd{~} key pops a
9006 number from the stack and uses it as the prefix argument for the
9009 Using algebraic entry: @kbd{C-x ( ' index($) @key{RET} C-x )}.
9011 @node Programming Answer 4, Programming Answer 5, Programming Answer 3, Answers to Exercises
9012 @subsection Programming Tutorial Exercise 4
9015 Here's one way: @kbd{C-x ( @key{RET} V R + @key{TAB} v l / C-x )}.
9017 @node Programming Answer 5, Programming Answer 6, Programming Answer 4, Answers to Exercises
9018 @subsection Programming Tutorial Exercise 5
9022 2: 1 1: 1.61803398502 2: 1.61803398502
9023 1: 20 . 1: 1.61803398875
9026 1 @key{RET} 20 Z < & 1 + Z > I H P
9031 This answer is quite accurate.
9033 @node Programming Answer 6, Programming Answer 7, Programming Answer 5, Answers to Exercises
9034 @subsection Programming Tutorial Exercise 6
9040 [ [ 0, 1 ] * [a, b] = [b, a + b]
9045 Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @expr{n+1}
9046 and @expr{n+2}. Here's one program that does the job:
9049 C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] @key{RET} v u @key{DEL} C-x )
9053 This program is quite efficient because Calc knows how to raise a
9054 matrix (or other value) to the power @expr{n} in only
9055 @texline @math{\log_2 n}
9056 @infoline @expr{log(n,2)}
9057 steps. For example, this program can compute the 1000th Fibonacci
9058 number (a 209-digit integer!) in about 10 steps; even though the
9059 @kbd{Z < ... Z >} solution had much simpler steps, it would have
9060 required so many steps that it would not have been practical.
9062 @node Programming Answer 7, Programming Answer 8, Programming Answer 6, Answers to Exercises
9063 @subsection Programming Tutorial Exercise 7
9066 The trick here is to compute the harmonic numbers differently, so that
9067 the loop counter itself accumulates the sum of reciprocals. We use
9068 a separate variable to hold the integer counter.
9076 1 t 1 1 @key{RET} 4 Z ( t 2 r 1 1 + s 1 & Z )
9081 The body of the loop goes as follows: First save the harmonic sum
9082 so far in variable 2. Then delete it from the stack; the for loop
9083 itself will take care of remembering it for us. Next, recall the
9084 count from variable 1, add one to it, and feed its reciprocal to
9085 the for loop to use as the step value. The for loop will increase
9086 the ``loop counter'' by that amount and keep going until the
9087 loop counter exceeds 4.
9092 1: 3.99498713092 2: 3.99498713092
9096 r 1 r 2 @key{RET} 31 & +
9100 Thus we find that the 30th harmonic number is 3.99, and the 31st
9101 harmonic number is 4.02.
9103 @node Programming Answer 8, Programming Answer 9, Programming Answer 7, Answers to Exercises
9104 @subsection Programming Tutorial Exercise 8
9107 The first step is to compute the derivative @expr{f'(x)} and thus
9109 @texline @math{\displaystyle{x - {f(x) \over f'(x)}}}.
9110 @infoline @expr{x - f(x)/f'(x)}.
9112 (Because this definition is long, it will be repeated in concise form
9113 below. You can use @w{@kbd{C-x * m}} to load it from there. While you are
9114 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9115 keystrokes without executing them. In the following diagrams we'll
9116 pretend Calc actually executed the keystrokes as you typed them,
9117 just for purposes of illustration.)
9121 2: sin(cos(x)) - 0.5 3: 4.5
9122 1: 4.5 2: sin(cos(x)) - 0.5
9123 . 1: -(sin(x) cos(cos(x)))
9126 ' sin(cos(x))-0.5 @key{RET} 4.5 m r C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET}
9134 1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x))
9137 / ' x @key{RET} @key{TAB} - t 1
9141 Now, we enter the loop. We'll use a repeat loop with a 20-repetition
9142 limit just in case the method fails to converge for some reason.
9143 (Normally, the @w{@kbd{Z /}} command will stop the loop before all 20
9144 repetitions are done.)
9148 1: 4.5 3: 4.5 2: 4.5
9149 . 2: x + (sin(cos(x)) ... 1: 5.24196456928
9153 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9157 This is the new guess for @expr{x}. Now we compare it with the
9158 old one to see if we've converged.
9162 3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348
9167 @key{RET} M-@key{TAB} a = Z / Z > Z ' C-x )
9171 The loop converges in just a few steps to this value. To check
9172 the result, we can simply substitute it back into the equation.
9180 @key{RET} ' sin(cos($)) @key{RET}
9184 Let's test the new definition again:
9192 ' x^2-9 @key{RET} 1 X
9196 Once again, here's the full Newton's Method definition:
9200 C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET} / ' x @key{RET} @key{TAB} - t 1
9201 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9202 @key{RET} M-@key{TAB} a = Z /
9209 @c [fix-ref Nesting and Fixed Points]
9210 It turns out that Calc has a built-in command for applying a formula
9211 repeatedly until it converges to a number. @xref{Nesting and Fixed Points},
9212 to see how to use it.
9214 @c [fix-ref Root Finding]
9215 Also, of course, @kbd{a R} is a built-in command that uses Newton's
9216 method (among others) to look for numerical solutions to any equation.
9217 @xref{Root Finding}.
9219 @node Programming Answer 9, Programming Answer 10, Programming Answer 8, Answers to Exercises
9220 @subsection Programming Tutorial Exercise 9
9223 The first step is to adjust @expr{z} to be greater than 5. A simple
9224 ``for'' loop will do the job here. If @expr{z} is less than 5, we
9225 reduce the problem using
9226 @texline @math{\psi(z) = \psi(z+1) - 1/z}.
9227 @infoline @expr{psi(z) = psi(z+1) - 1/z}. We go
9229 @texline @math{\psi(z+1)},
9230 @infoline @expr{psi(z+1)},
9231 and remember to add back a factor of @expr{-1/z} when we're done. This
9232 step is repeated until @expr{z > 5}.
9234 (Because this definition is long, it will be repeated in concise form
9235 below. You can use @w{@kbd{C-x * m}} to load it from there. While you are
9236 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9237 keystrokes without executing them. In the following diagrams we'll
9238 pretend Calc actually executed the keystrokes as you typed them,
9239 just for purposes of illustration.)
9246 1.0 @key{RET} C-x ( Z ` s 1 0 t 2
9250 Here, variable 1 holds @expr{z} and variable 2 holds the adjustment
9251 factor. If @expr{z < 5}, we use a loop to increase it.
9253 (By the way, we started with @samp{1.0} instead of the integer 1 because
9254 otherwise the calculation below will try to do exact fractional arithmetic,
9255 and will never converge because fractions compare equal only if they
9256 are exactly equal, not just equal to within the current precision.)
9265 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9269 Now we compute the initial part of the sum:
9270 @texline @math{\ln z - {1 \over 2z}}
9271 @infoline @expr{ln(z) - 1/2z}
9272 minus the adjustment factor.
9276 2: 1.79175946923 2: 1.7084261359 1: -0.57490719743
9277 1: 0.0833333333333 1: 2.28333333333 .
9284 Now we evaluate the series. We'll use another ``for'' loop counting
9285 up the value of @expr{2 n}. (Calc does have a summation command,
9286 @kbd{a +}, but we'll use loops just to get more practice with them.)
9290 3: -0.5749 3: -0.5749 4: -0.5749 2: -0.5749
9291 2: 2 2: 1:6 3: 1:6 1: 2.3148e-3
9296 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9303 3: -0.5749 3: -0.5772 2: -0.5772 1: -0.577215664892
9304 2: -0.5749 2: -0.5772 1: 0 .
9305 1: 2.3148e-3 1: -0.5749 .
9308 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z / 2 Z ) Z ' C-x )
9312 This is the value of
9313 @texline @math{-\gamma},
9314 @infoline @expr{- gamma},
9315 with a slight bit of roundoff error. To get a full 12 digits, let's use
9320 2: -0.577215664892 2: -0.577215664892
9321 1: 1. 1: -0.577215664901532
9323 1. @key{RET} p 16 @key{RET} X
9327 Here's the complete sequence of keystrokes:
9332 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9334 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9335 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z /
9342 @node Programming Answer 10, Programming Answer 11, Programming Answer 9, Answers to Exercises
9343 @subsection Programming Tutorial Exercise 10
9346 Taking the derivative of a term of the form @expr{x^n} will produce
9348 @texline @math{n x^{n-1}}.
9349 @infoline @expr{n x^(n-1)}.
9350 Taking the derivative of a constant
9351 produces zero. From this it is easy to see that the @expr{n}th
9352 derivative of a polynomial, evaluated at @expr{x = 0}, will equal the
9353 coefficient on the @expr{x^n} term times @expr{n!}.
9355 (Because this definition is long, it will be repeated in concise form
9356 below. You can use @w{@kbd{C-x * m}} to load it from there. While you are
9357 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9358 keystrokes without executing them. In the following diagrams we'll
9359 pretend Calc actually executed the keystrokes as you typed them,
9360 just for purposes of illustration.)
9364 2: 5 x^4 + (x + 1)^2 3: 5 x^4 + (x + 1)^2
9369 ' 5 x^4 + (x+1)^2 @key{RET} 6 C-x ( Z ` [ ] t 1 0 @key{TAB}
9374 Variable 1 will accumulate the vector of coefficients.
9378 2: 0 3: 0 2: 5 x^4 + ...
9379 1: 5 x^4 + ... 2: 5 x^4 + ... 1: 1
9383 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9388 Note that @kbd{s | 1} appends the top-of-stack value to the vector
9389 in a variable; it is completely analogous to @kbd{s + 1}. We could
9390 have written instead, @kbd{r 1 @key{TAB} | t 1}.
9394 1: 20 x^3 + 2 x + 2 1: 0 1: [1, 2, 1, 0, 5, 0, 0]
9397 a d x @key{RET} 1 Z ) @key{DEL} r 1 Z ' C-x )
9401 To convert back, a simple method is just to map the coefficients
9402 against a table of powers of @expr{x}.
9406 2: [1, 2, 1, 0, 5, 0, 0] 2: [1, 2, 1, 0, 5, 0, 0]
9407 1: 6 1: [0, 1, 2, 3, 4, 5, 6]
9410 6 @key{RET} 1 + 0 @key{RET} 1 C-u v x
9417 2: [1, 2, 1, 0, 5, 0, 0] 2: 1 + 2 x + x^2 + 5 x^4
9418 1: [1, x, x^2, x^3, ... ] .
9421 ' x @key{RET} @key{TAB} V M ^ *
9425 Once again, here are the whole polynomial to/from vector programs:
9429 C-x ( Z ` [ ] t 1 0 @key{TAB}
9430 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9436 C-x ( 1 + 0 @key{RET} 1 C-u v x ' x @key{RET} @key{TAB} V M ^ * C-x )
9440 @node Programming Answer 11, Programming Answer 12, Programming Answer 10, Answers to Exercises
9441 @subsection Programming Tutorial Exercise 11
9444 First we define a dummy program to go on the @kbd{z s} key. The true
9445 @w{@kbd{z s}} key is supposed to take two numbers from the stack and
9446 return one number, so @key{DEL} as a dummy definition will make
9447 sure the stack comes out right.
9455 4 @key{RET} 2 C-x ( @key{DEL} C-x ) Z K s @key{RET} 2
9459 The last step replaces the 2 that was eaten during the creation
9460 of the dummy @kbd{z s} command. Now we move on to the real
9461 definition. The recurrence needs to be rewritten slightly,
9462 to the form @expr{s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)}.
9464 (Because this definition is long, it will be repeated in concise form
9465 below. You can use @kbd{C-x * m} to load it from there.)
9475 C-x ( M-2 @key{RET} a = Z [ @key{DEL} @key{DEL} 1 Z :
9482 4: 4 2: 4 2: 3 4: 3 4: 3 3: 3
9483 3: 2 1: 2 1: 2 3: 2 3: 2 2: 2
9484 2: 2 . . 2: 3 2: 3 1: 3
9488 @key{RET} 0 a = Z [ @key{DEL} @key{DEL} 0 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9493 (Note that the value 3 that our dummy @kbd{z s} produces is not correct;
9494 it is merely a placeholder that will do just as well for now.)
9498 3: 3 4: 3 3: 3 2: 3 1: -6
9499 2: 3 3: 3 2: 3 1: 9 .
9504 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9511 1: -6 2: 4 1: 11 2: 11
9515 Z ] Z ] C-x ) Z K s @key{RET} @key{DEL} 4 @key{RET} 2 z s M-@key{RET} k s
9519 Even though the result that we got during the definition was highly
9520 bogus, once the definition is complete the @kbd{z s} command gets
9523 Here's the full program once again:
9527 C-x ( M-2 @key{RET} a =
9528 Z [ @key{DEL} @key{DEL} 1
9530 Z [ @key{DEL} @key{DEL} 0
9531 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9532 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9539 You can read this definition using @kbd{C-x * m} (@code{read-kbd-macro})
9540 followed by @kbd{Z K s}, without having to make a dummy definition
9541 first, because @code{read-kbd-macro} doesn't need to execute the
9542 definition as it reads it in. For this reason, @code{C-x * m} is often
9543 the easiest way to create recursive programs in Calc.
9545 @node Programming Answer 12, , Programming Answer 11, Answers to Exercises
9546 @subsection Programming Tutorial Exercise 12
9549 This turns out to be a much easier way to solve the problem. Let's
9550 denote Stirling numbers as calls of the function @samp{s}.
9552 First, we store the rewrite rules corresponding to the definition of
9553 Stirling numbers in a convenient variable:
9556 s e StirlingRules @key{RET}
9557 [ s(n,n) := 1 :: n >= 0,
9558 s(n,0) := 0 :: n > 0,
9559 s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ]
9563 Now, it's just a matter of applying the rules:
9567 2: 4 1: s(4, 2) 1: 11
9571 4 @key{RET} 2 C-x ( ' s($$,$) @key{RET} a r StirlingRules @key{RET} C-x )
9575 As in the case of the @code{fib} rules, it would be useful to put these
9576 rules in @code{EvalRules} and to add a @samp{:: remember} condition to
9579 @c This ends the table-of-contents kludge from above:
9581 \global\let\chapternofonts=\oldchapternofonts
9586 @node Introduction, Data Types, Tutorial, Top
9587 @chapter Introduction
9590 This chapter is the beginning of the Calc reference manual.
9591 It covers basic concepts such as the stack, algebraic and
9592 numeric entry, undo, numeric prefix arguments, etc.
9595 @c (Chapter 2, the Tutorial, has been printed in a separate volume.)
9603 * Quick Calculator::
9604 * Prefix Arguments::
9607 * Multiple Calculators::
9608 * Troubleshooting Commands::
9611 @node Basic Commands, Help Commands, Introduction, Introduction
9612 @section Basic Commands
9617 @cindex Starting the Calculator
9618 @cindex Running the Calculator
9619 To start the Calculator in its standard interface, type @kbd{M-x calc}.
9620 By default this creates a pair of small windows, @samp{*Calculator*}
9621 and @samp{*Calc Trail*}. The former displays the contents of the
9622 Calculator stack and is manipulated exclusively through Calc commands.
9623 It is possible (though not usually necessary) to create several Calc
9624 mode buffers each of which has an independent stack, undo list, and
9625 mode settings. There is exactly one Calc Trail buffer; it records a
9626 list of the results of all calculations that have been done. The
9627 Calc Trail buffer uses a variant of Calc mode, so Calculator commands
9628 still work when the trail buffer's window is selected. It is possible
9629 to turn the trail window off, but the @samp{*Calc Trail*} buffer itself
9630 still exists and is updated silently. @xref{Trail Commands}.
9637 In most installations, the @kbd{C-x * c} key sequence is a more
9638 convenient way to start the Calculator. Also, @kbd{C-x * *}
9639 is a synonym for @kbd{C-x * c} unless you last used Calc
9644 @pindex calc-execute-extended-command
9645 Most Calc commands use one or two keystrokes. Lower- and upper-case
9646 letters are distinct. Commands may also be entered in full @kbd{M-x} form;
9647 for some commands this is the only form. As a convenience, the @kbd{x}
9648 key (@code{calc-execute-extended-command})
9649 is like @kbd{M-x} except that it enters the initial string @samp{calc-}
9650 for you. For example, the following key sequences are equivalent:
9651 @kbd{S}, @kbd{M-x calc-sin @key{RET}}, @kbd{x sin @key{RET}}.
9653 Although Calc is designed to be used from the keyboard, some of
9654 Calc's more common commands are available from a menu. In the menu, the
9655 arguments to the functions are given by referring to their stack level
9658 @cindex Extensions module
9659 @cindex @file{calc-ext} module
9660 The Calculator exists in many parts. When you type @kbd{C-x * c}, the
9661 Emacs ``auto-load'' mechanism will bring in only the first part, which
9662 contains the basic arithmetic functions. The other parts will be
9663 auto-loaded the first time you use the more advanced commands like trig
9664 functions or matrix operations. This is done to improve the response time
9665 of the Calculator in the common case when all you need to do is a
9666 little arithmetic. If for some reason the Calculator fails to load an
9667 extension module automatically, you can force it to load all the
9668 extensions by using the @kbd{C-x * L} (@code{calc-load-everything})
9669 command. @xref{Mode Settings}.
9671 If you type @kbd{M-x calc} or @kbd{C-x * c} with any numeric prefix argument,
9672 the Calculator is loaded if necessary, but it is not actually started.
9673 If the argument is positive, the @file{calc-ext} extensions are also
9674 loaded if necessary. User-written Lisp code that wishes to make use
9675 of Calc's arithmetic routines can use @samp{(calc 0)} or @samp{(calc 1)}
9676 to auto-load the Calculator.
9680 If you type @kbd{C-x * b}, then next time you use @kbd{C-x * c} you
9681 will get a Calculator that uses the full height of the Emacs screen.
9682 When full-screen mode is on, @kbd{C-x * c} runs the @code{full-calc}
9683 command instead of @code{calc}. From the Unix shell you can type
9684 @samp{emacs -f full-calc} to start a new Emacs specifically for use
9685 as a calculator. When Calc is started from the Emacs command line
9686 like this, Calc's normal ``quit'' commands actually quit Emacs itself.
9689 @pindex calc-other-window
9690 The @kbd{C-x * o} command is like @kbd{C-x * c} except that the Calc
9691 window is not actually selected. If you are already in the Calc
9692 window, @kbd{C-x * o} switches you out of it. (The regular Emacs
9693 @kbd{C-x o} command would also work for this, but it has a
9694 tendency to drop you into the Calc Trail window instead, which
9695 @kbd{C-x * o} takes care not to do.)
9700 For one quick calculation, you can type @kbd{C-x * q} (@code{quick-calc})
9701 which prompts you for a formula (like @samp{2+3/4}). The result is
9702 displayed at the bottom of the Emacs screen without ever creating
9703 any special Calculator windows. @xref{Quick Calculator}.
9708 Finally, if you are using the X window system you may want to try
9709 @kbd{C-x * k} (@code{calc-keypad}) which runs Calc with a
9710 ``calculator keypad'' picture as well as a stack display. Click on
9711 the keys with the mouse to operate the calculator. @xref{Keypad Mode}.
9715 @cindex Quitting the Calculator
9716 @cindex Exiting the Calculator
9717 The @kbd{q} key (@code{calc-quit}) exits Calc mode and closes the
9718 Calculator's window(s). It does not delete the Calculator buffers.
9719 If you type @kbd{M-x calc} again, the Calculator will reappear with the
9720 contents of the stack intact. Typing @kbd{C-x * c} or @kbd{C-x * *}
9721 again from inside the Calculator buffer is equivalent to executing
9722 @code{calc-quit}; you can think of @kbd{C-x * *} as toggling the
9723 Calculator on and off.
9726 The @kbd{C-x * x} command also turns the Calculator off, no matter which
9727 user interface (standard, Keypad, or Embedded) is currently active.
9728 It also cancels @code{calc-edit} mode if used from there.
9731 @pindex calc-refresh
9732 @cindex Refreshing a garbled display
9733 @cindex Garbled displays, refreshing
9734 The @kbd{d @key{SPC}} key sequence (@code{calc-refresh}) redraws the contents
9735 of the Calculator buffer from memory. Use this if the contents of the
9736 buffer have been damaged somehow.
9741 The @kbd{o} key (@code{calc-realign}) moves the cursor back to its
9742 ``home'' position at the bottom of the Calculator buffer.
9746 @pindex calc-scroll-left
9747 @pindex calc-scroll-right
9748 @cindex Horizontal scrolling
9750 @cindex Wide text, scrolling
9751 The @kbd{<} and @kbd{>} keys are bound to @code{calc-scroll-left} and
9752 @code{calc-scroll-right}. These are just like the normal horizontal
9753 scrolling commands except that they scroll one half-screen at a time by
9754 default. (Calc formats its output to fit within the bounds of the
9755 window whenever it can.)
9759 @pindex calc-scroll-down
9760 @pindex calc-scroll-up
9761 @cindex Vertical scrolling
9762 The @kbd{@{} and @kbd{@}} keys are bound to @code{calc-scroll-down}
9763 and @code{calc-scroll-up}. They scroll up or down by one-half the
9764 height of the Calc window.
9768 The @kbd{C-x * 0} command (@code{calc-reset}; that's @kbd{C-x *} followed
9769 by a zero) resets the Calculator to its initial state. This clears
9770 the stack, resets all the modes to their initial values (the values
9771 that were saved with @kbd{m m} (@code{calc-save-modes})), clears the
9772 caches (@pxref{Caches}), and so on. (It does @emph{not} erase the
9773 values of any variables.) With an argument of 0, Calc will be reset to
9774 its default state; namely, the modes will be given their default values.
9775 With a positive prefix argument, @kbd{C-x * 0} preserves the contents of
9776 the stack but resets everything else to its initial state; with a
9777 negative prefix argument, @kbd{C-x * 0} preserves the contents of the
9778 stack but resets everything else to its default state.
9780 @node Help Commands, Stack Basics, Basic Commands, Introduction
9781 @section Help Commands
9784 @cindex Help commands
9804 The @kbd{?} key (@code{calc-help}) displays a series of brief help messages.
9805 Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs's
9806 @key{ESC} and @kbd{C-x} prefixes. You can type
9807 @kbd{?} after a prefix to see a list of commands beginning with that
9808 prefix. (If the message includes @samp{[MORE]}, press @kbd{?} again
9809 to see additional commands for that prefix.)
9812 @pindex calc-full-help
9813 The @kbd{h h} (@code{calc-full-help}) command displays all the @kbd{?}
9814 responses at once. When printed, this makes a nice, compact (three pages)
9815 summary of Calc keystrokes.
9817 In general, the @kbd{h} key prefix introduces various commands that
9818 provide help within Calc. Many of the @kbd{h} key functions are
9819 Calc-specific analogues to the @kbd{C-h} functions for Emacs help.
9825 The @kbd{h i} (@code{calc-info}) command runs the Emacs Info system
9826 to read this manual on-line. This is basically the same as typing
9827 @kbd{C-h i} (the regular way to run the Info system), then, if Info
9828 is not already in the Calc manual, selecting the beginning of the
9829 manual. The @kbd{C-x * i} command is another way to read the Calc
9830 manual; it is different from @kbd{h i} in that it works any time,
9831 not just inside Calc. The plain @kbd{i} key is also equivalent to
9832 @kbd{h i}, though this key is obsolete and may be replaced with a
9833 different command in a future version of Calc.
9837 @pindex calc-tutorial
9838 The @kbd{h t} (@code{calc-tutorial}) command runs the Info system on
9839 the Tutorial section of the Calc manual. It is like @kbd{h i},
9840 except that it selects the starting node of the tutorial rather
9841 than the beginning of the whole manual. (It actually selects the
9842 node ``Interactive Tutorial'' which tells a few things about
9843 using the Info system before going on to the actual tutorial.)
9844 The @kbd{C-x * t} key is equivalent to @kbd{h t} (but it works at
9849 @pindex calc-info-summary
9850 The @kbd{h s} (@code{calc-info-summary}) command runs the Info system
9851 on the Summary node of the Calc manual. @xref{Summary}. The @kbd{C-x * s}
9852 key is equivalent to @kbd{h s}.
9855 @pindex calc-describe-key
9856 The @kbd{h k} (@code{calc-describe-key}) command looks up a key
9857 sequence in the Calc manual. For example, @kbd{h k H a S} looks
9858 up the documentation on the @kbd{H a S} (@code{calc-solve-for})
9859 command. This works by looking up the textual description of
9860 the key(s) in the Key Index of the manual, then jumping to the
9861 node indicated by the index.
9863 Most Calc commands do not have traditional Emacs documentation
9864 strings, since the @kbd{h k} command is both more convenient and
9865 more instructive. This means the regular Emacs @kbd{C-h k}
9866 (@code{describe-key}) command will not be useful for Calc keystrokes.
9869 @pindex calc-describe-key-briefly
9870 The @kbd{h c} (@code{calc-describe-key-briefly}) command reads a
9871 key sequence and displays a brief one-line description of it at
9872 the bottom of the screen. It looks for the key sequence in the
9873 Summary node of the Calc manual; if it doesn't find the sequence
9874 there, it acts just like its regular Emacs counterpart @kbd{C-h c}
9875 (@code{describe-key-briefly}). For example, @kbd{h c H a S}
9876 gives the description:
9879 H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes)
9883 which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for}
9884 takes a value @expr{a} from the stack, prompts for a value @expr{v},
9885 then applies the algebraic function @code{fsolve} to these values.
9886 The @samp{?=notes} message means you can now type @kbd{?} to see
9887 additional notes from the summary that apply to this command.
9890 @pindex calc-describe-function
9891 The @kbd{h f} (@code{calc-describe-function}) command looks up an
9892 algebraic function or a command name in the Calc manual. Enter an
9893 algebraic function name to look up that function in the Function
9894 Index or enter a command name beginning with @samp{calc-} to look it
9895 up in the Command Index. This command will also look up operator
9896 symbols that can appear in algebraic formulas, like @samp{%} and
9900 @pindex calc-describe-variable
9901 The @kbd{h v} (@code{calc-describe-variable}) command looks up a
9902 variable in the Calc manual. Enter a variable name like @code{pi} or
9906 @pindex describe-bindings
9907 The @kbd{h b} (@code{calc-describe-bindings}) command is just like
9908 @kbd{C-h b}, except that only local (Calc-related) key bindings are
9912 The @kbd{h n} or @kbd{h C-n} (@code{calc-view-news}) command displays
9913 the ``news'' or change history of Calc. This is kept in the file
9914 @file{README}, which Calc looks for in the same directory as the Calc
9920 The @kbd{h C-c}, @kbd{h C-d}, and @kbd{h C-w} keys display copying,
9921 distribution, and warranty information about Calc. These work by
9922 pulling up the appropriate parts of the ``Copying'' or ``Reporting
9923 Bugs'' sections of the manual.
9925 @node Stack Basics, Numeric Entry, Help Commands, Introduction
9926 @section Stack Basics
9929 @cindex Stack basics
9930 @c [fix-tut RPN Calculations and the Stack]
9931 Calc uses RPN notation. If you are not familiar with RPN, @pxref{RPN
9934 To add the numbers 1 and 2 in Calc you would type the keys:
9935 @kbd{1 @key{RET} 2 +}.
9936 (@key{RET} corresponds to the @key{ENTER} key on most calculators.)
9937 The first three keystrokes ``push'' the numbers 1 and 2 onto the stack. The
9938 @kbd{+} key always ``pops'' the top two numbers from the stack, adds them,
9939 and pushes the result (3) back onto the stack. This number is ready for
9940 further calculations: @kbd{5 -} pushes 5 onto the stack, then pops the
9941 3 and 5, subtracts them, and pushes the result (@mathit{-2}).
9943 Note that the ``top'' of the stack actually appears at the @emph{bottom}
9944 of the buffer. A line containing a single @samp{.} character signifies
9945 the end of the buffer; Calculator commands operate on the number(s)
9946 directly above this line. The @kbd{d t} (@code{calc-truncate-stack})
9947 command allows you to move the @samp{.} marker up and down in the stack;
9948 @pxref{Truncating the Stack}.
9951 @pindex calc-line-numbering
9952 Stack elements are numbered consecutively, with number 1 being the top of
9953 the stack. These line numbers are ordinarily displayed on the lefthand side
9954 of the window. The @kbd{d l} (@code{calc-line-numbering}) command controls
9955 whether these numbers appear. (Line numbers may be turned off since they
9956 slow the Calculator down a bit and also clutter the display.)
9959 @pindex calc-realign
9960 The unshifted letter @kbd{o} (@code{calc-realign}) command repositions
9961 the cursor to its top-of-stack ``home'' position. It also undoes any
9962 horizontal scrolling in the window. If you give it a numeric prefix
9963 argument, it instead moves the cursor to the specified stack element.
9965 The @key{RET} (or equivalent @key{SPC}) key is only required to separate
9966 two consecutive numbers.
9967 (After all, if you typed @kbd{1 2} by themselves the Calculator
9968 would enter the number 12.) If you press @key{RET} or @key{SPC} @emph{not}
9969 right after typing a number, the key duplicates the number on the top of
9970 the stack. @kbd{@key{RET} *} is thus a handy way to square a number.
9972 The @key{DEL} key pops and throws away the top number on the stack.
9973 The @key{TAB} key swaps the top two objects on the stack.
9974 @xref{Stack and Trail}, for descriptions of these and other stack-related
9977 @node Numeric Entry, Algebraic Entry, Stack Basics, Introduction
9978 @section Numeric Entry
9984 @cindex Numeric entry
9985 @cindex Entering numbers
9986 Pressing a digit or other numeric key begins numeric entry using the
9987 minibuffer. The number is pushed on the stack when you press the @key{RET}
9988 or @key{SPC} keys. If you press any other non-numeric key, the number is
9989 pushed onto the stack and the appropriate operation is performed. If
9990 you press a numeric key which is not valid, the key is ignored.
9993 @cindex Negative numbers, entering
9995 There are three different concepts corresponding to the word ``minus,''
9996 typified by @expr{a-b} (subtraction), @expr{-x}
9997 (change-sign), and @expr{-5} (negative number). Calc uses three
9998 different keys for these operations, respectively:
9999 @kbd{-}, @kbd{n}, and @kbd{_} (the underscore). The @kbd{-} key subtracts
10000 the two numbers on the top of the stack. The @kbd{n} key changes the sign
10001 of the number on the top of the stack or the number currently being entered.
10002 The @kbd{_} key begins entry of a negative number or changes the sign of
10003 the number currently being entered. The following sequences all enter the
10004 number @mathit{-5} onto the stack: @kbd{0 @key{RET} 5 -}, @kbd{5 n @key{RET}},
10005 @kbd{5 @key{RET} n}, @kbd{_ 5 @key{RET}}, @kbd{5 _ @key{RET}}.
10007 Some other keys are active during numeric entry, such as @kbd{#} for
10008 non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms.
10009 These notations are described later in this manual with the corresponding
10010 data types. @xref{Data Types}.
10012 During numeric entry, the only editing key available is @key{DEL}.
10014 @node Algebraic Entry, Quick Calculator, Numeric Entry, Introduction
10015 @section Algebraic Entry
10019 @pindex calc-algebraic-entry
10020 @cindex Algebraic notation
10021 @cindex Formulas, entering
10022 The @kbd{'} (@code{calc-algebraic-entry}) command can be used to enter
10023 calculations in algebraic form. This is accomplished by typing the
10024 apostrophe key, ', followed by the expression in standard format:
10032 @texline @math{2+(3\times4) = 14}
10033 @infoline @expr{2+(3*4) = 14}
10034 and push it on the stack. If you wish you can
10035 ignore the RPN aspect of Calc altogether and simply enter algebraic
10036 expressions in this way. You may want to use @key{DEL} every so often to
10037 clear previous results off the stack.
10039 You can press the apostrophe key during normal numeric entry to switch
10040 the half-entered number into Algebraic entry mode. One reason to do
10041 this would be to fix a typo, as the full Emacs cursor motion and editing
10042 keys are available during algebraic entry but not during numeric entry.
10044 In the same vein, during either numeric or algebraic entry you can
10045 press @kbd{`} (backquote) to switch to @code{calc-edit} mode, where
10046 you complete your half-finished entry in a separate buffer.
10047 @xref{Editing Stack Entries}.
10050 @pindex calc-algebraic-mode
10051 @cindex Algebraic Mode
10052 If you prefer algebraic entry, you can use the command @kbd{m a}
10053 (@code{calc-algebraic-mode}) to set Algebraic mode. In this mode,
10054 digits and other keys that would normally start numeric entry instead
10055 start full algebraic entry; as long as your formula begins with a digit
10056 you can omit the apostrophe. Open parentheses and square brackets also
10057 begin algebraic entry. You can still do RPN calculations in this mode,
10058 but you will have to press @key{RET} to terminate every number:
10059 @kbd{2 @key{RET} 3 @key{RET} * 4 @key{RET} +} would accomplish the same
10060 thing as @kbd{2*3+4 @key{RET}}.
10062 @cindex Incomplete Algebraic Mode
10063 If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a}
10064 command, it enables Incomplete Algebraic mode; this is like regular
10065 Algebraic mode except that it applies to the @kbd{(} and @kbd{[} keys
10066 only. Numeric keys still begin a numeric entry in this mode.
10069 @pindex calc-total-algebraic-mode
10070 @cindex Total Algebraic Mode
10071 The @kbd{m t} (@code{calc-total-algebraic-mode}) gives you an even
10072 stronger algebraic-entry mode, in which @emph{all} regular letter and
10073 punctuation keys begin algebraic entry. Use this if you prefer typing
10074 @w{@kbd{sqrt( )}} instead of @kbd{Q}, @w{@kbd{factor( )}} instead of
10075 @kbd{a f}, and so on. To type regular Calc commands when you are in
10076 Total Algebraic mode, hold down the @key{META} key. Thus @kbd{M-q}
10077 is the command to quit Calc, @kbd{M-p} sets the precision, and
10078 @kbd{M-m t} (or @kbd{M-m M-t}, if you prefer) turns Total Algebraic
10079 mode back off again. Meta keys also terminate algebraic entry, so
10080 that @kbd{2+3 M-S} is equivalent to @kbd{2+3 @key{RET} M-S}. The symbol
10081 @samp{Alg*} will appear in the mode line whenever you are in this mode.
10083 Pressing @kbd{'} (the apostrophe) a second time re-enters the previous
10084 algebraic formula. You can then use the normal Emacs editing keys to
10085 modify this formula to your liking before pressing @key{RET}.
10088 @cindex Formulas, referring to stack
10089 Within a formula entered from the keyboard, the symbol @kbd{$}
10090 represents the number on the top of the stack. If an entered formula
10091 contains any @kbd{$} characters, the Calculator replaces the top of
10092 stack with that formula rather than simply pushing the formula onto the
10093 stack. Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{$*2
10094 @key{RET}} replaces it with 6. Note that the @kbd{$} key always
10095 initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{$} is the
10096 first character in the new formula.
10098 Higher stack elements can be accessed from an entered formula with the
10099 symbols @kbd{$$}, @kbd{$$$}, and so on. The number of stack elements
10100 removed (to be replaced by the entered values) equals the number of dollar
10101 signs in the longest such symbol in the formula. For example, @samp{$$+$$$}
10102 adds the second and third stack elements, replacing the top three elements
10103 with the answer. (All information about the top stack element is thus lost
10104 since no single @samp{$} appears in this formula.)
10106 A slightly different way to refer to stack elements is with a dollar
10107 sign followed by a number: @samp{$1}, @samp{$2}, and so on are much
10108 like @samp{$}, @samp{$$}, etc., except that stack entries referred
10109 to numerically are not replaced by the algebraic entry. That is, while
10110 @samp{$+1} replaces 5 on the stack with 6, @samp{$1+1} leaves the 5
10111 on the stack and pushes an additional 6.
10113 If a sequence of formulas are entered separated by commas, each formula
10114 is pushed onto the stack in turn. For example, @samp{1,2,3} pushes
10115 those three numbers onto the stack (leaving the 3 at the top), and
10116 @samp{$+1,$-1} replaces a 5 on the stack with 4 followed by 6. Also,
10117 @samp{$,$$} exchanges the top two elements of the stack, just like the
10120 You can finish an algebraic entry with @kbd{M-=} or @kbd{M-@key{RET}} instead
10121 of @key{RET}. This uses @kbd{=} to evaluate the variables in each
10122 formula that goes onto the stack. (Thus @kbd{' pi @key{RET}} pushes
10123 the variable @samp{pi}, but @kbd{' pi M-@key{RET}} pushes 3.1415.)
10125 If you finish your algebraic entry by pressing @key{LFD} (or @kbd{C-j})
10126 instead of @key{RET}, Calc disables simplification
10127 (as if by @kbd{m O}; @pxref{Simplification Modes}) while the entry
10128 is being pushed on the stack. Thus @kbd{' 1+2 @key{RET}} pushes 3
10129 on the stack, but @kbd{' 1+2 @key{LFD}} pushes the formula @expr{1+2};
10130 you might then press @kbd{=} when it is time to evaluate this formula.
10132 @node Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction
10133 @section ``Quick Calculator'' Mode
10138 @cindex Quick Calculator
10139 There is another way to invoke the Calculator if all you need to do
10140 is make one or two quick calculations. Type @kbd{C-x * q} (or
10141 @kbd{M-x quick-calc}), then type any formula as an algebraic entry.
10142 The Calculator will compute the result and display it in the echo
10143 area, without ever actually putting up a Calc window.
10145 You can use the @kbd{$} character in a Quick Calculator formula to
10146 refer to the previous Quick Calculator result. Older results are
10147 not retained; the Quick Calculator has no effect on the full
10148 Calculator's stack or trail. If you compute a result and then
10149 forget what it was, just run @code{C-x * q} again and enter
10150 @samp{$} as the formula.
10152 If this is the first time you have used the Calculator in this Emacs
10153 session, the @kbd{C-x * q} command will create the @code{*Calculator*}
10154 buffer and perform all the usual initializations; it simply will
10155 refrain from putting that buffer up in a new window. The Quick
10156 Calculator refers to the @code{*Calculator*} buffer for all mode
10157 settings. Thus, for example, to set the precision that the Quick
10158 Calculator uses, simply run the full Calculator momentarily and use
10159 the regular @kbd{p} command.
10161 If you use @code{C-x * q} from inside the Calculator buffer, the
10162 effect is the same as pressing the apostrophe key (algebraic entry).
10164 The result of a Quick calculation is placed in the Emacs ``kill ring''
10165 as well as being displayed. A subsequent @kbd{C-y} command will
10166 yank the result into the editing buffer. You can also use this
10167 to yank the result into the next @kbd{C-x * q} input line as a more
10168 explicit alternative to @kbd{$} notation, or to yank the result
10169 into the Calculator stack after typing @kbd{C-x * c}.
10171 If you finish your formula by typing @key{LFD} (or @kbd{C-j}) instead
10172 of @key{RET}, the result is inserted immediately into the current
10173 buffer rather than going into the kill ring.
10175 Quick Calculator results are actually evaluated as if by the @kbd{=}
10176 key (which replaces variable names by their stored values, if any).
10177 If the formula you enter is an assignment to a variable using the
10178 @samp{:=} operator, say, @samp{foo := 2 + 3} or @samp{foo := foo + 1},
10179 then the result of the evaluation is stored in that Calc variable.
10180 @xref{Store and Recall}.
10182 If the result is an integer and the current display radix is decimal,
10183 the number will also be displayed in hex, octal and binary formats. If
10184 the integer is in the range from 1 to 126, it will also be displayed as
10185 an ASCII character.
10187 For example, the quoted character @samp{"x"} produces the vector
10188 result @samp{[120]} (because 120 is the ASCII code of the lower-case
10189 `x'; @pxref{Strings}). Since this is a vector, not an integer, it
10190 is displayed only according to the current mode settings. But
10191 running Quick Calc again and entering @samp{120} will produce the
10192 result @samp{120 (16#78, 8#170, x)} which shows the number in its
10193 decimal, hexadecimal, octal, and ASCII forms.
10195 Please note that the Quick Calculator is not any faster at loading
10196 or computing the answer than the full Calculator; the name ``quick''
10197 merely refers to the fact that it's much less hassle to use for
10198 small calculations.
10200 @node Prefix Arguments, Undo, Quick Calculator, Introduction
10201 @section Numeric Prefix Arguments
10204 Many Calculator commands use numeric prefix arguments. Some, such as
10205 @kbd{d s} (@code{calc-sci-notation}), set a parameter to the value of
10206 the prefix argument or use a default if you don't use a prefix.
10207 Others (like @kbd{d f} (@code{calc-fix-notation})) require an argument
10208 and prompt for a number if you don't give one as a prefix.
10210 As a rule, stack-manipulation commands accept a numeric prefix argument
10211 which is interpreted as an index into the stack. A positive argument
10212 operates on the top @var{n} stack entries; a negative argument operates
10213 on the @var{n}th stack entry in isolation; and a zero argument operates
10214 on the entire stack.
10216 Most commands that perform computations (such as the arithmetic and
10217 scientific functions) accept a numeric prefix argument that allows the
10218 operation to be applied across many stack elements. For unary operations
10219 (that is, functions of one argument like absolute value or complex
10220 conjugate), a positive prefix argument applies that function to the top
10221 @var{n} stack entries simultaneously, and a negative argument applies it
10222 to the @var{n}th stack entry only. For binary operations (functions of
10223 two arguments like addition, GCD, and vector concatenation), a positive
10224 prefix argument ``reduces'' the function across the top @var{n}
10225 stack elements (for example, @kbd{C-u 5 +} sums the top 5 stack entries;
10226 @pxref{Reducing and Mapping}), and a negative argument maps the next-to-top
10227 @var{n} stack elements with the top stack element as a second argument
10228 (for example, @kbd{7 c-u -5 +} adds 7 to the top 5 stack elements).
10229 This feature is not available for operations which use the numeric prefix
10230 argument for some other purpose.
10232 Numeric prefixes are specified the same way as always in Emacs: Press
10233 a sequence of @key{META}-digits, or press @key{ESC} followed by digits,
10234 or press @kbd{C-u} followed by digits. Some commands treat plain
10235 @kbd{C-u} (without any actual digits) specially.
10238 @pindex calc-num-prefix
10239 You can type @kbd{~} (@code{calc-num-prefix}) to pop an integer from the
10240 top of the stack and enter it as the numeric prefix for the next command.
10241 For example, @kbd{C-u 16 p} sets the precision to 16 digits; an alternate
10242 (silly) way to do this would be @kbd{2 @key{RET} 4 ^ ~ p}, i.e., compute 2
10243 to the fourth power and set the precision to that value.
10245 Conversely, if you have typed a numeric prefix argument the @kbd{~} key
10246 pushes it onto the stack in the form of an integer.
10248 @node Undo, Error Messages, Prefix Arguments, Introduction
10249 @section Undoing Mistakes
10255 @cindex Mistakes, undoing
10256 @cindex Undoing mistakes
10257 @cindex Errors, undoing
10258 The shift-@kbd{U} key (@code{calc-undo}) undoes the most recent operation.
10259 If that operation added or dropped objects from the stack, those objects
10260 are removed or restored. If it was a ``store'' operation, you are
10261 queried whether or not to restore the variable to its original value.
10262 The @kbd{U} key may be pressed any number of times to undo successively
10263 farther back in time; with a numeric prefix argument it undoes a
10264 specified number of operations. When the Calculator is quit, as with
10265 the @kbd{q} (@code{calc-quit}) command, the undo history will be
10266 truncated to the length of the customizable variable
10267 @code{calc-undo-length} (@pxref{Customizing Calc}), which by default
10268 is @expr{100}. (Recall that @kbd{C-x * c} is synonymous with
10269 @code{calc-quit} while inside the Calculator; this also truncates the
10272 Currently the mode-setting commands (like @code{calc-precision}) are not
10273 undoable. You can undo past a point where you changed a mode, but you
10274 will need to reset the mode yourself.
10278 @cindex Redoing after an Undo
10279 The shift-@kbd{D} key (@code{calc-redo}) redoes an operation that was
10280 mistakenly undone. Pressing @kbd{U} with a negative prefix argument is
10281 equivalent to executing @code{calc-redo}. You can redo any number of
10282 times, up to the number of recent consecutive undo commands. Redo
10283 information is cleared whenever you give any command that adds new undo
10284 information, i.e., if you undo, then enter a number on the stack or make
10285 any other change, then it will be too late to redo.
10287 @kindex M-@key{RET}
10288 @pindex calc-last-args
10289 @cindex Last-arguments feature
10290 @cindex Arguments, restoring
10291 The @kbd{M-@key{RET}} key (@code{calc-last-args}) is like undo in that
10292 it restores the arguments of the most recent command onto the stack;
10293 however, it does not remove the result of that command. Given a numeric
10294 prefix argument, this command applies to the @expr{n}th most recent
10295 command which removed items from the stack; it pushes those items back
10298 The @kbd{K} (@code{calc-keep-args}) command provides a related function
10299 to @kbd{M-@key{RET}}. @xref{Stack and Trail}.
10301 It is also possible to recall previous results or inputs using the trail.
10302 @xref{Trail Commands}.
10304 The standard Emacs @kbd{C-_} undo key is recognized as a synonym for @kbd{U}.
10306 @node Error Messages, Multiple Calculators, Undo, Introduction
10307 @section Error Messages
10312 @cindex Errors, messages
10313 @cindex Why did an error occur?
10314 Many situations that would produce an error message in other calculators
10315 simply create unsimplified formulas in the Emacs Calculator. For example,
10316 @kbd{1 @key{RET} 0 /} pushes the formula @expr{1 / 0}; @w{@kbd{0 L}} pushes
10317 the formula @samp{ln(0)}. Floating-point overflow and underflow are also
10318 reasons for this to happen.
10320 When a function call must be left in symbolic form, Calc usually
10321 produces a message explaining why. Messages that are probably
10322 surprising or indicative of user errors are displayed automatically.
10323 Other messages are simply kept in Calc's memory and are displayed only
10324 if you type @kbd{w} (@code{calc-why}). You can also press @kbd{w} if
10325 the same computation results in several messages. (The first message
10326 will end with @samp{[w=more]} in this case.)
10329 @pindex calc-auto-why
10330 The @kbd{d w} (@code{calc-auto-why}) command controls when error messages
10331 are displayed automatically. (Calc effectively presses @kbd{w} for you
10332 after your computation finishes.) By default, this occurs only for
10333 ``important'' messages. The other possible modes are to report
10334 @emph{all} messages automatically, or to report none automatically (so
10335 that you must always press @kbd{w} yourself to see the messages).
10337 @node Multiple Calculators, Troubleshooting Commands, Error Messages, Introduction
10338 @section Multiple Calculators
10341 @pindex another-calc
10342 It is possible to have any number of Calc mode buffers at once.
10343 Usually this is done by executing @kbd{M-x another-calc}, which
10344 is similar to @kbd{C-x * c} except that if a @samp{*Calculator*}
10345 buffer already exists, a new, independent one with a name of the
10346 form @samp{*Calculator*<@var{n}>} is created. You can also use the
10347 command @code{calc-mode} to put any buffer into Calculator mode, but
10348 this would ordinarily never be done.
10350 The @kbd{q} (@code{calc-quit}) command does not destroy a Calculator buffer;
10351 it only closes its window. Use @kbd{M-x kill-buffer} to destroy a
10354 Each Calculator buffer keeps its own stack, undo list, and mode settings
10355 such as precision, angular mode, and display formats. In Emacs terms,
10356 variables such as @code{calc-stack} are buffer-local variables. The
10357 global default values of these variables are used only when a new
10358 Calculator buffer is created. The @code{calc-quit} command saves
10359 the stack and mode settings of the buffer being quit as the new defaults.
10361 There is only one trail buffer, @samp{*Calc Trail*}, used by all
10362 Calculator buffers.
10364 @node Troubleshooting Commands, , Multiple Calculators, Introduction
10365 @section Troubleshooting Commands
10368 This section describes commands you can use in case a computation
10369 incorrectly fails or gives the wrong answer.
10371 @xref{Reporting Bugs}, if you find a problem that appears to be due
10372 to a bug or deficiency in Calc.
10375 * Autoloading Problems::
10376 * Recursion Depth::
10381 @node Autoloading Problems, Recursion Depth, Troubleshooting Commands, Troubleshooting Commands
10382 @subsection Autoloading Problems
10385 The Calc program is split into many component files; components are
10386 loaded automatically as you use various commands that require them.
10387 Occasionally Calc may lose track of when a certain component is
10388 necessary; typically this means you will type a command and it won't
10389 work because some function you've never heard of was undefined.
10392 @pindex calc-load-everything
10393 If this happens, the easiest workaround is to type @kbd{C-x * L}
10394 (@code{calc-load-everything}) to force all the parts of Calc to be
10395 loaded right away. This will cause Emacs to take up a lot more
10396 memory than it would otherwise, but it's guaranteed to fix the problem.
10398 @node Recursion Depth, Caches, Autoloading Problems, Troubleshooting Commands
10399 @subsection Recursion Depth
10404 @pindex calc-more-recursion-depth
10405 @pindex calc-less-recursion-depth
10406 @cindex Recursion depth
10407 @cindex ``Computation got stuck'' message
10408 @cindex @code{max-lisp-eval-depth}
10409 @cindex @code{max-specpdl-size}
10410 Calc uses recursion in many of its calculations. Emacs Lisp keeps a
10411 variable @code{max-lisp-eval-depth} which limits the amount of recursion
10412 possible in an attempt to recover from program bugs. If a calculation
10413 ever halts incorrectly with the message ``Computation got stuck or
10414 ran too long,'' use the @kbd{M} command (@code{calc-more-recursion-depth})
10415 to increase this limit. (Of course, this will not help if the
10416 calculation really did get stuck due to some problem inside Calc.)
10418 The limit is always increased (multiplied) by a factor of two. There
10419 is also an @kbd{I M} (@code{calc-less-recursion-depth}) command which
10420 decreases this limit by a factor of two, down to a minimum value of 200.
10421 The default value is 1000.
10423 These commands also double or halve @code{max-specpdl-size}, another
10424 internal Lisp recursion limit. The minimum value for this limit is 600.
10426 @node Caches, Debugging Calc, Recursion Depth, Troubleshooting Commands
10431 @cindex Flushing caches
10432 Calc saves certain values after they have been computed once. For
10433 example, the @kbd{P} (@code{calc-pi}) command initially ``knows'' the
10434 constant @cpi{} to about 20 decimal places; if the current precision
10435 is greater than this, it will recompute @cpi{} using a series
10436 approximation. This value will not need to be recomputed ever again
10437 unless you raise the precision still further. Many operations such as
10438 logarithms and sines make use of similarly cached values such as
10440 @texline @math{\ln 2}.
10441 @infoline @expr{ln(2)}.
10442 The visible effect of caching is that
10443 high-precision computations may seem to do extra work the first time.
10444 Other things cached include powers of two (for the binary arithmetic
10445 functions), matrix inverses and determinants, symbolic integrals, and
10446 data points computed by the graphing commands.
10448 @pindex calc-flush-caches
10449 If you suspect a Calculator cache has become corrupt, you can use the
10450 @code{calc-flush-caches} command to reset all caches to the empty state.
10451 (This should only be necessary in the event of bugs in the Calculator.)
10452 The @kbd{C-x * 0} (with the zero key) command also resets caches along
10453 with all other aspects of the Calculator's state.
10455 @node Debugging Calc, , Caches, Troubleshooting Commands
10456 @subsection Debugging Calc
10459 A few commands exist to help in the debugging of Calc commands.
10460 @xref{Programming}, to see the various ways that you can write
10461 your own Calc commands.
10464 @pindex calc-timing
10465 The @kbd{Z T} (@code{calc-timing}) command turns on and off a mode
10466 in which the timing of slow commands is reported in the Trail.
10467 Any Calc command that takes two seconds or longer writes a line
10468 to the Trail showing how many seconds it took. This value is
10469 accurate only to within one second.
10471 All steps of executing a command are included; in particular, time
10472 taken to format the result for display in the stack and trail is
10473 counted. Some prompts also count time taken waiting for them to
10474 be answered, while others do not; this depends on the exact
10475 implementation of the command. For best results, if you are timing
10476 a sequence that includes prompts or multiple commands, define a
10477 keyboard macro to run the whole sequence at once. Calc's @kbd{X}
10478 command (@pxref{Keyboard Macros}) will then report the time taken
10479 to execute the whole macro.
10481 Another advantage of the @kbd{X} command is that while it is
10482 executing, the stack and trail are not updated from step to step.
10483 So if you expect the output of your test sequence to leave a result
10484 that may take a long time to format and you don't wish to count
10485 this formatting time, end your sequence with a @key{DEL} keystroke
10486 to clear the result from the stack. When you run the sequence with
10487 @kbd{X}, Calc will never bother to format the large result.
10489 Another thing @kbd{Z T} does is to increase the Emacs variable
10490 @code{gc-cons-threshold} to a much higher value (two million; the
10491 usual default in Calc is 250,000) for the duration of each command.
10492 This generally prevents garbage collection during the timing of
10493 the command, though it may cause your Emacs process to grow
10494 abnormally large. (Garbage collection time is a major unpredictable
10495 factor in the timing of Emacs operations.)
10497 Another command that is useful when debugging your own Lisp
10498 extensions to Calc is @kbd{M-x calc-pass-errors}, which disables
10499 the error handler that changes the ``@code{max-lisp-eval-depth}
10500 exceeded'' message to the much more friendly ``Computation got
10501 stuck or ran too long.'' This handler interferes with the Emacs
10502 Lisp debugger's @code{debug-on-error} mode. Errors are reported
10503 in the handler itself rather than at the true location of the
10504 error. After you have executed @code{calc-pass-errors}, Lisp
10505 errors will be reported correctly but the user-friendly message
10508 @node Data Types, Stack and Trail, Introduction, Top
10509 @chapter Data Types
10512 This chapter discusses the various types of objects that can be placed
10513 on the Calculator stack, how they are displayed, and how they are
10514 entered. (@xref{Data Type Formats}, for information on how these data
10515 types are represented as underlying Lisp objects.)
10517 Integers, fractions, and floats are various ways of describing real
10518 numbers. HMS forms also for many purposes act as real numbers. These
10519 types can be combined to form complex numbers, modulo forms, error forms,
10520 or interval forms. (But these last four types cannot be combined
10521 arbitrarily: error forms may not contain modulo forms, for example.)
10522 Finally, all these types of numbers may be combined into vectors,
10523 matrices, or algebraic formulas.
10526 * Integers:: The most basic data type.
10527 * Fractions:: This and above are called @dfn{rationals}.
10528 * Floats:: This and above are called @dfn{reals}.
10529 * Complex Numbers:: This and above are called @dfn{numbers}.
10531 * Vectors and Matrices::
10538 * Incomplete Objects::
10543 @node Integers, Fractions, Data Types, Data Types
10548 The Calculator stores integers to arbitrary precision. Addition,
10549 subtraction, and multiplication of integers always yields an exact
10550 integer result. (If the result of a division or exponentiation of
10551 integers is not an integer, it is expressed in fractional or
10552 floating-point form according to the current Fraction mode.
10553 @xref{Fraction Mode}.)
10555 A decimal integer is represented as an optional sign followed by a
10556 sequence of digits. Grouping (@pxref{Grouping Digits}) can be used to
10557 insert a comma at every third digit for display purposes, but you
10558 must not type commas during the entry of numbers.
10561 A non-decimal integer is represented as an optional sign, a radix
10562 between 2 and 36, a @samp{#} symbol, and one or more digits. For radix 11
10563 and above, the letters A through Z (upper- or lower-case) count as
10564 digits and do not terminate numeric entry mode. @xref{Radix Modes}, for how
10565 to set the default radix for display of integers. Numbers of any radix
10566 may be entered at any time. If you press @kbd{#} at the beginning of a
10567 number, the current display radix is used.
10569 @node Fractions, Floats, Integers, Data Types
10574 A @dfn{fraction} is a ratio of two integers. Fractions are traditionally
10575 written ``2/3'' but Calc uses the notation @samp{2:3}. (The @kbd{/} key
10576 performs RPN division; the following two sequences push the number
10577 @samp{2:3} on the stack: @kbd{2 :@: 3 @key{RET}}, or @kbd{2 @key{RET} 3 /}
10578 assuming Fraction mode has been enabled.)
10579 When the Calculator produces a fractional result it always reduces it to
10580 simplest form, which may in fact be an integer.
10582 Fractions may also be entered in a three-part form, where @samp{2:3:4}
10583 represents two-and-three-quarters. @xref{Fraction Formats}, for fraction
10586 Non-decimal fractions are entered and displayed as
10587 @samp{@var{radix}#@var{num}:@var{denom}} (or in the analogous three-part
10588 form). The numerator and denominator always use the same radix.
10590 @node Floats, Complex Numbers, Fractions, Data Types
10594 @cindex Floating-point numbers
10595 A floating-point number or @dfn{float} is a number stored in scientific
10596 notation. The number of significant digits in the fractional part is
10597 governed by the current floating precision (@pxref{Precision}). The
10598 range of acceptable values is from
10599 @texline @math{10^{-3999999}}
10600 @infoline @expr{10^-3999999}
10602 @texline @math{10^{4000000}}
10603 @infoline @expr{10^4000000}
10604 (exclusive), plus the corresponding negative values and zero.
10606 Calculations that would exceed the allowable range of values (such
10607 as @samp{exp(exp(20))}) are left in symbolic form by Calc. The
10608 messages ``floating-point overflow'' or ``floating-point underflow''
10609 indicate that during the calculation a number would have been produced
10610 that was too large or too close to zero, respectively, to be represented
10611 by Calc. This does not necessarily mean the final result would have
10612 overflowed, just that an overflow occurred while computing the result.
10613 (In fact, it could report an underflow even though the final result
10614 would have overflowed!)
10616 If a rational number and a float are mixed in a calculation, the result
10617 will in general be expressed as a float. Commands that require an integer
10618 value (such as @kbd{k g} [@code{gcd}]) will also accept integer-valued
10619 floats, i.e., floating-point numbers with nothing after the decimal point.
10621 Floats are identified by the presence of a decimal point and/or an
10622 exponent. In general a float consists of an optional sign, digits
10623 including an optional decimal point, and an optional exponent consisting
10624 of an @samp{e}, an optional sign, and up to seven exponent digits.
10625 For example, @samp{23.5e-2} is 23.5 times ten to the minus-second power,
10628 Floating-point numbers are normally displayed in decimal notation with
10629 all significant figures shown. Exceedingly large or small numbers are
10630 displayed in scientific notation. Various other display options are
10631 available. @xref{Float Formats}.
10633 @cindex Accuracy of calculations
10634 Floating-point numbers are stored in decimal, not binary. The result
10635 of each operation is rounded to the nearest value representable in the
10636 number of significant digits specified by the current precision,
10637 rounding away from zero in the case of a tie. Thus (in the default
10638 display mode) what you see is exactly what you get. Some operations such
10639 as square roots and transcendental functions are performed with several
10640 digits of extra precision and then rounded down, in an effort to make the
10641 final result accurate to the full requested precision. However,
10642 accuracy is not rigorously guaranteed. If you suspect the validity of a
10643 result, try doing the same calculation in a higher precision. The
10644 Calculator's arithmetic is not intended to be IEEE-conformant in any
10647 While floats are always @emph{stored} in decimal, they can be entered
10648 and displayed in any radix just like integers and fractions. Since a
10649 float that is entered in a radix other that 10 will be converted to
10650 decimal, the number that Calc stores may not be exactly the number that
10651 was entered, it will be the closest decimal approximation given the
10652 current precision. The notation @samp{@var{radix}#@var{ddd}.@var{ddd}}
10653 is a floating-point number whose digits are in the specified radix.
10654 Note that the @samp{.} is more aptly referred to as a ``radix point''
10655 than as a decimal point in this case. The number @samp{8#123.4567} is
10656 defined as @samp{8#1234567 * 8^-4}. If the radix is 14 or less, you can
10657 use @samp{e} notation to write a non-decimal number in scientific
10658 notation. The exponent is written in decimal, and is considered to be a
10659 power of the radix: @samp{8#1234567e-4}. If the radix is 15 or above,
10660 the letter @samp{e} is a digit, so scientific notation must be written
10661 out, e.g., @samp{16#123.4567*16^2}. The first two exercises of the
10662 Modes Tutorial explore some of the properties of non-decimal floats.
10664 @node Complex Numbers, Infinities, Floats, Data Types
10665 @section Complex Numbers
10668 @cindex Complex numbers
10669 There are two supported formats for complex numbers: rectangular and
10670 polar. The default format is rectangular, displayed in the form
10671 @samp{(@var{real},@var{imag})} where @var{real} is the real part and
10672 @var{imag} is the imaginary part, each of which may be any real number.
10673 Rectangular complex numbers can also be displayed in @samp{@var{a}+@var{b}i}
10674 notation; @pxref{Complex Formats}.
10676 Polar complex numbers are displayed in the form
10677 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'
10678 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'
10679 where @var{r} is the nonnegative magnitude and
10680 @texline @math{\theta}
10681 @infoline @var{theta}
10682 is the argument or phase angle. The range of
10683 @texline @math{\theta}
10684 @infoline @var{theta}
10685 depends on the current angular mode (@pxref{Angular Modes}); it is
10686 generally between @mathit{-180} and @mathit{+180} degrees or the equivalent range
10689 Complex numbers are entered in stages using incomplete objects.
10690 @xref{Incomplete Objects}.
10692 Operations on rectangular complex numbers yield rectangular complex
10693 results, and similarly for polar complex numbers. Where the two types
10694 are mixed, or where new complex numbers arise (as for the square root of
10695 a negative real), the current @dfn{Polar mode} is used to determine the
10696 type. @xref{Polar Mode}.
10698 A complex result in which the imaginary part is zero (or the phase angle
10699 is 0 or 180 degrees or @cpi{} radians) is automatically converted to a real
10702 @node Infinities, Vectors and Matrices, Complex Numbers, Data Types
10703 @section Infinities
10707 @cindex @code{inf} variable
10708 @cindex @code{uinf} variable
10709 @cindex @code{nan} variable
10713 The word @code{inf} represents the mathematical concept of @dfn{infinity}.
10714 Calc actually has three slightly different infinity-like values:
10715 @code{inf}, @code{uinf}, and @code{nan}. These are just regular
10716 variable names (@pxref{Variables}); you should avoid using these
10717 names for your own variables because Calc gives them special
10718 treatment. Infinities, like all variable names, are normally
10719 entered using algebraic entry.
10721 Mathematically speaking, it is not rigorously correct to treat
10722 ``infinity'' as if it were a number, but mathematicians often do
10723 so informally. When they say that @samp{1 / inf = 0}, what they
10724 really mean is that @expr{1 / x}, as @expr{x} becomes larger and
10725 larger, becomes arbitrarily close to zero. So you can imagine
10726 that if @expr{x} got ``all the way to infinity,'' then @expr{1 / x}
10727 would go all the way to zero. Similarly, when they say that
10728 @samp{exp(inf) = inf}, they mean that
10729 @texline @math{e^x}
10730 @infoline @expr{exp(x)}
10731 grows without bound as @expr{x} grows. The symbol @samp{-inf} likewise
10732 stands for an infinitely negative real value; for example, we say that
10733 @samp{exp(-inf) = 0}. You can have an infinity pointing in any
10734 direction on the complex plane: @samp{sqrt(-inf) = i inf}.
10736 The same concept of limits can be used to define @expr{1 / 0}. We
10737 really want the value that @expr{1 / x} approaches as @expr{x}
10738 approaches zero. But if all we have is @expr{1 / 0}, we can't
10739 tell which direction @expr{x} was coming from. If @expr{x} was
10740 positive and decreasing toward zero, then we should say that
10741 @samp{1 / 0 = inf}. But if @expr{x} was negative and increasing
10742 toward zero, the answer is @samp{1 / 0 = -inf}. In fact, @expr{x}
10743 could be an imaginary number, giving the answer @samp{i inf} or
10744 @samp{-i inf}. Calc uses the special symbol @samp{uinf} to mean
10745 @dfn{undirected infinity}, i.e., a value which is infinitely
10746 large but with an unknown sign (or direction on the complex plane).
10748 Calc actually has three modes that say how infinities are handled.
10749 Normally, infinities never arise from calculations that didn't
10750 already have them. Thus, @expr{1 / 0} is treated simply as an
10751 error and left unevaluated. The @kbd{m i} (@code{calc-infinite-mode})
10752 command (@pxref{Infinite Mode}) enables a mode in which
10753 @expr{1 / 0} evaluates to @code{uinf} instead. There is also
10754 an alternative type of infinite mode which says to treat zeros
10755 as if they were positive, so that @samp{1 / 0 = inf}. While this
10756 is less mathematically correct, it may be the answer you want in
10759 Since all infinities are ``as large'' as all others, Calc simplifies,
10760 e.g., @samp{5 inf} to @samp{inf}. Another example is
10761 @samp{5 - inf = -inf}, where the @samp{-inf} is so large that
10762 adding a finite number like five to it does not affect it.
10763 Note that @samp{a - inf} also results in @samp{-inf}; Calc assumes
10764 that variables like @code{a} always stand for finite quantities.
10765 Just to show that infinities really are all the same size,
10766 note that @samp{sqrt(inf) = inf^2 = exp(inf) = inf} in Calc's
10769 It's not so easy to define certain formulas like @samp{0 * inf} and
10770 @samp{inf / inf}. Depending on where these zeros and infinities
10771 came from, the answer could be literally anything. The latter
10772 formula could be the limit of @expr{x / x} (giving a result of one),
10773 or @expr{2 x / x} (giving two), or @expr{x^2 / x} (giving @code{inf}),
10774 or @expr{x / x^2} (giving zero). Calc uses the symbol @code{nan}
10775 to represent such an @dfn{indeterminate} value. (The name ``nan''
10776 comes from analogy with the ``NAN'' concept of IEEE standard
10777 arithmetic; it stands for ``Not A Number.'' This is somewhat of a
10778 misnomer, since @code{nan} @emph{does} stand for some number or
10779 infinity, it's just that @emph{which} number it stands for
10780 cannot be determined.) In Calc's notation, @samp{0 * inf = nan}
10781 and @samp{inf / inf = nan}. A few other common indeterminate
10782 expressions are @samp{inf - inf} and @samp{inf ^ 0}. Also,
10783 @samp{0 / 0 = nan} if you have turned on Infinite mode
10784 (as described above).
10786 Infinities are especially useful as parts of @dfn{intervals}.
10787 @xref{Interval Forms}.
10789 @node Vectors and Matrices, Strings, Infinities, Data Types
10790 @section Vectors and Matrices
10794 @cindex Plain vectors
10796 The @dfn{vector} data type is flexible and general. A vector is simply a
10797 list of zero or more data objects. When these objects are numbers, the
10798 whole is a vector in the mathematical sense. When these objects are
10799 themselves vectors of equal (nonzero) length, the whole is a @dfn{matrix}.
10800 A vector which is not a matrix is referred to here as a @dfn{plain vector}.
10802 A vector is displayed as a list of values separated by commas and enclosed
10803 in square brackets: @samp{[1, 2, 3]}. Thus the following is a 2 row by
10804 3 column matrix: @samp{[[1, 2, 3], [4, 5, 6]]}. Vectors, like complex
10805 numbers, are entered as incomplete objects. @xref{Incomplete Objects}.
10806 During algebraic entry, vectors are entered all at once in the usual
10807 brackets-and-commas form. Matrices may be entered algebraically as nested
10808 vectors, or using the shortcut notation @w{@samp{[1, 2, 3; 4, 5, 6]}},
10809 with rows separated by semicolons. The commas may usually be omitted
10810 when entering vectors: @samp{[1 2 3]}. Curly braces may be used in
10811 place of brackets: @samp{@{1, 2, 3@}}, but the commas are required in
10814 Traditional vector and matrix arithmetic is also supported;
10815 @pxref{Basic Arithmetic} and @pxref{Matrix Functions}.
10816 Many other operations are applied to vectors element-wise. For example,
10817 the complex conjugate of a vector is a vector of the complex conjugates
10824 Algebraic functions for building vectors include @samp{vec(a, b, c)}
10825 to build @samp{[a, b, c]}, @samp{cvec(a, n, m)} to build an
10826 @texline @math{n\times m}
10827 @infoline @var{n}x@var{m}
10828 matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers
10829 from 1 to @samp{n}.
10831 @node Strings, HMS Forms, Vectors and Matrices, Data Types
10837 @cindex Character strings
10838 Character strings are not a special data type in the Calculator.
10839 Rather, a string is represented simply as a vector all of whose
10840 elements are integers in the range 0 to 255 (ASCII codes). You can
10841 enter a string at any time by pressing the @kbd{"} key. Quotation
10842 marks and backslashes are written @samp{\"} and @samp{\\}, respectively,
10843 inside strings. Other notations introduced by backslashes are:
10859 Finally, a backslash followed by three octal digits produces any
10860 character from its ASCII code.
10863 @pindex calc-display-strings
10864 Strings are normally displayed in vector-of-integers form. The
10865 @w{@kbd{d "}} (@code{calc-display-strings}) command toggles a mode in
10866 which any vectors of small integers are displayed as quoted strings
10869 The backslash notations shown above are also used for displaying
10870 strings. Characters 128 and above are not translated by Calc; unless
10871 you have an Emacs modified for 8-bit fonts, these will show up in
10872 backslash-octal-digits notation. For characters below 32, and
10873 for character 127, Calc uses the backslash-letter combination if
10874 there is one, or otherwise uses a @samp{\^} sequence.
10876 The only Calc feature that uses strings is @dfn{compositions};
10877 @pxref{Compositions}. Strings also provide a convenient
10878 way to do conversions between ASCII characters and integers.
10884 There is a @code{string} function which provides a different display
10885 format for strings. Basically, @samp{string(@var{s})}, where @var{s}
10886 is a vector of integers in the proper range, is displayed as the
10887 corresponding string of characters with no surrounding quotation
10888 marks or other modifications. Thus @samp{string("ABC")} (or
10889 @samp{string([65 66 67])}) will look like @samp{ABC} on the stack.
10890 This happens regardless of whether @w{@kbd{d "}} has been used. The
10891 only way to turn it off is to use @kbd{d U} (unformatted language
10892 mode) which will display @samp{string("ABC")} instead.
10894 Control characters are displayed somewhat differently by @code{string}.
10895 Characters below 32, and character 127, are shown using @samp{^} notation
10896 (same as shown above, but without the backslash). The quote and
10897 backslash characters are left alone, as are characters 128 and above.
10903 The @code{bstring} function is just like @code{string} except that
10904 the resulting string is breakable across multiple lines if it doesn't
10905 fit all on one line. Potential break points occur at every space
10906 character in the string.
10908 @node HMS Forms, Date Forms, Strings, Data Types
10912 @cindex Hours-minutes-seconds forms
10913 @cindex Degrees-minutes-seconds forms
10914 @dfn{HMS} stands for Hours-Minutes-Seconds; when used as an angular
10915 argument, the interpretation is Degrees-Minutes-Seconds. All functions
10916 that operate on angles accept HMS forms. These are interpreted as
10917 degrees regardless of the current angular mode. It is also possible to
10918 use HMS as the angular mode so that calculated angles are expressed in
10919 degrees, minutes, and seconds.
10925 @kindex ' (HMS forms)
10929 @kindex " (HMS forms)
10933 @kindex h (HMS forms)
10937 @kindex o (HMS forms)
10941 @kindex m (HMS forms)
10945 @kindex s (HMS forms)
10946 The default format for HMS values is
10947 @samp{@var{hours}@@ @var{mins}' @var{secs}"}. During entry, the letters
10948 @samp{h} (for ``hours'') or
10949 @samp{o} (approximating the ``degrees'' symbol) are accepted as well as
10950 @samp{@@}, @samp{m} is accepted in place of @samp{'}, and @samp{s} is
10951 accepted in place of @samp{"}.
10952 The @var{hours} value is an integer (or integer-valued float).
10953 The @var{mins} value is an integer or integer-valued float between 0 and 59.
10954 The @var{secs} value is a real number between 0 (inclusive) and 60
10955 (exclusive). A positive HMS form is interpreted as @var{hours} +
10956 @var{mins}/60 + @var{secs}/3600. A negative HMS form is interpreted
10957 as @mathit{- @var{hours}} @mathit{-} @var{mins}/60 @mathit{-} @var{secs}/3600.
10958 Display format for HMS forms is quite flexible. @xref{HMS Formats}.
10960 HMS forms can be added and subtracted. When they are added to numbers,
10961 the numbers are interpreted according to the current angular mode. HMS
10962 forms can also be multiplied and divided by real numbers. Dividing
10963 two HMS forms produces a real-valued ratio of the two angles.
10966 @cindex Time of day
10967 Just for kicks, @kbd{M-x calc-time} pushes the current time of day on
10968 the stack as an HMS form.
10970 @node Date Forms, Modulo Forms, HMS Forms, Data Types
10971 @section Date Forms
10975 A @dfn{date form} represents a date and possibly an associated time.
10976 Simple date arithmetic is supported: Adding a number to a date
10977 produces a new date shifted by that many days; adding an HMS form to
10978 a date shifts it by that many hours. Subtracting two date forms
10979 computes the number of days between them (represented as a simple
10980 number). Many other operations, such as multiplying two date forms,
10981 are nonsensical and are not allowed by Calc.
10983 Date forms are entered and displayed enclosed in @samp{< >} brackets.
10984 The default format is, e.g., @samp{<Wed Jan 9, 1991>} for dates,
10985 or @samp{<3:32:20pm Wed Jan 9, 1991>} for dates with times.
10986 Input is flexible; date forms can be entered in any of the usual
10987 notations for dates and times. @xref{Date Formats}.
10989 Date forms are stored internally as numbers, specifically the number
10990 of days since midnight on the morning of December 31 of the year 1 BC@.
10991 If the internal number is an integer, the form represents a date only;
10992 if the internal number is a fraction or float, the form represents
10993 a date and time. For example, @samp{<6:00am Thu Jan 10, 1991>}
10994 is represented by the number 726842.25. The standard precision of
10995 12 decimal digits is enough to ensure that a (reasonable) date and
10996 time can be stored without roundoff error.
10998 If the current precision is greater than 12, date forms will keep
10999 additional digits in the seconds position. For example, if the
11000 precision is 15, the seconds will keep three digits after the
11001 decimal point. Decreasing the precision below 12 may cause the
11002 time part of a date form to become inaccurate. This can also happen
11003 if astronomically high years are used, though this will not be an
11004 issue in everyday (or even everymillennium) use. Note that date
11005 forms without times are stored as exact integers, so roundoff is
11006 never an issue for them.
11008 You can use the @kbd{v p} (@code{calc-pack}) and @kbd{v u}
11009 (@code{calc-unpack}) commands to get at the numerical representation
11010 of a date form. @xref{Packing and Unpacking}.
11012 Date forms can go arbitrarily far into the future or past. Negative
11013 year numbers represent years BC@. There is no ``year 0''; the day
11014 before @samp{<Mon Jan 1, +1>} is @samp{<Sun Dec 31, -1>}. These are
11015 days 1 and 0 respectively in Calc's internal numbering scheme. The
11016 Gregorian calendar is used for all dates, including dates before the
11017 Gregorian calendar was invented (although that can be configured; see
11018 below). Thus Calc's use of the day number @mathit{-10000} to
11019 represent August 15, 28 BC should be taken with a grain of salt.
11021 @cindex Julian calendar
11022 @cindex Gregorian calendar
11023 Some historical background: The Julian calendar was created by
11024 Julius Caesar in the year 46 BC as an attempt to fix the confusion
11025 caused by the irregular Roman calendar that was used before that time.
11026 The Julian calendar introduced an extra day in all years divisible by
11027 four. After some initial confusion, the calendar was adopted around
11028 the year we call 8 AD@. Some centuries later it became
11029 apparent that the Julian year of 365.25 days was itself not quite
11030 right. In 1582 Pope Gregory XIII introduced the Gregorian calendar,
11031 which added the new rule that years divisible by 100, but not by 400,
11032 were not to be considered leap years despite being divisible by four.
11033 Many countries delayed adoption of the Gregorian calendar
11034 because of religious differences. For example, Great Britain and the
11035 British colonies switched to the Gregorian calendar in September
11036 1752, when the Julian calendar was eleven days behind the
11037 Gregorian calendar. That year in Britain, the day after September 2
11038 was September 14. To take another example, Russia did not adopt the
11039 Gregorian calendar until 1918, and that year in Russia the day after
11040 January 31 was February 14. Calc's reckoning therefore matches English
11041 practice starting in 1752 and Russian practice starting in 1918, but
11042 disagrees with earlier dates in both countries.
11044 When the Julian calendar was introduced, it had January 1 as the first
11045 day of the year. By the Middle Ages, many European countries
11046 had changed the beginning of a new year to a different date, often to
11047 a religious festival. Almost all countries reverted to using January 1
11048 as the beginning of the year by the time they adopted the Gregorian
11051 Some calendars attempt to mimic the historical situation by using the
11052 Gregorian calendar for recent dates and the Julian calendar for older
11053 dates. The @code{cal} program in most Unix implementations does this,
11054 for example. While January 1 wasn't always the beginning of a calendar
11055 year, these hybrid calendars still use January 1 as the beginning of
11056 the year even for older dates. The customizable variable
11057 @code{calc-gregorian-switch} (@pxref{Customizing Calc}) can be set to
11058 have Calc's date forms switch from the Julian to Gregorian calendar at
11059 any specified date.
11061 Today's timekeepers introduce an occasional ``leap second''.
11062 These do not occur regularly and Calc does not take these minor
11063 effects into account. (If it did, it would have to report a
11064 non-integer number of days between, say,
11065 @samp{<12:00am Mon Jan 1, 1900>} and
11066 @samp{<12:00am Sat Jan 1, 2000>}.)
11068 @cindex Julian day counting
11069 Another day counting system in common use is, confusingly, also called
11070 ``Julian.'' Julian days go from noon to noon. The Julian day number
11071 is the numbers of days since 12:00 noon (GMT) on November 24, 4714 BC
11072 in the Gregorian calendar (i.e., January 1, 4713 BC in the Julian
11073 calendar). In Calc's scheme (in GMT) the Julian day origin is
11074 @mathit{-1721422.5}, because Calc starts at midnight instead of noon.
11075 Thus to convert a Calc date code obtained by unpacking a
11076 date form into a Julian day number, simply add 1721422.5 after
11077 compensating for the time zone difference. The built-in @kbd{t J}
11078 command performs this conversion for you.
11080 The Julian day number is based on the Julian cycle, which was invented
11081 in 1583 by Joseph Justus Scaliger. Scaliger named it the Julian cycle
11082 since it involves the Julian calendar, but some have suggested that
11083 Scaliger named it in honor of his father, Julius Caesar Scaliger. The
11084 Julian cycle is based on three other cycles: the indiction cycle, the
11085 Metonic cycle, and the solar cycle. The indiction cycle is a 15 year
11086 cycle originally used by the Romans for tax purposes but later used to
11087 date medieval documents. The Metonic cycle is a 19 year cycle; 19
11088 years is close to being a common multiple of a solar year and a lunar
11089 month, and so every 19 years the phases of the moon will occur on the
11090 same days of the year. The solar cycle is a 28 year cycle; the Julian
11091 calendar repeats itself every 28 years. The smallest time period
11092 which contains multiples of all three cycles is the least common
11093 multiple of 15 years, 19 years and 28 years, which (since they're
11094 pairwise relatively prime) is
11095 @texline @math{15\times 19\times 28 = 7980} years.
11096 @infoline 15*19*28 = 7980 years.
11097 This is the length of a Julian cycle. Working backwards, the previous
11098 year in which all three cycles began was 4713 BC, and so Scaliger
11099 chose that year as the beginning of a Julian cycle. Since at the time
11100 there were no historical records from before 4713 BC, using this year
11101 as a starting point had the advantage of avoiding negative year
11102 numbers. In 1849, the astronomer John Herschel (son of William
11103 Herschel) suggested using the number of days since the beginning of
11104 the Julian cycle as an astronomical dating system; this idea was taken
11105 up by other astronomers. (At the time, noon was the start of the
11106 astronomical day. Herschel originally suggested counting the days
11107 since Jan 1, 4713 BC at noon Alexandria time; this was later amended to
11108 noon GMT@.) Julian day numbering is largely used in astronomy.
11110 @cindex Unix time format
11111 The Unix operating system measures time as an integer number of
11112 seconds since midnight, Jan 1, 1970. To convert a Calc date
11113 value into a Unix time stamp, first subtract 719164 (the code
11114 for @samp{<Jan 1, 1970>}), then multiply by 86400 (the number of
11115 seconds in a day) and press @kbd{R} to round to the nearest
11116 integer. If you have a date form, you can simply subtract the
11117 day @samp{<Jan 1, 1970>} instead of unpacking and subtracting
11118 719164. Likewise, divide by 86400 and add @samp{<Jan 1, 1970>}
11119 to convert from Unix time to a Calc date form. (Note that
11120 Unix normally maintains the time in the GMT time zone; you may
11121 need to subtract five hours to get New York time, or eight hours
11122 for California time. The same is usually true of Julian day
11123 counts.) The built-in @kbd{t U} command performs these
11126 @node Modulo Forms, Error Forms, Date Forms, Data Types
11127 @section Modulo Forms
11130 @cindex Modulo forms
11131 A @dfn{modulo form} is a real number which is taken modulo (i.e., within
11132 an integer multiple of) some value @var{M}. Arithmetic modulo @var{M}
11133 often arises in number theory. Modulo forms are written
11134 `@var{a} @tfn{mod} @var{M}',
11135 where @var{a} and @var{M} are real numbers or HMS forms, and
11136 @texline @math{0 \le a < M}.
11137 @infoline @expr{0 <= a < @var{M}}.
11138 In many applications @expr{a} and @expr{M} will be
11139 integers but this is not required.
11144 @kindex M (modulo forms)
11148 @tindex mod (operator)
11149 To create a modulo form during numeric entry, press the shift-@kbd{M}
11150 key to enter the word @samp{mod}. As a special convenience, pressing
11151 shift-@kbd{M} a second time automatically enters the value of @expr{M}
11152 that was most recently used before. During algebraic entry, either
11153 type @samp{mod} by hand or press @kbd{M-m} (that's @kbd{@key{META}-m}).
11154 Once again, pressing this a second time enters the current modulo.
11156 Modulo forms are not to be confused with the modulo operator @samp{%}.
11157 The expression @samp{27 % 10} means to compute 27 modulo 10 to produce
11158 the result 7. Further computations treat this 7 as just a regular integer.
11159 The expression @samp{27 mod 10} produces the result @samp{7 mod 10};
11160 further computations with this value are again reduced modulo 10 so that
11161 the result always lies in the desired range.
11163 When two modulo forms with identical @expr{M}'s are added or multiplied,
11164 the Calculator simply adds or multiplies the values, then reduces modulo
11165 @expr{M}. If one argument is a modulo form and the other a plain number,
11166 the plain number is treated like a compatible modulo form. It is also
11167 possible to raise modulo forms to powers; the result is the value raised
11168 to the power, then reduced modulo @expr{M}. (When all values involved
11169 are integers, this calculation is done much more efficiently than
11170 actually computing the power and then reducing.)
11172 @cindex Modulo division
11173 Two modulo forms `@var{a} @tfn{mod} @var{M}' and `@var{b} @tfn{mod} @var{M}'
11174 can be divided if @expr{a}, @expr{b}, and @expr{M} are all
11175 integers. The result is the modulo form which, when multiplied by
11176 `@var{b} @tfn{mod} @var{M}', produces `@var{a} @tfn{mod} @var{M}'. If
11177 there is no solution to this equation (which can happen only when
11178 @expr{M} is non-prime), or if any of the arguments are non-integers, the
11179 division is left in symbolic form. Other operations, such as square
11180 roots, are not yet supported for modulo forms. (Note that, although
11181 @w{`@tfn{(}@var{a} @tfn{mod} @var{M}@tfn{)^.5}'} will compute a ``modulo square root''
11182 in the sense of reducing
11183 @texline @math{\sqrt a}
11184 @infoline @expr{sqrt(a)}
11185 modulo @expr{M}, this is not a useful definition from the
11186 number-theoretical point of view.)
11188 It is possible to mix HMS forms and modulo forms. For example, an
11189 HMS form modulo 24 could be used to manipulate clock times; an HMS
11190 form modulo 360 would be suitable for angles. Making the modulo @expr{M}
11191 also be an HMS form eliminates troubles that would arise if the angular
11192 mode were inadvertently set to Radians, in which case
11193 @w{@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees modulo
11196 Modulo forms cannot have variables or formulas for components. If you
11197 enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus
11198 to each of the coefficients: @samp{(1 mod 5) x + (2 mod 5)}.
11200 You can use @kbd{v p} and @kbd{%} to modify modulo forms.
11201 @xref{Packing and Unpacking}. @xref{Basic Arithmetic}.
11207 The algebraic function @samp{makemod(a, m)} builds the modulo form
11208 @w{@samp{a mod m}}.
11210 @node Error Forms, Interval Forms, Modulo Forms, Data Types
11211 @section Error Forms
11214 @cindex Error forms
11215 @cindex Standard deviations
11216 An @dfn{error form} is a number with an associated standard
11217 deviation, as in @samp{2.3 +/- 0.12}. The notation
11218 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11219 @infoline `@var{x} @tfn{+/-} sigma'
11220 stands for an uncertain value which follows
11221 a normal or Gaussian distribution of mean @expr{x} and standard
11222 deviation or ``error''
11223 @texline @math{\sigma}.
11224 @infoline @expr{sigma}.
11225 Both the mean and the error can be either numbers or
11226 formulas. Generally these are real numbers but the mean may also be
11227 complex. If the error is negative or complex, it is changed to its
11228 absolute value. An error form with zero error is converted to a
11229 regular number by the Calculator.
11231 All arithmetic and transcendental functions accept error forms as input.
11232 Operations on the mean-value part work just like operations on regular
11233 numbers. The error part for any function @expr{f(x)} (such as
11234 @texline @math{\sin x}
11235 @infoline @expr{sin(x)})
11236 is defined by the error of @expr{x} times the derivative of @expr{f}
11237 evaluated at the mean value of @expr{x}. For a two-argument function
11238 @expr{f(x,y)} (such as addition) the error is the square root of the sum
11239 of the squares of the errors due to @expr{x} and @expr{y}.
11242 f(x \hbox{\code{ +/- }} \sigma)
11243 &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr
11244 f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y)
11245 &= f(x,y) \hbox{\code{ +/- }}
11246 \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x}
11248 +\left(\sigma_y \left| {\partial f(x,y) \over \partial y}
11249 \right| \right)^2 } \cr
11253 definition assumes the errors in @expr{x} and @expr{y} are uncorrelated.
11254 A side effect of this definition is that @samp{(2 +/- 1) * (2 +/- 1)}
11255 is not the same as @samp{(2 +/- 1)^2}; the former represents the product
11256 of two independent values which happen to have the same probability
11257 distributions, and the latter is the product of one random value with itself.
11258 The former will produce an answer with less error, since on the average
11259 the two independent errors can be expected to cancel out.
11261 Consult a good text on error analysis for a discussion of the proper use
11262 of standard deviations. Actual errors often are neither Gaussian-distributed
11263 nor uncorrelated, and the above formulas are valid only when errors
11264 are small. As an example, the error arising from
11265 @texline `@tfn{sin(}@var{x} @tfn{+/-} @math{\sigma}@tfn{)}'
11266 @infoline `@tfn{sin(}@var{x} @tfn{+/-} @var{sigma}@tfn{)}'
11268 @texline `@math{\sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11269 @infoline `@var{sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11270 When @expr{x} is close to zero,
11271 @texline @math{\cos x}
11272 @infoline @expr{cos(x)}
11273 is close to one so the error in the sine is close to
11274 @texline @math{\sigma};
11275 @infoline @expr{sigma};
11276 this makes sense, since
11277 @texline @math{\sin x}
11278 @infoline @expr{sin(x)}
11279 is approximately @expr{x} near zero, so a given error in @expr{x} will
11280 produce about the same error in the sine. Likewise, near 90 degrees
11281 @texline @math{\cos x}
11282 @infoline @expr{cos(x)}
11283 is nearly zero and so the computed error is
11284 small: The sine curve is nearly flat in that region, so an error in @expr{x}
11285 has relatively little effect on the value of
11286 @texline @math{\sin x}.
11287 @infoline @expr{sin(x)}.
11288 However, consider @samp{sin(90 +/- 1000)}. The cosine of 90 is zero, so
11289 Calc will report zero error! We get an obviously wrong result because
11290 we have violated the small-error approximation underlying the error
11291 analysis. If the error in @expr{x} had been small, the error in
11292 @texline @math{\sin x}
11293 @infoline @expr{sin(x)}
11294 would indeed have been negligible.
11299 @kindex p (error forms)
11301 To enter an error form during regular numeric entry, use the @kbd{p}
11302 (``plus-or-minus'') key to type the @samp{+/-} symbol. (If you try actually
11303 typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's
11304 @kbd{+} command!) Within an algebraic formula, you can press @kbd{M-+} to
11305 type the @samp{+/-} symbol, or type it out by hand.
11307 Error forms and complex numbers can be mixed; the formulas shown above
11308 are used for complex numbers, too; note that if the error part evaluates
11309 to a complex number its absolute value (or the square root of the sum of
11310 the squares of the absolute values of the two error contributions) is
11311 used. Mathematically, this corresponds to a radially symmetric Gaussian
11312 distribution of numbers on the complex plane. However, note that Calc
11313 considers an error form with real components to represent a real number,
11314 not a complex distribution around a real mean.
11316 Error forms may also be composed of HMS forms. For best results, both
11317 the mean and the error should be HMS forms if either one is.
11323 The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}.
11325 @node Interval Forms, Incomplete Objects, Error Forms, Data Types
11326 @section Interval Forms
11329 @cindex Interval forms
11330 An @dfn{interval} is a subset of consecutive real numbers. For example,
11331 the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4,
11332 inclusive. If you multiply it by the interval @samp{[0.5 ..@: 2]} you
11333 obtain @samp{[1 ..@: 8]}. This calculation represents the fact that if
11334 you multiply some number in the range @samp{[2 ..@: 4]} by some other
11335 number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range
11336 from 1 to 8. Interval arithmetic is used to get a worst-case estimate
11337 of the possible range of values a computation will produce, given the
11338 set of possible values of the input.
11341 Calc supports several varieties of intervals, including @dfn{closed}
11342 intervals of the type shown above, @dfn{open} intervals such as
11343 @samp{(2 ..@: 4)}, which represents the range of numbers from 2 to 4
11344 @emph{exclusive}, and @dfn{semi-open} intervals in which one end
11345 uses a round parenthesis and the other a square bracket. In mathematical
11347 @samp{[2 ..@: 4]} means @expr{2 <= x <= 4}, whereas
11348 @samp{[2 ..@: 4)} represents @expr{2 <= x < 4},
11349 @samp{(2 ..@: 4]} represents @expr{2 < x <= 4}, and
11350 @samp{(2 ..@: 4)} represents @expr{2 < x < 4}.
11353 Calc supports several varieties of intervals, including \dfn{closed}
11354 intervals of the type shown above, \dfn{open} intervals such as
11355 \samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4
11356 \emph{exclusive}, and \dfn{semi-open} intervals in which one end
11357 uses a round parenthesis and the other a square bracket. In mathematical
11360 [2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 \le x \le 4 \cr
11361 [2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 \le x < 4 \cr
11362 (2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 < x \le 4 \cr
11363 (2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 < x < 4 \cr
11367 The lower and upper limits of an interval must be either real numbers
11368 (or HMS or date forms), or symbolic expressions which are assumed to be
11369 real-valued, or @samp{-inf} and @samp{inf}. In general the lower limit
11370 must be less than the upper limit. A closed interval containing only
11371 one value, @samp{[3 ..@: 3]}, is converted to a plain number (3)
11372 automatically. An interval containing no values at all (such as
11373 @samp{[3 ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not
11374 guaranteed to behave well when used in arithmetic. Note that the
11375 interval @samp{[3 .. inf)} represents all real numbers greater than
11376 or equal to 3, and @samp{(-inf .. inf)} represents all real numbers.
11377 In fact, @samp{[-inf .. inf]} represents all real numbers including
11378 the real infinities.
11380 Intervals are entered in the notation shown here, either as algebraic
11381 formulas, or using incomplete forms. (@xref{Incomplete Objects}.)
11382 In algebraic formulas, multiple periods in a row are collected from
11383 left to right, so that @samp{1...1e2} is interpreted as @samp{1.0 ..@: 1e2}
11384 rather than @samp{1 ..@: 0.1e2}. Add spaces or zeros if you want to
11385 get the other interpretation. If you omit the lower or upper limit,
11386 a default of @samp{-inf} or @samp{inf} (respectively) is furnished.
11388 Infinite mode also affects operations on intervals
11389 (@pxref{Infinities}). Calc will always introduce an open infinity,
11390 as in @samp{1 / (0 .. 2] = [0.5 .. inf)}. But closed infinities,
11391 @w{@samp{1 / [0 .. 2] = [0.5 .. inf]}}, arise only in Infinite mode;
11392 otherwise they are left unevaluated. Note that the ``direction'' of
11393 a zero is not an issue in this case since the zero is always assumed
11394 to be continuous with the rest of the interval. For intervals that
11395 contain zero inside them Calc is forced to give the result,
11396 @samp{1 / (-2 .. 2) = [-inf .. inf]}.
11398 While it may seem that intervals and error forms are similar, they are
11399 based on entirely different concepts of inexact quantities. An error
11401 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11402 @infoline `@var{x} @tfn{+/-} @var{sigma}'
11403 means a variable is random, and its value could
11404 be anything but is ``probably'' within one
11405 @texline @math{\sigma}
11406 @infoline @var{sigma}
11407 of the mean value @expr{x}. An interval
11408 `@tfn{[}@var{a} @tfn{..@:} @var{b}@tfn{]}' means a
11409 variable's value is unknown, but guaranteed to lie in the specified
11410 range. Error forms are statistical or ``average case'' approximations;
11411 interval arithmetic tends to produce ``worst case'' bounds on an
11414 Intervals may not contain complex numbers, but they may contain
11415 HMS forms or date forms.
11417 @xref{Set Operations}, for commands that interpret interval forms
11418 as subsets of the set of real numbers.
11424 The algebraic function @samp{intv(n, a, b)} builds an interval form
11425 from @samp{a} to @samp{b}; @samp{n} is an integer code which must
11426 be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or
11429 Please note that in fully rigorous interval arithmetic, care would be
11430 taken to make sure that the computation of the lower bound rounds toward
11431 minus infinity, while upper bound computations round toward plus
11432 infinity. Calc's arithmetic always uses a round-to-nearest mode,
11433 which means that roundoff errors could creep into an interval
11434 calculation to produce intervals slightly smaller than they ought to
11435 be. For example, entering @samp{[1..2]} and pressing @kbd{Q 2 ^}
11436 should yield the interval @samp{[1..2]} again, but in fact it yields the
11437 (slightly too small) interval @samp{[1..1.9999999]} due to roundoff
11440 @node Incomplete Objects, Variables, Interval Forms, Data Types
11441 @section Incomplete Objects
11461 @cindex Incomplete vectors
11462 @cindex Incomplete complex numbers
11463 @cindex Incomplete interval forms
11464 When @kbd{(} or @kbd{[} is typed to begin entering a complex number or
11465 vector, respectively, the effect is to push an @dfn{incomplete} complex
11466 number or vector onto the stack. The @kbd{,} key adds the value(s) at
11467 the top of the stack onto the current incomplete object. The @kbd{)}
11468 and @kbd{]} keys ``close'' the incomplete object after adding any values
11469 on the top of the stack in front of the incomplete object.
11471 As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]}
11472 pushes the vector @samp{[2, 6, 9]} onto the stack. Likewise, @kbd{( 1 , 2 Q )}
11473 pushes the complex number @samp{(1, 1.414)} (approximately).
11475 If several values lie on the stack in front of the incomplete object,
11476 all are collected and appended to the object. Thus the @kbd{,} key
11477 is redundant: @kbd{[ 2 @key{RET} 3 @key{RET} 2 * 9 ]}. Some people
11478 prefer the equivalent @key{SPC} key to @key{RET}.
11480 As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or
11481 @kbd{,} adds a zero or duplicates the preceding value in the list being
11482 formed. Typing @key{DEL} during incomplete entry removes the last item
11486 The @kbd{;} key is used in the same way as @kbd{,} to create polar complex
11487 numbers: @kbd{( 1 ; 2 )}. When entering a vector, @kbd{;} is useful for
11488 creating a matrix. In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is
11489 equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}.
11493 Incomplete entry is also used to enter intervals. For example,
11494 @kbd{[ 2 ..@: 4 )} enters a semi-open interval. Note that when you type
11495 the first period, it will be interpreted as a decimal point, but when
11496 you type a second period immediately afterward, it is re-interpreted as
11497 part of the interval symbol. Typing @kbd{..} corresponds to executing
11498 the @code{calc-dots} command.
11500 If you find incomplete entry distracting, you may wish to enter vectors
11501 and complex numbers as algebraic formulas by pressing the apostrophe key.
11503 @node Variables, Formulas, Incomplete Objects, Data Types
11507 @cindex Variables, in formulas
11508 A @dfn{variable} is somewhere between a storage register on a conventional
11509 calculator, and a variable in a programming language. (In fact, a Calc
11510 variable is really just an Emacs Lisp variable that contains a Calc number
11511 or formula.) A variable's name is normally composed of letters and digits.
11512 Calc also allows apostrophes and @code{#} signs in variable names.
11513 (The Calc variable @code{foo} corresponds to the Emacs Lisp variable
11514 @code{var-foo}, but unless you access the variable from within Emacs
11515 Lisp, you don't need to worry about it. Variable names in algebraic
11516 formulas implicitly have @samp{var-} prefixed to their names. The
11517 @samp{#} character in variable names used in algebraic formulas
11518 corresponds to a dash @samp{-} in the Lisp variable name. If the name
11519 contains any dashes, the prefix @samp{var-} is @emph{not} automatically
11520 added. Thus the two formulas @samp{foo + 1} and @samp{var#foo + 1} both
11521 refer to the same variable.)
11523 In a command that takes a variable name, you can either type the full
11524 name of a variable, or type a single digit to use one of the special
11525 convenience variables @code{q0} through @code{q9}. For example,
11526 @kbd{3 s s 2} stores the number 3 in variable @code{q2}, and
11527 @w{@kbd{3 s s foo @key{RET}}} stores that number in variable
11530 To push a variable itself (as opposed to the variable's value) on the
11531 stack, enter its name as an algebraic expression using the apostrophe
11535 @pindex calc-evaluate
11536 @cindex Evaluation of variables in a formula
11537 @cindex Variables, evaluation
11538 @cindex Formulas, evaluation
11539 The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by
11540 replacing all variables in the formula which have been given values by a
11541 @code{calc-store} or @code{calc-let} command by their stored values.
11542 Other variables are left alone. Thus a variable that has not been
11543 stored acts like an abstract variable in algebra; a variable that has
11544 been stored acts more like a register in a traditional calculator.
11545 With a positive numeric prefix argument, @kbd{=} evaluates the top
11546 @var{n} stack entries; with a negative argument, @kbd{=} evaluates
11547 the @var{n}th stack entry.
11549 @cindex @code{e} variable
11550 @cindex @code{pi} variable
11551 @cindex @code{i} variable
11552 @cindex @code{phi} variable
11553 @cindex @code{gamma} variable
11559 A few variables are called @dfn{special constants}. Their names are
11560 @samp{e}, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}.
11561 (@xref{Scientific Functions}.) When they are evaluated with @kbd{=},
11562 their values are calculated if necessary according to the current precision
11563 or complex polar mode. If you wish to use these symbols for other purposes,
11564 simply undefine or redefine them using @code{calc-store}.
11566 The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for
11567 infinite or indeterminate values. It's best not to use them as
11568 regular variables, since Calc uses special algebraic rules when
11569 it manipulates them. Calc displays a warning message if you store
11570 a value into any of these special variables.
11572 @xref{Store and Recall}, for a discussion of commands dealing with variables.
11574 @node Formulas, , Variables, Data Types
11579 @cindex Expressions
11580 @cindex Operators in formulas
11581 @cindex Precedence of operators
11582 When you press the apostrophe key you may enter any expression or formula
11583 in algebraic form. (Calc uses the terms ``expression'' and ``formula''
11584 interchangeably.) An expression is built up of numbers, variable names,
11585 and function calls, combined with various arithmetic operators.
11587 be used to indicate grouping. Spaces are ignored within formulas, except
11588 that spaces are not permitted within variable names or numbers.
11589 Arithmetic operators, in order from highest to lowest precedence, and
11590 with their equivalent function names, are:
11592 @samp{_} [@code{subscr}] (subscripts);
11594 postfix @samp{%} [@code{percent}] (as in @samp{25% = 0.25});
11596 prefix @samp{!} [@code{lnot}] (logical ``not,'' as in @samp{!x});
11598 @samp{+/-} [@code{sdev}] (the standard deviation symbol) and
11599 @samp{mod} [@code{makemod}] (the symbol for modulo forms);
11601 postfix @samp{!} [@code{fact}] (factorial, as in @samp{n!})
11602 and postfix @samp{!!} [@code{dfact}] (double factorial);
11604 @samp{^} [@code{pow}] (raised-to-the-power-of);
11606 prefix @samp{+} and @samp{-} [@code{neg}] (as in @samp{-x});
11608 @samp{*} [@code{mul}];
11610 @samp{/} [@code{div}], @samp{%} [@code{mod}] (modulo), and
11611 @samp{\} [@code{idiv}] (integer division);
11613 infix @samp{+} [@code{add}] and @samp{-} [@code{sub}] (as in @samp{x-y});
11615 @samp{|} [@code{vconcat}] (vector concatenation);
11617 relations @samp{=} [@code{eq}], @samp{!=} [@code{neq}], @samp{<} [@code{lt}],
11618 @samp{>} [@code{gt}], @samp{<=} [@code{leq}], and @samp{>=} [@code{geq}];
11620 @samp{&&} [@code{land}] (logical ``and'');
11622 @samp{||} [@code{lor}] (logical ``or'');
11624 the C-style ``if'' operator @samp{a?b:c} [@code{if}];
11626 @samp{!!!} [@code{pnot}] (rewrite pattern ``not'');
11628 @samp{&&&} [@code{pand}] (rewrite pattern ``and'');
11630 @samp{|||} [@code{por}] (rewrite pattern ``or'');
11632 @samp{:=} [@code{assign}] (for assignments and rewrite rules);
11634 @samp{::} [@code{condition}] (rewrite pattern condition);
11636 @samp{=>} [@code{evalto}].
11638 Note that, unlike in usual computer notation, multiplication binds more
11639 strongly than division: @samp{a*b/c*d} is equivalent to
11640 @texline @math{a b \over c d}.
11641 @infoline @expr{(a*b)/(c*d)}.
11643 @cindex Multiplication, implicit
11644 @cindex Implicit multiplication
11645 The multiplication sign @samp{*} may be omitted in many cases. In particular,
11646 if the righthand side is a number, variable name, or parenthesized
11647 expression, the @samp{*} may be omitted. Implicit multiplication has the
11648 same precedence as the explicit @samp{*} operator. The one exception to
11649 the rule is that a variable name followed by a parenthesized expression,
11651 is interpreted as a function call, not an implicit @samp{*}. In many
11652 cases you must use a space if you omit the @samp{*}: @samp{2a} is the
11653 same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab}
11654 is a variable called @code{ab}, @emph{not} the product of @samp{a} and
11655 @samp{b}! Also note that @samp{f (x)} is still a function call.
11657 @cindex Implicit comma in vectors
11658 The rules are slightly different for vectors written with square brackets.
11659 In vectors, the space character is interpreted (like the comma) as a
11660 separator of elements of the vector. Thus @w{@samp{[ 2a b+c d ]}} is
11661 equivalent to @samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent
11662 to @samp{2*a*b + c*d}.
11663 Note that spaces around the brackets, and around explicit commas, are
11664 ignored. To force spaces to be interpreted as multiplication you can
11665 enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is
11666 interpreted as @samp{[a*b, 2*c*d]}. An implicit comma is also inserted
11667 between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.
11669 Vectors that contain commas (not embedded within nested parentheses or
11670 brackets) do not treat spaces specially: @samp{[a b, 2 c d]} is a vector
11671 of two elements. Also, if it would be an error to treat spaces as
11672 separators, but not otherwise, then Calc will ignore spaces:
11673 @w{@samp{[a - b]}} is a vector of one element, but @w{@samp{[a -b]}} is
11674 a vector of two elements. Finally, vectors entered with curly braces
11675 instead of square brackets do not give spaces any special treatment.
11676 When Calc displays a vector that does not contain any commas, it will
11677 insert parentheses if necessary to make the meaning clear:
11678 @w{@samp{[(a b)]}}.
11680 The expression @samp{5%-2} is ambiguous; is this five-percent minus two,
11681 or five modulo minus-two? Calc always interprets the leftmost symbol as
11682 an infix operator preferentially (modulo, in this case), so you would
11683 need to write @samp{(5%)-2} to get the former interpretation.
11685 @cindex Function call notation
11686 A function call is, e.g., @samp{sin(1+x)}. (The Calc algebraic function
11687 @code{foo} corresponds to the Emacs Lisp function @code{calcFunc-foo},
11688 but unless you access the function from within Emacs Lisp, you don't
11689 need to worry about it.) Most mathematical Calculator commands like
11690 @code{calc-sin} have function equivalents like @code{sin}.
11691 If no Lisp function is defined for a function called by a formula, the
11692 call is left as it is during algebraic manipulation: @samp{f(x+y)} is
11693 left alone. Beware that many innocent-looking short names like @code{in}
11694 and @code{re} have predefined meanings which could surprise you; however,
11695 single letters or single letters followed by digits are always safe to
11696 use for your own function names. @xref{Function Index}.
11698 In the documentation for particular commands, the notation @kbd{H S}
11699 (@code{calc-sinh}) [@code{sinh}] means that the key sequence @kbd{H S}, the
11700 command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all
11701 represent the same operation.
11703 Commands that interpret (``parse'') text as algebraic formulas include
11704 algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse
11705 the contents of the editing buffer when you finish, the @kbd{C-x * g}
11706 and @w{@kbd{C-x * r}} commands, the @kbd{C-y} command, the X window system
11707 ``paste'' mouse operation, and Embedded mode. All of these operations
11708 use the same rules for parsing formulas; in particular, language modes
11709 (@pxref{Language Modes}) affect them all in the same way.
11711 When you read a large amount of text into the Calculator (say a vector
11712 which represents a big set of rewrite rules; @pxref{Rewrite Rules}),
11713 you may wish to include comments in the text. Calc's formula parser
11714 ignores the symbol @samp{%%} and anything following it on a line:
11717 [ a + b, %% the sum of "a" and "b"
11719 %% last line is coming up:
11724 This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}.
11726 @xref{Syntax Tables}, for a way to create your own operators and other
11727 input notations. @xref{Compositions}, for a way to create new display
11730 @xref{Algebra}, for commands for manipulating formulas symbolically.
11732 @node Stack and Trail, Mode Settings, Data Types, Top
11733 @chapter Stack and Trail Commands
11736 This chapter describes the Calc commands for manipulating objects on the
11737 stack and in the trail buffer. (These commands operate on objects of any
11738 type, such as numbers, vectors, formulas, and incomplete objects.)
11741 * Stack Manipulation::
11742 * Editing Stack Entries::
11747 @node Stack Manipulation, Editing Stack Entries, Stack and Trail, Stack and Trail
11748 @section Stack Manipulation Commands
11754 @cindex Duplicating stack entries
11755 To duplicate the top object on the stack, press @key{RET} or @key{SPC}
11756 (two equivalent keys for the @code{calc-enter} command).
11757 Given a positive numeric prefix argument, these commands duplicate
11758 several elements at the top of the stack.
11759 Given a negative argument,
11760 these commands duplicate the specified element of the stack.
11761 Given an argument of zero, they duplicate the entire stack.
11762 For example, with @samp{10 20 30} on the stack,
11763 @key{RET} creates @samp{10 20 30 30},
11764 @kbd{C-u 2 @key{RET}} creates @samp{10 20 30 20 30},
11765 @kbd{C-u - 2 @key{RET}} creates @samp{10 20 30 20}, and
11766 @kbd{C-u 0 @key{RET}} creates @samp{10 20 30 10 20 30}.
11770 The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you
11771 have it, else on @kbd{C-j}) is like @code{calc-enter}
11772 except that the sign of the numeric prefix argument is interpreted
11773 oppositely. Also, with no prefix argument the default argument is 2.
11774 Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}}
11775 are both equivalent to @kbd{C-u - 2 @key{RET}}, producing
11776 @samp{10 20 30 20}.
11781 @cindex Removing stack entries
11782 @cindex Deleting stack entries
11783 To remove the top element from the stack, press @key{DEL} (@code{calc-pop}).
11784 The @kbd{C-d} key is a synonym for @key{DEL}.
11785 (If the top element is an incomplete object with at least one element, the
11786 last element is removed from it.) Given a positive numeric prefix argument,
11787 several elements are removed. Given a negative argument, the specified
11788 element of the stack is deleted. Given an argument of zero, the entire
11790 For example, with @samp{10 20 30} on the stack,
11791 @key{DEL} leaves @samp{10 20},
11792 @kbd{C-u 2 @key{DEL}} leaves @samp{10},
11793 @kbd{C-u - 2 @key{DEL}} leaves @samp{10 30}, and
11794 @kbd{C-u 0 @key{DEL}} leaves an empty stack.
11796 @kindex M-@key{DEL}
11797 @pindex calc-pop-above
11798 The @kbd{M-@key{DEL}} (@code{calc-pop-above}) command is to @key{DEL} what
11799 @key{LFD} is to @key{RET}: It interprets the sign of the numeric
11800 prefix argument in the opposite way, and the default argument is 2.
11801 Thus @kbd{M-@key{DEL}} by itself removes the second-from-top stack element,
11802 leaving the first, third, fourth, and so on; @kbd{M-3 M-@key{DEL}} deletes
11803 the third stack element.
11806 @pindex calc-roll-down
11807 To exchange the top two elements of the stack, press @key{TAB}
11808 (@code{calc-roll-down}). Given a positive numeric prefix argument, the
11809 specified number of elements at the top of the stack are rotated downward.
11810 Given a negative argument, the entire stack is rotated downward the specified
11811 number of times. Given an argument of zero, the entire stack is reversed
11813 For example, with @samp{10 20 30 40 50} on the stack,
11814 @key{TAB} creates @samp{10 20 30 50 40},
11815 @kbd{C-u 3 @key{TAB}} creates @samp{10 20 50 30 40},
11816 @kbd{C-u - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and
11817 @kbd{C-u 0 @key{TAB}} creates @samp{50 40 30 20 10}.
11819 @kindex M-@key{TAB}
11820 @pindex calc-roll-up
11821 The command @kbd{M-@key{TAB}} (@code{calc-roll-up}) is analogous to @key{TAB}
11822 except that it rotates upward instead of downward. Also, the default
11823 with no prefix argument is to rotate the top 3 elements.
11824 For example, with @samp{10 20 30 40 50} on the stack,
11825 @kbd{M-@key{TAB}} creates @samp{10 20 40 50 30},
11826 @kbd{C-u 4 M-@key{TAB}} creates @samp{10 30 40 50 20},
11827 @kbd{C-u - 2 M-@key{TAB}} creates @samp{30 40 50 10 20}, and
11828 @kbd{C-u 0 M-@key{TAB}} creates @samp{50 40 30 20 10}.
11830 A good way to view the operation of @key{TAB} and @kbd{M-@key{TAB}} is in
11831 terms of moving a particular element to a new position in the stack.
11832 With a positive argument @var{n}, @key{TAB} moves the top stack
11833 element down to level @var{n}, making room for it by pulling all the
11834 intervening stack elements toward the top. @kbd{M-@key{TAB}} moves the
11835 element at level @var{n} up to the top. (Compare with @key{LFD},
11836 which copies instead of moving the element in level @var{n}.)
11838 With a negative argument @mathit{-@var{n}}, @key{TAB} rotates the stack
11839 to move the object in level @var{n} to the deepest place in the
11840 stack, and the object in level @mathit{@var{n}+1} to the top. @kbd{M-@key{TAB}}
11841 rotates the deepest stack element to be in level @var{n}, also
11842 putting the top stack element in level @mathit{@var{n}+1}.
11844 @xref{Selecting Subformulas}, for a way to apply these commands to
11845 any portion of a vector or formula on the stack.
11848 @pindex calc-transpose-lines
11849 @cindex Moving stack entries
11850 The command @kbd{C-x C-t} (@code{calc-transpose-lines}) will transpose
11851 the stack object determined by the point with the stack object at the
11852 next higher level. For example, with @samp{10 20 30 40 50} on the
11853 stack and the point on the line containing @samp{30}, @kbd{C-x C-t}
11854 creates @samp{10 20 40 30 50}. More generally, @kbd{C-x C-t} acts on
11855 the stack objects determined by the current point (and mark) similar
11856 to how the text-mode command @code{transpose-lines} acts on
11857 lines. With argument @var{n}, @kbd{C-x C-t} will move the stack object
11858 at the level above the current point and move it past N other objects;
11859 for example, with @samp{10 20 30 40 50} on the stack and the point on
11860 the line containing @samp{30}, @kbd{C-u 2 C-x C-t} creates
11861 @samp{10 40 20 30 50}. With an argument of 0, @kbd{C-x C-t} will switch
11862 the stack objects at the levels determined by the point and the mark.
11864 @node Editing Stack Entries, Trail Commands, Stack Manipulation, Stack and Trail
11865 @section Editing Stack Entries
11870 @pindex calc-edit-finish
11871 @cindex Editing the stack with Emacs
11872 The @kbd{`} (@code{calc-edit}) command creates a temporary buffer
11873 (@samp{*Calc Edit*}) for editing the top-of-stack value using regular
11874 Emacs commands. Note that @kbd{`} is a backquote, not a quote. With a
11875 numeric prefix argument, it edits the specified number of stack entries
11876 at once. (An argument of zero edits the entire stack; a negative
11877 argument edits one specific stack entry.)
11879 When you are done editing, press @kbd{C-c C-c} to finish and return
11880 to Calc. The @key{RET} and @key{LFD} keys also work to finish most
11881 sorts of editing, though in some cases Calc leaves @key{RET} with its
11882 usual meaning (``insert a newline'') if it's a situation where you
11883 might want to insert new lines into the editing buffer.
11885 When you finish editing, the Calculator parses the lines of text in
11886 the @samp{*Calc Edit*} buffer as numbers or formulas, replaces the
11887 original stack elements in the original buffer with these new values,
11888 then kills the @samp{*Calc Edit*} buffer. The original Calculator buffer
11889 continues to exist during editing, but for best results you should be
11890 careful not to change it until you have finished the edit. You can
11891 also cancel the edit by killing the buffer with @kbd{C-x k}.
11893 The formula is normally reevaluated as it is put onto the stack.
11894 For example, editing @samp{a + 2} to @samp{3 + 2} and pressing
11895 @kbd{C-c C-c} will push 5 on the stack. If you use @key{LFD} to
11896 finish, Calc will put the result on the stack without evaluating it.
11898 If you give a prefix argument to @kbd{C-c C-c},
11899 Calc will not kill the @samp{*Calc Edit*} buffer. You can switch
11900 back to that buffer and continue editing if you wish. However, you
11901 should understand that if you initiated the edit with @kbd{`}, the
11902 @kbd{C-c C-c} operation will be programmed to replace the top of the
11903 stack with the new edited value, and it will do this even if you have
11904 rearranged the stack in the meanwhile. This is not so much of a problem
11905 with other editing commands, though, such as @kbd{s e}
11906 (@code{calc-edit-variable}; @pxref{Operations on Variables}).
11908 If the @code{calc-edit} command involves more than one stack entry,
11909 each line of the @samp{*Calc Edit*} buffer is interpreted as a
11910 separate formula. Otherwise, the entire buffer is interpreted as
11911 one formula, with line breaks ignored. (You can use @kbd{C-o} or
11912 @kbd{C-q C-j} to insert a newline in the buffer without pressing @key{RET}.)
11914 The @kbd{`} key also works during numeric or algebraic entry. The
11915 text entered so far is moved to the @code{*Calc Edit*} buffer for
11916 more extensive editing than is convenient in the minibuffer.
11918 @node Trail Commands, Keep Arguments, Editing Stack Entries, Stack and Trail
11919 @section Trail Commands
11922 @cindex Trail buffer
11923 The commands for manipulating the Calc Trail buffer are two-key sequences
11924 beginning with the @kbd{t} prefix.
11927 @pindex calc-trail-display
11928 The @kbd{t d} (@code{calc-trail-display}) command turns display of the
11929 trail on and off. Normally the trail display is toggled on if it was off,
11930 off if it was on. With a numeric prefix of zero, this command always
11931 turns the trail off; with a prefix of one, it always turns the trail on.
11932 The other trail-manipulation commands described here automatically turn
11933 the trail on. Note that when the trail is off values are still recorded
11934 there; they are simply not displayed. To set Emacs to turn the trail
11935 off by default, type @kbd{t d} and then save the mode settings with
11936 @kbd{m m} (@code{calc-save-modes}).
11939 @pindex calc-trail-in
11941 @pindex calc-trail-out
11942 The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o}
11943 (@code{calc-trail-out}) commands switch the cursor into and out of the
11944 Calc Trail window. In practice they are rarely used, since the commands
11945 shown below are a more convenient way to move around in the
11946 trail, and they work ``by remote control'' when the cursor is still
11947 in the Calculator window.
11949 @cindex Trail pointer
11950 There is a @dfn{trail pointer} which selects some entry of the trail at
11951 any given time. The trail pointer looks like a @samp{>} symbol right
11952 before the selected number. The following commands operate on the
11953 trail pointer in various ways.
11956 @pindex calc-trail-yank
11957 @cindex Retrieving previous results
11958 The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in
11959 the trail and pushes it onto the Calculator stack. It allows you to
11960 re-use any previously computed value without retyping. With a numeric
11961 prefix argument @var{n}, it yanks the value @var{n} lines above the current
11965 @pindex calc-trail-scroll-left
11967 @pindex calc-trail-scroll-right
11968 The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >}
11969 (@code{calc-trail-scroll-right}) commands horizontally scroll the trail
11970 window left or right by one half of its width.
11973 @pindex calc-trail-next
11975 @pindex calc-trail-previous
11977 @pindex calc-trail-forward
11979 @pindex calc-trail-backward
11980 The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p}
11981 (@code{calc-trail-previous)} commands move the trail pointer down or up
11982 one line. The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b}
11983 (@code{calc-trail-backward}) commands move the trail pointer down or up
11984 one screenful at a time. All of these commands accept numeric prefix
11985 arguments to move several lines or screenfuls at a time.
11988 @pindex calc-trail-first
11990 @pindex calc-trail-last
11992 @pindex calc-trail-here
11993 The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]}
11994 (@code{calc-trail-last}) commands move the trail pointer to the first or
11995 last line of the trail. The @kbd{t h} (@code{calc-trail-here}) command
11996 moves the trail pointer to the cursor position; unlike the other trail
11997 commands, @kbd{t h} works only when Calc Trail is the selected window.
12000 @pindex calc-trail-isearch-forward
12002 @pindex calc-trail-isearch-backward
12004 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
12005 (@code{calc-trail-isearch-backward}) commands perform an incremental
12006 search forward or backward through the trail. You can press @key{RET}
12007 to terminate the search; the trail pointer moves to the current line.
12008 If you cancel the search with @kbd{C-g}, the trail pointer stays where
12009 it was when the search began.
12012 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
12013 (@code{calc-trail-isearch-backward}) com\-mands perform an incremental
12014 search forward or backward through the trail. You can press @key{RET}
12015 to terminate the search; the trail pointer moves to the current line.
12016 If you cancel the search with @kbd{C-g}, the trail pointer stays where
12017 it was when the search began.
12021 @pindex calc-trail-marker
12022 The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a
12023 line of text of your own choosing into the trail. The text is inserted
12024 after the line containing the trail pointer; this usually means it is
12025 added to the end of the trail. Trail markers are useful mainly as the
12026 targets for later incremental searches in the trail.
12029 @pindex calc-trail-kill
12030 The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line
12031 from the trail. The line is saved in the Emacs kill ring suitable for
12032 yanking into another buffer, but it is not easy to yank the text back
12033 into the trail buffer. With a numeric prefix argument, this command
12034 kills the @var{n} lines below or above the selected one.
12036 The @kbd{t .} (@code{calc-full-trail-vectors}) command is described
12037 elsewhere; @pxref{Vector and Matrix Formats}.
12039 @node Keep Arguments, , Trail Commands, Stack and Trail
12040 @section Keep Arguments
12044 @pindex calc-keep-args
12045 The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for
12046 the following command. It prevents that command from removing its
12047 arguments from the stack. For example, after @kbd{2 @key{RET} 3 +},
12048 the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +},
12049 the stack contains the arguments and the result: @samp{2 3 5}.
12051 With the exception of keyboard macros, this works for all commands that
12052 take arguments off the stack. (To avoid potentially unpleasant behavior,
12053 a @kbd{K} prefix before a keyboard macro will be ignored. A @kbd{K}
12054 prefix called @emph{within} the keyboard macro will still take effect.)
12055 As another example, @kbd{K a s} simplifies a formula, pushing the
12056 simplified version of the formula onto the stack after the original
12057 formula (rather than replacing the original formula). Note that you
12058 could get the same effect by typing @kbd{@key{RET} a s}, copying the
12059 formula and then simplifying the copy. One difference is that for a very
12060 large formula the time taken to format the intermediate copy in
12061 @kbd{@key{RET} a s} could be noticeable; @kbd{K a s} would avoid this
12064 Even stack manipulation commands are affected. @key{TAB} works by
12065 popping two values and pushing them back in the opposite order,
12066 so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}.
12068 A few Calc commands provide other ways of doing the same thing.
12069 For example, @kbd{' sin($)} replaces the number on the stack with
12070 its sine using algebraic entry; to push the sine and keep the
12071 original argument you could use either @kbd{' sin($1)} or
12072 @kbd{K ' sin($)}. @xref{Algebraic Entry}. Also, the @kbd{s s}
12073 command is effectively the same as @kbd{K s t}. @xref{Storing Variables}.
12075 If you execute a command and then decide you really wanted to keep
12076 the argument, you can press @kbd{M-@key{RET}} (@code{calc-last-args}).
12077 This command pushes the last arguments that were popped by any command
12078 onto the stack. Note that the order of things on the stack will be
12079 different than with @kbd{K}: @kbd{2 @key{RET} 3 + M-@key{RET}} leaves
12080 @samp{5 2 3} on the stack instead of @samp{2 3 5}. @xref{Undo}.
12082 @node Mode Settings, Arithmetic, Stack and Trail, Top
12083 @chapter Mode Settings
12086 This chapter describes commands that set modes in the Calculator.
12087 They do not affect the contents of the stack, although they may change
12088 the @emph{appearance} or @emph{interpretation} of the stack's contents.
12091 * General Mode Commands::
12093 * Inverse and Hyperbolic::
12094 * Calculation Modes::
12095 * Simplification Modes::
12103 @node General Mode Commands, Precision, Mode Settings, Mode Settings
12104 @section General Mode Commands
12108 @pindex calc-save-modes
12109 @cindex Continuous memory
12110 @cindex Saving mode settings
12111 @cindex Permanent mode settings
12112 @cindex Calc init file, mode settings
12113 You can save all of the current mode settings in your Calc init file
12114 (the file given by the variable @code{calc-settings-file}, typically
12115 @file{~/.emacs.d/calc.el}) with the @kbd{m m} (@code{calc-save-modes})
12116 command. This will cause Emacs to reestablish these modes each time
12117 it starts up. The modes saved in the file include everything
12118 controlled by the @kbd{m} and @kbd{d} prefix keys, the current
12119 precision and binary word size, whether or not the trail is displayed,
12120 the current height of the Calc window, and more. The current
12121 interface (used when you type @kbd{C-x * *}) is also saved. If there
12122 were already saved mode settings in the file, they are replaced.
12123 Otherwise, the new mode information is appended to the end of the
12127 @pindex calc-mode-record-mode
12128 The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to
12129 record all the mode settings (as if by pressing @kbd{m m}) every
12130 time a mode setting changes. If the modes are saved this way, then this
12131 ``automatic mode recording'' mode is also saved.
12132 Type @kbd{m R} again to disable this method of recording the mode
12133 settings. To turn it off permanently, the @kbd{m m} command will also be
12134 necessary. (If Embedded mode is enabled, other options for recording
12135 the modes are available; @pxref{Mode Settings in Embedded Mode}.)
12138 @pindex calc-settings-file-name
12139 The @kbd{m F} (@code{calc-settings-file-name}) command allows you to
12140 choose a different file than the current value of @code{calc-settings-file}
12141 for @kbd{m m}, @kbd{Z P}, and similar commands to save permanent information.
12142 You are prompted for a file name. All Calc modes are then reset to
12143 their default values, then settings from the file you named are loaded
12144 if this file exists, and this file becomes the one that Calc will
12145 use in the future for commands like @kbd{m m}. The default settings
12146 file name is @file{~/.emacs.d/calc.el}. You can see the current file name by
12147 giving a blank response to the @kbd{m F} prompt. See also the
12148 discussion of the @code{calc-settings-file} variable; @pxref{Customizing Calc}.
12150 If the file name you give is your user init file (typically
12151 @file{~/.emacs}), @kbd{m F} will not automatically load the new file. This
12152 is because your user init file may contain other things you don't want
12153 to reread. You can give
12154 a numeric prefix argument of 1 to @kbd{m F} to force it to read the
12155 file no matter what. Conversely, an argument of @mathit{-1} tells
12156 @kbd{m F} @emph{not} to read the new file. An argument of 2 or @mathit{-2}
12157 tells @kbd{m F} not to reset the modes to their defaults beforehand,
12158 which is useful if you intend your new file to have a variant of the
12159 modes present in the file you were using before.
12162 @pindex calc-always-load-extensions
12163 The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode
12164 in which the first use of Calc loads the entire program, including all
12165 extensions modules. Otherwise, the extensions modules will not be loaded
12166 until the various advanced Calc features are used. Since this mode only
12167 has effect when Calc is first loaded, @kbd{m x} is usually followed by
12168 @kbd{m m} to make the mode-setting permanent. To load all of Calc just
12169 once, rather than always in the future, you can press @kbd{C-x * L}.
12172 @pindex calc-shift-prefix
12173 The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which
12174 all of Calc's letter prefix keys may be typed shifted as well as unshifted.
12175 If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often
12176 you might find it easier to turn this mode on so that you can type
12177 @kbd{A S} instead. When this mode is enabled, the commands that used to
12178 be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can
12179 now be invoked by pressing the shifted letter twice: @kbd{A A}. Note
12180 that the @kbd{v} prefix key always works both shifted and unshifted, and
12181 the @kbd{z} and @kbd{Z} prefix keys are always distinct. Also, the @kbd{h}
12182 prefix is not affected by this mode. Press @kbd{m S} again to disable
12183 shifted-prefix mode.
12185 @node Precision, Inverse and Hyperbolic, General Mode Commands, Mode Settings
12190 @pindex calc-precision
12191 @cindex Precision of calculations
12192 The @kbd{p} (@code{calc-precision}) command controls the precision to
12193 which floating-point calculations are carried. The precision must be
12194 at least 3 digits and may be arbitrarily high, within the limits of
12195 memory and time. This affects only floats: Integer and rational
12196 calculations are always carried out with as many digits as necessary.
12198 The @kbd{p} key prompts for the current precision. If you wish you
12199 can instead give the precision as a numeric prefix argument.
12201 Many internal calculations are carried to one or two digits higher
12202 precision than normal. Results are rounded down afterward to the
12203 current precision. Unless a special display mode has been selected,
12204 floats are always displayed with their full stored precision, i.e.,
12205 what you see is what you get. Reducing the current precision does not
12206 round values already on the stack, but those values will be rounded
12207 down before being used in any calculation. The @kbd{c 0} through
12208 @kbd{c 9} commands (@pxref{Conversions}) can be used to round an
12209 existing value to a new precision.
12211 @cindex Accuracy of calculations
12212 It is important to distinguish the concepts of @dfn{precision} and
12213 @dfn{accuracy}. In the normal usage of these words, the number
12214 123.4567 has a precision of 7 digits but an accuracy of 4 digits.
12215 The precision is the total number of digits not counting leading
12216 or trailing zeros (regardless of the position of the decimal point).
12217 The accuracy is simply the number of digits after the decimal point
12218 (again not counting trailing zeros). In Calc you control the precision,
12219 not the accuracy of computations. If you were to set the accuracy
12220 instead, then calculations like @samp{exp(100)} would generate many
12221 more digits than you would typically need, while @samp{exp(-100)} would
12222 probably round to zero! In Calc, both these computations give you
12223 exactly 12 (or the requested number of) significant digits.
12225 The only Calc features that deal with accuracy instead of precision
12226 are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}),
12227 and the rounding functions like @code{floor} and @code{round}
12228 (@pxref{Integer Truncation}). Also, @kbd{c 0} through @kbd{c 9}
12229 deal with both precision and accuracy depending on the magnitudes
12230 of the numbers involved.
12232 If you need to work with a particular fixed accuracy (say, dollars and
12233 cents with two digits after the decimal point), one solution is to work
12234 with integers and an ``implied'' decimal point. For example, $8.99
12235 divided by 6 would be entered @kbd{899 @key{RET} 6 /}, yielding 149.833
12236 (actually $1.49833 with our implied decimal point); pressing @kbd{R}
12237 would round this to 150 cents, i.e., $1.50.
12239 @xref{Floats}, for still more on floating-point precision and related
12242 @node Inverse and Hyperbolic, Calculation Modes, Precision, Mode Settings
12243 @section Inverse and Hyperbolic Flags
12247 @pindex calc-inverse
12248 There is no single-key equivalent to the @code{calc-arcsin} function.
12249 Instead, you must first press @kbd{I} (@code{calc-inverse}) to set
12250 the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}).
12251 The @kbd{I} key actually toggles the Inverse Flag. When this flag
12252 is set, the word @samp{Inv} appears in the mode line.
12255 @pindex calc-hyperbolic
12256 Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the
12257 Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}.
12258 If both of these flags are set at once, the effect will be
12259 @code{calc-arcsinh}. (The Hyperbolic flag is also used by some
12260 non-trigonometric commands; for example @kbd{H L} computes a base-10,
12261 instead of base-@mathit{e}, logarithm.)
12263 Command names like @code{calc-arcsin} are provided for completeness, and
12264 may be executed with @kbd{x} or @kbd{M-x}. Their effect is simply to
12265 toggle the Inverse and/or Hyperbolic flags and then execute the
12266 corresponding base command (@code{calc-sin} in this case).
12269 @pindex calc-option
12270 The @kbd{O} key (@code{calc-option}) sets another flag, the
12271 @dfn{Option Flag}, which also can alter the subsequent Calc command in
12274 The Inverse, Hyperbolic and Option flags apply only to the next
12275 Calculator command, after which they are automatically cleared. (They
12276 are also cleared if the next keystroke is not a Calc command.) Digits
12277 you type after @kbd{I}, @kbd{H} or @kbd{O} (or @kbd{K}) are treated as
12278 prefix arguments for the next command, not as numeric entries. The
12279 same is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means
12280 to subtract and keep arguments).
12282 Another Calc prefix flag, @kbd{K} (keep-arguments), is discussed
12283 elsewhere. @xref{Keep Arguments}.
12285 @node Calculation Modes, Simplification Modes, Inverse and Hyperbolic, Mode Settings
12286 @section Calculation Modes
12289 The commands in this section are two-key sequences beginning with
12290 the @kbd{m} prefix. (That's the letter @kbd{m}, not the @key{META} key.)
12291 The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere
12292 (@pxref{Algebraic Entry}).
12301 * Automatic Recomputation::
12302 * Working Message::
12305 @node Angular Modes, Polar Mode, Calculation Modes, Calculation Modes
12306 @subsection Angular Modes
12309 @cindex Angular mode
12310 The Calculator supports three notations for angles: radians, degrees,
12311 and degrees-minutes-seconds. When a number is presented to a function
12312 like @code{sin} that requires an angle, the current angular mode is
12313 used to interpret the number as either radians or degrees. If an HMS
12314 form is presented to @code{sin}, it is always interpreted as
12315 degrees-minutes-seconds.
12317 Functions that compute angles produce a number in radians, a number in
12318 degrees, or an HMS form depending on the current angular mode. If the
12319 result is a complex number and the current mode is HMS, the number is
12320 instead expressed in degrees. (Complex-number calculations would
12321 normally be done in Radians mode, though. Complex numbers are converted
12322 to degrees by calculating the complex result in radians and then
12323 multiplying by 180 over @cpi{}.)
12326 @pindex calc-radians-mode
12328 @pindex calc-degrees-mode
12330 @pindex calc-hms-mode
12331 The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}),
12332 and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode.
12333 The current angular mode is displayed on the Emacs mode line.
12334 The default angular mode is Degrees.
12336 @node Polar Mode, Fraction Mode, Angular Modes, Calculation Modes
12337 @subsection Polar Mode
12341 The Calculator normally ``prefers'' rectangular complex numbers in the
12342 sense that rectangular form is used when the proper form can not be
12343 decided from the input. This might happen by multiplying a rectangular
12344 number by a polar one, by taking the square root of a negative real
12345 number, or by entering @kbd{( 2 @key{SPC} 3 )}.
12348 @pindex calc-polar-mode
12349 The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number
12350 preference between rectangular and polar forms. In Polar mode, all
12351 of the above example situations would produce polar complex numbers.
12353 @node Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes
12354 @subsection Fraction Mode
12357 @cindex Fraction mode
12358 @cindex Division of integers
12359 Division of two integers normally yields a floating-point number if the
12360 result cannot be expressed as an integer. In some cases you would
12361 rather get an exact fractional answer. One way to accomplish this is
12362 to use the @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command, which
12363 divides the two integers on the top of the stack to produce a fraction:
12364 @kbd{6 @key{RET} 4 :} produces @expr{3:2} even though
12365 @kbd{6 @key{RET} 4 /} produces @expr{1.5}.
12368 @pindex calc-frac-mode
12369 To set the Calculator to produce fractional results for normal integer
12370 divisions, use the @kbd{m f} (@code{calc-frac-mode}) command.
12371 For example, @expr{8/4} produces @expr{2} in either mode,
12372 but @expr{6/4} produces @expr{3:2} in Fraction mode, @expr{1.5} in
12375 At any time you can use @kbd{c f} (@code{calc-float}) to convert a
12376 fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a
12377 float to a fraction. @xref{Conversions}.
12379 @node Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes
12380 @subsection Infinite Mode
12383 @cindex Infinite mode
12384 The Calculator normally treats results like @expr{1 / 0} as errors;
12385 formulas like this are left in unsimplified form. But Calc can be
12386 put into a mode where such calculations instead produce ``infinite''
12390 @pindex calc-infinite-mode
12391 The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode
12392 on and off. When the mode is off, infinities do not arise except
12393 in calculations that already had infinities as inputs. (One exception
12394 is that infinite open intervals like @samp{[0 .. inf)} can be
12395 generated; however, intervals closed at infinity (@samp{[0 .. inf]})
12396 will not be generated when Infinite mode is off.)
12398 With Infinite mode turned on, @samp{1 / 0} will generate @code{uinf},
12399 an undirected infinity. @xref{Infinities}, for a discussion of the
12400 difference between @code{inf} and @code{uinf}. Also, @expr{0 / 0}
12401 evaluates to @code{nan}, the ``indeterminate'' symbol. Various other
12402 functions can also return infinities in this mode; for example,
12403 @samp{ln(0) = -inf}, and @samp{gamma(-7) = uinf}. Once again,
12404 note that @samp{exp(inf) = inf} regardless of Infinite mode because
12405 this calculation has infinity as an input.
12407 @cindex Positive Infinite mode
12408 The @kbd{m i} command with a numeric prefix argument of zero,
12409 i.e., @kbd{C-u 0 m i}, turns on a Positive Infinite mode in
12410 which zero is treated as positive instead of being directionless.
12411 Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode.
12412 Note that zero never actually has a sign in Calc; there are no
12413 separate representations for @mathit{+0} and @mathit{-0}. Positive
12414 Infinite mode merely changes the interpretation given to the
12415 single symbol, @samp{0}. One consequence of this is that, while
12416 you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0}
12417 is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}.
12419 @node Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes
12420 @subsection Symbolic Mode
12423 @cindex Symbolic mode
12424 @cindex Inexact results
12425 Calculations are normally performed numerically wherever possible.
12426 For example, the @code{calc-sqrt} command, or @code{sqrt} function in an
12427 algebraic expression, produces a numeric answer if the argument is a
12428 number or a symbolic expression if the argument is an expression:
12429 @kbd{2 Q} pushes 1.4142 but @kbd{@key{'} x+1 @key{RET} Q} pushes @samp{sqrt(x+1)}.
12432 @pindex calc-symbolic-mode
12433 In @dfn{Symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode})
12434 command, functions which would produce inexact, irrational results are
12435 left in symbolic form. Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes
12439 @pindex calc-eval-num
12440 The shift-@kbd{N} (@code{calc-eval-num}) command evaluates numerically
12441 the expression at the top of the stack, by temporarily disabling
12442 @code{calc-symbolic-mode} and executing @kbd{=} (@code{calc-evaluate}).
12443 Given a numeric prefix argument, it also
12444 sets the floating-point precision to the specified value for the duration
12447 To evaluate a formula numerically without expanding the variables it
12448 contains, you can use the key sequence @kbd{m s a v m s} (this uses
12449 @code{calc-alg-evaluate}, which resimplifies but doesn't evaluate
12452 @node Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation Modes
12453 @subsection Matrix and Scalar Modes
12456 @cindex Matrix mode
12457 @cindex Scalar mode
12458 Calc sometimes makes assumptions during algebraic manipulation that
12459 are awkward or incorrect when vectors and matrices are involved.
12460 Calc has two modes, @dfn{Matrix mode} and @dfn{Scalar mode}, which
12461 modify its behavior around vectors in useful ways.
12464 @pindex calc-matrix-mode
12465 Press @kbd{m v} (@code{calc-matrix-mode}) once to enter Matrix mode.
12466 In this mode, all objects are assumed to be matrices unless provably
12467 otherwise. One major effect is that Calc will no longer consider
12468 multiplication to be commutative. (Recall that in matrix arithmetic,
12469 @samp{A*B} is not the same as @samp{B*A}.) This assumption affects
12470 rewrite rules and algebraic simplification. Another effect of this
12471 mode is that calculations that would normally produce constants like
12472 0 and 1 (e.g., @expr{a - a} and @expr{a / a}, respectively) will now
12473 produce function calls that represent ``generic'' zero or identity
12474 matrices: @samp{idn(0)}, @samp{idn(1)}. The @code{idn} function
12475 @samp{idn(@var{a},@var{n})} returns @var{a} times an @var{n}x@var{n}
12476 identity matrix; if @var{n} is omitted, it doesn't know what
12477 dimension to use and so the @code{idn} call remains in symbolic
12478 form. However, if this generic identity matrix is later combined
12479 with a matrix whose size is known, it will be converted into
12480 a true identity matrix of the appropriate size. On the other hand,
12481 if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc
12482 will assume it really was a scalar after all and produce, e.g., 3.
12484 Press @kbd{m v} a second time to get Scalar mode. Here, objects are
12485 assumed @emph{not} to be vectors or matrices unless provably so.
12486 For example, normally adding a variable to a vector, as in
12487 @samp{[x, y, z] + a}, will leave the sum in symbolic form because
12488 as far as Calc knows, @samp{a} could represent either a number or
12489 another 3-vector. In Scalar mode, @samp{a} is assumed to be a
12490 non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}.
12492 Press @kbd{m v} a third time to return to the normal mode of operation.
12494 If you press @kbd{m v} with a numeric prefix argument @var{n}, you
12495 get a special ``dimensioned'' Matrix mode in which matrices of
12496 unknown size are assumed to be @var{n}x@var{n} square matrices.
12497 Then, the function call @samp{idn(1)} will expand into an actual
12498 matrix rather than representing a ``generic'' matrix. Simply typing
12499 @kbd{C-u m v} will get you a square Matrix mode, in which matrices of
12500 unknown size are assumed to be square matrices of unspecified size.
12502 @cindex Declaring scalar variables
12503 Of course these modes are approximations to the true state of
12504 affairs, which is probably that some quantities will be matrices
12505 and others will be scalars. One solution is to ``declare''
12506 certain variables or functions to be scalar-valued.
12507 @xref{Declarations}, to see how to make declarations in Calc.
12509 There is nothing stopping you from declaring a variable to be
12510 scalar and then storing a matrix in it; however, if you do, the
12511 results you get from Calc may not be valid. Suppose you let Calc
12512 get the result @samp{[x+a, y+a, z+a]} shown above, and then stored
12513 @samp{[1, 2, 3]} in @samp{a}. The result would not be the same as
12514 for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken
12515 your earlier promise to Calc that @samp{a} would be scalar.
12517 Another way to mix scalars and matrices is to use selections
12518 (@pxref{Selecting Subformulas}). Use Matrix mode when operating on
12519 your formula normally; then, to apply Scalar mode to a certain part
12520 of the formula without affecting the rest just select that part,
12521 change into Scalar mode and press @kbd{=} to resimplify the part
12522 under this mode, then change back to Matrix mode before deselecting.
12524 @node Automatic Recomputation, Working Message, Matrix Mode, Calculation Modes
12525 @subsection Automatic Recomputation
12528 The @dfn{evaluates-to} operator, @samp{=>}, has the special
12529 property that any @samp{=>} formulas on the stack are recomputed
12530 whenever variable values or mode settings that might affect them
12531 are changed. @xref{Evaluates-To Operator}.
12534 @pindex calc-auto-recompute
12535 The @kbd{m C} (@code{calc-auto-recompute}) command turns this
12536 automatic recomputation on and off. If you turn it off, Calc will
12537 not update @samp{=>} operators on the stack (nor those in the
12538 attached Embedded mode buffer, if there is one). They will not
12539 be updated unless you explicitly do so by pressing @kbd{=} or until
12540 you press @kbd{m C} to turn recomputation back on. (While automatic
12541 recomputation is off, you can think of @kbd{m C m C} as a command
12542 to update all @samp{=>} operators while leaving recomputation off.)
12544 To update @samp{=>} operators in an Embedded buffer while
12545 automatic recomputation is off, use @w{@kbd{C-x * u}}.
12546 @xref{Embedded Mode}.
12548 @node Working Message, , Automatic Recomputation, Calculation Modes
12549 @subsection Working Messages
12552 @cindex Performance
12553 @cindex Working messages
12554 Since the Calculator is written entirely in Emacs Lisp, which is not
12555 designed for heavy numerical work, many operations are quite slow.
12556 The Calculator normally displays the message @samp{Working...} in the
12557 echo area during any command that may be slow. In addition, iterative
12558 operations such as square roots and trigonometric functions display the
12559 intermediate result at each step. Both of these types of messages can
12560 be disabled if you find them distracting.
12563 @pindex calc-working
12564 Type @kbd{m w} (@code{calc-working}) with a numeric prefix of 0 to
12565 disable all ``working'' messages. Use a numeric prefix of 1 to enable
12566 only the plain @samp{Working...} message. Use a numeric prefix of 2 to
12567 see intermediate results as well. With no numeric prefix this displays
12570 While it may seem that the ``working'' messages will slow Calc down
12571 considerably, experiments have shown that their impact is actually
12572 quite small. But if your terminal is slow you may find that it helps
12573 to turn the messages off.
12575 @node Simplification Modes, Declarations, Calculation Modes, Mode Settings
12576 @section Simplification Modes
12579 The current @dfn{simplification mode} controls how numbers and formulas
12580 are ``normalized'' when being taken from or pushed onto the stack.
12581 Some normalizations are unavoidable, such as rounding floating-point
12582 results to the current precision, and reducing fractions to simplest
12583 form. Others, such as simplifying a formula like @expr{a+a} (or @expr{2+3}),
12584 are done automatically but can be turned off when necessary.
12586 When you press a key like @kbd{+} when @expr{2} and @expr{3} are on the
12587 stack, Calc pops these numbers, normalizes them, creates the formula
12588 @expr{2+3}, normalizes it, and pushes the result. Of course the standard
12589 rules for normalizing @expr{2+3} will produce the result @expr{5}.
12591 Simplification mode commands consist of the lower-case @kbd{m} prefix key
12592 followed by a shifted letter.
12595 @pindex calc-no-simplify-mode
12596 The @kbd{m O} (@code{calc-no-simplify-mode}) command turns off all optional
12597 simplifications. These would leave a formula like @expr{2+3} alone. In
12598 fact, nothing except simple numbers are ever affected by normalization
12599 in this mode. Explicit simplification commands, such as @kbd{=} or
12600 @kbd{a s}, can still be given to simplify any formulas.
12601 @xref{Algebraic Definitions}, for a sample use of
12602 No-Simplification mode.
12606 @pindex calc-num-simplify-mode
12607 The @kbd{m N} (@code{calc-num-simplify-mode}) command turns off simplification
12608 of any formulas except those for which all arguments are constants. For
12609 example, @expr{1+2} is simplified to @expr{3}, and @expr{a+(2-2)} is
12610 simplified to @expr{a+0} but no further, since one argument of the sum
12611 is not a constant. Unfortunately, @expr{(a+2)-2} is @emph{not} simplified
12612 because the top-level @samp{-} operator's arguments are not both
12613 constant numbers (one of them is the formula @expr{a+2}).
12614 A constant is a number or other numeric object (such as a constant
12615 error form or modulo form), or a vector all of whose
12616 elements are constant.
12619 @pindex calc-basic-simplify-mode
12620 The @kbd{m I} (@code{calc-basic-simplify-mode}) command does some basic
12621 simplifications for all formulas. This includes many easy and
12622 fast algebraic simplifications such as @expr{a+0} to @expr{a}, and
12623 @expr{a + 2 a} to @expr{3 a}, as well as evaluating functions like
12624 @expr{@tfn{deriv}(x^2, x)} to @expr{2 x}.
12627 @pindex calc-bin-simplify-mode
12628 The @kbd{m B} (@code{calc-bin-simplify-mode}) mode applies the basic
12629 simplifications to a result and then, if the result is an integer,
12630 uses the @kbd{b c} (@code{calc-clip}) command to clip the integer according
12631 to the current binary word size. @xref{Binary Functions}. Real numbers
12632 are rounded to the nearest integer and then clipped; other kinds of
12633 results (after the basic simplifications) are left alone.
12636 @pindex calc-alg-simplify-mode
12637 The @kbd{m A} (@code{calc-alg-simplify-mode}) mode does standard
12638 algebraic simplifications. @xref{Algebraic Simplifications}.
12641 @pindex calc-ext-simplify-mode
12642 The @kbd{m E} (@code{calc-ext-simplify-mode}) mode does ``extended'', or
12643 ``unsafe'', algebraic simplification. @xref{Unsafe Simplifications}.
12646 @pindex calc-units-simplify-mode
12647 The @kbd{m U} (@code{calc-units-simplify-mode}) mode does units
12648 simplification. @xref{Simplification of Units}. These include the
12649 algebraic simplifications, plus variable names which
12650 are identifiable as unit names (like @samp{mm} for ``millimeters'')
12651 are simplified with their unit definitions in mind.
12653 A common technique is to set the simplification mode down to the lowest
12654 amount of simplification you will allow to be applied automatically, then
12655 use manual commands like @kbd{a s} and @kbd{c c} (@code{calc-clean}) to
12656 perform higher types of simplifications on demand.
12657 @node Declarations, Display Modes, Simplification Modes, Mode Settings
12658 @section Declarations
12661 A @dfn{declaration} is a statement you make that promises you will
12662 use a certain variable or function in a restricted way. This may
12663 give Calc the freedom to do things that it couldn't do if it had to
12664 take the fully general situation into account.
12667 * Declaration Basics::
12668 * Kinds of Declarations::
12669 * Functions for Declarations::
12672 @node Declaration Basics, Kinds of Declarations, Declarations, Declarations
12673 @subsection Declaration Basics
12677 @pindex calc-declare-variable
12678 The @kbd{s d} (@code{calc-declare-variable}) command is the easiest
12679 way to make a declaration for a variable. This command prompts for
12680 the variable name, then prompts for the declaration. The default
12681 at the declaration prompt is the previous declaration, if any.
12682 You can edit this declaration, or press @kbd{C-k} to erase it and
12683 type a new declaration. (Or, erase it and press @key{RET} to clear
12684 the declaration, effectively ``undeclaring'' the variable.)
12686 A declaration is in general a vector of @dfn{type symbols} and
12687 @dfn{range} values. If there is only one type symbol or range value,
12688 you can write it directly rather than enclosing it in a vector.
12689 For example, @kbd{s d foo @key{RET} real @key{RET}} declares @code{foo} to
12690 be a real number, and @kbd{s d bar @key{RET} [int, const, [1..6]] @key{RET}}
12691 declares @code{bar} to be a constant integer between 1 and 6.
12692 (Actually, you can omit the outermost brackets and Calc will
12693 provide them for you: @kbd{s d bar @key{RET} int, const, [1..6] @key{RET}}.)
12695 @cindex @code{Decls} variable
12697 Declarations in Calc are kept in a special variable called @code{Decls}.
12698 This variable encodes the set of all outstanding declarations in
12699 the form of a matrix. Each row has two elements: A variable or
12700 vector of variables declared by that row, and the declaration
12701 specifier as described above. You can use the @kbd{s D} command to
12702 edit this variable if you wish to see all the declarations at once.
12703 @xref{Operations on Variables}, for a description of this command
12704 and the @kbd{s p} command that allows you to save your declarations
12705 permanently if you wish.
12707 Items being declared can also be function calls. The arguments in
12708 the call are ignored; the effect is to say that this function returns
12709 values of the declared type for any valid arguments. The @kbd{s d}
12710 command declares only variables, so if you wish to make a function
12711 declaration you will have to edit the @code{Decls} matrix yourself.
12713 For example, the declaration matrix
12719 [ f(1,2,3), [0 .. inf) ] ]
12724 declares that @code{foo} represents a real number, @code{j}, @code{k}
12725 and @code{n} represent integers, and the function @code{f} always
12726 returns a real number in the interval shown.
12729 If there is a declaration for the variable @code{All}, then that
12730 declaration applies to all variables that are not otherwise declared.
12731 It does not apply to function names. For example, using the row
12732 @samp{[All, real]} says that all your variables are real unless they
12733 are explicitly declared without @code{real} in some other row.
12734 The @kbd{s d} command declares @code{All} if you give a blank
12735 response to the variable-name prompt.
12737 @node Kinds of Declarations, Functions for Declarations, Declaration Basics, Declarations
12738 @subsection Kinds of Declarations
12741 The type-specifier part of a declaration (that is, the second prompt
12742 in the @kbd{s d} command) can be a type symbol, an interval, or a
12743 vector consisting of zero or more type symbols followed by zero or
12744 more intervals or numbers that represent the set of possible values
12749 [ [ a, [1, 2, 3, 4, 5] ]
12751 [ c, [int, 1 .. 5] ] ]
12755 Here @code{a} is declared to contain one of the five integers shown;
12756 @code{b} is any number in the interval from 1 to 5 (any real number
12757 since we haven't specified), and @code{c} is any integer in that
12758 interval. Thus the declarations for @code{a} and @code{c} are
12759 nearly equivalent (see below).
12761 The type-specifier can be the empty vector @samp{[]} to say that
12762 nothing is known about a given variable's value. This is the same
12763 as not declaring the variable at all except that it overrides any
12764 @code{All} declaration which would otherwise apply.
12766 The initial value of @code{Decls} is the empty vector @samp{[]}.
12767 If @code{Decls} has no stored value or if the value stored in it
12768 is not valid, it is ignored and there are no declarations as far
12769 as Calc is concerned. (The @kbd{s d} command will replace such a
12770 malformed value with a fresh empty matrix, @samp{[]}, before recording
12771 the new declaration.) Unrecognized type symbols are ignored.
12773 The following type symbols describe what sorts of numbers will be
12774 stored in a variable:
12780 Numerical integers. (Integers or integer-valued floats.)
12782 Fractions. (Rational numbers which are not integers.)
12784 Rational numbers. (Either integers or fractions.)
12786 Floating-point numbers.
12788 Real numbers. (Integers, fractions, or floats. Actually,
12789 intervals and error forms with real components also count as
12792 Positive real numbers. (Strictly greater than zero.)
12794 Nonnegative real numbers. (Greater than or equal to zero.)
12796 Numbers. (Real or complex.)
12799 Calc uses this information to determine when certain simplifications
12800 of formulas are safe. For example, @samp{(x^y)^z} cannot be
12801 simplified to @samp{x^(y z)} in general; for example,
12802 @samp{((-3)^2)^1:2} is 3, but @samp{(-3)^(2*1:2) = (-3)^1} is @mathit{-3}.
12803 However, this simplification @emph{is} safe if @code{z} is known
12804 to be an integer, or if @code{x} is known to be a nonnegative
12805 real number. If you have given declarations that allow Calc to
12806 deduce either of these facts, Calc will perform this simplification
12809 Calc can apply a certain amount of logic when using declarations.
12810 For example, @samp{(x^y)^(2n+1)} will be simplified if @code{n}
12811 has been declared @code{int}; Calc knows that an integer times an
12812 integer, plus an integer, must always be an integer. (In fact,
12813 Calc would simplify @samp{(-x)^(2n+1)} to @samp{-(x^(2n+1))} since
12814 it is able to determine that @samp{2n+1} must be an odd integer.)
12816 Similarly, @samp{(abs(x)^y)^z} will be simplified to @samp{abs(x)^(y z)}
12817 because Calc knows that the @code{abs} function always returns a
12818 nonnegative real. If you had a @code{myabs} function that also had
12819 this property, you could get Calc to recognize it by adding the row
12820 @samp{[myabs(), nonneg]} to the @code{Decls} matrix.
12822 One instance of this simplification is @samp{sqrt(x^2)} (since the
12823 @code{sqrt} function is effectively a one-half power). Normally
12824 Calc leaves this formula alone. After the command
12825 @kbd{s d x @key{RET} real @key{RET}}, however, it can simplify the formula to
12826 @samp{abs(x)}. And after @kbd{s d x @key{RET} nonneg @key{RET}}, Calc can
12827 simplify this formula all the way to @samp{x}.
12829 If there are any intervals or real numbers in the type specifier,
12830 they comprise the set of possible values that the variable or
12831 function being declared can have. In particular, the type symbol
12832 @code{real} is effectively the same as the range @samp{[-inf .. inf]}
12833 (note that infinity is included in the range of possible values);
12834 @code{pos} is the same as @samp{(0 .. inf]}, and @code{nonneg} is
12835 the same as @samp{[0 .. inf]}. Saying @samp{[real, [-5 .. 5]]} is
12836 redundant because the fact that the variable is real can be
12837 deduced just from the interval, but @samp{[int, [-5 .. 5]]} and
12838 @samp{[rat, [-5 .. 5]]} are useful combinations.
12840 Note that the vector of intervals or numbers is in the same format
12841 used by Calc's set-manipulation commands. @xref{Set Operations}.
12843 The type specifier @samp{[1, 2, 3]} is equivalent to
12844 @samp{[numint, 1, 2, 3]}, @emph{not} to @samp{[int, 1, 2, 3]}.
12845 In other words, the range of possible values means only that
12846 the variable's value must be numerically equal to a number in
12847 that range, but not that it must be equal in type as well.
12848 Calc's set operations act the same way; @samp{in(2, [1., 2., 3.])}
12849 and @samp{in(1.5, [1:2, 3:2, 5:2])} both report ``true.''
12851 If you use a conflicting combination of type specifiers, the
12852 results are unpredictable. An example is @samp{[pos, [0 .. 5]]},
12853 where the interval does not lie in the range described by the
12856 ``Real'' declarations mostly affect simplifications involving powers
12857 like the one described above. Another case where they are used
12858 is in the @kbd{a P} command which returns a list of all roots of a
12859 polynomial; if the variable has been declared real, only the real
12860 roots (if any) will be included in the list.
12862 ``Integer'' declarations are used for simplifications which are valid
12863 only when certain values are integers (such as @samp{(x^y)^z}
12866 Calc's algebraic simplifications also make use of declarations when
12867 simplifying equations and inequalities. They will cancel @code{x}
12868 from both sides of @samp{a x = b x} only if it is sure @code{x}
12869 is non-zero, say, because it has a @code{pos} declaration.
12870 To declare specifically that @code{x} is real and non-zero,
12871 use @samp{[[-inf .. 0), (0 .. inf]]}. (There is no way in the
12872 current notation to say that @code{x} is nonzero but not necessarily
12873 real.) The @kbd{a e} command does ``unsafe'' simplifications,
12874 including canceling @samp{x} from the equation when @samp{x} is
12875 not known to be nonzero.
12877 Another set of type symbols distinguish between scalars and vectors.
12881 The value is not a vector.
12883 The value is a vector.
12885 The value is a matrix (a rectangular vector of vectors).
12887 The value is a square matrix.
12890 These type symbols can be combined with the other type symbols
12891 described above; @samp{[int, matrix]} describes an object which
12892 is a matrix of integers.
12894 Scalar/vector declarations are used to determine whether certain
12895 algebraic operations are safe. For example, @samp{[a, b, c] + x}
12896 is normally not simplified to @samp{[a + x, b + x, c + x]}, but
12897 it will be if @code{x} has been declared @code{scalar}. On the
12898 other hand, multiplication is usually assumed to be commutative,
12899 but the terms in @samp{x y} will never be exchanged if both @code{x}
12900 and @code{y} are known to be vectors or matrices. (Calc currently
12901 never distinguishes between @code{vector} and @code{matrix}
12904 @xref{Matrix Mode}, for a discussion of Matrix mode and
12905 Scalar mode, which are similar to declaring @samp{[All, matrix]}
12906 or @samp{[All, scalar]} but much more convenient.
12908 One more type symbol that is recognized is used with the @kbd{H a d}
12909 command for taking total derivatives of a formula. @xref{Calculus}.
12913 The value is a constant with respect to other variables.
12916 Calc does not check the declarations for a variable when you store
12917 a value in it. However, storing @mathit{-3.5} in a variable that has
12918 been declared @code{pos}, @code{int}, or @code{matrix} may have
12919 unexpected effects; Calc may evaluate @samp{sqrt(x^2)} to @expr{3.5}
12920 if it substitutes the value first, or to @expr{-3.5} if @code{x}
12921 was declared @code{pos} and the formula @samp{sqrt(x^2)} is
12922 simplified to @samp{x} before the value is substituted. Before
12923 using a variable for a new purpose, it is best to use @kbd{s d}
12924 or @kbd{s D} to check to make sure you don't still have an old
12925 declaration for the variable that will conflict with its new meaning.
12927 @node Functions for Declarations, , Kinds of Declarations, Declarations
12928 @subsection Functions for Declarations
12931 Calc has a set of functions for accessing the current declarations
12932 in a convenient manner. These functions return 1 if the argument
12933 can be shown to have the specified property, or 0 if the argument
12934 can be shown @emph{not} to have that property; otherwise they are
12935 left unevaluated. These functions are suitable for use with rewrite
12936 rules (@pxref{Conditional Rewrite Rules}) or programming constructs
12937 (@pxref{Conditionals in Macros}). They can be entered only using
12938 algebraic notation. @xref{Logical Operations}, for functions
12939 that perform other tests not related to declarations.
12941 For example, @samp{dint(17)} returns 1 because 17 is an integer, as
12942 do @samp{dint(n)} and @samp{dint(2 n - 3)} if @code{n} has been declared
12943 @code{int}, but @samp{dint(2.5)} and @samp{dint(n + 0.5)} return 0.
12944 Calc consults knowledge of its own built-in functions as well as your
12945 own declarations: @samp{dint(floor(x))} returns 1.
12959 The @code{dint} function checks if its argument is an integer.
12960 The @code{dnatnum} function checks if its argument is a natural
12961 number, i.e., a nonnegative integer. The @code{dnumint} function
12962 checks if its argument is numerically an integer, i.e., either an
12963 integer or an integer-valued float. Note that these and the other
12964 data type functions also accept vectors or matrices composed of
12965 suitable elements, and that real infinities @samp{inf} and @samp{-inf}
12966 are considered to be integers for the purposes of these functions.
12972 The @code{drat} function checks if its argument is rational, i.e.,
12973 an integer or fraction. Infinities count as rational, but intervals
12974 and error forms do not.
12980 The @code{dreal} function checks if its argument is real. This
12981 includes integers, fractions, floats, real error forms, and intervals.
12987 The @code{dimag} function checks if its argument is imaginary,
12988 i.e., is mathematically equal to a real number times @expr{i}.
13002 The @code{dpos} function checks for positive (but nonzero) reals.
13003 The @code{dneg} function checks for negative reals. The @code{dnonneg}
13004 function checks for nonnegative reals, i.e., reals greater than or
13005 equal to zero. Note that Calc's algebraic simplifications, which are
13006 effectively applied to all conditions in rewrite rules, can simplify
13007 an expression like @expr{x > 0} to 1 or 0 using @code{dpos}.
13008 So the actual functions @code{dpos}, @code{dneg}, and @code{dnonneg}
13009 are rarely necessary.
13015 The @code{dnonzero} function checks that its argument is nonzero.
13016 This includes all nonzero real or complex numbers, all intervals that
13017 do not include zero, all nonzero modulo forms, vectors all of whose
13018 elements are nonzero, and variables or formulas whose values can be
13019 deduced to be nonzero. It does not include error forms, since they
13020 represent values which could be anything including zero. (This is
13021 also the set of objects considered ``true'' in conditional contexts.)
13031 The @code{deven} function returns 1 if its argument is known to be
13032 an even integer (or integer-valued float); it returns 0 if its argument
13033 is known not to be even (because it is known to be odd or a non-integer).
13034 Calc's algebraic simplifications use this to simplify a test of the form
13035 @samp{x % 2 = 0}. There is also an analogous @code{dodd} function.
13041 The @code{drange} function returns a set (an interval or a vector
13042 of intervals and/or numbers; @pxref{Set Operations}) that describes
13043 the set of possible values of its argument. If the argument is
13044 a variable or a function with a declaration, the range is copied
13045 from the declaration. Otherwise, the possible signs of the
13046 expression are determined using a method similar to @code{dpos},
13047 etc., and a suitable set like @samp{[0 .. inf]} is returned. If
13048 the expression is not provably real, the @code{drange} function
13049 remains unevaluated.
13055 The @code{dscalar} function returns 1 if its argument is provably
13056 scalar, or 0 if its argument is provably non-scalar. It is left
13057 unevaluated if this cannot be determined. (If Matrix mode or Scalar
13058 mode is in effect, this function returns 1 or 0, respectively,
13059 if it has no other information.) When Calc interprets a condition
13060 (say, in a rewrite rule) it considers an unevaluated formula to be
13061 ``false.'' Thus, @samp{dscalar(a)} is ``true'' only if @code{a} is
13062 provably scalar, and @samp{!dscalar(a)} is ``true'' only if @code{a}
13063 is provably non-scalar; both are ``false'' if there is insufficient
13064 information to tell.
13066 @node Display Modes, Language Modes, Declarations, Mode Settings
13067 @section Display Modes
13070 The commands in this section are two-key sequences beginning with the
13071 @kbd{d} prefix. The @kbd{d l} (@code{calc-line-numbering}) and @kbd{d b}
13072 (@code{calc-line-breaking}) commands are described elsewhere;
13073 @pxref{Stack Basics} and @pxref{Normal Language Modes}, respectively.
13074 Display formats for vectors and matrices are also covered elsewhere;
13075 @pxref{Vector and Matrix Formats}.
13077 One thing all display modes have in common is their treatment of the
13078 @kbd{H} prefix. This prefix causes any mode command that would normally
13079 refresh the stack to leave the stack display alone. The word ``Dirty''
13080 will appear in the mode line when Calc thinks the stack display may not
13081 reflect the latest mode settings.
13083 @kindex d @key{RET}
13084 @pindex calc-refresh-top
13085 The @kbd{d @key{RET}} (@code{calc-refresh-top}) command reformats the
13086 top stack entry according to all the current modes. Positive prefix
13087 arguments reformat the top @var{n} entries; negative prefix arguments
13088 reformat the specified entry, and a prefix of zero is equivalent to
13089 @kbd{d @key{SPC}} (@code{calc-refresh}), which reformats the entire stack.
13090 For example, @kbd{H d s M-2 d @key{RET}} changes to scientific notation
13091 but reformats only the top two stack entries in the new mode.
13093 The @kbd{I} prefix has another effect on the display modes. The mode
13094 is set only temporarily; the top stack entry is reformatted according
13095 to that mode, then the original mode setting is restored. In other
13096 words, @kbd{I d s} is equivalent to @kbd{H d s d @key{RET} H d (@var{old mode})}.
13100 * Grouping Digits::
13102 * Complex Formats::
13103 * Fraction Formats::
13106 * Truncating the Stack::
13111 @node Radix Modes, Grouping Digits, Display Modes, Display Modes
13112 @subsection Radix Modes
13115 @cindex Radix display
13116 @cindex Non-decimal numbers
13117 @cindex Decimal and non-decimal numbers
13118 Calc normally displays numbers in decimal (@dfn{base-10} or @dfn{radix-10})
13119 notation. Calc can actually display in any radix from two (binary) to 36.
13120 When the radix is above 10, the letters @code{A} to @code{Z} are used as
13121 digits. When entering such a number, letter keys are interpreted as
13122 potential digits rather than terminating numeric entry mode.
13128 @cindex Hexadecimal integers
13129 @cindex Octal integers
13130 The key sequences @kbd{d 2}, @kbd{d 8}, @kbd{d 6}, and @kbd{d 0} select
13131 binary, octal, hexadecimal, and decimal as the current display radix,
13132 respectively. Numbers can always be entered in any radix, though the
13133 current radix is used as a default if you press @kbd{#} without any initial
13134 digits. A number entered without a @kbd{#} is @emph{always} interpreted
13139 To set the radix generally, use @kbd{d r} (@code{calc-radix}) and enter
13140 an integer from 2 to 36. You can specify the radix as a numeric prefix
13141 argument; otherwise you will be prompted for it.
13144 @pindex calc-leading-zeros
13145 @cindex Leading zeros
13146 Integers normally are displayed with however many digits are necessary to
13147 represent the integer and no more. The @kbd{d z} (@code{calc-leading-zeros})
13148 command causes integers to be padded out with leading zeros according to the
13149 current binary word size. (@xref{Binary Functions}, for a discussion of
13150 word size.) If the absolute value of the word size is @expr{w}, all integers
13151 are displayed with at least enough digits to represent
13152 @texline @math{2^w-1}
13153 @infoline @expr{(2^w)-1}
13154 in the current radix. (Larger integers will still be displayed in their
13157 @cindex Two's complements
13158 Calc can display @expr{w}-bit integers using two's complement
13159 notation, although this is most useful with the binary, octal and
13160 hexadecimal display modes. This option is selected by using the
13161 @kbd{O} option prefix before setting the display radix, and a negative word
13162 size might be appropriate (@pxref{Binary Functions}). In two's
13163 complement notation, the integers in the (nearly) symmetric interval
13165 @texline @math{-2^{w-1}}
13166 @infoline @expr{-2^(w-1)}
13168 @texline @math{2^{w-1}-1}
13169 @infoline @expr{2^(w-1)-1}
13170 are represented by the integers from @expr{0} to @expr{2^w-1}:
13171 the integers from @expr{0} to
13172 @texline @math{2^{w-1}-1}
13173 @infoline @expr{2^(w-1)-1}
13174 are represented by themselves and the integers from
13175 @texline @math{-2^{w-1}}
13176 @infoline @expr{-2^(w-1)}
13177 to @expr{-1} are represented by the integers from
13178 @texline @math{2^{w-1}}
13179 @infoline @expr{2^(w-1)}
13180 to @expr{2^w-1} (the integer @expr{k} is represented by @expr{k+2^w}).
13181 Calc will display a two's complement integer by the radix (either
13182 @expr{2}, @expr{8} or @expr{16}), two @kbd{#} symbols, and then its
13183 representation (including any leading zeros necessary to include all
13184 @expr{w} bits). In a two's complement display mode, numbers that
13185 are not displayed in two's complement notation (i.e., that aren't
13187 @texline @math{-2^{w-1}}
13188 @infoline @expr{-2^(w-1)}
13191 @texline @math{2^{w-1}-1})
13192 @infoline @expr{2^(w-1)-1})
13193 will be represented using Calc's usual notation (in the appropriate
13196 @node Grouping Digits, Float Formats, Radix Modes, Display Modes
13197 @subsection Grouping Digits
13201 @pindex calc-group-digits
13202 @cindex Grouping digits
13203 @cindex Digit grouping
13204 Long numbers can be hard to read if they have too many digits. For
13205 example, the factorial of 30 is 33 digits long! Press @kbd{d g}
13206 (@code{calc-group-digits}) to enable @dfn{Grouping} mode, in which digits
13207 are displayed in clumps of 3 or 4 (depending on the current radix)
13208 separated by commas.
13210 The @kbd{d g} command toggles grouping on and off.
13211 With a numeric prefix of 0, this command displays the current state of
13212 the grouping flag; with an argument of minus one it disables grouping;
13213 with a positive argument @expr{N} it enables grouping on every @expr{N}
13214 digits. For floating-point numbers, grouping normally occurs only
13215 before the decimal point. A negative prefix argument @expr{-N} enables
13216 grouping every @expr{N} digits both before and after the decimal point.
13219 @pindex calc-group-char
13220 The @kbd{d ,} (@code{calc-group-char}) command allows you to choose any
13221 character as the grouping separator. The default is the comma character.
13222 If you find it difficult to read vectors of large integers grouped with
13223 commas, you may wish to use spaces or some other character instead.
13224 This command takes the next character you type, whatever it is, and
13225 uses it as the digit separator. As a special case, @kbd{d , \} selects
13226 @samp{\,} (@TeX{}'s thin-space symbol) as the digit separator.
13228 Please note that grouped numbers will not generally be parsed correctly
13229 if re-read in textual form, say by the use of @kbd{C-x * y} and @kbd{C-x * g}.
13230 (@xref{Kill and Yank}, for details on these commands.) One exception is
13231 the @samp{\,} separator, which doesn't interfere with parsing because it
13232 is ignored by @TeX{} language mode.
13234 @node Float Formats, Complex Formats, Grouping Digits, Display Modes
13235 @subsection Float Formats
13238 Floating-point quantities are normally displayed in standard decimal
13239 form, with scientific notation used if the exponent is especially high
13240 or low. All significant digits are normally displayed. The commands
13241 in this section allow you to choose among several alternative display
13242 formats for floats.
13245 @pindex calc-normal-notation
13246 The @kbd{d n} (@code{calc-normal-notation}) command selects the normal
13247 display format. All significant figures in a number are displayed.
13248 With a positive numeric prefix, numbers are rounded if necessary to
13249 that number of significant digits. With a negative numerix prefix,
13250 the specified number of significant digits less than the current
13251 precision is used. (Thus @kbd{C-u -2 d n} displays 10 digits if the
13252 current precision is 12.)
13255 @pindex calc-fix-notation
13256 The @kbd{d f} (@code{calc-fix-notation}) command selects fixed-point
13257 notation. The numeric argument is the number of digits after the
13258 decimal point, zero or more. This format will relax into scientific
13259 notation if a nonzero number would otherwise have been rounded all the
13260 way to zero. Specifying a negative number of digits is the same as
13261 for a positive number, except that small nonzero numbers will be rounded
13262 to zero rather than switching to scientific notation.
13265 @pindex calc-sci-notation
13266 @cindex Scientific notation, display of
13267 The @kbd{d s} (@code{calc-sci-notation}) command selects scientific
13268 notation. A positive argument sets the number of significant figures
13269 displayed, of which one will be before and the rest after the decimal
13270 point. A negative argument works the same as for @kbd{d n} format.
13271 The default is to display all significant digits.
13274 @pindex calc-eng-notation
13275 @cindex Engineering notation, display of
13276 The @kbd{d e} (@code{calc-eng-notation}) command selects engineering
13277 notation. This is similar to scientific notation except that the
13278 exponent is rounded down to a multiple of three, with from one to three
13279 digits before the decimal point. An optional numeric prefix sets the
13280 number of significant digits to display, as for @kbd{d s}.
13282 It is important to distinguish between the current @emph{precision} and
13283 the current @emph{display format}. After the commands @kbd{C-u 10 p}
13284 and @kbd{C-u 6 d n} the Calculator computes all results to ten
13285 significant figures but displays only six. (In fact, intermediate
13286 calculations are often carried to one or two more significant figures,
13287 but values placed on the stack will be rounded down to ten figures.)
13288 Numbers are never actually rounded to the display precision for storage,
13289 except by commands like @kbd{C-k} and @kbd{C-x * y} which operate on the
13290 actual displayed text in the Calculator buffer.
13293 @pindex calc-point-char
13294 The @kbd{d .} (@code{calc-point-char}) command selects the character used
13295 as a decimal point. Normally this is a period; users in some countries
13296 may wish to change this to a comma. Note that this is only a display
13297 style; on entry, periods must always be used to denote floating-point
13298 numbers, and commas to separate elements in a list.
13300 @node Complex Formats, Fraction Formats, Float Formats, Display Modes
13301 @subsection Complex Formats
13305 @pindex calc-complex-notation
13306 There are three supported notations for complex numbers in rectangular
13307 form. The default is as a pair of real numbers enclosed in parentheses
13308 and separated by a comma: @samp{(a,b)}. The @kbd{d c}
13309 (@code{calc-complex-notation}) command selects this style.
13312 @pindex calc-i-notation
13314 @pindex calc-j-notation
13315 The other notations are @kbd{d i} (@code{calc-i-notation}), in which
13316 numbers are displayed in @samp{a+bi} form, and @kbd{d j}
13317 (@code{calc-j-notation}) which displays the form @samp{a+bj} preferred
13318 in some disciplines.
13320 @cindex @code{i} variable
13322 Complex numbers are normally entered in @samp{(a,b)} format.
13323 If you enter @samp{2+3i} as an algebraic formula, it will be stored as
13324 the formula @samp{2 + 3 * i}. However, if you use @kbd{=} to evaluate
13325 this formula and you have not changed the variable @samp{i}, the @samp{i}
13326 will be interpreted as @samp{(0,1)} and the formula will be simplified
13327 to @samp{(2,3)}. Other commands (like @code{calc-sin}) will @emph{not}
13328 interpret the formula @samp{2 + 3 * i} as a complex number.
13329 @xref{Variables}, under ``special constants.''
13331 @node Fraction Formats, HMS Formats, Complex Formats, Display Modes
13332 @subsection Fraction Formats
13336 @pindex calc-over-notation
13337 Display of fractional numbers is controlled by the @kbd{d o}
13338 (@code{calc-over-notation}) command. By default, a number like
13339 eight thirds is displayed in the form @samp{8:3}. The @kbd{d o} command
13340 prompts for a one- or two-character format. If you give one character,
13341 that character is used as the fraction separator. Common separators are
13342 @samp{:} and @samp{/}. (During input of numbers, the @kbd{:} key must be
13343 used regardless of the display format; in particular, the @kbd{/} is used
13344 for RPN-style division, @emph{not} for entering fractions.)
13346 If you give two characters, fractions use ``integer-plus-fractional-part''
13347 notation. For example, the format @samp{+/} would display eight thirds
13348 as @samp{2+2/3}. If two colons are present in a number being entered,
13349 the number is interpreted in this form (so that the entries @kbd{2:2:3}
13350 and @kbd{8:3} are equivalent).
13352 It is also possible to follow the one- or two-character format with
13353 a number. For example: @samp{:10} or @samp{+/3}. In this case,
13354 Calc adjusts all fractions that are displayed to have the specified
13355 denominator, if possible. Otherwise it adjusts the denominator to
13356 be a multiple of the specified value. For example, in @samp{:6} mode
13357 the fraction @expr{1:6} will be unaffected, but @expr{2:3} will be
13358 displayed as @expr{4:6}, @expr{1:2} will be displayed as @expr{3:6},
13359 and @expr{1:8} will be displayed as @expr{3:24}. Integers are also
13360 affected by this mode: 3 is displayed as @expr{18:6}. Note that the
13361 format @samp{:1} writes fractions the same as @samp{:}, but it writes
13362 integers as @expr{n:1}.
13364 The fraction format does not affect the way fractions or integers are
13365 stored, only the way they appear on the screen. The fraction format
13366 never affects floats.
13368 @node HMS Formats, Date Formats, Fraction Formats, Display Modes
13369 @subsection HMS Formats
13373 @pindex calc-hms-notation
13374 The @kbd{d h} (@code{calc-hms-notation}) command controls the display of
13375 HMS (hours-minutes-seconds) forms. It prompts for a string which
13376 consists basically of an ``hours'' marker, optional punctuation, a
13377 ``minutes'' marker, more optional punctuation, and a ``seconds'' marker.
13378 Punctuation is zero or more spaces, commas, or semicolons. The hours
13379 marker is one or more non-punctuation characters. The minutes and
13380 seconds markers must be single non-punctuation characters.
13382 The default HMS format is @samp{@@ ' "}, producing HMS values of the form
13383 @samp{23@@ 30' 15.75"}. The format @samp{deg, ms} would display this same
13384 value as @samp{23deg, 30m15.75s}. During numeric entry, the @kbd{h} or @kbd{o}
13385 keys are recognized as synonyms for @kbd{@@} regardless of display format.
13386 The @kbd{m} and @kbd{s} keys are recognized as synonyms for @kbd{'} and
13387 @kbd{"}, respectively, but only if an @kbd{@@} (or @kbd{h} or @kbd{o}) has
13388 already been typed; otherwise, they have their usual meanings
13389 (@kbd{m-} prefix and @kbd{s-} prefix). Thus, @kbd{5 "}, @kbd{0 @@ 5 "}, and
13390 @kbd{0 h 5 s} are some of the ways to enter the quantity ``five seconds.''
13391 The @kbd{'} key is recognized as ``minutes'' only if @kbd{@@} (or @kbd{h} or
13392 @kbd{o}) has already been pressed; otherwise it means to switch to algebraic
13395 @node Date Formats, Truncating the Stack, HMS Formats, Display Modes
13396 @subsection Date Formats
13400 @pindex calc-date-notation
13401 The @kbd{d d} (@code{calc-date-notation}) command controls the display
13402 of date forms (@pxref{Date Forms}). It prompts for a string which
13403 contains letters that represent the various parts of a date and time.
13404 To show which parts should be omitted when the form represents a pure
13405 date with no time, parts of the string can be enclosed in @samp{< >}
13406 marks. If you don't include @samp{< >} markers in the format, Calc
13407 guesses at which parts, if any, should be omitted when formatting
13410 The default format is: @samp{<H:mm:SSpp >Www Mmm D, YYYY}.
13411 An example string in this format is @samp{3:32pm Wed Jan 9, 1991}.
13412 If you enter a blank format string, this default format is
13415 Calc uses @samp{< >} notation for nameless functions as well as for
13416 dates. @xref{Specifying Operators}. To avoid confusion with nameless
13417 functions, your date formats should avoid using the @samp{#} character.
13421 * Date Formatting Codes::
13422 * Free-Form Dates::
13423 * Standard Date Formats::
13426 @node ISO-8601, Date Formatting Codes, Date Formats, Date Formats
13427 @subsubsection ISO-8601
13431 The same date can be written down in different formats and Calc tries
13432 to allow you to choose your preferred format. Some common formats are
13433 ambiguous, however; for example, 10/11/2012 means October 11,
13434 2012 in the United States but it means November 10, 2012 in
13435 Europe. To help avoid such ambiguities, the International Organization
13436 for Standardization (ISO) provides the ISO-8601 standard, which
13437 provides three different but easily distinguishable and unambiguous
13438 ways to represent a date.
13440 The ISO-8601 calendar date representation is
13443 @var{YYYY}-@var{MM}-@var{DD}
13447 where @var{YYYY} is the four digit year, @var{MM} is the two-digit month
13448 number (01 for January to 12 for December), and @var{DD} is the
13449 two-digit day of the month (01 to 31). (Note that @var{YYYY} does not
13450 correspond to Calc's date formatting code, which will be introduced
13451 later.) The year, which should be padded with zeros to ensure it has at
13452 least four digits, is the Gregorian year, except that the year before
13453 0001 (1 AD) is the year 0000 (1 BC). The date October 11, 2012 is
13454 written 2012-10-11 in this representation and November 10, 2012 is
13455 written 2012-11-10.
13457 The ISO-8601 ordinal date representation is
13460 @var{YYYY}-@var{DDD}
13464 where @var{YYYY} is the year, as above, and @var{DDD} is the day of the year.
13465 The date December 31, 2011 is written 2011-365 in this representation
13466 and January 1, 2012 is written 2012-001.
13468 The ISO-8601 week date representation is
13471 @var{YYYY}-W@var{ww}-@var{D}
13475 where @var{YYYY} is the ISO week-numbering year, @var{ww} is the two
13476 digit week number (preceded by a literal ``W''), and @var{D} is the day
13477 of the week (1 for Monday through 7 for Sunday). The ISO week-numbering
13478 year is based on the Gregorian year but can differ slightly. The first
13479 week of an ISO week-numbering year is the week with the Gregorian year's
13480 first Thursday in it (equivalently, the week containing January 4);
13481 any day of that week (Monday through Sunday) is part of the same ISO
13482 week-numbering year, any day from the previous week is part of the
13483 previous year. For example, January 4, 2013 is on a Friday, and so
13484 the first week for the ISO week-numbering year 2013 starts on
13485 Monday, December 31, 2012. The day December 31, 2012 is then part of the
13486 Gregorian year 2012 but ISO week-numbering year 2013. In the week
13487 date representation, this week goes from 2013-W01-1 (December 31,
13488 2012) to 2013-W01-7 (January 6, 2013).
13490 All three ISO-8601 representations arrange the numbers from most
13491 significant to least significant; as well as being unambiguous
13492 representations, they are easy to sort since chronological order in
13493 this formats corresponds to lexicographical order. The hyphens are
13496 The ISO-8601 standard uses a 24 hour clock; a particular time is
13497 represented by @var{hh}:@var{mm}:@var{ss} where @var{hh} is the
13498 two-digit hour (from 00 to 24), @var{mm} is the two-digit minute (from
13499 00 to 59) and @var{ss} is the two-digit second. The seconds or minutes
13500 and seconds can be omitted, and decimals can be added. If a date with a
13501 time is represented, they should be separated by a literal ``T'', so noon
13502 on December 13, 2012 can be represented as 2012-12-13T12:00
13504 @node Date Formatting Codes, Free-Form Dates, ISO-8601, Date Formats
13505 @subsubsection Date Formatting Codes
13508 When displaying a date, the current date format is used. All
13509 characters except for letters and @samp{<} and @samp{>} are
13510 copied literally when dates are formatted. The portion between
13511 @samp{< >} markers is omitted for pure dates, or included for
13512 date/time forms. Letters are interpreted according to the table
13515 When dates are read in during algebraic entry, Calc first tries to
13516 match the input string to the current format either with or without
13517 the time part. The punctuation characters (including spaces) must
13518 match exactly; letter fields must correspond to suitable text in
13519 the input. If this doesn't work, Calc checks if the input is a
13520 simple number; if so, the number is interpreted as a number of days
13521 since Dec 31, 1 BC@. Otherwise, Calc tries a much more relaxed and
13522 flexible algorithm which is described in the next section.
13524 Weekday names are ignored during reading.
13526 Two-digit year numbers are interpreted as lying in the range
13527 from 1941 to 2039. Years outside that range are always
13528 entered and displayed in full. Year numbers with a leading
13529 @samp{+} sign are always interpreted exactly, allowing the
13530 entry and display of the years 1 through 99 AD.
13532 Here is a complete list of the formatting codes for dates:
13536 Year: ``91'' for 1991, ``7'' for 2007, ``+23'' for 23 AD.
13538 Year: ``91'' for 1991, ``07'' for 2007, ``+23'' for 23 AD.
13540 Year: ``91'' for 1991, `` 7'' for 2007, ``+23'' for 23 AD.
13542 Year: ``1991'' for 1991, ``23'' for 23 AD.
13544 Year: ``1991'' for 1991, ``+23'' for 23 AD.
13546 Year: ``1991'' for 1991, ``0023'' for 23 AD., ``0000'' for 1 BC.
13548 Year: ISO-8601 week-numbering year.
13550 Year: ``ad'' or blank.
13552 Year: ``AD'' or blank.
13554 Year: ``ad '' or blank. (Note trailing space.)
13556 Year: ``AD '' or blank.
13558 Year: ``a.d.@:'' or blank.
13560 Year: ``A.D.'' or blank.
13562 Year: ``bc'' or blank.
13564 Year: ``BC'' or blank.
13566 Year: `` bc'' or blank. (Note leading space.)
13568 Year: `` BC'' or blank.
13570 Year: ``b.c.@:'' or blank.
13572 Year: ``B.C.'' or blank.
13574 Month: ``8'' for August.
13576 Month: ``08'' for August.
13578 Month: `` 8'' for August.
13580 Month: ``AUG'' for August.
13582 Month: ``Aug'' for August.
13584 Month: ``aug'' for August.
13586 Month: ``AUGUST'' for August.
13588 Month: ``August'' for August.
13590 Day: ``7'' for 7th day of month.
13592 Day: ``07'' for 7th day of month.
13594 Day: `` 7'' for 7th day of month.
13596 Weekday: ``0'' for Sunday, ``6'' for Saturday.
13598 Weekday: ``1'' for Monday, ``7'' for Sunday.
13600 Weekday: ``SUN'' for Sunday.
13602 Weekday: ``Sun'' for Sunday.
13604 Weekday: ``sun'' for Sunday.
13606 Weekday: ``SUNDAY'' for Sunday.
13608 Weekday: ``Sunday'' for Sunday.
13610 Week number: ISO-8601 week number, ``W01'' for week 1.
13612 Day of year: ``34'' for Feb. 3.
13614 Day of year: ``034'' for Feb. 3.
13616 Day of year: `` 34'' for Feb. 3.
13618 Letter: Literal ``T''.
13620 Hour: ``5'' for 5 AM; ``17'' for 5 PM.
13622 Hour: ``05'' for 5 AM; ``17'' for 5 PM.
13624 Hour: `` 5'' for 5 AM; ``17'' for 5 PM.
13626 Hour: ``5'' for 5 AM and 5 PM.
13628 Hour: ``05'' for 5 AM and 5 PM.
13630 Hour: `` 5'' for 5 AM and 5 PM.
13632 AM/PM: ``a'' or ``p''.
13634 AM/PM: ``A'' or ``P''.
13636 AM/PM: ``am'' or ``pm''.
13638 AM/PM: ``AM'' or ``PM''.
13640 AM/PM: ``a.m.@:'' or ``p.m.''.
13642 AM/PM: ``A.M.'' or ``P.M.''.
13644 Minutes: ``7'' for 7.
13646 Minutes: ``07'' for 7.
13648 Minutes: `` 7'' for 7.
13650 Seconds: ``7'' for 7; ``7.23'' for 7.23.
13652 Seconds: ``07'' for 7; ``07.23'' for 7.23.
13654 Seconds: `` 7'' for 7; `` 7.23'' for 7.23.
13656 Optional seconds: ``07'' for 7; blank for 0.
13658 Optional seconds: `` 7'' for 7; blank for 0.
13660 Numeric date/time: ``726842.25'' for 6:00am Wed Jan 9, 1991.
13662 Numeric date: ``726842'' for any time on Wed Jan 9, 1991.
13664 Julian date/time: ``2448265.75'' for 6:00am Wed Jan 9, 1991.
13666 Julian date: ``2448266'' for any time on Wed Jan 9, 1991.
13668 Unix time: ``663400800'' for 6:00am Wed Jan 9, 1991.
13670 Brackets suppression. An ``X'' at the front of the format
13671 causes the surrounding @w{@samp{< >}} delimiters to be omitted
13672 when formatting dates. Note that the brackets are still
13673 required for algebraic entry.
13676 If ``SS'' or ``BS'' (optional seconds) is preceded by a colon, the
13677 colon is also omitted if the seconds part is zero.
13679 If ``bb,'' ``bbb'' or ``bbbb'' or their upper-case equivalents
13680 appear in the format, then negative year numbers are displayed
13681 without a minus sign. Note that ``aa'' and ``bb'' are mutually
13682 exclusive. Some typical usages would be @samp{YYYY AABB};
13683 @samp{AAAYYYYBBB}; @samp{YYYYBBB}.
13685 The formats ``YY,'' ``YYYY,'' ``MM,'' ``DD,'' ``ddd,'' ``hh,'' ``HH,''
13686 ``mm,'' ``ss,'' and ``SS'' actually match any number of digits during
13687 reading unless several of these codes are strung together with no
13688 punctuation in between, in which case the input must have exactly as
13689 many digits as there are letters in the format.
13691 The ``j,'' ``J,'' and ``U'' formats do not make any time zone
13692 adjustment. They effectively use @samp{julian(x,0)} and
13693 @samp{unixtime(x,0)} to make the conversion; @pxref{Date Arithmetic}.
13695 @node Free-Form Dates, Standard Date Formats, Date Formatting Codes, Date Formats
13696 @subsubsection Free-Form Dates
13699 When reading a date form during algebraic entry, Calc falls back
13700 on the algorithm described here if the input does not exactly
13701 match the current date format. This algorithm generally
13702 ``does the right thing'' and you don't have to worry about it,
13703 but it is described here in full detail for the curious.
13705 Calc does not distinguish between upper- and lower-case letters
13706 while interpreting dates.
13708 First, the time portion, if present, is located somewhere in the
13709 text and then removed. The remaining text is then interpreted as
13712 A time is of the form @samp{hh:mm:ss}, possibly with the seconds
13713 part omitted and possibly with an AM/PM indicator added to indicate
13714 12-hour time. If the AM/PM is present, the minutes may also be
13715 omitted. The AM/PM part may be any of the words @samp{am},
13716 @samp{pm}, @samp{noon}, or @samp{midnight}; each of these may be
13717 abbreviated to one letter, and the alternate forms @samp{a.m.},
13718 @samp{p.m.}, and @samp{mid} are also understood. Obviously
13719 @samp{noon} and @samp{midnight} are allowed only on 12:00:00.
13720 The words @samp{noon}, @samp{mid}, and @samp{midnight} are also
13721 recognized with no number attached.
13723 If there is no AM/PM indicator, the time is interpreted in 24-hour
13726 To read the date portion, all words and numbers are isolated
13727 from the string; other characters are ignored. All words must
13728 be either month names or day-of-week names (the latter of which
13729 are ignored). Names can be written in full or as three-letter
13732 Large numbers, or numbers with @samp{+} or @samp{-} signs,
13733 are interpreted as years. If one of the other numbers is
13734 greater than 12, then that must be the day and the remaining
13735 number in the input is therefore the month. Otherwise, Calc
13736 assumes the month, day and year are in the same order that they
13737 appear in the current date format. If the year is omitted, the
13738 current year is taken from the system clock.
13740 If there are too many or too few numbers, or any unrecognizable
13741 words, then the input is rejected.
13743 If there are any large numbers (of five digits or more) other than
13744 the year, they are ignored on the assumption that they are something
13745 like Julian dates that were included along with the traditional
13746 date components when the date was formatted.
13748 One of the words @samp{ad}, @samp{a.d.}, @samp{bc}, or @samp{b.c.}
13749 may optionally be used; the latter two are equivalent to a
13750 minus sign on the year value.
13752 If you always enter a four-digit year, and use a name instead
13753 of a number for the month, there is no danger of ambiguity.
13755 @node Standard Date Formats, , Free-Form Dates, Date Formats
13756 @subsubsection Standard Date Formats
13759 There are actually ten standard date formats, numbered 0 through 9.
13760 Entering a blank line at the @kbd{d d} command's prompt gives
13761 you format number 1, Calc's usual format. You can enter any digit
13762 to select the other formats.
13764 To create your own standard date formats, give a numeric prefix
13765 argument from 0 to 9 to the @w{@kbd{d d}} command. The format you
13766 enter will be recorded as the new standard format of that
13767 number, as well as becoming the new current date format.
13768 You can save your formats permanently with the @w{@kbd{m m}}
13769 command (@pxref{Mode Settings}).
13773 @samp{N} (Numerical format)
13775 @samp{<H:mm:SSpp >Www Mmm D, YYYY} (American format)
13777 @samp{D Mmm YYYY<, h:mm:SS>} (European format)
13779 @samp{Www Mmm BD< hh:mm:ss> YYYY} (Unix written date format)
13781 @samp{M/D/Y< H:mm:SSpp>} (American slashed format)
13783 @samp{D.M.Y< h:mm:SS>} (European dotted format)
13785 @samp{M-D-Y< H:mm:SSpp>} (American dashed format)
13787 @samp{D-M-Y< h:mm:SS>} (European dashed format)
13789 @samp{j<, h:mm:ss>} (Julian day plus time)
13791 @samp{YYddd< hh:mm:ss>} (Year-day format)
13793 @samp{ZYYY-MM-DD Www< hh:mm>} (Org mode format)
13795 @samp{IYYY-Iww-w< Thh:mm:ss>} (ISO-8601 week numbering format)
13798 @node Truncating the Stack, Justification, Date Formats, Display Modes
13799 @subsection Truncating the Stack
13803 @pindex calc-truncate-stack
13804 @cindex Truncating the stack
13805 @cindex Narrowing the stack
13806 The @kbd{d t} (@code{calc-truncate-stack}) command moves the @samp{.}@:
13807 line that marks the top-of-stack up or down in the Calculator buffer.
13808 The number right above that line is considered to the be at the top of
13809 the stack. Any numbers below that line are ``hidden'' from all stack
13810 operations (although still visible to the user). This is similar to the
13811 Emacs ``narrowing'' feature, except that the values below the @samp{.}
13812 are @emph{visible}, just temporarily frozen. This feature allows you to
13813 keep several independent calculations running at once in different parts
13814 of the stack, or to apply a certain command to an element buried deep in
13817 Pressing @kbd{d t} by itself moves the @samp{.} to the line the cursor
13818 is on. Thus, this line and all those below it become hidden. To un-hide
13819 these lines, move down to the end of the buffer and press @w{@kbd{d t}}.
13820 With a positive numeric prefix argument @expr{n}, @kbd{d t} hides the
13821 bottom @expr{n} values in the buffer. With a negative argument, it hides
13822 all but the top @expr{n} values. With an argument of zero, it hides zero
13823 values, i.e., moves the @samp{.} all the way down to the bottom.
13826 @pindex calc-truncate-up
13828 @pindex calc-truncate-down
13829 The @kbd{d [} (@code{calc-truncate-up}) and @kbd{d ]}
13830 (@code{calc-truncate-down}) commands move the @samp{.} up or down one
13831 line at a time (or several lines with a prefix argument).
13833 @node Justification, Labels, Truncating the Stack, Display Modes
13834 @subsection Justification
13838 @pindex calc-left-justify
13840 @pindex calc-center-justify
13842 @pindex calc-right-justify
13843 Values on the stack are normally left-justified in the window. You can
13844 control this arrangement by typing @kbd{d <} (@code{calc-left-justify}),
13845 @kbd{d >} (@code{calc-right-justify}), or @kbd{d =}
13846 (@code{calc-center-justify}). For example, in Right-Justification mode,
13847 stack entries are displayed flush-right against the right edge of the
13850 If you change the width of the Calculator window you may have to type
13851 @kbd{d @key{SPC}} (@code{calc-refresh}) to re-align right-justified or centered
13854 Right-justification is especially useful together with fixed-point
13855 notation (see @code{d f}; @code{calc-fix-notation}). With these modes
13856 together, the decimal points on numbers will always line up.
13858 With a numeric prefix argument, the justification commands give you
13859 a little extra control over the display. The argument specifies the
13860 horizontal ``origin'' of a display line. It is also possible to
13861 specify a maximum line width using the @kbd{d b} command (@pxref{Normal
13862 Language Modes}). For reference, the precise rules for formatting and
13863 breaking lines are given below. Notice that the interaction between
13864 origin and line width is slightly different in each justification
13867 In Left-Justified mode, the line is indented by a number of spaces
13868 given by the origin (default zero). If the result is longer than the
13869 maximum line width, if given, or too wide to fit in the Calc window
13870 otherwise, then it is broken into lines which will fit; each broken
13871 line is indented to the origin.
13873 In Right-Justified mode, lines are shifted right so that the rightmost
13874 character is just before the origin, or just before the current
13875 window width if no origin was specified. If the line is too long
13876 for this, then it is broken; the current line width is used, if
13877 specified, or else the origin is used as a width if that is
13878 specified, or else the line is broken to fit in the window.
13880 In Centering mode, the origin is the column number of the center of
13881 each stack entry. If a line width is specified, lines will not be
13882 allowed to go past that width; Calc will either indent less or
13883 break the lines if necessary. If no origin is specified, half the
13884 line width or Calc window width is used.
13886 Note that, in each case, if line numbering is enabled the display
13887 is indented an additional four spaces to make room for the line
13888 number. The width of the line number is taken into account when
13889 positioning according to the current Calc window width, but not
13890 when positioning by explicit origins and widths. In the latter
13891 case, the display is formatted as specified, and then uniformly
13892 shifted over four spaces to fit the line numbers.
13894 @node Labels, , Justification, Display Modes
13899 @pindex calc-left-label
13900 The @kbd{d @{} (@code{calc-left-label}) command prompts for a string,
13901 then displays that string to the left of every stack entry. If the
13902 entries are left-justified (@pxref{Justification}), then they will
13903 appear immediately after the label (unless you specified an origin
13904 greater than the length of the label). If the entries are centered
13905 or right-justified, the label appears on the far left and does not
13906 affect the horizontal position of the stack entry.
13908 Give a blank string (with @kbd{d @{ @key{RET}}) to turn the label off.
13911 @pindex calc-right-label
13912 The @kbd{d @}} (@code{calc-right-label}) command similarly adds a
13913 label on the righthand side. It does not affect positioning of
13914 the stack entries unless they are right-justified. Also, if both
13915 a line width and an origin are given in Right-Justified mode, the
13916 stack entry is justified to the origin and the righthand label is
13917 justified to the line width.
13919 One application of labels would be to add equation numbers to
13920 formulas you are manipulating in Calc and then copying into a
13921 document (possibly using Embedded mode). The equations would
13922 typically be centered, and the equation numbers would be on the
13923 left or right as you prefer.
13925 @node Language Modes, Modes Variable, Display Modes, Mode Settings
13926 @section Language Modes
13929 The commands in this section change Calc to use a different notation for
13930 entry and display of formulas, corresponding to the conventions of some
13931 other common language such as Pascal or @LaTeX{}. Objects displayed on the
13932 stack or yanked from the Calculator to an editing buffer will be formatted
13933 in the current language; objects entered in algebraic entry or yanked from
13934 another buffer will be interpreted according to the current language.
13936 The current language has no effect on things written to or read from the
13937 trail buffer, nor does it affect numeric entry. Only algebraic entry is
13938 affected. You can make even algebraic entry ignore the current language
13939 and use the standard notation by giving a numeric prefix, e.g., @kbd{C-u '}.
13941 For example, suppose the formula @samp{2*a[1] + atan(a[2])} occurs in a C
13942 program; elsewhere in the program you need the derivatives of this formula
13943 with respect to @samp{a[1]} and @samp{a[2]}. First, type @kbd{d C}
13944 to switch to C notation. Now use @code{C-u C-x * g} to grab the formula
13945 into the Calculator, @kbd{a d a[1] @key{RET}} to differentiate with respect
13946 to the first variable, and @kbd{C-x * y} to yank the formula for the derivative
13947 back into your C program. Press @kbd{U} to undo the differentiation and
13948 repeat with @kbd{a d a[2] @key{RET}} for the other derivative.
13950 Without being switched into C mode first, Calc would have misinterpreted
13951 the brackets in @samp{a[1]} and @samp{a[2]}, would not have known that
13952 @code{atan} was equivalent to Calc's built-in @code{arctan} function,
13953 and would have written the formula back with notations (like implicit
13954 multiplication) which would not have been valid for a C program.
13956 As another example, suppose you are maintaining a C program and a @LaTeX{}
13957 document, each of which needs a copy of the same formula. You can grab the
13958 formula from the program in C mode, switch to @LaTeX{} mode, and yank the
13959 formula into the document in @LaTeX{} math-mode format.
13961 Language modes are selected by typing the letter @kbd{d} followed by a
13962 shifted letter key.
13965 * Normal Language Modes::
13966 * C FORTRAN Pascal::
13967 * TeX and LaTeX Language Modes::
13968 * Eqn Language Mode::
13969 * Yacas Language Mode::
13970 * Maxima Language Mode::
13971 * Giac Language Mode::
13972 * Mathematica Language Mode::
13973 * Maple Language Mode::
13978 @node Normal Language Modes, C FORTRAN Pascal, Language Modes, Language Modes
13979 @subsection Normal Language Modes
13983 @pindex calc-normal-language
13984 The @kbd{d N} (@code{calc-normal-language}) command selects the usual
13985 notation for Calc formulas, as described in the rest of this manual.
13986 Matrices are displayed in a multi-line tabular format, but all other
13987 objects are written in linear form, as they would be typed from the
13991 @pindex calc-flat-language
13992 @cindex Matrix display
13993 The @kbd{d O} (@code{calc-flat-language}) command selects a language
13994 identical with the normal one, except that matrices are written in
13995 one-line form along with everything else. In some applications this
13996 form may be more suitable for yanking data into other buffers.
13999 @pindex calc-line-breaking
14000 @cindex Line breaking
14001 @cindex Breaking up long lines
14002 Even in one-line mode, long formulas or vectors will still be split
14003 across multiple lines if they exceed the width of the Calculator window.
14004 The @kbd{d b} (@code{calc-line-breaking}) command turns this line-breaking
14005 feature on and off. (It works independently of the current language.)
14006 If you give a numeric prefix argument of five or greater to the @kbd{d b}
14007 command, that argument will specify the line width used when breaking
14011 @pindex calc-big-language
14012 The @kbd{d B} (@code{calc-big-language}) command selects a language
14013 which uses textual approximations to various mathematical notations,
14014 such as powers, quotients, and square roots:
14024 in place of @samp{sqrt((a+1)/b + c^2)}.
14026 Subscripts like @samp{a_i} are displayed as actual subscripts in Big
14027 mode. Double subscripts, @samp{a_i_j} (@samp{subscr(subscr(a, i), j)})
14028 are displayed as @samp{a} with subscripts separated by commas:
14029 @samp{i, j}. They must still be entered in the usual underscore
14032 One slight ambiguity of Big notation is that
14041 can represent either the negative rational number @expr{-3:4}, or the
14042 actual expression @samp{-(3/4)}; but the latter formula would normally
14043 never be displayed because it would immediately be evaluated to
14044 @expr{-3:4} or @expr{-0.75}, so this ambiguity is not a problem in
14047 Non-decimal numbers are displayed with subscripts. Thus there is no
14048 way to tell the difference between @samp{16#C2} and @samp{C2_16},
14049 though generally you will know which interpretation is correct.
14050 Logarithms @samp{log(x,b)} and @samp{log10(x)} also use subscripts
14053 In Big mode, stack entries often take up several lines. To aid
14054 readability, stack entries are separated by a blank line in this mode.
14055 You may find it useful to expand the Calc window's height using
14056 @kbd{C-x ^} (@code{enlarge-window}) or to make the Calc window the only
14057 one on the screen with @kbd{C-x 1} (@code{delete-other-windows}).
14059 Long lines are currently not rearranged to fit the window width in
14060 Big mode, so you may need to use the @kbd{<} and @kbd{>} keys
14061 to scroll across a wide formula. For really big formulas, you may
14062 even need to use @kbd{@{} and @kbd{@}} to scroll up and down.
14065 @pindex calc-unformatted-language
14066 The @kbd{d U} (@code{calc-unformatted-language}) command altogether disables
14067 the use of operator notation in formulas. In this mode, the formula
14068 shown above would be displayed:
14071 sqrt(add(div(add(a, 1), b), pow(c, 2)))
14074 These four modes differ only in display format, not in the format
14075 expected for algebraic entry. The standard Calc operators work in
14076 all four modes, and unformatted notation works in any language mode
14077 (except that Mathematica mode expects square brackets instead of
14080 @node C FORTRAN Pascal, TeX and LaTeX Language Modes, Normal Language Modes, Language Modes
14081 @subsection C, FORTRAN, and Pascal Modes
14085 @pindex calc-c-language
14087 The @kbd{d C} (@code{calc-c-language}) command selects the conventions
14088 of the C language for display and entry of formulas. This differs from
14089 the normal language mode in a variety of (mostly minor) ways. In
14090 particular, C language operators and operator precedences are used in
14091 place of Calc's usual ones. For example, @samp{a^b} means @samp{xor(a,b)}
14092 in C mode; a value raised to a power is written as a function call,
14095 In C mode, vectors and matrices use curly braces instead of brackets.
14096 Octal and hexadecimal values are written with leading @samp{0} or @samp{0x}
14097 rather than using the @samp{#} symbol. Array subscripting is
14098 translated into @code{subscr} calls, so that @samp{a[i]} in C
14099 mode is the same as @samp{a_i} in Normal mode. Assignments
14100 turn into the @code{assign} function, which Calc normally displays
14101 using the @samp{:=} symbol.
14103 The variables @code{pi} and @code{e} would be displayed @samp{pi}
14104 and @samp{e} in Normal mode, but in C mode they are displayed as
14105 @samp{M_PI} and @samp{M_E}, corresponding to the names of constants
14106 typically provided in the @file{<math.h>} header. Functions whose
14107 names are different in C are translated automatically for entry and
14108 display purposes. For example, entering @samp{asin(x)} will push the
14109 formula @samp{arcsin(x)} onto the stack; this formula will be displayed
14110 as @samp{asin(x)} as long as C mode is in effect.
14113 @pindex calc-pascal-language
14114 @cindex Pascal language
14115 The @kbd{d P} (@code{calc-pascal-language}) command selects Pascal
14116 conventions. Like C mode, Pascal mode interprets array brackets and uses
14117 a different table of operators. Hexadecimal numbers are entered and
14118 displayed with a preceding dollar sign. (Thus the regular meaning of
14119 @kbd{$2} during algebraic entry does not work in Pascal mode, though
14120 @kbd{$} (and @kbd{$$}, etc.) not followed by digits works the same as
14121 always.) No special provisions are made for other non-decimal numbers,
14122 vectors, and so on, since there is no universally accepted standard way
14123 of handling these in Pascal.
14126 @pindex calc-fortran-language
14127 @cindex FORTRAN language
14128 The @kbd{d F} (@code{calc-fortran-language}) command selects FORTRAN
14129 conventions. Various function names are transformed into FORTRAN
14130 equivalents. Vectors are written as @samp{/1, 2, 3/}, and may be
14131 entered this way or using square brackets. Since FORTRAN uses round
14132 parentheses for both function calls and array subscripts, Calc displays
14133 both in the same way; @samp{a(i)} is interpreted as a function call
14134 upon reading, and subscripts must be entered as @samp{subscr(a, i)}.
14135 If the variable @code{a} has been declared to have type
14136 @code{vector} or @code{matrix}, however, then @samp{a(i)} will be
14137 parsed as a subscript. (@xref{Declarations}.) Usually it doesn't
14138 matter, though; if you enter the subscript expression @samp{a(i)} and
14139 Calc interprets it as a function call, you'll never know the difference
14140 unless you switch to another language mode or replace @code{a} with an
14141 actual vector (or unless @code{a} happens to be the name of a built-in
14144 Underscores are allowed in variable and function names in all of these
14145 language modes. The underscore here is equivalent to the @samp{#} in
14146 Normal mode, or to hyphens in the underlying Emacs Lisp variable names.
14148 FORTRAN and Pascal modes normally do not adjust the case of letters in
14149 formulas. Most built-in Calc names use lower-case letters. If you use a
14150 positive numeric prefix argument with @kbd{d P} or @kbd{d F}, these
14151 modes will use upper-case letters exclusively for display, and will
14152 convert to lower-case on input. With a negative prefix, these modes
14153 convert to lower-case for display and input.
14155 @node TeX and LaTeX Language Modes, Eqn Language Mode, C FORTRAN Pascal, Language Modes
14156 @subsection @TeX{} and @LaTeX{} Language Modes
14160 @pindex calc-tex-language
14161 @cindex TeX language
14163 @pindex calc-latex-language
14164 @cindex LaTeX language
14165 The @kbd{d T} (@code{calc-tex-language}) command selects the conventions
14166 of ``math mode'' in Donald Knuth's @TeX{} typesetting language,
14167 and the @kbd{d L} (@code{calc-latex-language}) command selects the
14168 conventions of ``math mode'' in @LaTeX{}, a typesetting language that
14169 uses @TeX{} as its formatting engine. Calc's @LaTeX{} language mode can
14170 read any formula that the @TeX{} language mode can, although @LaTeX{}
14171 mode may display it differently.
14173 Formulas are entered and displayed in the appropriate notation;
14174 @texline @math{\sin(a/b)}
14175 @infoline @expr{sin(a/b)}
14176 will appear as @samp{\sin\left( @{a \over b@} \right)} in @TeX{} mode and
14177 @samp{\sin\left(\frac@{a@}@{b@}\right)} in @LaTeX{} mode.
14178 Math formulas are often enclosed by @samp{$ $} signs in @TeX{} and
14179 @LaTeX{}; these should be omitted when interfacing with Calc. To Calc,
14180 the @samp{$} sign has the same meaning it always does in algebraic
14181 formulas (a reference to an existing entry on the stack).
14183 Complex numbers are displayed as in @samp{3 + 4i}. Fractions and
14184 quotients are written using @code{\over} in @TeX{} mode (as in
14185 @code{@{a \over b@}}) and @code{\frac} in @LaTeX{} mode (as in
14186 @code{\frac@{a@}@{b@}}); binomial coefficients are written with
14187 @code{\choose} in @TeX{} mode (as in @code{@{a \choose b@}}) and
14188 @code{\binom} in @LaTeX{} mode (as in @code{\binom@{a@}@{b@}}).
14189 Interval forms are written with @code{\ldots}, and error forms are
14190 written with @code{\pm}. Absolute values are written as in
14191 @samp{|x + 1|}, and the floor and ceiling functions are written with
14192 @code{\lfloor}, @code{\rfloor}, etc. The words @code{\left} and
14193 @code{\right} are ignored when reading formulas in @TeX{} and @LaTeX{}
14194 modes. Both @code{inf} and @code{uinf} are written as @code{\infty};
14195 when read, @code{\infty} always translates to @code{inf}.
14197 Function calls are written the usual way, with the function name followed
14198 by the arguments in parentheses. However, functions for which @TeX{}
14199 and @LaTeX{} have special names (like @code{\sin}) will use curly braces
14200 instead of parentheses for very simple arguments. During input, curly
14201 braces and parentheses work equally well for grouping, but when the
14202 document is formatted the curly braces will be invisible. Thus the
14204 @texline @math{\sin{2 x}}
14205 @infoline @expr{sin 2x}
14207 @texline @math{\sin(2 + x)}.
14208 @infoline @expr{sin(2 + x)}.
14210 The @TeX{} specific unit names (@pxref{Predefined Units}) will not use
14211 the @samp{tex} prefix; the unit name for a @TeX{} point will be
14212 @samp{pt} instead of @samp{texpt}, for example.
14214 Function and variable names not treated specially by @TeX{} and @LaTeX{}
14215 are simply written out as-is, which will cause them to come out in
14216 italic letters in the printed document. If you invoke @kbd{d T} or
14217 @kbd{d L} with a positive numeric prefix argument, names of more than
14218 one character will instead be enclosed in a protective commands that
14219 will prevent them from being typeset in the math italics; they will be
14220 written @samp{\hbox@{@var{name}@}} in @TeX{} mode and
14221 @samp{\text@{@var{name}@}} in @LaTeX{} mode. The
14222 @samp{\hbox@{ @}} and @samp{\text@{ @}} notations are ignored during
14223 reading. If you use a negative prefix argument, such function names are
14224 written @samp{\@var{name}}, and function names that begin with @code{\} during
14225 reading have the @code{\} removed. (Note that in this mode, long
14226 variable names are still written with @code{\hbox} or @code{\text}.
14227 However, you can always make an actual variable name like @code{\bar} in
14230 During reading, text of the form @samp{\matrix@{ ...@: @}} is replaced
14231 by @samp{[ ...@: ]}. The same also applies to @code{\pmatrix} and
14232 @code{\bmatrix}. In @LaTeX{} mode this also applies to
14233 @samp{\begin@{matrix@} ... \end@{matrix@}},
14234 @samp{\begin@{bmatrix@} ... \end@{bmatrix@}},
14235 @samp{\begin@{pmatrix@} ... \end@{pmatrix@}}, as well as
14236 @samp{\begin@{smallmatrix@} ... \end@{smallmatrix@}}.
14237 The symbol @samp{&} is interpreted as a comma,
14238 and the symbols @samp{\cr} and @samp{\\} are interpreted as semicolons.
14239 During output, matrices are displayed in @samp{\matrix@{ a & b \\ c & d@}}
14240 format in @TeX{} mode and in
14241 @samp{\begin@{pmatrix@} a & b \\ c & d \end@{pmatrix@}} format in
14242 @LaTeX{} mode; you may need to edit this afterwards to change to your
14243 preferred matrix form. If you invoke @kbd{d T} or @kbd{d L} with an
14244 argument of 2 or -2, then matrices will be displayed in two-dimensional
14255 This may be convenient for isolated matrices, but could lead to
14256 expressions being displayed like
14259 \begin@{pmatrix@} \times x
14266 While this wouldn't bother Calc, it is incorrect @LaTeX{}.
14267 (Similarly for @TeX{}.)
14269 Accents like @code{\tilde} and @code{\bar} translate into function
14270 calls internally (@samp{tilde(x)}, @samp{bar(x)}). The @code{\underline}
14271 sequence is treated as an accent. The @code{\vec} accent corresponds
14272 to the function name @code{Vec}, because @code{vec} is the name of
14273 a built-in Calc function. The following table shows the accents
14274 in Calc, @TeX{}, @LaTeX{} and @dfn{eqn} (described in the next section):
14279 @let@calcindexershow=@calcindexernoshow @c Suppress marginal notes
14280 @let@calcindexersh=@calcindexernoshow
14389 acute \acute \acute
14393 breve \breve \breve
14395 check \check \check
14401 dotdot \ddot \ddot dotdot
14404 grave \grave \grave
14409 tilde \tilde \tilde tilde
14411 under \underline \underline under
14416 The @samp{=>} (evaluates-to) operator appears as a @code{\to} symbol:
14417 @samp{@{@var{a} \to @var{b}@}}. @TeX{} defines @code{\to} as an
14418 alias for @code{\rightarrow}. However, if the @samp{=>} is the
14419 top-level expression being formatted, a slightly different notation
14420 is used: @samp{\evalto @var{a} \to @var{b}}. The @code{\evalto}
14421 word is ignored by Calc's input routines, and is undefined in @TeX{}.
14422 You will typically want to include one of the following definitions
14423 at the top of a @TeX{} file that uses @code{\evalto}:
14427 \def\evalto#1\to@{@}
14430 The first definition formats evaluates-to operators in the usual
14431 way. The second causes only the @var{b} part to appear in the
14432 printed document; the @var{a} part and the arrow are hidden.
14433 Another definition you may wish to use is @samp{\let\to=\Rightarrow}
14434 which causes @code{\to} to appear more like Calc's @samp{=>} symbol.
14435 @xref{Evaluates-To Operator}, for a discussion of @code{evalto}.
14437 The complete set of @TeX{} control sequences that are ignored during
14441 \hbox \mbox \text \left \right
14442 \, \> \: \; \! \quad \qquad \hfil \hfill
14443 \displaystyle \textstyle \dsize \tsize
14444 \scriptstyle \scriptscriptstyle \ssize \ssize
14445 \rm \bf \it \sl \roman \bold \italic \slanted
14446 \cal \mit \Cal \Bbb \frak \goth
14450 Note that, because these symbols are ignored, reading a @TeX{} or
14451 @LaTeX{} formula into Calc and writing it back out may lose spacing and
14454 Also, the ``discretionary multiplication sign'' @samp{\*} is read
14455 the same as @samp{*}.
14458 The @TeX{} version of this manual includes some printed examples at the
14459 end of this section.
14462 Here are some examples of how various Calc formulas are formatted in @TeX{}:
14467 \sin\left( {a^2 \over b_i} \right)
14471 $$ \sin\left( a^2 \over b_i \right) $$
14477 [(3, 4), 3:4, 3 +/- 4, [3 .. inf)]
14478 [3 + 4i, @{3 \over 4@}, 3 \pm 4, [3 \ldots \infty)]
14482 $$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
14488 [abs(a), abs(a / b), floor(a), ceil(a / b)]
14489 [|a|, \left| a \over b \right|,
14490 \lfloor a \rfloor, \left\lceil a \over b \right\rceil]
14494 $$ [|a|, \left| a \over b \right|,
14495 \lfloor a \rfloor, \left\lceil a \over b \right\rceil] $$
14501 [sin(a), sin(2 a), sin(2 + a), sin(a / b)]
14502 [\sin@{a@}, \sin@{2 a@}, \sin(2 + a),
14503 \sin\left( @{a \over b@} \right)]
14507 $$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
14511 First with plain @kbd{d T}, then with @kbd{C-u d T}, then finally with
14512 @kbd{C-u - d T} (using the example definition
14513 @samp{\def\foo#1@{\tilde F(#1)@}}:
14517 [f(a), foo(bar), sin(pi)]
14518 [f(a), foo(bar), \sin{\pi}]
14519 [f(a), \hbox@{foo@}(\hbox@{bar@}), \sin@{\pi@}]
14520 [f(a), \foo@{\hbox@{bar@}@}, \sin@{\pi@}]
14524 $$ [f(a), foo(bar), \sin{\pi}] $$
14525 $$ [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}] $$
14526 $$ [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$
14530 First with @samp{\def\evalto@{@}}, then with @samp{\def\evalto#1\to@{@}}:
14535 \evalto 2 + 3 \to 5
14544 First with standard @code{\to}, then with @samp{\let\to\Rightarrow}:
14548 [2 + 3 => 5, a / 2 => (b + c) / 2]
14549 [@{2 + 3 \to 5@}, @{@{a \over 2@} \to @{b + c \over 2@}@}]
14553 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
14554 {\let\to\Rightarrow
14555 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
14559 Matrices normally, then changing @code{\matrix} to @code{\pmatrix}:
14563 [ [ a / b, 0 ], [ 0, 2^(x + 1) ] ]
14564 \matrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14565 \pmatrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14569 $$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14570 $$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14575 @node Eqn Language Mode, Yacas Language Mode, TeX and LaTeX Language Modes, Language Modes
14576 @subsection Eqn Language Mode
14580 @pindex calc-eqn-language
14581 @dfn{Eqn} is another popular formatter for math formulas. It is
14582 designed for use with the TROFF text formatter, and comes standard
14583 with many versions of Unix. The @kbd{d E} (@code{calc-eqn-language})
14584 command selects @dfn{eqn} notation.
14586 The @dfn{eqn} language's main idiosyncrasy is that whitespace plays
14587 a significant part in the parsing of the language. For example,
14588 @samp{sqrt x+1 + y} treats @samp{x+1} as the argument of the
14589 @code{sqrt} operator. @dfn{Eqn} also understands more conventional
14590 grouping using curly braces: @samp{sqrt@{x+1@} + y}. Braces are
14591 required only when the argument contains spaces.
14593 In Calc's @dfn{eqn} mode, however, curly braces are required to
14594 delimit arguments of operators like @code{sqrt}. The first of the
14595 above examples would treat only the @samp{x} as the argument of
14596 @code{sqrt}, and in fact @samp{sin x+1} would be interpreted as
14597 @samp{sin * x + 1}, because @code{sin} is not a special operator
14598 in the @dfn{eqn} language. If you always surround the argument
14599 with curly braces, Calc will never misunderstand.
14601 Calc also understands parentheses as grouping characters. Another
14602 peculiarity of @dfn{eqn}'s syntax makes it advisable to separate
14603 words with spaces from any surrounding characters that aren't curly
14604 braces, so Calc writes @samp{sin ( x + y )} in @dfn{eqn} mode.
14605 (The spaces around @code{sin} are important to make @dfn{eqn}
14606 recognize that @code{sin} should be typeset in a roman font, and
14607 the spaces around @code{x} and @code{y} are a good idea just in
14608 case the @dfn{eqn} document has defined special meanings for these
14611 Powers and subscripts are written with the @code{sub} and @code{sup}
14612 operators, respectively. Note that the caret symbol @samp{^} is
14613 treated the same as a space in @dfn{eqn} mode, as is the @samp{~}
14614 symbol (these are used to introduce spaces of various widths into
14615 the typeset output of @dfn{eqn}).
14617 As in @LaTeX{} mode, Calc's formatter omits parentheses around the
14618 arguments of functions like @code{ln} and @code{sin} if they are
14619 ``simple-looking''; in this case Calc surrounds the argument with
14620 braces, separated by a @samp{~} from the function name: @samp{sin~@{x@}}.
14622 Font change codes (like @samp{roman @var{x}}) and positioning codes
14623 (like @samp{~} and @samp{down @var{n} @var{x}}) are ignored by the
14624 @dfn{eqn} reader. Also ignored are the words @code{left}, @code{right},
14625 @code{mark}, and @code{lineup}. Quotation marks in @dfn{eqn} mode input
14626 are treated the same as curly braces: @samp{sqrt "1+x"} is equivalent to
14627 @samp{sqrt @{1+x@}}; this is only an approximation to the true meaning
14628 of quotes in @dfn{eqn}, but it is good enough for most uses.
14630 Accent codes (@samp{@var{x} dot}) are handled by treating them as
14631 function calls (@samp{dot(@var{x})}) internally.
14632 @xref{TeX and LaTeX Language Modes}, for a table of these accent
14633 functions. The @code{prime} accent is treated specially if it occurs on
14634 a variable or function name: @samp{f prime prime @w{( x prime )}} is
14635 stored internally as @samp{f'@w{'}(x')}. For example, taking the
14636 derivative of @samp{f(2 x)} with @kbd{a d x} will produce @samp{2 f'(2
14637 x)}, which @dfn{eqn} mode will display as @samp{2 f prime ( 2 x )}.
14639 Assignments are written with the @samp{<-} (left-arrow) symbol,
14640 and @code{evalto} operators are written with @samp{->} or
14641 @samp{evalto ... ->} (@pxref{TeX and LaTeX Language Modes}, for a discussion
14642 of this). The regular Calc symbols @samp{:=} and @samp{=>} are also
14643 recognized for these operators during reading.
14645 Vectors in @dfn{eqn} mode use regular Calc square brackets, but
14646 matrices are formatted as @samp{matrix @{ ccol @{ a above b @} ... @}}.
14647 The words @code{lcol} and @code{rcol} are recognized as synonyms
14648 for @code{ccol} during input, and are generated instead of @code{ccol}
14649 if the matrix justification mode so specifies.
14651 @node Yacas Language Mode, Maxima Language Mode, Eqn Language Mode, Language Modes
14652 @subsection Yacas Language Mode
14656 @pindex calc-yacas-language
14657 @cindex Yacas language
14658 The @kbd{d Y} (@code{calc-yacas-language}) command selects the
14659 conventions of Yacas, a free computer algebra system. While the
14660 operators and functions in Yacas are similar to those of Calc, the names
14661 of built-in functions in Yacas are capitalized. The Calc formula
14662 @samp{sin(2 x)}, for example, is entered and displayed @samp{Sin(2 x)}
14663 in Yacas mode, and `@samp{arcsin(x^2)} is @samp{ArcSin(x^2)} in Yacas
14664 mode. Complex numbers are written are written @samp{3 + 4 I}.
14665 The standard special constants are written @code{Pi}, @code{E},
14666 @code{I}, @code{GoldenRatio} and @code{Gamma}. @code{Infinity}
14667 represents both @code{inf} and @code{uinf}, and @code{Undefined}
14668 represents @code{nan}.
14670 Certain operators on functions, such as @code{D} for differentiation
14671 and @code{Integrate} for integration, take a prefix form in Yacas. For
14672 example, the derivative of @w{@samp{e^x sin(x)}} can be computed with
14673 @w{@samp{D(x) Exp(x)*Sin(x)}}.
14675 Other notable differences between Yacas and standard Calc expressions
14676 are that vectors and matrices use curly braces in Yacas, and subscripts
14677 use square brackets. If, for example, @samp{A} represents the list
14678 @samp{@{a,2,c,4@}}, then @samp{A[3]} would equal @samp{c}.
14681 @node Maxima Language Mode, Giac Language Mode, Yacas Language Mode, Language Modes
14682 @subsection Maxima Language Mode
14686 @pindex calc-maxima-language
14687 @cindex Maxima language
14688 The @kbd{d X} (@code{calc-maxima-language}) command selects the
14689 conventions of Maxima, another free computer algebra system. The
14690 function names in Maxima are similar, but not always identical, to Calc.
14691 For example, instead of @samp{arcsin(x)}, Maxima will use
14692 @samp{asin(x)}. Complex numbers are written @samp{3 + 4 %i}. The
14693 standard special constants are written @code{%pi}, @code{%e},
14694 @code{%i}, @code{%phi} and @code{%gamma}. In Maxima, @code{inf} means
14695 the same as in Calc, but @code{infinity} represents Calc's @code{uinf}.
14697 Underscores as well as percent signs are allowed in function and
14698 variable names in Maxima mode. The underscore again is equivalent to
14699 the @samp{#} in Normal mode, and the percent sign is equivalent to
14702 Maxima uses square brackets for lists and vectors, and matrices are
14703 written as calls to the function @code{matrix}, given the row vectors of
14704 the matrix as arguments. Square brackets are also used as subscripts.
14706 @node Giac Language Mode, Mathematica Language Mode, Maxima Language Mode, Language Modes
14707 @subsection Giac Language Mode
14711 @pindex calc-giac-language
14712 @cindex Giac language
14713 The @kbd{d A} (@code{calc-giac-language}) command selects the
14714 conventions of Giac, another free computer algebra system. The function
14715 names in Giac are similar to Maxima. Complex numbers are written
14716 @samp{3 + 4 i}. The standard special constants in Giac are the same as
14717 in Calc, except that @code{infinity} represents both Calc's @code{inf}
14720 Underscores are allowed in function and variable names in Giac mode.
14721 Brackets are used for subscripts. In Giac, indexing of lists begins at
14722 0, instead of 1 as in Calc. So if @samp{A} represents the list
14723 @samp{[a,2,c,4]}, then @samp{A[2]} would equal @samp{c}. In general,
14724 @samp{A[n]} in Giac mode corresponds to @samp{A_(n+1)} in Normal mode.
14726 The Giac interval notation @samp{2 .. 3} has no surrounding brackets;
14727 Calc reads @samp{2 .. 3} as the closed interval @samp{[2 .. 3]} and
14728 writes any kind of interval as @samp{2 .. 3}. This means you cannot see
14729 the difference between an open and a closed interval while in Giac mode.
14731 @node Mathematica Language Mode, Maple Language Mode, Giac Language Mode, Language Modes
14732 @subsection Mathematica Language Mode
14736 @pindex calc-mathematica-language
14737 @cindex Mathematica language
14738 The @kbd{d M} (@code{calc-mathematica-language}) command selects the
14739 conventions of Mathematica. Notable differences in Mathematica mode
14740 are that the names of built-in functions are capitalized, and function
14741 calls use square brackets instead of parentheses. Thus the Calc
14742 formula @samp{sin(2 x)} is entered and displayed @w{@samp{Sin[2 x]}} in
14745 Vectors and matrices use curly braces in Mathematica. Complex numbers
14746 are written @samp{3 + 4 I}. The standard special constants in Calc are
14747 written @code{Pi}, @code{E}, @code{I}, @code{GoldenRatio}, @code{EulerGamma},
14748 @code{Infinity}, @code{ComplexInfinity}, and @code{Indeterminate} in
14750 Non-decimal numbers are written, e.g., @samp{16^^7fff}. Floating-point
14751 numbers in scientific notation are written @samp{1.23*10.^3}.
14752 Subscripts use double square brackets: @samp{a[[i]]}.
14754 @node Maple Language Mode, Compositions, Mathematica Language Mode, Language Modes
14755 @subsection Maple Language Mode
14759 @pindex calc-maple-language
14760 @cindex Maple language
14761 The @kbd{d W} (@code{calc-maple-language}) command selects the
14762 conventions of Maple.
14764 Maple's language is much like C@. Underscores are allowed in symbol
14765 names; square brackets are used for subscripts; explicit @samp{*}s for
14766 multiplications are required. Use either @samp{^} or @samp{**} to
14769 Maple uses square brackets for lists and curly braces for sets. Calc
14770 interprets both notations as vectors, and displays vectors with square
14771 brackets. This means Maple sets will be converted to lists when they
14772 pass through Calc. As a special case, matrices are written as calls
14773 to the function @code{matrix}, given a list of lists as the argument,
14774 and can be read in this form or with all-capitals @code{MATRIX}.
14776 The Maple interval notation @samp{2 .. 3} is like Giac's interval
14777 notation, and is handled the same by Calc.
14779 Maple writes complex numbers as @samp{3 + 4*I}. Its special constants
14780 are @code{Pi}, @code{E}, @code{I}, and @code{infinity} (all three of
14781 @code{inf}, @code{uinf}, and @code{nan} display as @code{infinity}).
14782 Floating-point numbers are written @samp{1.23*10.^3}.
14784 Among things not currently handled by Calc's Maple mode are the
14785 various quote symbols, procedures and functional operators, and
14786 inert (@samp{&}) operators.
14788 @node Compositions, Syntax Tables, Maple Language Mode, Language Modes
14789 @subsection Compositions
14792 @cindex Compositions
14793 There are several @dfn{composition functions} which allow you to get
14794 displays in a variety of formats similar to those in Big language
14795 mode. Most of these functions do not evaluate to anything; they are
14796 placeholders which are left in symbolic form by Calc's evaluator but
14797 are recognized by Calc's display formatting routines.
14799 Two of these, @code{string} and @code{bstring}, are described elsewhere.
14800 @xref{Strings}. For example, @samp{string("ABC")} is displayed as
14801 @samp{ABC}. When viewed on the stack it will be indistinguishable from
14802 the variable @code{ABC}, but internally it will be stored as
14803 @samp{string([65, 66, 67])} and can still be manipulated this way; for
14804 example, the selection and vector commands @kbd{j 1 v v j u} would
14805 select the vector portion of this object and reverse the elements, then
14806 deselect to reveal a string whose characters had been reversed.
14808 The composition functions do the same thing in all language modes
14809 (although their components will of course be formatted in the current
14810 language mode). The one exception is Unformatted mode (@kbd{d U}),
14811 which does not give the composition functions any special treatment.
14812 The functions are discussed here because of their relationship to
14813 the language modes.
14816 * Composition Basics::
14817 * Horizontal Compositions::
14818 * Vertical Compositions::
14819 * Other Compositions::
14820 * Information about Compositions::
14821 * User-Defined Compositions::
14824 @node Composition Basics, Horizontal Compositions, Compositions, Compositions
14825 @subsubsection Composition Basics
14828 Compositions are generally formed by stacking formulas together
14829 horizontally or vertically in various ways. Those formulas are
14830 themselves compositions. @TeX{} users will find this analogous
14831 to @TeX{}'s ``boxes.'' Each multi-line composition has a
14832 @dfn{baseline}; horizontal compositions use the baselines to
14833 decide how formulas should be positioned relative to one another.
14834 For example, in the Big mode formula
14846 the second term of the sum is four lines tall and has line three as
14847 its baseline. Thus when the term is combined with 17, line three
14848 is placed on the same level as the baseline of 17.
14854 Another important composition concept is @dfn{precedence}. This is
14855 an integer that represents the binding strength of various operators.
14856 For example, @samp{*} has higher precedence (195) than @samp{+} (180),
14857 which means that @samp{(a * b) + c} will be formatted without the
14858 parentheses, but @samp{a * (b + c)} will keep the parentheses.
14860 The operator table used by normal and Big language modes has the
14861 following precedences:
14864 _ 1200 @r{(subscripts)}
14865 % 1100 @r{(as in n}%@r{)}
14866 ! 1000 @r{(as in }!@r{n)}
14869 !! 210 @r{(as in n}!!@r{)}
14870 ! 210 @r{(as in n}!@r{)}
14872 - 197 @r{(as in }-@r{n)}
14873 * 195 @r{(or implicit multiplication)}
14875 + - 180 @r{(as in a}+@r{b)}
14877 < = 160 @r{(and other relations)}
14889 The general rule is that if an operator with precedence @expr{n}
14890 occurs as an argument to an operator with precedence @expr{m}, then
14891 the argument is enclosed in parentheses if @expr{n < m}. Top-level
14892 expressions and expressions which are function arguments, vector
14893 components, etc., are formatted with precedence zero (so that they
14894 normally never get additional parentheses).
14896 For binary left-associative operators like @samp{+}, the righthand
14897 argument is actually formatted with one-higher precedence than shown
14898 in the table. This makes sure @samp{(a + b) + c} omits the parentheses,
14899 but the unnatural form @samp{a + (b + c)} keeps its parentheses.
14900 Right-associative operators like @samp{^} format the lefthand argument
14901 with one-higher precedence.
14907 The @code{cprec} function formats an expression with an arbitrary
14908 precedence. For example, @samp{cprec(abc, 185)} will combine into
14909 sums and products as follows: @samp{7 + abc}, @samp{7 (abc)} (because
14910 this @code{cprec} form has higher precedence than addition, but lower
14911 precedence than multiplication).
14917 A final composition issue is @dfn{line breaking}. Calc uses two
14918 different strategies for ``flat'' and ``non-flat'' compositions.
14919 A non-flat composition is anything that appears on multiple lines
14920 (not counting line breaking). Examples would be matrices and Big
14921 mode powers and quotients. Non-flat compositions are displayed
14922 exactly as specified. If they come out wider than the current
14923 window, you must use horizontal scrolling (@kbd{<} and @kbd{>}) to
14926 Flat compositions, on the other hand, will be broken across several
14927 lines if they are too wide to fit the window. Certain points in a
14928 composition are noted internally as @dfn{break points}. Calc's
14929 general strategy is to fill each line as much as possible, then to
14930 move down to the next line starting at the first break point that
14931 didn't fit. However, the line breaker understands the hierarchical
14932 structure of formulas. It will not break an ``inner'' formula if
14933 it can use an earlier break point from an ``outer'' formula instead.
14934 For example, a vector of sums might be formatted as:
14938 [ a + b + c, d + e + f,
14939 g + h + i, j + k + l, m ]
14944 If the @samp{m} can fit, then so, it seems, could the @samp{g}.
14945 But Calc prefers to break at the comma since the comma is part
14946 of a ``more outer'' formula. Calc would break at a plus sign
14947 only if it had to, say, if the very first sum in the vector had
14948 itself been too large to fit.
14950 Of the composition functions described below, only @code{choriz}
14951 generates break points. The @code{bstring} function (@pxref{Strings})
14952 also generates breakable items: A break point is added after every
14953 space (or group of spaces) except for spaces at the very beginning or
14956 Composition functions themselves count as levels in the formula
14957 hierarchy, so a @code{choriz} that is a component of a larger
14958 @code{choriz} will be less likely to be broken. As a special case,
14959 if a @code{bstring} occurs as a component of a @code{choriz} or
14960 @code{choriz}-like object (such as a vector or a list of arguments
14961 in a function call), then the break points in that @code{bstring}
14962 will be on the same level as the break points of the surrounding
14965 @node Horizontal Compositions, Vertical Compositions, Composition Basics, Compositions
14966 @subsubsection Horizontal Compositions
14973 The @code{choriz} function takes a vector of objects and composes
14974 them horizontally. For example, @samp{choriz([17, a b/c, d])} formats
14975 as @w{@samp{17a b / cd}} in Normal language mode, or as
14986 in Big language mode. This is actually one case of the general
14987 function @samp{choriz(@var{vec}, @var{sep}, @var{prec})}, where
14988 either or both of @var{sep} and @var{prec} may be omitted.
14989 @var{Prec} gives the @dfn{precedence} to use when formatting
14990 each of the components of @var{vec}. The default precedence is
14991 the precedence from the surrounding environment.
14993 @var{Sep} is a string (i.e., a vector of character codes as might
14994 be entered with @code{" "} notation) which should separate components
14995 of the composition. Also, if @var{sep} is given, the line breaker
14996 will allow lines to be broken after each occurrence of @var{sep}.
14997 If @var{sep} is omitted, the composition will not be breakable
14998 (unless any of its component compositions are breakable).
15000 For example, @samp{2 choriz([a, b c, d = e], " + ", 180)} is
15001 formatted as @samp{2 a + b c + (d = e)}. To get the @code{choriz}
15002 to have precedence 180 ``outwards'' as well as ``inwards,''
15003 enclose it in a @code{cprec} form: @samp{2 cprec(choriz(...), 180)}
15004 formats as @samp{2 (a + b c + (d = e))}.
15006 The baseline of a horizontal composition is the same as the
15007 baselines of the component compositions, which are all aligned.
15009 @node Vertical Compositions, Other Compositions, Horizontal Compositions, Compositions
15010 @subsubsection Vertical Compositions
15017 The @code{cvert} function makes a vertical composition. Each
15018 component of the vector is centered in a column. The baseline of
15019 the result is by default the top line of the resulting composition.
15020 For example, @samp{f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))}
15021 formats in Big mode as
15036 There are several special composition functions that work only as
15037 components of a vertical composition. The @code{cbase} function
15038 controls the baseline of the vertical composition; the baseline
15039 will be the same as the baseline of whatever component is enclosed
15040 in @code{cbase}. Thus @samp{f(cvert([a, cbase(bb), ccc]),
15041 cvert([a^2 + 1, cbase(b^2)]))} displays as
15061 There are also @code{ctbase} and @code{cbbase} functions which
15062 make the baseline of the vertical composition equal to the top
15063 or bottom line (rather than the baseline) of that component.
15064 Thus @samp{cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) +
15065 cvert([cbbase(a / b)])} gives
15077 There should be only one @code{cbase}, @code{ctbase}, or @code{cbbase}
15078 function in a given vertical composition. These functions can also
15079 be written with no arguments: @samp{ctbase()} is a zero-height object
15080 which means the baseline is the top line of the following item, and
15081 @samp{cbbase()} means the baseline is the bottom line of the preceding
15088 The @code{crule} function builds a ``rule,'' or horizontal line,
15089 across a vertical composition. By itself @samp{crule()} uses @samp{-}
15090 characters to build the rule. You can specify any other character,
15091 e.g., @samp{crule("=")}. The argument must be a character code or
15092 vector of exactly one character code. It is repeated to match the
15093 width of the widest item in the stack. For example, a quotient
15094 with a thick line is @samp{cvert([a + 1, cbase(crule("=")), b^2])}:
15113 Finally, the functions @code{clvert} and @code{crvert} act exactly
15114 like @code{cvert} except that the items are left- or right-justified
15115 in the stack. Thus @samp{clvert([a, bb, ccc]) + crvert([a, bb, ccc])}
15126 Like @code{choriz}, the vertical compositions accept a second argument
15127 which gives the precedence to use when formatting the components.
15128 Vertical compositions do not support separator strings.
15130 @node Other Compositions, Information about Compositions, Vertical Compositions, Compositions
15131 @subsubsection Other Compositions
15138 The @code{csup} function builds a superscripted expression. For
15139 example, @samp{csup(a, b)} looks the same as @samp{a^b} does in Big
15140 language mode. This is essentially a horizontal composition of
15141 @samp{a} and @samp{b}, where @samp{b} is shifted up so that its
15142 bottom line is one above the baseline.
15148 Likewise, the @code{csub} function builds a subscripted expression.
15149 This shifts @samp{b} down so that its top line is one below the
15150 bottom line of @samp{a} (note that this is not quite analogous to
15151 @code{csup}). Other arrangements can be obtained by using
15152 @code{choriz} and @code{cvert} directly.
15158 The @code{cflat} function formats its argument in ``flat'' mode,
15159 as obtained by @samp{d O}, if the current language mode is normal
15160 or Big. It has no effect in other language modes. For example,
15161 @samp{a^(b/c)} is formatted by Big mode like @samp{csup(a, cflat(b/c))}
15162 to improve its readability.
15168 The @code{cspace} function creates horizontal space. For example,
15169 @samp{cspace(4)} is effectively the same as @samp{string(" ")}.
15170 A second string (i.e., vector of characters) argument is repeated
15171 instead of the space character. For example, @samp{cspace(4, "ab")}
15172 looks like @samp{abababab}. If the second argument is not a string,
15173 it is formatted in the normal way and then several copies of that
15174 are composed together: @samp{cspace(4, a^2)} yields
15184 If the number argument is zero, this is a zero-width object.
15190 The @code{cvspace} function creates vertical space, or a vertical
15191 stack of copies of a certain string or formatted object. The
15192 baseline is the center line of the resulting stack. A numerical
15193 argument of zero will produce an object which contributes zero
15194 height if used in a vertical composition.
15204 There are also @code{ctspace} and @code{cbspace} functions which
15205 create vertical space with the baseline the same as the baseline
15206 of the top or bottom copy, respectively, of the second argument.
15207 Thus @samp{cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)}
15224 @node Information about Compositions, User-Defined Compositions, Other Compositions, Compositions
15225 @subsubsection Information about Compositions
15228 The functions in this section are actual functions; they compose their
15229 arguments according to the current language and other display modes,
15230 then return a certain measurement of the composition as an integer.
15236 The @code{cwidth} function measures the width, in characters, of a
15237 composition. For example, @samp{cwidth(a + b)} is 5, and
15238 @samp{cwidth(a / b)} is 5 in Normal mode, 1 in Big mode, and 11 in
15239 @TeX{} mode (for @samp{@{a \over b@}}). The argument may involve
15240 the composition functions described in this section.
15246 The @code{cheight} function measures the height of a composition.
15247 This is the total number of lines in the argument's printed form.
15257 The functions @code{cascent} and @code{cdescent} measure the amount
15258 of the height that is above (and including) the baseline, or below
15259 the baseline, respectively. Thus @samp{cascent(@var{x}) + cdescent(@var{x})}
15260 always equals @samp{cheight(@var{x})}. For a one-line formula like
15261 @samp{a + b}, @code{cascent} returns 1 and @code{cdescent} returns 0.
15262 For @samp{a / b} in Big mode, @code{cascent} returns 2 and @code{cdescent}
15263 returns 1. The only formula for which @code{cascent} will return zero
15264 is @samp{cvspace(0)} or equivalents.
15266 @node User-Defined Compositions, , Information about Compositions, Compositions
15267 @subsubsection User-Defined Compositions
15271 @pindex calc-user-define-composition
15272 The @kbd{Z C} (@code{calc-user-define-composition}) command lets you
15273 define the display format for any algebraic function. You provide a
15274 formula containing a certain number of argument variables on the stack.
15275 Any time Calc formats a call to the specified function in the current
15276 language mode and with that number of arguments, Calc effectively
15277 replaces the function call with that formula with the arguments
15280 Calc builds the default argument list by sorting all the variable names
15281 that appear in the formula into alphabetical order. You can edit this
15282 argument list before pressing @key{RET} if you wish. Any variables in
15283 the formula that do not appear in the argument list will be displayed
15284 literally; any arguments that do not appear in the formula will not
15285 affect the display at all.
15287 You can define formats for built-in functions, for functions you have
15288 defined with @kbd{Z F} (@pxref{Algebraic Definitions}), or for functions
15289 which have no definitions but are being used as purely syntactic objects.
15290 You can define different formats for each language mode, and for each
15291 number of arguments, using a succession of @kbd{Z C} commands. When
15292 Calc formats a function call, it first searches for a format defined
15293 for the current language mode (and number of arguments); if there is
15294 none, it uses the format defined for the Normal language mode. If
15295 neither format exists, Calc uses its built-in standard format for that
15296 function (usually just @samp{@var{func}(@var{args})}).
15298 If you execute @kbd{Z C} with the number 0 on the stack instead of a
15299 formula, any defined formats for the function in the current language
15300 mode will be removed. The function will revert to its standard format.
15302 For example, the default format for the binomial coefficient function
15303 @samp{choose(n, m)} in the Big language mode is
15314 You might prefer the notation,
15324 To define this notation, first make sure you are in Big mode,
15325 then put the formula
15328 choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])
15332 on the stack and type @kbd{Z C}. Answer the first prompt with
15333 @code{choose}. The second prompt will be the default argument list
15334 of @samp{(C m n)}. Edit this list to be @samp{(n m)} and press
15335 @key{RET}. Now, try it out: For example, turn simplification
15336 off with @kbd{m O} and enter @samp{choose(a,b) + choose(7,3)}
15337 as an algebraic entry.
15346 As another example, let's define the usual notation for Stirling
15347 numbers of the first kind, @samp{stir1(n, m)}. This is just like
15348 the regular format for binomial coefficients but with square brackets
15349 instead of parentheses.
15352 choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])
15355 Now type @kbd{Z C stir1 @key{RET}}, edit the argument list to
15356 @samp{(n m)}, and type @key{RET}.
15358 The formula provided to @kbd{Z C} usually will involve composition
15359 functions, but it doesn't have to. Putting the formula @samp{a + b + c}
15360 onto the stack and typing @kbd{Z C foo @key{RET} @key{RET}} would define
15361 the function @samp{foo(x,y,z)} to display like @samp{x + y + z}.
15362 This ``sum'' will act exactly like a real sum for all formatting
15363 purposes (it will be parenthesized the same, and so on). However
15364 it will be computationally unrelated to a sum. For example, the
15365 formula @samp{2 * foo(1, 2, 3)} will display as @samp{2 (1 + 2 + 3)}.
15366 Operator precedences have caused the ``sum'' to be written in
15367 parentheses, but the arguments have not actually been summed.
15368 (Generally a display format like this would be undesirable, since
15369 it can easily be confused with a real sum.)
15371 The special function @code{eval} can be used inside a @kbd{Z C}
15372 composition formula to cause all or part of the formula to be
15373 evaluated at display time. For example, if the formula is
15374 @samp{a + eval(b + c)}, then @samp{foo(1, 2, 3)} will be displayed
15375 as @samp{1 + 5}. Evaluation will use the default simplifications,
15376 regardless of the current simplification mode. There are also
15377 @code{evalsimp} and @code{evalextsimp} which simplify as if by
15378 @kbd{a s} and @kbd{a e} (respectively). Note that these ``functions''
15379 operate only in the context of composition formulas (and also in
15380 rewrite rules, where they serve a similar purpose; @pxref{Rewrite
15381 Rules}). On the stack, a call to @code{eval} will be left in
15384 It is not a good idea to use @code{eval} except as a last resort.
15385 It can cause the display of formulas to be extremely slow. For
15386 example, while @samp{eval(a + b)} might seem quite fast and simple,
15387 there are several situations where it could be slow. For example,
15388 @samp{a} and/or @samp{b} could be polar complex numbers, in which
15389 case doing the sum requires trigonometry. Or, @samp{a} could be
15390 the factorial @samp{fact(100)} which is unevaluated because you
15391 have typed @kbd{m O}; @code{eval} will evaluate it anyway to
15392 produce a large, unwieldy integer.
15394 You can save your display formats permanently using the @kbd{Z P}
15395 command (@pxref{Creating User Keys}).
15397 @node Syntax Tables, , Compositions, Language Modes
15398 @subsection Syntax Tables
15401 @cindex Syntax tables
15402 @cindex Parsing formulas, customized
15403 Syntax tables do for input what compositions do for output: They
15404 allow you to teach custom notations to Calc's formula parser.
15405 Calc keeps a separate syntax table for each language mode.
15407 (Note that the Calc ``syntax tables'' discussed here are completely
15408 unrelated to the syntax tables described in the Emacs manual.)
15411 @pindex calc-edit-user-syntax
15412 The @kbd{Z S} (@code{calc-edit-user-syntax}) command edits the
15413 syntax table for the current language mode. If you want your
15414 syntax to work in any language, define it in the Normal language
15415 mode. Type @kbd{C-c C-c} to finish editing the syntax table, or
15416 @kbd{C-x k} to cancel the edit. The @kbd{m m} command saves all
15417 the syntax tables along with the other mode settings;
15418 @pxref{General Mode Commands}.
15421 * Syntax Table Basics::
15422 * Precedence in Syntax Tables::
15423 * Advanced Syntax Patterns::
15424 * Conditional Syntax Rules::
15427 @node Syntax Table Basics, Precedence in Syntax Tables, Syntax Tables, Syntax Tables
15428 @subsubsection Syntax Table Basics
15431 @dfn{Parsing} is the process of converting a raw string of characters,
15432 such as you would type in during algebraic entry, into a Calc formula.
15433 Calc's parser works in two stages. First, the input is broken down
15434 into @dfn{tokens}, such as words, numbers, and punctuation symbols
15435 like @samp{+}, @samp{:=}, and @samp{+/-}. Space between tokens is
15436 ignored (except when it serves to separate adjacent words). Next,
15437 the parser matches this string of tokens against various built-in
15438 syntactic patterns, such as ``an expression followed by @samp{+}
15439 followed by another expression'' or ``a name followed by @samp{(},
15440 zero or more expressions separated by commas, and @samp{)}.''
15442 A @dfn{syntax table} is a list of user-defined @dfn{syntax rules},
15443 which allow you to specify new patterns to define your own
15444 favorite input notations. Calc's parser always checks the syntax
15445 table for the current language mode, then the table for the Normal
15446 language mode, before it uses its built-in rules to parse an
15447 algebraic formula you have entered. Each syntax rule should go on
15448 its own line; it consists of a @dfn{pattern}, a @samp{:=} symbol,
15449 and a Calc formula with an optional @dfn{condition}. (Syntax rules
15450 resemble algebraic rewrite rules, but the notation for patterns is
15451 completely different.)
15453 A syntax pattern is a list of tokens, separated by spaces.
15454 Except for a few special symbols, tokens in syntax patterns are
15455 matched literally, from left to right. For example, the rule,
15462 would cause Calc to parse the formula @samp{4+foo()*5} as if it
15463 were @samp{4+(2+3)*5}. Notice that the parentheses were written
15464 as two separate tokens in the rule. As a result, the rule works
15465 for both @samp{foo()} and @w{@samp{foo ( )}}. If we had written
15466 the rule as @samp{foo () := 2+3}, then Calc would treat @samp{()}
15467 as a single, indivisible token, so that @w{@samp{foo( )}} would
15468 not be recognized by the rule. (It would be parsed as a regular
15469 zero-argument function call instead.) In fact, this rule would
15470 also make trouble for the rest of Calc's parser: An unrelated
15471 formula like @samp{bar()} would now be tokenized into @samp{bar ()}
15472 instead of @samp{bar ( )}, so that the standard parser for function
15473 calls would no longer recognize it!
15475 While it is possible to make a token with a mixture of letters
15476 and punctuation symbols, this is not recommended. It is better to
15477 break it into several tokens, as we did with @samp{foo()} above.
15479 The symbol @samp{#} in a syntax pattern matches any Calc expression.
15480 On the righthand side, the things that matched the @samp{#}s can
15481 be referred to as @samp{#1}, @samp{#2}, and so on (where @samp{#1}
15482 matches the leftmost @samp{#} in the pattern). For example, these
15483 rules match a user-defined function, prefix operator, infix operator,
15484 and postfix operator, respectively:
15487 foo ( # ) := myfunc(#1)
15488 foo # := myprefix(#1)
15489 # foo # := myinfix(#1,#2)
15490 # foo := mypostfix(#1)
15493 Thus @samp{foo(3)} will parse as @samp{myfunc(3)}, and @samp{2+3 foo}
15494 will parse as @samp{mypostfix(2+3)}.
15496 It is important to write the first two rules in the order shown,
15497 because Calc tries rules in order from first to last. If the
15498 pattern @samp{foo #} came first, it would match anything that could
15499 match the @samp{foo ( # )} rule, since an expression in parentheses
15500 is itself a valid expression. Thus the @w{@samp{foo ( # )}} rule would
15501 never get to match anything. Likewise, the last two rules must be
15502 written in the order shown or else @samp{3 foo 4} will be parsed as
15503 @samp{mypostfix(3) * 4}. (Of course, the best way to avoid these
15504 ambiguities is not to use the same symbol in more than one way at
15505 the same time! In case you're not convinced, try the following
15506 exercise: How will the above rules parse the input @samp{foo(3,4)},
15507 if at all? Work it out for yourself, then try it in Calc and see.)
15509 Calc is quite flexible about what sorts of patterns are allowed.
15510 The only rule is that every pattern must begin with a literal
15511 token (like @samp{foo} in the first two patterns above), or with
15512 a @samp{#} followed by a literal token (as in the last two
15513 patterns). After that, any mixture is allowed, although putting
15514 two @samp{#}s in a row will not be very useful since two
15515 expressions with nothing between them will be parsed as one
15516 expression that uses implicit multiplication.
15518 As a more practical example, Maple uses the notation
15519 @samp{sum(a(i), i=1..10)} for sums, which Calc's Maple mode doesn't
15520 recognize at present. To handle this syntax, we simply add the
15524 sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)
15528 to the Maple mode syntax table. As another example, C mode can't
15529 read assignment operators like @samp{++} and @samp{*=}. We can
15530 define these operators quite easily:
15533 # *= # := muleq(#1,#2)
15534 # ++ := postinc(#1)
15539 To complete the job, we would use corresponding composition functions
15540 and @kbd{Z C} to cause these functions to display in their respective
15541 Maple and C notations. (Note that the C example ignores issues of
15542 operator precedence, which are discussed in the next section.)
15544 You can enclose any token in quotes to prevent its usual
15545 interpretation in syntax patterns:
15548 # ":=" # := becomes(#1,#2)
15551 Quotes also allow you to include spaces in a token, although once
15552 again it is generally better to use two tokens than one token with
15553 an embedded space. To include an actual quotation mark in a quoted
15554 token, precede it with a backslash. (This also works to include
15555 backslashes in tokens.)
15558 # "bad token" # "/\"\\" # := silly(#1,#2,#3)
15562 This will parse @samp{3 bad token 4 /"\ 5} to @samp{silly(3,4,5)}.
15564 The token @kbd{#} has a predefined meaning in Calc's formula parser;
15565 it is not valid to use @samp{"#"} in a syntax rule. However, longer
15566 tokens that include the @samp{#} character are allowed. Also, while
15567 @samp{"$"} and @samp{"\""} are allowed as tokens, their presence in
15568 the syntax table will prevent those characters from working in their
15569 usual ways (referring to stack entries and quoting strings,
15572 Finally, the notation @samp{%%} anywhere in a syntax table causes
15573 the rest of the line to be ignored as a comment.
15575 @node Precedence in Syntax Tables, Advanced Syntax Patterns, Syntax Table Basics, Syntax Tables
15576 @subsubsection Precedence
15579 Different operators are generally assigned different @dfn{precedences}.
15580 By default, an operator defined by a rule like
15583 # foo # := foo(#1,#2)
15587 will have an extremely low precedence, so that @samp{2*3+4 foo 5 == 6}
15588 will be parsed as @samp{(2*3+4) foo (5 == 6)}. To change the
15589 precedence of an operator, use the notation @samp{#/@var{p}} in
15590 place of @samp{#}, where @var{p} is an integer precedence level.
15591 For example, 185 lies between the precedences for @samp{+} and
15592 @samp{*}, so if we change this rule to
15595 #/185 foo #/186 := foo(#1,#2)
15599 then @samp{2+3 foo 4*5} will be parsed as @samp{2+(3 foo (4*5))}.
15600 Also, because we've given the righthand expression slightly higher
15601 precedence, our new operator will be left-associative:
15602 @samp{1 foo 2 foo 3} will be parsed as @samp{(1 foo 2) foo 3}.
15603 By raising the precedence of the lefthand expression instead, we
15604 can create a right-associative operator.
15606 @xref{Composition Basics}, for a table of precedences of the
15607 standard Calc operators. For the precedences of operators in other
15608 language modes, look in the Calc source file @file{calc-lang.el}.
15610 @node Advanced Syntax Patterns, Conditional Syntax Rules, Precedence in Syntax Tables, Syntax Tables
15611 @subsubsection Advanced Syntax Patterns
15614 To match a function with a variable number of arguments, you could
15618 foo ( # ) := myfunc(#1)
15619 foo ( # , # ) := myfunc(#1,#2)
15620 foo ( # , # , # ) := myfunc(#1,#2,#3)
15624 but this isn't very elegant. To match variable numbers of items,
15625 Calc uses some notations inspired regular expressions and the
15626 ``extended BNF'' style used by some language designers.
15629 foo ( @{ # @}*, ) := apply(myfunc,#1)
15632 The token @samp{@{} introduces a repeated or optional portion.
15633 One of the three tokens @samp{@}*}, @samp{@}+}, or @samp{@}?}
15634 ends the portion. These will match zero or more, one or more,
15635 or zero or one copies of the enclosed pattern, respectively.
15636 In addition, @samp{@}*} and @samp{@}+} can be followed by a
15637 separator token (with no space in between, as shown above).
15638 Thus @samp{@{ # @}*,} matches nothing, or one expression, or
15639 several expressions separated by commas.
15641 A complete @samp{@{ ... @}} item matches as a vector of the
15642 items that matched inside it. For example, the above rule will
15643 match @samp{foo(1,2,3)} to get @samp{apply(myfunc,[1,2,3])}.
15644 The Calc @code{apply} function takes a function name and a vector
15645 of arguments and builds a call to the function with those
15646 arguments, so the net result is the formula @samp{myfunc(1,2,3)}.
15648 If the body of a @samp{@{ ... @}} contains several @samp{#}s
15649 (or nested @samp{@{ ... @}} constructs), then the items will be
15650 strung together into the resulting vector. If the body
15651 does not contain anything but literal tokens, the result will
15652 always be an empty vector.
15655 foo ( @{ # , # @}+, ) := bar(#1)
15656 foo ( @{ @{ # @}*, @}*; ) := matrix(#1)
15660 will parse @samp{foo(1, 2, 3, 4)} as @samp{bar([1, 2, 3, 4])}, and
15661 @samp{foo(1, 2; 3, 4)} as @samp{matrix([[1, 2], [3, 4]])}. Also, after
15662 some thought it's easy to see how this pair of rules will parse
15663 @samp{foo(1, 2, 3)} as @samp{matrix([[1, 2, 3]])}, since the first
15664 rule will only match an even number of arguments. The rule
15667 foo ( # @{ , # , # @}? ) := bar(#1,#2)
15671 will parse @samp{foo(2,3,4)} as @samp{bar(2,[3,4])}, and
15672 @samp{foo(2)} as @samp{bar(2,[])}.
15674 The notation @samp{@{ ... @}?.} (note the trailing period) works
15675 just the same as regular @samp{@{ ... @}?}, except that it does not
15676 count as an argument; the following two rules are equivalent:
15679 foo ( # , @{ also @}? # ) := bar(#1,#3)
15680 foo ( # , @{ also @}?. # ) := bar(#1,#2)
15684 Note that in the first case the optional text counts as @samp{#2},
15685 which will always be an empty vector, but in the second case no
15686 empty vector is produced.
15688 Another variant is @samp{@{ ... @}?$}, which means the body is
15689 optional only at the end of the input formula. All built-in syntax
15690 rules in Calc use this for closing delimiters, so that during
15691 algebraic entry you can type @kbd{[sqrt(2), sqrt(3 @key{RET}}, omitting
15692 the closing parenthesis and bracket. Calc does this automatically
15693 for trailing @samp{)}, @samp{]}, and @samp{>} tokens in syntax
15694 rules, but you can use @samp{@{ ... @}?$} explicitly to get
15695 this effect with any token (such as @samp{"@}"} or @samp{end}).
15696 Like @samp{@{ ... @}?.}, this notation does not count as an
15697 argument. Conversely, you can use quotes, as in @samp{")"}, to
15698 prevent a closing-delimiter token from being automatically treated
15701 Calc's parser does not have full backtracking, which means some
15702 patterns will not work as you might expect:
15705 foo ( @{ # , @}? # , # ) := bar(#1,#2,#3)
15709 Here we are trying to make the first argument optional, so that
15710 @samp{foo(2,3)} parses as @samp{bar([],2,3)}. Unfortunately, Calc
15711 first tries to match @samp{2,} against the optional part of the
15712 pattern, finds a match, and so goes ahead to match the rest of the
15713 pattern. Later on it will fail to match the second comma, but it
15714 doesn't know how to go back and try the other alternative at that
15715 point. One way to get around this would be to use two rules:
15718 foo ( # , # , # ) := bar([#1],#2,#3)
15719 foo ( # , # ) := bar([],#1,#2)
15722 More precisely, when Calc wants to match an optional or repeated
15723 part of a pattern, it scans forward attempting to match that part.
15724 If it reaches the end of the optional part without failing, it
15725 ``finalizes'' its choice and proceeds. If it fails, though, it
15726 backs up and tries the other alternative. Thus Calc has ``partial''
15727 backtracking. A fully backtracking parser would go on to make sure
15728 the rest of the pattern matched before finalizing the choice.
15730 @node Conditional Syntax Rules, , Advanced Syntax Patterns, Syntax Tables
15731 @subsubsection Conditional Syntax Rules
15734 It is possible to attach a @dfn{condition} to a syntax rule. For
15738 foo ( # ) := ifoo(#1) :: integer(#1)
15739 foo ( # ) := gfoo(#1)
15743 will parse @samp{foo(3)} as @samp{ifoo(3)}, but will parse
15744 @samp{foo(3.5)} and @samp{foo(x)} as calls to @code{gfoo}. Any
15745 number of conditions may be attached; all must be true for the
15746 rule to succeed. A condition is ``true'' if it evaluates to a
15747 nonzero number. @xref{Logical Operations}, for a list of Calc
15748 functions like @code{integer} that perform logical tests.
15750 The exact sequence of events is as follows: When Calc tries a
15751 rule, it first matches the pattern as usual. It then substitutes
15752 @samp{#1}, @samp{#2}, etc., in the conditions, if any. Next, the
15753 conditions are simplified and evaluated in order from left to right,
15754 using the algebraic simplifications (@pxref{Simplifying Formulas}).
15755 Each result is true if it is a nonzero number, or an expression
15756 that can be proven to be nonzero (@pxref{Declarations}). If the
15757 results of all conditions are true, the expression (such as
15758 @samp{ifoo(#1)}) has its @samp{#}s substituted, and that is the
15759 result of the parse. If the result of any condition is false, Calc
15760 goes on to try the next rule in the syntax table.
15762 Syntax rules also support @code{let} conditions, which operate in
15763 exactly the same way as they do in algebraic rewrite rules.
15764 @xref{Other Features of Rewrite Rules}, for details. A @code{let}
15765 condition is always true, but as a side effect it defines a
15766 variable which can be used in later conditions, and also in the
15767 expression after the @samp{:=} sign:
15770 foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)
15774 The @code{dnumint} function tests if a value is numerically an
15775 integer, i.e., either a true integer or an integer-valued float.
15776 This rule will parse @code{foo} with a half-integer argument,
15777 like @samp{foo(3.5)}, to a call like @samp{hifoo(4.)}.
15779 The lefthand side of a syntax rule @code{let} must be a simple
15780 variable, not the arbitrary pattern that is allowed in rewrite
15783 The @code{matches} function is also treated specially in syntax
15784 rule conditions (again, in the same way as in rewrite rules).
15785 @xref{Matching Commands}. If the matching pattern contains
15786 meta-variables, then those meta-variables may be used in later
15787 conditions and in the result expression. The arguments to
15788 @code{matches} are not evaluated in this situation.
15791 sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])
15795 This is another way to implement the Maple mode @code{sum} notation.
15796 In this approach, we allow @samp{#2} to equal the whole expression
15797 @samp{i=1..10}. Then, we use @code{matches} to break it apart into
15798 its components. If the expression turns out not to match the pattern,
15799 the syntax rule will fail. Note that @kbd{Z S} always uses Calc's
15800 Normal language mode for editing expressions in syntax rules, so we
15801 must use regular Calc notation for the interval @samp{[b..c]} that
15802 will correspond to the Maple mode interval @samp{1..10}.
15804 @node Modes Variable, Calc Mode Line, Language Modes, Mode Settings
15805 @section The @code{Modes} Variable
15809 @pindex calc-get-modes
15810 The @kbd{m g} (@code{calc-get-modes}) command pushes onto the stack
15811 a vector of numbers that describes the various mode settings that
15812 are in effect. With a numeric prefix argument, it pushes only the
15813 @var{n}th mode, i.e., the @var{n}th element of this vector. Keyboard
15814 macros can use the @kbd{m g} command to modify their behavior based
15815 on the current mode settings.
15817 @cindex @code{Modes} variable
15819 The modes vector is also available in the special variable
15820 @code{Modes}. In other words, @kbd{m g} is like @kbd{s r Modes @key{RET}}.
15821 It will not work to store into this variable; in fact, if you do,
15822 @code{Modes} will cease to track the current modes. (The @kbd{m g}
15823 command will continue to work, however.)
15825 In general, each number in this vector is suitable as a numeric
15826 prefix argument to the associated mode-setting command. (Recall
15827 that the @kbd{~} key takes a number from the stack and gives it as
15828 a numeric prefix to the next command.)
15830 The elements of the modes vector are as follows:
15834 Current precision. Default is 12; associated command is @kbd{p}.
15837 Binary word size. Default is 32; associated command is @kbd{b w}.
15840 Stack size (not counting the value about to be pushed by @kbd{m g}).
15841 This is zero if @kbd{m g} is executed with an empty stack.
15844 Number radix. Default is 10; command is @kbd{d r}.
15847 Floating-point format. This is the number of digits, plus the
15848 constant 0 for normal notation, 10000 for scientific notation,
15849 20000 for engineering notation, or 30000 for fixed-point notation.
15850 These codes are acceptable as prefix arguments to the @kbd{d n}
15851 command, but note that this may lose information: For example,
15852 @kbd{d s} and @kbd{C-u 12 d s} have similar (but not quite
15853 identical) effects if the current precision is 12, but they both
15854 produce a code of 10012, which will be treated by @kbd{d n} as
15855 @kbd{C-u 12 d s}. If the precision then changes, the float format
15856 will still be frozen at 12 significant figures.
15859 Angular mode. Default is 1 (degrees). Other values are 2 (radians)
15860 and 3 (HMS). The @kbd{m d} command accepts these prefixes.
15863 Symbolic mode. Value is 0 or 1; default is 0. Command is @kbd{m s}.
15866 Fraction mode. Value is 0 or 1; default is 0. Command is @kbd{m f}.
15869 Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0.
15870 Command is @kbd{m p}.
15873 Matrix/Scalar mode. Default value is @mathit{-1}. Value is 0 for Scalar
15874 mode, @mathit{-2} for Matrix mode, @mathit{-3} for square Matrix mode,
15876 @texline @math{N\times N}
15877 @infoline @var{N}x@var{N}
15878 Matrix mode. Command is @kbd{m v}.
15881 Simplification mode. Default is 1. Value is @mathit{-1} for off (@kbd{m O}),
15882 0 for @kbd{m N}, 2 for @kbd{m B}, 3 for @kbd{m A}, 4 for @kbd{m E},
15883 or 5 for @w{@kbd{m U}}. The @kbd{m D} command accepts these prefixes.
15886 Infinite mode. Default is @mathit{-1} (off). Value is 1 if the mode is on,
15887 or 0 if the mode is on with positive zeros. Command is @kbd{m i}.
15890 For example, the sequence @kbd{M-1 m g @key{RET} 2 + ~ p} increases the
15891 precision by two, leaving a copy of the old precision on the stack.
15892 Later, @kbd{~ p} will restore the original precision using that
15893 stack value. (This sequence might be especially useful inside a
15896 As another example, @kbd{M-3 m g 1 - ~ @key{DEL}} deletes all but the
15897 oldest (bottommost) stack entry.
15899 Yet another example: The HP-48 ``round'' command rounds a number
15900 to the current displayed precision. You could roughly emulate this
15901 in Calc with the sequence @kbd{M-5 m g 10000 % ~ c c}. (This
15902 would not work for fixed-point mode, but it wouldn't be hard to
15903 do a full emulation with the help of the @kbd{Z [} and @kbd{Z ]}
15904 programming commands. @xref{Conditionals in Macros}.)
15906 @node Calc Mode Line, , Modes Variable, Mode Settings
15907 @section The Calc Mode Line
15910 @cindex Mode line indicators
15911 This section is a summary of all symbols that can appear on the
15912 Calc mode line, the highlighted bar that appears under the Calc
15913 stack window (or under an editing window in Embedded mode).
15915 The basic mode line format is:
15918 --%*-Calc: 12 Deg @var{other modes} (Calculator)
15921 The @samp{%*} indicates that the buffer is ``read-only''; it shows that
15922 regular Emacs commands are not allowed to edit the stack buffer
15923 as if it were text.
15925 The word @samp{Calc:} changes to @samp{CalcEmbed:} if Embedded mode
15926 is enabled. The words after this describe the various Calc modes
15927 that are in effect.
15929 The first mode is always the current precision, an integer.
15930 The second mode is always the angular mode, either @code{Deg},
15931 @code{Rad}, or @code{Hms}.
15933 Here is a complete list of the remaining symbols that can appear
15938 Algebraic mode (@kbd{m a}; @pxref{Algebraic Entry}).
15941 Incomplete algebraic mode (@kbd{C-u m a}).
15944 Total algebraic mode (@kbd{m t}).
15947 Symbolic mode (@kbd{m s}; @pxref{Symbolic Mode}).
15950 Matrix mode (@kbd{m v}; @pxref{Matrix Mode}).
15952 @item Matrix@var{n}
15953 Dimensioned Matrix mode (@kbd{C-u @var{n} m v}; @pxref{Matrix Mode}).
15956 Square Matrix mode (@kbd{C-u m v}; @pxref{Matrix Mode}).
15959 Scalar mode (@kbd{m v}; @pxref{Matrix Mode}).
15962 Polar complex mode (@kbd{m p}; @pxref{Polar Mode}).
15965 Fraction mode (@kbd{m f}; @pxref{Fraction Mode}).
15968 Infinite mode (@kbd{m i}; @pxref{Infinite Mode}).
15971 Positive Infinite mode (@kbd{C-u 0 m i}).
15974 Default simplifications off (@kbd{m O}; @pxref{Simplification Modes}).
15977 Default simplifications for numeric arguments only (@kbd{m N}).
15979 @item BinSimp@var{w}
15980 Binary-integer simplification mode; word size @var{w} (@kbd{m B}, @kbd{b w}).
15983 Basic simplification mode (@kbd{m I}).
15986 Extended algebraic simplification mode (@kbd{m E}).
15989 Units simplification mode (@kbd{m U}).
15992 Current radix is 2 (@kbd{d 2}; @pxref{Radix Modes}).
15995 Current radix is 8 (@kbd{d 8}).
15998 Current radix is 16 (@kbd{d 6}).
16001 Current radix is @var{n} (@kbd{d r}).
16004 Leading zeros (@kbd{d z}; @pxref{Radix Modes}).
16007 Big language mode (@kbd{d B}; @pxref{Normal Language Modes}).
16010 One-line normal language mode (@kbd{d O}).
16013 Unformatted language mode (@kbd{d U}).
16016 C language mode (@kbd{d C}; @pxref{C FORTRAN Pascal}).
16019 Pascal language mode (@kbd{d P}).
16022 FORTRAN language mode (@kbd{d F}).
16025 @TeX{} language mode (@kbd{d T}; @pxref{TeX and LaTeX Language Modes}).
16028 @LaTeX{} language mode (@kbd{d L}; @pxref{TeX and LaTeX Language Modes}).
16031 @dfn{Eqn} language mode (@kbd{d E}; @pxref{Eqn Language Mode}).
16034 Mathematica language mode (@kbd{d M}; @pxref{Mathematica Language Mode}).
16037 Maple language mode (@kbd{d W}; @pxref{Maple Language Mode}).
16040 Normal float mode with @var{n} digits (@kbd{d n}; @pxref{Float Formats}).
16043 Fixed point mode with @var{n} digits after the point (@kbd{d f}).
16046 Scientific notation mode (@kbd{d s}).
16049 Scientific notation with @var{n} digits (@kbd{d s}).
16052 Engineering notation mode (@kbd{d e}).
16055 Engineering notation with @var{n} digits (@kbd{d e}).
16058 Left-justified display indented by @var{n} (@kbd{d <}; @pxref{Justification}).
16061 Right-justified display (@kbd{d >}).
16064 Right-justified display with width @var{n} (@kbd{d >}).
16067 Centered display (@kbd{d =}).
16069 @item Center@var{n}
16070 Centered display with center column @var{n} (@kbd{d =}).
16073 Line breaking with width @var{n} (@kbd{d b}; @pxref{Normal Language Modes}).
16076 No line breaking (@kbd{d b}).
16079 Selections show deep structure (@kbd{j b}; @pxref{Making Selections}).
16082 Record modes in @file{~/.emacs.d/calc.el} (@kbd{m R}; @pxref{General Mode Commands}).
16085 Record modes in Embedded buffer (@kbd{m R}).
16088 Record modes as editing-only in Embedded buffer (@kbd{m R}).
16091 Record modes as permanent-only in Embedded buffer (@kbd{m R}).
16094 Record modes as global in Embedded buffer (@kbd{m R}).
16097 Automatic recomputation turned off (@kbd{m C}; @pxref{Automatic
16101 GNUPLOT process is alive in background (@pxref{Graphics}).
16104 Top-of-stack has a selection (Embedded only; @pxref{Making Selections}).
16107 The stack display may not be up-to-date (@pxref{Display Modes}).
16110 ``Inverse'' prefix was pressed (@kbd{I}; @pxref{Inverse and Hyperbolic}).
16113 ``Hyperbolic'' prefix was pressed (@kbd{H}).
16116 ``Keep-arguments'' prefix was pressed (@kbd{K}).
16119 Stack is truncated (@kbd{d t}; @pxref{Truncating the Stack}).
16122 In addition, the symbols @code{Active} and @code{~Active} can appear
16123 as minor modes on an Embedded buffer's mode line. @xref{Embedded Mode}.
16125 @node Arithmetic, Scientific Functions, Mode Settings, Top
16126 @chapter Arithmetic Functions
16129 This chapter describes the Calc commands for doing simple calculations
16130 on numbers, such as addition, absolute value, and square roots. These
16131 commands work by removing the top one or two values from the stack,
16132 performing the desired operation, and pushing the result back onto the
16133 stack. If the operation cannot be performed, the result pushed is a
16134 formula instead of a number, such as @samp{2/0} (because division by zero
16135 is invalid) or @samp{sqrt(x)} (because the argument @samp{x} is a formula).
16137 Most of the commands described here can be invoked by a single keystroke.
16138 Some of the more obscure ones are two-letter sequences beginning with
16139 the @kbd{f} (``functions'') prefix key.
16141 @xref{Prefix Arguments}, for a discussion of the effect of numeric
16142 prefix arguments on commands in this chapter which do not otherwise
16143 interpret a prefix argument.
16146 * Basic Arithmetic::
16147 * Integer Truncation::
16148 * Complex Number Functions::
16150 * Date Arithmetic::
16151 * Financial Functions::
16152 * Binary Functions::
16155 @node Basic Arithmetic, Integer Truncation, Arithmetic, Arithmetic
16156 @section Basic Arithmetic
16165 The @kbd{+} (@code{calc-plus}) command adds two numbers. The numbers may
16166 be any of the standard Calc data types. The resulting sum is pushed back
16169 If both arguments of @kbd{+} are vectors or matrices (of matching dimensions),
16170 the result is a vector or matrix sum. If one argument is a vector and the
16171 other a scalar (i.e., a non-vector), the scalar is added to each of the
16172 elements of the vector to form a new vector. If the scalar is not a
16173 number, the operation is left in symbolic form: Suppose you added @samp{x}
16174 to the vector @samp{[1,2]}. You may want the result @samp{[1+x,2+x]}, or
16175 you may plan to substitute a 2-vector for @samp{x} in the future. Since
16176 the Calculator can't tell which interpretation you want, it makes the
16177 safest assumption. @xref{Reducing and Mapping}, for a way to add @samp{x}
16178 to every element of a vector.
16180 If either argument of @kbd{+} is a complex number, the result will in general
16181 be complex. If one argument is in rectangular form and the other polar,
16182 the current Polar mode determines the form of the result. If Symbolic
16183 mode is enabled, the sum may be left as a formula if the necessary
16184 conversions for polar addition are non-trivial.
16186 If both arguments of @kbd{+} are HMS forms, the forms are added according to
16187 the usual conventions of hours-minutes-seconds notation. If one argument
16188 is an HMS form and the other is a number, that number is converted from
16189 degrees or radians (depending on the current Angular mode) to HMS format
16190 and then the two HMS forms are added.
16192 If one argument of @kbd{+} is a date form, the other can be either a
16193 real number, which advances the date by a certain number of days, or
16194 an HMS form, which advances the date by a certain amount of time.
16195 Subtracting two date forms yields the number of days between them.
16196 Adding two date forms is meaningless, but Calc interprets it as the
16197 subtraction of one date form and the negative of the other. (The
16198 negative of a date form can be understood by remembering that dates
16199 are stored as the number of days before or after Jan 1, 1 AD.)
16201 If both arguments of @kbd{+} are error forms, the result is an error form
16202 with an appropriately computed standard deviation. If one argument is an
16203 error form and the other is a number, the number is taken to have zero error.
16204 Error forms may have symbolic formulas as their mean and/or error parts;
16205 adding these will produce a symbolic error form result. However, adding an
16206 error form to a plain symbolic formula (as in @samp{(a +/- b) + c}) will not
16207 work, for the same reasons just mentioned for vectors. Instead you must
16208 write @samp{(a +/- b) + (c +/- 0)}.
16210 If both arguments of @kbd{+} are modulo forms with equal values of @expr{M},
16211 or if one argument is a modulo form and the other a plain number, the
16212 result is a modulo form which represents the sum, modulo @expr{M}, of
16215 If both arguments of @kbd{+} are intervals, the result is an interval
16216 which describes all possible sums of the possible input values. If
16217 one argument is a plain number, it is treated as the interval
16218 @w{@samp{[x ..@: x]}}.
16220 If one argument of @kbd{+} is an infinity and the other is not, the
16221 result is that same infinity. If both arguments are infinite and in
16222 the same direction, the result is the same infinity, but if they are
16223 infinite in different directions the result is @code{nan}.
16231 The @kbd{-} (@code{calc-minus}) command subtracts two values. The top
16232 number on the stack is subtracted from the one behind it, so that the
16233 computation @kbd{5 @key{RET} 2 -} produces 3, not @mathit{-3}. All options
16234 available for @kbd{+} are available for @kbd{-} as well.
16242 The @kbd{*} (@code{calc-times}) command multiplies two numbers. If one
16243 argument is a vector and the other a scalar, the scalar is multiplied by
16244 the elements of the vector to produce a new vector. If both arguments
16245 are vectors, the interpretation depends on the dimensions of the
16246 vectors: If both arguments are matrices, a matrix multiplication is
16247 done. If one argument is a matrix and the other a plain vector, the
16248 vector is interpreted as a row vector or column vector, whichever is
16249 dimensionally correct. If both arguments are plain vectors, the result
16250 is a single scalar number which is the dot product of the two vectors.
16252 If one argument of @kbd{*} is an HMS form and the other a number, the
16253 HMS form is multiplied by that amount. It is an error to multiply two
16254 HMS forms together, or to attempt any multiplication involving date
16255 forms. Error forms, modulo forms, and intervals can be multiplied;
16256 see the comments for addition of those forms. When two error forms
16257 or intervals are multiplied they are considered to be statistically
16258 independent; thus, @samp{[-2 ..@: 3] * [-2 ..@: 3]} is @samp{[-6 ..@: 9]},
16259 whereas @w{@samp{[-2 ..@: 3] ^ 2}} is @samp{[0 ..@: 9]}.
16262 @pindex calc-divide
16267 The @kbd{/} (@code{calc-divide}) command divides two numbers.
16269 When combining multiplication and division in an algebraic formula, it
16270 is good style to use parentheses to distinguish between possible
16271 interpretations; the expression @samp{a/b*c} should be written
16272 @samp{(a/b)*c} or @samp{a/(b*c)}, as appropriate. Without the
16273 parentheses, Calc will interpret @samp{a/b*c} as @samp{a/(b*c)}, since
16274 in algebraic entry Calc gives division a lower precedence than
16275 multiplication. (This is not standard across all computer languages, and
16276 Calc may change the precedence depending on the language mode being used.
16277 @xref{Language Modes}.) This default ordering can be changed by setting
16278 the customizable variable @code{calc-multiplication-has-precedence} to
16279 @code{nil} (@pxref{Customizing Calc}); this will give multiplication and
16280 division equal precedences. Note that Calc's default choice of
16281 precedence allows @samp{a b / c d} to be used as a shortcut for
16290 When dividing a scalar @expr{B} by a square matrix @expr{A}, the
16291 computation performed is @expr{B} times the inverse of @expr{A}. This
16292 also occurs if @expr{B} is itself a vector or matrix, in which case the
16293 effect is to solve the set of linear equations represented by @expr{B}.
16294 If @expr{B} is a matrix with the same number of rows as @expr{A}, or a
16295 plain vector (which is interpreted here as a column vector), then the
16296 equation @expr{A X = B} is solved for the vector or matrix @expr{X}.
16297 Otherwise, if @expr{B} is a non-square matrix with the same number of
16298 @emph{columns} as @expr{A}, the equation @expr{X A = B} is solved. If
16299 you wish a vector @expr{B} to be interpreted as a row vector to be
16300 solved as @expr{X A = B}, make it into a one-row matrix with @kbd{C-u 1
16301 v p} first. To force a left-handed solution with a square matrix
16302 @expr{B}, transpose @expr{A} and @expr{B} before dividing, then
16303 transpose the result.
16305 HMS forms can be divided by real numbers or by other HMS forms. Error
16306 forms can be divided in any combination of ways. Modulo forms where both
16307 values and the modulo are integers can be divided to get an integer modulo
16308 form result. Intervals can be divided; dividing by an interval that
16309 encompasses zero or has zero as a limit will result in an infinite
16318 The @kbd{^} (@code{calc-power}) command raises a number to a power. If
16319 the power is an integer, an exact result is computed using repeated
16320 multiplications. For non-integer powers, Calc uses Newton's method or
16321 logarithms and exponentials. Square matrices can be raised to integer
16322 powers. If either argument is an error (or interval or modulo) form,
16323 the result is also an error (or interval or modulo) form.
16327 If you press the @kbd{I} (inverse) key first, the @kbd{I ^} command
16328 computes an Nth root: @kbd{125 @key{RET} 3 I ^} computes the number 5.
16329 (This is entirely equivalent to @kbd{125 @key{RET} 1:3 ^}.)
16338 The @kbd{\} (@code{calc-idiv}) command divides two numbers on the stack
16339 to produce an integer result. It is equivalent to dividing with
16340 @key{/}, then rounding down with @kbd{F} (@code{calc-floor}), only a bit
16341 more convenient and efficient. Also, since it is an all-integer
16342 operation when the arguments are integers, it avoids problems that
16343 @kbd{/ F} would have with floating-point roundoff.
16351 The @kbd{%} (@code{calc-mod}) command performs a ``modulo'' (or ``remainder'')
16352 operation. Mathematically, @samp{a%b = a - (a\b)*b}, and is defined
16353 for all real numbers @expr{a} and @expr{b} (except @expr{b=0}). For
16354 positive @expr{b}, the result will always be between 0 (inclusive) and
16355 @expr{b} (exclusive). Modulo does not work for HMS forms and error forms.
16356 If @expr{a} is a modulo form, its modulo is changed to @expr{b}, which
16357 must be positive real number.
16362 The @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command
16363 divides the two integers on the top of the stack to produce a fractional
16364 result. This is a convenient shorthand for enabling Fraction mode (with
16365 @kbd{m f}) temporarily and using @samp{/}. Note that during numeric entry
16366 the @kbd{:} key is interpreted as a fraction separator, so to divide 8 by 6
16367 you would have to type @kbd{8 @key{RET} 6 @key{RET} :}. (Of course, in
16368 this case, it would be much easier simply to enter the fraction directly
16369 as @kbd{8:6 @key{RET}}!)
16372 @pindex calc-change-sign
16373 The @kbd{n} (@code{calc-change-sign}) command negates the number on the top
16374 of the stack. It works on numbers, vectors and matrices, HMS forms, date
16375 forms, error forms, intervals, and modulo forms.
16380 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the absolute
16381 value of a number. The result of @code{abs} is always a nonnegative
16382 real number: With a complex argument, it computes the complex magnitude.
16383 With a vector or matrix argument, it computes the Frobenius norm, i.e.,
16384 the square root of the sum of the squares of the absolute values of the
16385 elements. The absolute value of an error form is defined by replacing
16386 the mean part with its absolute value and leaving the error part the same.
16387 The absolute value of a modulo form is undefined. The absolute value of
16388 an interval is defined in the obvious way.
16391 @pindex calc-abssqr
16393 The @kbd{f A} (@code{calc-abssqr}) [@code{abssqr}] command computes the
16394 absolute value squared of a number, vector or matrix, or error form.
16399 The @kbd{f s} (@code{calc-sign}) [@code{sign}] command returns 1 if its
16400 argument is positive, @mathit{-1} if its argument is negative, or 0 if its
16401 argument is zero. In algebraic form, you can also write @samp{sign(a,x)}
16402 which evaluates to @samp{x * sign(a)}, i.e., either @samp{x}, @samp{-x}, or
16403 zero depending on the sign of @samp{a}.
16409 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
16410 reciprocal of a number, i.e., @expr{1 / x}. Operating on a square
16411 matrix, it computes the inverse of that matrix.
16416 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] command computes the square
16417 root of a number. For a negative real argument, the result will be a
16418 complex number whose form is determined by the current Polar mode.
16423 The @kbd{f h} (@code{calc-hypot}) [@code{hypot}] command computes the square
16424 root of the sum of the squares of two numbers. That is, @samp{hypot(a,b)}
16425 is the length of the hypotenuse of a right triangle with sides @expr{a}
16426 and @expr{b}. If the arguments are complex numbers, their squared
16427 magnitudes are used.
16432 The @kbd{f Q} (@code{calc-isqrt}) [@code{isqrt}] command computes the
16433 integer square root of an integer. This is the true square root of the
16434 number, rounded down to an integer. For example, @samp{isqrt(10)}
16435 produces 3. Note that, like @kbd{\} [@code{idiv}], this uses exact
16436 integer arithmetic throughout to avoid roundoff problems. If the input
16437 is a floating-point number or other non-integer value, this is exactly
16438 the same as @samp{floor(sqrt(x))}.
16446 The @kbd{f n} (@code{calc-min}) [@code{min}] and @kbd{f x} (@code{calc-max})
16447 [@code{max}] commands take the minimum or maximum of two real numbers,
16448 respectively. These commands also work on HMS forms, date forms,
16449 intervals, and infinities. (In algebraic expressions, these functions
16450 take any number of arguments and return the maximum or minimum among
16451 all the arguments.)
16455 @pindex calc-mant-part
16457 @pindex calc-xpon-part
16459 The @kbd{f M} (@code{calc-mant-part}) [@code{mant}] function extracts
16460 the ``mantissa'' part @expr{m} of its floating-point argument; @kbd{f X}
16461 (@code{calc-xpon-part}) [@code{xpon}] extracts the ``exponent'' part
16462 @expr{e}. The original number is equal to
16463 @texline @math{m \times 10^e},
16464 @infoline @expr{m * 10^e},
16465 where @expr{m} is in the interval @samp{[1.0 ..@: 10.0)} except that
16466 @expr{m=e=0} if the original number is zero. For integers
16467 and fractions, @code{mant} returns the number unchanged and @code{xpon}
16468 returns zero. The @kbd{v u} (@code{calc-unpack}) command can also be
16469 used to ``unpack'' a floating-point number; this produces an integer
16470 mantissa and exponent, with the constraint that the mantissa is not
16471 a multiple of ten (again except for the @expr{m=e=0} case).
16474 @pindex calc-scale-float
16476 The @kbd{f S} (@code{calc-scale-float}) [@code{scf}] function scales a number
16477 by a given power of ten. Thus, @samp{scf(mant(x), xpon(x)) = x} for any
16478 real @samp{x}. The second argument must be an integer, but the first
16479 may actually be any numeric value. For example, @samp{scf(5,-2) = 0.05}
16480 or @samp{1:20} depending on the current Fraction mode.
16484 @pindex calc-decrement
16485 @pindex calc-increment
16488 The @kbd{f [} (@code{calc-decrement}) [@code{decr}] and @kbd{f ]}
16489 (@code{calc-increment}) [@code{incr}] functions decrease or increase
16490 a number by one unit. For integers, the effect is obvious. For
16491 floating-point numbers, the change is by one unit in the last place.
16492 For example, incrementing @samp{12.3456} when the current precision
16493 is 6 digits yields @samp{12.3457}. If the current precision had been
16494 8 digits, the result would have been @samp{12.345601}. Incrementing
16495 @samp{0.0} produces
16496 @texline @math{10^{-p}},
16497 @infoline @expr{10^-p},
16498 where @expr{p} is the current
16499 precision. These operations are defined only on integers and floats.
16500 With numeric prefix arguments, they change the number by @expr{n} units.
16502 Note that incrementing followed by decrementing, or vice-versa, will
16503 almost but not quite always cancel out. Suppose the precision is
16504 6 digits and the number @samp{9.99999} is on the stack. Incrementing
16505 will produce @samp{10.0000}; decrementing will produce @samp{9.9999}.
16506 One digit has been dropped. This is an unavoidable consequence of the
16507 way floating-point numbers work.
16509 Incrementing a date/time form adjusts it by a certain number of seconds.
16510 Incrementing a pure date form adjusts it by a certain number of days.
16512 @node Integer Truncation, Complex Number Functions, Basic Arithmetic, Arithmetic
16513 @section Integer Truncation
16516 There are four commands for truncating a real number to an integer,
16517 differing mainly in their treatment of negative numbers. All of these
16518 commands have the property that if the argument is an integer, the result
16519 is the same integer. An integer-valued floating-point argument is converted
16522 If you press @kbd{H} (@code{calc-hyperbolic}) first, the result will be
16523 expressed as an integer-valued floating-point number.
16525 @cindex Integer part of a number
16534 The @kbd{F} (@code{calc-floor}) [@code{floor} or @code{ffloor}] command
16535 truncates a real number to the next lower integer, i.e., toward minus
16536 infinity. Thus @kbd{3.6 F} produces 3, but @kbd{_3.6 F} produces
16540 @pindex calc-ceiling
16547 The @kbd{I F} (@code{calc-ceiling}) [@code{ceil} or @code{fceil}]
16548 command truncates toward positive infinity. Thus @kbd{3.6 I F} produces
16549 4, and @kbd{_3.6 I F} produces @mathit{-3}.
16559 The @kbd{R} (@code{calc-round}) [@code{round} or @code{fround}] command
16560 rounds to the nearest integer. When the fractional part is .5 exactly,
16561 this command rounds away from zero. (All other rounding in the
16562 Calculator uses this convention as well.) Thus @kbd{3.5 R} produces 4
16563 but @kbd{3.4 R} produces 3; @kbd{_3.5 R} produces @mathit{-4}.
16573 The @kbd{I R} (@code{calc-trunc}) [@code{trunc} or @code{ftrunc}]
16574 command truncates toward zero. In other words, it ``chops off''
16575 everything after the decimal point. Thus @kbd{3.6 I R} produces 3 and
16576 @kbd{_3.6 I R} produces @mathit{-3}.
16578 These functions may not be applied meaningfully to error forms, but they
16579 do work for intervals. As a convenience, applying @code{floor} to a
16580 modulo form floors the value part of the form. Applied to a vector,
16581 these functions operate on all elements of the vector one by one.
16582 Applied to a date form, they operate on the internal numerical
16583 representation of dates, converting a date/time form into a pure date.
16601 There are two more rounding functions which can only be entered in
16602 algebraic notation. The @code{roundu} function is like @code{round}
16603 except that it rounds up, toward plus infinity, when the fractional
16604 part is .5. This distinction matters only for negative arguments.
16605 Also, @code{rounde} rounds to an even number in the case of a tie,
16606 rounding up or down as necessary. For example, @samp{rounde(3.5)} and
16607 @samp{rounde(4.5)} both return 4, but @samp{rounde(5.5)} returns 6.
16608 The advantage of round-to-even is that the net error due to rounding
16609 after a long calculation tends to cancel out to zero. An important
16610 subtle point here is that the number being fed to @code{rounde} will
16611 already have been rounded to the current precision before @code{rounde}
16612 begins. For example, @samp{rounde(2.500001)} with a current precision
16613 of 6 will incorrectly, or at least surprisingly, yield 2 because the
16614 argument will first have been rounded down to @expr{2.5} (which
16615 @code{rounde} sees as an exact tie between 2 and 3).
16617 Each of these functions, when written in algebraic formulas, allows
16618 a second argument which specifies the number of digits after the
16619 decimal point to keep. For example, @samp{round(123.4567, 2)} will
16620 produce the answer 123.46, and @samp{round(123.4567, -1)} will
16621 produce 120 (i.e., the cutoff is one digit to the @emph{left} of
16622 the decimal point). A second argument of zero is equivalent to
16623 no second argument at all.
16625 @cindex Fractional part of a number
16626 To compute the fractional part of a number (i.e., the amount which, when
16627 added to `@tfn{floor(}@var{n}@tfn{)}', will produce @var{n}) just take @var{n}
16628 modulo 1 using the @code{%} command.
16630 Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm),
16631 and @kbd{f Q} (integer square root) commands, which are analogous to
16632 @kbd{/}, @kbd{B}, and @kbd{Q}, respectively, except that they take integer
16633 arguments and return the result rounded down to an integer.
16635 @node Complex Number Functions, Conversions, Integer Truncation, Arithmetic
16636 @section Complex Number Functions
16642 The @kbd{J} (@code{calc-conj}) [@code{conj}] command computes the
16643 complex conjugate of a number. For complex number @expr{a+bi}, the
16644 complex conjugate is @expr{a-bi}. If the argument is a real number,
16645 this command leaves it the same. If the argument is a vector or matrix,
16646 this command replaces each element by its complex conjugate.
16649 @pindex calc-argument
16651 The @kbd{G} (@code{calc-argument}) [@code{arg}] command computes the
16652 ``argument'' or polar angle of a complex number. For a number in polar
16653 notation, this is simply the second component of the pair
16654 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'.
16655 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'.
16656 The result is expressed according to the current angular mode and will
16657 be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees
16658 (inclusive), or the equivalent range in radians.
16660 @pindex calc-imaginary
16661 The @code{calc-imaginary} command multiplies the number on the
16662 top of the stack by the imaginary number @expr{i = (0,1)}. This
16663 command is not normally bound to a key in Calc, but it is available
16664 on the @key{IMAG} button in Keypad mode.
16669 The @kbd{f r} (@code{calc-re}) [@code{re}] command replaces a complex number
16670 by its real part. This command has no effect on real numbers. (As an
16671 added convenience, @code{re} applied to a modulo form extracts
16677 The @kbd{f i} (@code{calc-im}) [@code{im}] command replaces a complex number
16678 by its imaginary part; real numbers are converted to zero. With a vector
16679 or matrix argument, these functions operate element-wise.
16684 @kindex v p (complex)
16685 @kindex V p (complex)
16687 The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on
16688 the stack into a composite object such as a complex number. With
16689 a prefix argument of @mathit{-1}, it produces a rectangular complex number;
16690 with an argument of @mathit{-2}, it produces a polar complex number.
16691 (Also, @pxref{Building Vectors}.)
16696 @kindex v u (complex)
16697 @kindex V u (complex)
16698 @pindex calc-unpack
16699 The @kbd{v u} (@code{calc-unpack}) command takes the complex number
16700 (or other composite object) on the top of the stack and unpacks it
16701 into its separate components.
16703 @node Conversions, Date Arithmetic, Complex Number Functions, Arithmetic
16704 @section Conversions
16707 The commands described in this section convert numbers from one form
16708 to another; they are two-key sequences beginning with the letter @kbd{c}.
16713 The @kbd{c f} (@code{calc-float}) [@code{pfloat}] command converts the
16714 number on the top of the stack to floating-point form. For example,
16715 @expr{23} is converted to @expr{23.0}, @expr{3:2} is converted to
16716 @expr{1.5}, and @expr{2.3} is left the same. If the value is a composite
16717 object such as a complex number or vector, each of the components is
16718 converted to floating-point. If the value is a formula, all numbers
16719 in the formula are converted to floating-point. Note that depending
16720 on the current floating-point precision, conversion to floating-point
16721 format may lose information.
16723 As a special exception, integers which appear as powers or subscripts
16724 are not floated by @kbd{c f}. If you really want to float a power,
16725 you can use a @kbd{j s} command to select the power followed by @kbd{c f}.
16726 Because @kbd{c f} cannot examine the formula outside of the selection,
16727 it does not notice that the thing being floated is a power.
16728 @xref{Selecting Subformulas}.
16730 The normal @kbd{c f} command is ``pervasive'' in the sense that it
16731 applies to all numbers throughout the formula. The @code{pfloat}
16732 algebraic function never stays around in a formula; @samp{pfloat(a + 1)}
16733 changes to @samp{a + 1.0} as soon as it is evaluated.
16737 With the Hyperbolic flag, @kbd{H c f} [@code{float}] operates
16738 only on the number or vector of numbers at the top level of its
16739 argument. Thus, @samp{float(1)} is 1.0, but @samp{float(a + 1)}
16740 is left unevaluated because its argument is not a number.
16742 You should use @kbd{H c f} if you wish to guarantee that the final
16743 value, once all the variables have been assigned, is a float; you
16744 would use @kbd{c f} if you wish to do the conversion on the numbers
16745 that appear right now.
16748 @pindex calc-fraction
16750 The @kbd{c F} (@code{calc-fraction}) [@code{pfrac}] command converts a
16751 floating-point number into a fractional approximation. By default, it
16752 produces a fraction whose decimal representation is the same as the
16753 input number, to within the current precision. You can also give a
16754 numeric prefix argument to specify a tolerance, either directly, or,
16755 if the prefix argument is zero, by using the number on top of the stack
16756 as the tolerance. If the tolerance is a positive integer, the fraction
16757 is correct to within that many significant figures. If the tolerance is
16758 a non-positive integer, it specifies how many digits fewer than the current
16759 precision to use. If the tolerance is a floating-point number, the
16760 fraction is correct to within that absolute amount.
16764 The @code{pfrac} function is pervasive, like @code{pfloat}.
16765 There is also a non-pervasive version, @kbd{H c F} [@code{frac}],
16766 which is analogous to @kbd{H c f} discussed above.
16769 @pindex calc-to-degrees
16771 The @kbd{c d} (@code{calc-to-degrees}) [@code{deg}] command converts a
16772 number into degrees form. The value on the top of the stack may be an
16773 HMS form (interpreted as degrees-minutes-seconds), or a real number which
16774 will be interpreted in radians regardless of the current angular mode.
16777 @pindex calc-to-radians
16779 The @kbd{c r} (@code{calc-to-radians}) [@code{rad}] command converts an
16780 HMS form or angle in degrees into an angle in radians.
16783 @pindex calc-to-hms
16785 The @kbd{c h} (@code{calc-to-hms}) [@code{hms}] command converts a real
16786 number, interpreted according to the current angular mode, to an HMS
16787 form describing the same angle. In algebraic notation, the @code{hms}
16788 function also accepts three arguments: @samp{hms(@var{h}, @var{m}, @var{s})}.
16789 (The three-argument version is independent of the current angular mode.)
16791 @pindex calc-from-hms
16792 The @code{calc-from-hms} command converts the HMS form on the top of the
16793 stack into a real number according to the current angular mode.
16800 The @kbd{c p} (@code{calc-polar}) command converts the complex number on
16801 the top of the stack from polar to rectangular form, or from rectangular
16802 to polar form, whichever is appropriate. Real numbers are left the same.
16803 This command is equivalent to the @code{rect} or @code{polar}
16804 functions in algebraic formulas, depending on the direction of
16805 conversion. (It uses @code{polar}, except that if the argument is
16806 already a polar complex number, it uses @code{rect} instead. The
16807 @kbd{I c p} command always uses @code{rect}.)
16812 The @kbd{c c} (@code{calc-clean}) [@code{pclean}] command ``cleans'' the
16813 number on the top of the stack. Floating point numbers are re-rounded
16814 according to the current precision. Polar numbers whose angular
16815 components have strayed from the @mathit{-180} to @mathit{+180} degree range
16816 are normalized. (Note that results will be undesirable if the current
16817 angular mode is different from the one under which the number was
16818 produced!) Integers and fractions are generally unaffected by this
16819 operation. Vectors and formulas are cleaned by cleaning each component
16820 number (i.e., pervasively).
16822 If the simplification mode is set below basic simplification, it is raised
16823 for the purposes of this command. Thus, @kbd{c c} applies the basic
16824 simplifications even if their automatic application is disabled.
16825 @xref{Simplification Modes}.
16827 @cindex Roundoff errors, correcting
16828 A numeric prefix argument to @kbd{c c} sets the floating-point precision
16829 to that value for the duration of the command. A positive prefix (of at
16830 least 3) sets the precision to the specified value; a negative or zero
16831 prefix decreases the precision by the specified amount.
16834 @pindex calc-clean-num
16835 The keystroke sequences @kbd{c 0} through @kbd{c 9} are equivalent
16836 to @kbd{c c} with the corresponding negative prefix argument. If roundoff
16837 errors have changed 2.0 into 1.999999, typing @kbd{c 1} to clip off one
16838 decimal place often conveniently does the trick.
16840 The @kbd{c c} command with a numeric prefix argument, and the @kbd{c 0}
16841 through @kbd{c 9} commands, also ``clip'' very small floating-point
16842 numbers to zero. If the exponent is less than or equal to the negative
16843 of the specified precision, the number is changed to 0.0. For example,
16844 if the current precision is 12, then @kbd{c 2} changes the vector
16845 @samp{[1e-8, 1e-9, 1e-10, 1e-11]} to @samp{[1e-8, 1e-9, 0, 0]}.
16846 Numbers this small generally arise from roundoff noise.
16848 If the numbers you are using really are legitimately this small,
16849 you should avoid using the @kbd{c 0} through @kbd{c 9} commands.
16850 (The plain @kbd{c c} command rounds to the current precision but
16851 does not clip small numbers.)
16853 One more property of @kbd{c 0} through @kbd{c 9}, and of @kbd{c c} with
16854 a prefix argument, is that integer-valued floats are converted to
16855 plain integers, so that @kbd{c 1} on @samp{[1., 1.5, 2., 2.5, 3.]}
16856 produces @samp{[1, 1.5, 2, 2.5, 3]}. This is not done for huge
16857 numbers (@samp{1e100} is technically an integer-valued float, but
16858 you wouldn't want it automatically converted to a 100-digit integer).
16863 With the Hyperbolic flag, @kbd{H c c} and @kbd{H c 0} through @kbd{H c 9}
16864 operate non-pervasively [@code{clean}].
16866 @node Date Arithmetic, Financial Functions, Conversions, Arithmetic
16867 @section Date Arithmetic
16870 @cindex Date arithmetic, additional functions
16871 The commands described in this section perform various conversions
16872 and calculations involving date forms (@pxref{Date Forms}). They
16873 use the @kbd{t} (for time/date) prefix key followed by shifted
16876 The simplest date arithmetic is done using the regular @kbd{+} and @kbd{-}
16877 commands. In particular, adding a number to a date form advances the
16878 date form by a certain number of days; adding an HMS form to a date
16879 form advances the date by a certain amount of time; and subtracting two
16880 date forms produces a difference measured in days. The commands
16881 described here provide additional, more specialized operations on dates.
16883 Many of these commands accept a numeric prefix argument; if you give
16884 plain @kbd{C-u} as the prefix, these commands will instead take the
16885 additional argument from the top of the stack.
16888 * Date Conversions::
16894 @node Date Conversions, Date Functions, Date Arithmetic, Date Arithmetic
16895 @subsection Date Conversions
16901 The @kbd{t D} (@code{calc-date}) [@code{date}] command converts a
16902 date form into a number, measured in days since Jan 1, 1 AD@. The
16903 result will be an integer if @var{date} is a pure date form, or a
16904 fraction or float if @var{date} is a date/time form. Or, if its
16905 argument is a number, it converts this number into a date form.
16907 With a numeric prefix argument, @kbd{t D} takes that many objects
16908 (up to six) from the top of the stack and interprets them in one
16909 of the following ways:
16911 The @samp{date(@var{year}, @var{month}, @var{day})} function
16912 builds a pure date form out of the specified year, month, and
16913 day, which must all be integers. @var{Year} is a year number,
16914 such as 1991 (@emph{not} the same as 91!). @var{Month} must be
16915 an integer in the range 1 to 12; @var{day} must be in the range
16916 1 to 31. If the specified month has fewer than 31 days and
16917 @var{day} is too large, the equivalent day in the following
16918 month will be used.
16920 The @samp{date(@var{month}, @var{day})} function builds a
16921 pure date form using the current year, as determined by the
16924 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hms})}
16925 function builds a date/time form using an @var{hms} form.
16927 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hour},
16928 @var{minute}, @var{second})} function builds a date/time form.
16929 @var{hour} should be an integer in the range 0 to 23;
16930 @var{minute} should be an integer in the range 0 to 59;
16931 @var{second} should be any real number in the range @samp{[0 .. 60)}.
16932 The last two arguments default to zero if omitted.
16935 @pindex calc-julian
16937 @cindex Julian day counts, conversions
16938 The @kbd{t J} (@code{calc-julian}) [@code{julian}] command converts
16939 a date form into a Julian day count, which is the number of days
16940 since noon (GMT) on Jan 1, 4713 BC@. A pure date is converted to an
16941 integer Julian count representing noon of that day. A date/time form
16942 is converted to an exact floating-point Julian count, adjusted to
16943 interpret the date form in the current time zone but the Julian
16944 day count in Greenwich Mean Time. A numeric prefix argument allows
16945 you to specify the time zone; @pxref{Time Zones}. Use a prefix of
16946 zero to suppress the time zone adjustment. Note that pure date forms
16947 are never time-zone adjusted.
16949 This command can also do the opposite conversion, from a Julian day
16950 count (either an integer day, or a floating-point day and time in
16951 the GMT zone), into a pure date form or a date/time form in the
16952 current or specified time zone.
16955 @pindex calc-unix-time
16957 @cindex Unix time format, conversions
16958 The @kbd{t U} (@code{calc-unix-time}) [@code{unixtime}] command
16959 converts a date form into a Unix time value, which is the number of
16960 seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result
16961 will be an integer if the current precision is 12 or less; for higher
16962 precision, the result may be a float with (@var{precision}@minus{}12)
16963 digits after the decimal. Just as for @kbd{t J}, the numeric time
16964 is interpreted in the GMT time zone and the date form is interpreted
16965 in the current or specified zone. Some systems use Unix-like
16966 numbering but with the local time zone; give a prefix of zero to
16967 suppress the adjustment if so.
16970 @pindex calc-convert-time-zones
16972 @cindex Time Zones, converting between
16973 The @kbd{t C} (@code{calc-convert-time-zones}) [@code{tzconv}]
16974 command converts a date form from one time zone to another. You
16975 are prompted for each time zone name in turn; you can answer with
16976 any suitable Calc time zone expression (@pxref{Time Zones}).
16977 If you answer either prompt with a blank line, the local time
16978 zone is used for that prompt. You can also answer the first
16979 prompt with @kbd{$} to take the two time zone names from the
16980 stack (and the date to be converted from the third stack level).
16982 @node Date Functions, Business Days, Date Conversions, Date Arithmetic
16983 @subsection Date Functions
16989 The @kbd{t N} (@code{calc-now}) [@code{now}] command pushes the
16990 current date and time on the stack as a date form. The time is
16991 reported in terms of the specified time zone; with no numeric prefix
16992 argument, @kbd{t N} reports for the current time zone.
16995 @pindex calc-date-part
16996 The @kbd{t P} (@code{calc-date-part}) command extracts one part
16997 of a date form. The prefix argument specifies the part; with no
16998 argument, this command prompts for a part code from 1 to 9.
16999 The various part codes are described in the following paragraphs.
17002 The @kbd{M-1 t P} [@code{year}] function extracts the year number
17003 from a date form as an integer, e.g., 1991. This and the
17004 following functions will also accept a real number for an
17005 argument, which is interpreted as a standard Calc day number.
17006 Note that this function will never return zero, since the year
17007 1 BC immediately precedes the year 1 AD.
17010 The @kbd{M-2 t P} [@code{month}] function extracts the month number
17011 from a date form as an integer in the range 1 to 12.
17014 The @kbd{M-3 t P} [@code{day}] function extracts the day number
17015 from a date form as an integer in the range 1 to 31.
17018 The @kbd{M-4 t P} [@code{hour}] function extracts the hour from
17019 a date form as an integer in the range 0 (midnight) to 23. Note
17020 that 24-hour time is always used. This returns zero for a pure
17021 date form. This function (and the following two) also accept
17022 HMS forms as input.
17025 The @kbd{M-5 t P} [@code{minute}] function extracts the minute
17026 from a date form as an integer in the range 0 to 59.
17029 The @kbd{M-6 t P} [@code{second}] function extracts the second
17030 from a date form. If the current precision is 12 or less,
17031 the result is an integer in the range 0 to 59. For higher
17032 precision, the result may instead be a floating-point number.
17035 The @kbd{M-7 t P} [@code{weekday}] function extracts the weekday
17036 number from a date form as an integer in the range 0 (Sunday)
17040 The @kbd{M-8 t P} [@code{yearday}] function extracts the day-of-year
17041 number from a date form as an integer in the range 1 (January 1)
17042 to 366 (December 31 of a leap year).
17045 The @kbd{M-9 t P} [@code{time}] function extracts the time portion
17046 of a date form as an HMS form. This returns @samp{0@@ 0' 0"}
17047 for a pure date form.
17050 @pindex calc-new-month
17052 The @kbd{t M} (@code{calc-new-month}) [@code{newmonth}] command
17053 computes a new date form that represents the first day of the month
17054 specified by the input date. The result is always a pure date
17055 form; only the year and month numbers of the input are retained.
17056 With a numeric prefix argument @var{n} in the range from 1 to 31,
17057 @kbd{t M} computes the @var{n}th day of the month. (If @var{n}
17058 is greater than the actual number of days in the month, or if
17059 @var{n} is zero, the last day of the month is used.)
17062 @pindex calc-new-year
17064 The @kbd{t Y} (@code{calc-new-year}) [@code{newyear}] command
17065 computes a new pure date form that represents the first day of
17066 the year specified by the input. The month, day, and time
17067 of the input date form are lost. With a numeric prefix argument
17068 @var{n} in the range from 1 to 366, @kbd{t Y} computes the
17069 @var{n}th day of the year (366 is treated as 365 in non-leap
17070 years). A prefix argument of 0 computes the last day of the
17071 year (December 31). A negative prefix argument from @mathit{-1} to
17072 @mathit{-12} computes the first day of the @var{n}th month of the year.
17075 @pindex calc-new-week
17077 The @kbd{t W} (@code{calc-new-week}) [@code{newweek}] command
17078 computes a new pure date form that represents the Sunday on or before
17079 the input date. With a numeric prefix argument, it can be made to
17080 use any day of the week as the starting day; the argument must be in
17081 the range from 0 (Sunday) to 6 (Saturday). This function always
17082 subtracts between 0 and 6 days from the input date.
17084 Here's an example use of @code{newweek}: Find the date of the next
17085 Wednesday after a given date. Using @kbd{M-3 t W} or @samp{newweek(d, 3)}
17086 will give you the @emph{preceding} Wednesday, so @samp{newweek(d+7, 3)}
17087 will give you the following Wednesday. A further look at the definition
17088 of @code{newweek} shows that if the input date is itself a Wednesday,
17089 this formula will return the Wednesday one week in the future. An
17090 exercise for the reader is to modify this formula to yield the same day
17091 if the input is already a Wednesday. Another interesting exercise is
17092 to preserve the time-of-day portion of the input (@code{newweek} resets
17093 the time to midnight; hint: how can @code{newweek} be defined in terms
17094 of the @code{weekday} function?).
17100 The @samp{pwday(@var{date})} function (not on any key) computes the
17101 day-of-month number of the Sunday on or before @var{date}. With
17102 two arguments, @samp{pwday(@var{date}, @var{day})} computes the day
17103 number of the Sunday on or before day number @var{day} of the month
17104 specified by @var{date}. The @var{day} must be in the range from
17105 7 to 31; if the day number is greater than the actual number of days
17106 in the month, the true number of days is used instead. Thus
17107 @samp{pwday(@var{date}, 7)} finds the first Sunday of the month, and
17108 @samp{pwday(@var{date}, 31)} finds the last Sunday of the month.
17109 With a third @var{weekday} argument, @code{pwday} can be made to look
17110 for any day of the week instead of Sunday.
17113 @pindex calc-inc-month
17115 The @kbd{t I} (@code{calc-inc-month}) [@code{incmonth}] command
17116 increases a date form by one month, or by an arbitrary number of
17117 months specified by a numeric prefix argument. The time portion,
17118 if any, of the date form stays the same. The day also stays the
17119 same, except that if the new month has fewer days the day
17120 number may be reduced to lie in the valid range. For example,
17121 @samp{incmonth(<Jan 31, 1991>)} produces @samp{<Feb 28, 1991>}.
17122 Because of this, @kbd{t I t I} and @kbd{M-2 t I} do not always give
17123 the same results (@samp{<Mar 28, 1991>} versus @samp{<Mar 31, 1991>}
17130 The @samp{incyear(@var{date}, @var{step})} function increases
17131 a date form by the specified number of years, which may be
17132 any positive or negative integer. Note that @samp{incyear(d, n)}
17133 is equivalent to @w{@samp{incmonth(d, 12*n)}}, but these do not have
17134 simple equivalents in terms of day arithmetic because
17135 months and years have varying lengths. If the @var{step}
17136 argument is omitted, 1 year is assumed. There is no keyboard
17137 command for this function; use @kbd{C-u 12 t I} instead.
17139 There is no @code{newday} function at all because @kbd{F} [@code{floor}]
17140 serves this purpose. Similarly, instead of @code{incday} and
17141 @code{incweek} simply use @expr{d + n} or @expr{d + 7 n}.
17143 @xref{Basic Arithmetic}, for the @kbd{f ]} [@code{incr}] command
17144 which can adjust a date/time form by a certain number of seconds.
17146 @node Business Days, Time Zones, Date Functions, Date Arithmetic
17147 @subsection Business Days
17150 Often time is measured in ``business days'' or ``working days,''
17151 where weekends and holidays are skipped. Calc's normal date
17152 arithmetic functions use calendar days, so that subtracting two
17153 consecutive Mondays will yield a difference of 7 days. By contrast,
17154 subtracting two consecutive Mondays would yield 5 business days
17155 (assuming two-day weekends and the absence of holidays).
17161 @pindex calc-business-days-plus
17162 @pindex calc-business-days-minus
17163 The @kbd{t +} (@code{calc-business-days-plus}) [@code{badd}]
17164 and @kbd{t -} (@code{calc-business-days-minus}) [@code{bsub}]
17165 commands perform arithmetic using business days. For @kbd{t +},
17166 one argument must be a date form and the other must be a real
17167 number (positive or negative). If the number is not an integer,
17168 then a certain amount of time is added as well as a number of
17169 days; for example, adding 0.5 business days to a time in Friday
17170 evening will produce a time in Monday morning. It is also
17171 possible to add an HMS form; adding @samp{12@@ 0' 0"} also adds
17172 half a business day. For @kbd{t -}, the arguments are either a
17173 date form and a number or HMS form, or two date forms, in which
17174 case the result is the number of business days between the two
17177 @cindex @code{Holidays} variable
17179 By default, Calc considers any day that is not a Saturday or
17180 Sunday to be a business day. You can define any number of
17181 additional holidays by editing the variable @code{Holidays}.
17182 (There is an @w{@kbd{s H}} convenience command for editing this
17183 variable.) Initially, @code{Holidays} contains the vector
17184 @samp{[sat, sun]}. Entries in the @code{Holidays} vector may
17185 be any of the following kinds of objects:
17189 Date forms (pure dates, not date/time forms). These specify
17190 particular days which are to be treated as holidays.
17193 Intervals of date forms. These specify a range of days, all of
17194 which are holidays (e.g., Christmas week). @xref{Interval Forms}.
17197 Nested vectors of date forms. Each date form in the vector is
17198 considered to be a holiday.
17201 Any Calc formula which evaluates to one of the above three things.
17202 If the formula involves the variable @expr{y}, it stands for a
17203 yearly repeating holiday; @expr{y} will take on various year
17204 numbers like 1992. For example, @samp{date(y, 12, 25)} specifies
17205 Christmas day, and @samp{newweek(date(y, 11, 7), 4) + 21} specifies
17206 Thanksgiving (which is held on the fourth Thursday of November).
17207 If the formula involves the variable @expr{m}, that variable
17208 takes on month numbers from 1 to 12: @samp{date(y, m, 15)} is
17209 a holiday that takes place on the 15th of every month.
17212 A weekday name, such as @code{sat} or @code{sun}. This is really
17213 a variable whose name is a three-letter, lower-case day name.
17216 An interval of year numbers (integers). This specifies the span of
17217 years over which this holiday list is to be considered valid. Any
17218 business-day arithmetic that goes outside this range will result
17219 in an error message. Use this if you are including an explicit
17220 list of holidays, rather than a formula to generate them, and you
17221 want to make sure you don't accidentally go beyond the last point
17222 where the holidays you entered are complete. If there is no
17223 limiting interval in the @code{Holidays} vector, the default
17224 @samp{[1 .. 2737]} is used. (This is the absolute range of years
17225 for which Calc's business-day algorithms will operate.)
17228 An interval of HMS forms. This specifies the span of hours that
17229 are to be considered one business day. For example, if this
17230 range is @samp{[9@@ 0' 0" .. 17@@ 0' 0"]} (i.e., 9am to 5pm), then
17231 the business day is only eight hours long, so that @kbd{1.5 t +}
17232 on @samp{<4:00pm Fri Dec 13, 1991>} will add one business day and
17233 four business hours to produce @samp{<12:00pm Tue Dec 17, 1991>}.
17234 Likewise, @kbd{t -} will now express differences in time as
17235 fractions of an eight-hour day. Times before 9am will be treated
17236 as 9am by business date arithmetic, and times at or after 5pm will
17237 be treated as 4:59:59pm. If there is no HMS interval in @code{Holidays},
17238 the full 24-hour day @samp{[0@ 0' 0" .. 24@ 0' 0"]} is assumed.
17239 (Regardless of the type of bounds you specify, the interval is
17240 treated as inclusive on the low end and exclusive on the high end,
17241 so that the work day goes from 9am up to, but not including, 5pm.)
17244 If the @code{Holidays} vector is empty, then @kbd{t +} and
17245 @kbd{t -} will act just like @kbd{+} and @kbd{-} because there will
17246 then be no difference between business days and calendar days.
17248 Calc expands the intervals and formulas you give into a complete
17249 list of holidays for internal use. This is done mainly to make
17250 sure it can detect multiple holidays. (For example,
17251 @samp{<Jan 1, 1989>} is both New Year's Day and a Sunday, but
17252 Calc's algorithms take care to count it only once when figuring
17253 the number of holidays between two dates.)
17255 Since the complete list of holidays for all the years from 1 to
17256 2737 would be huge, Calc actually computes only the part of the
17257 list between the smallest and largest years that have been involved
17258 in business-day calculations so far. Normally, you won't have to
17259 worry about this. Keep in mind, however, that if you do one
17260 calculation for 1992, and another for 1792, even if both involve
17261 only a small range of years, Calc will still work out all the
17262 holidays that fall in that 200-year span.
17264 If you add a (positive) number of days to a date form that falls on a
17265 weekend or holiday, the date form is treated as if it were the most
17266 recent business day. (Thus adding one business day to a Friday,
17267 Saturday, or Sunday will all yield the following Monday.) If you
17268 subtract a number of days from a weekend or holiday, the date is
17269 effectively on the following business day. (So subtracting one business
17270 day from Saturday, Sunday, or Monday yields the preceding Friday.) The
17271 difference between two dates one or both of which fall on holidays
17272 equals the number of actual business days between them. These
17273 conventions are consistent in the sense that, if you add @var{n}
17274 business days to any date, the difference between the result and the
17275 original date will come out to @var{n} business days. (It can't be
17276 completely consistent though; a subtraction followed by an addition
17277 might come out a bit differently, since @kbd{t +} is incapable of
17278 producing a date that falls on a weekend or holiday.)
17284 There is a @code{holiday} function, not on any keys, that takes
17285 any date form and returns 1 if that date falls on a weekend or
17286 holiday, as defined in @code{Holidays}, or 0 if the date is a
17289 @node Time Zones, , Business Days, Date Arithmetic
17290 @subsection Time Zones
17294 @cindex Daylight saving time
17295 Time zones and daylight saving time are a complicated business.
17296 The conversions to and from Julian and Unix-style dates automatically
17297 compute the correct time zone and daylight saving adjustment to use,
17298 provided they can figure out this information. This section describes
17299 Calc's time zone adjustment algorithm in detail, in case you want to
17300 do conversions in different time zones or in case Calc's algorithms
17301 can't determine the right correction to use.
17303 Adjustments for time zones and daylight saving time are done by
17304 @kbd{t U}, @kbd{t J}, @kbd{t N}, and @kbd{t C}, but not by any other
17305 commands. In particular, @samp{<may 1 1991> - <apr 1 1991>} evaluates
17306 to exactly 30 days even though there is a daylight-saving
17307 transition in between. This is also true for Julian pure dates:
17308 @samp{julian(<may 1 1991>) - julian(<apr 1 1991>)}. But Julian
17309 and Unix date/times will adjust for daylight saving time: using Calc's
17310 default daylight saving time rule (see the explanation below),
17311 @samp{julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)}
17312 evaluates to @samp{29.95833} (that's 29 days and 23 hours)
17313 because one hour was lost when daylight saving commenced on
17316 In brief, the idiom @samp{julian(@var{date1}) - julian(@var{date2})}
17317 computes the actual number of 24-hour periods between two dates, whereas
17318 @samp{@var{date1} - @var{date2}} computes the number of calendar
17319 days between two dates without taking daylight saving into account.
17321 @pindex calc-time-zone
17326 The @code{calc-time-zone} [@code{tzone}] command converts the time
17327 zone specified by its numeric prefix argument into a number of
17328 seconds difference from Greenwich mean time (GMT). If the argument
17329 is a number, the result is simply that value multiplied by 3600.
17330 Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If
17331 Daylight Saving time is in effect, one hour should be subtracted from
17332 the normal difference.
17334 If you give a prefix of plain @kbd{C-u}, @code{calc-time-zone} (like other
17335 date arithmetic commands that include a time zone argument) takes the
17336 zone argument from the top of the stack. (In the case of @kbd{t J}
17337 and @kbd{t U}, the normal argument is then taken from the second-to-top
17338 stack position.) This allows you to give a non-integer time zone
17339 adjustment. The time-zone argument can also be an HMS form, or
17340 it can be a variable which is a time zone name in upper- or lower-case.
17341 For example @samp{tzone(PST) = tzone(8)} and @samp{tzone(pdt) = tzone(7)}
17342 (for Pacific standard and daylight saving times, respectively).
17344 North American and European time zone names are defined as follows;
17345 note that for each time zone there is one name for standard time,
17346 another for daylight saving time, and a third for ``generalized'' time
17347 in which the daylight saving adjustment is computed from context.
17351 YST PST MST CST EST AST NST GMT WET MET MEZ
17352 9 8 7 6 5 4 3.5 0 -1 -2 -2
17354 YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ
17355 8 7 6 5 4 3 2.5 -1 -2 -3 -3
17357 YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ
17358 9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3
17362 @vindex math-tzone-names
17363 To define time zone names that do not appear in the above table,
17364 you must modify the Lisp variable @code{math-tzone-names}. This
17365 is a list of lists describing the different time zone names; its
17366 structure is best explained by an example. The three entries for
17367 Pacific Time look like this:
17371 ( ( "PST" 8 0 ) ; Name as an upper-case string, then standard
17372 ( "PDT" 8 -1 ) ; adjustment, then daylight saving adjustment.
17373 ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone.
17377 @cindex @code{TimeZone} variable
17379 With no arguments, @code{calc-time-zone} or @samp{tzone()} will by
17380 default get the time zone and daylight saving information from the
17381 calendar (@pxref{Daylight Saving,Calendar/Diary,The Calendar and the Diary,
17382 emacs,The GNU Emacs Manual}). To use a different time zone, or if the
17383 calendar does not give the desired result, you can set the Calc variable
17384 @code{TimeZone} (which is by default @code{nil}) to an appropriate
17385 time zone name. (The easiest way to do this is to edit the
17386 @code{TimeZone} variable using Calc's @kbd{s T} command, then use the
17387 @kbd{s p} (@code{calc-permanent-variable}) command to save the value of
17388 @code{TimeZone} permanently.)
17389 If the time zone given by @code{TimeZone} is a generalized time zone,
17390 e.g., @code{EGT}, Calc examines the date being converted to tell whether
17391 to use standard or daylight saving time. But if the current time zone
17392 is explicit, e.g., @code{EST} or @code{EDT}, then that adjustment is
17393 used exactly and Calc's daylight saving algorithm is not consulted.
17394 The special time zone name @code{local}
17395 is equivalent to no argument; i.e., it uses the information obtained
17398 The @kbd{t J} and @code{t U} commands with no numeric prefix
17399 arguments do the same thing as @samp{tzone()}; namely, use the
17400 information from the calendar if @code{TimeZone} is @code{nil},
17401 otherwise use the time zone given by @code{TimeZone}.
17403 @vindex math-daylight-savings-hook
17404 @findex math-std-daylight-savings
17405 When Calc computes the daylight saving information itself (i.e., when
17406 the @code{TimeZone} variable is set), it will by default consider
17407 daylight saving time to begin at 2 a.m.@: on the second Sunday of March
17408 (for years from 2007 on) or on the last Sunday in April (for years
17409 before 2007), and to end at 2 a.m.@: on the first Sunday of
17410 November. (for years from 2007 on) or the last Sunday in October (for
17411 years before 2007). These are the rules that have been in effect in
17412 much of North America since 1966 and take into account the rule change
17413 that began in 2007. If you are in a country that uses different rules
17414 for computing daylight saving time, you have two choices: Write your own
17415 daylight saving hook, or control time zones explicitly by setting the
17416 @code{TimeZone} variable and/or always giving a time-zone argument for
17417 the conversion functions.
17419 The Lisp variable @code{math-daylight-savings-hook} holds the
17420 name of a function that is used to compute the daylight saving
17421 adjustment for a given date. The default is
17422 @code{math-std-daylight-savings}, which computes an adjustment
17423 (either 0 or @mathit{-1}) using the North American rules given above.
17425 The daylight saving hook function is called with four arguments:
17426 The date, as a floating-point number in standard Calc format;
17427 a six-element list of the date decomposed into year, month, day,
17428 hour, minute, and second, respectively; a string which contains
17429 the generalized time zone name in upper-case, e.g., @code{"WEGT"};
17430 and a special adjustment to be applied to the hour value when
17431 converting into a generalized time zone (see below).
17433 @findex math-prev-weekday-in-month
17434 The Lisp function @code{math-prev-weekday-in-month} is useful for
17435 daylight saving computations. This is an internal version of
17436 the user-level @code{pwday} function described in the previous
17437 section. It takes four arguments: The floating-point date value,
17438 the corresponding six-element date list, the day-of-month number,
17439 and the weekday number (0-6).
17441 The default daylight saving hook ignores the time zone name, but a
17442 more sophisticated hook could use different algorithms for different
17443 time zones. It would also be possible to use different algorithms
17444 depending on the year number, but the default hook always uses the
17445 algorithm for 1987 and later. Here is a listing of the default
17446 daylight saving hook:
17449 (defun math-std-daylight-savings (date dt zone bump)
17450 (cond ((< (nth 1 dt) 4) 0)
17452 (let ((sunday (math-prev-weekday-in-month date dt 7 0)))
17453 (cond ((< (nth 2 dt) sunday) 0)
17454 ((= (nth 2 dt) sunday)
17455 (if (>= (nth 3 dt) (+ 3 bump)) -1 0))
17457 ((< (nth 1 dt) 10) -1)
17459 (let ((sunday (math-prev-weekday-in-month date dt 31 0)))
17460 (cond ((< (nth 2 dt) sunday) -1)
17461 ((= (nth 2 dt) sunday)
17462 (if (>= (nth 3 dt) (+ 2 bump)) 0 -1))
17469 The @code{bump} parameter is equal to zero when Calc is converting
17470 from a date form in a generalized time zone into a GMT date value.
17471 It is @mathit{-1} when Calc is converting in the other direction. The
17472 adjustments shown above ensure that the conversion behaves correctly
17473 and reasonably around the 2 a.m.@: transition in each direction.
17475 There is a ``missing'' hour between 2 a.m.@: and 3 a.m.@: at the
17476 beginning of daylight saving time; converting a date/time form that
17477 falls in this hour results in a time value for the following hour,
17478 from 3 a.m.@: to 4 a.m. At the end of daylight saving time, the
17479 hour from 1 a.m.@: to 2 a.m.@: repeats itself; converting a date/time
17480 form that falls in this hour results in a time value for the first
17481 manifestation of that time (@emph{not} the one that occurs one hour
17484 If @code{math-daylight-savings-hook} is @code{nil}, then the
17485 daylight saving adjustment is always taken to be zero.
17487 In algebraic formulas, @samp{tzone(@var{zone}, @var{date})}
17488 computes the time zone adjustment for a given zone name at a
17489 given date. The @var{date} is ignored unless @var{zone} is a
17490 generalized time zone. If @var{date} is a date form, the
17491 daylight saving computation is applied to it as it appears.
17492 If @var{date} is a numeric date value, it is adjusted for the
17493 daylight-saving version of @var{zone} before being given to
17494 the daylight saving hook. This odd-sounding rule ensures
17495 that the daylight-saving computation is always done in
17496 local time, not in the GMT time that a numeric @var{date}
17497 is typically represented in.
17503 The @samp{dsadj(@var{date}, @var{zone})} function computes the
17504 daylight saving adjustment that is appropriate for @var{date} in
17505 time zone @var{zone}. If @var{zone} is explicitly in or not in
17506 daylight saving time (e.g., @code{PDT} or @code{PST}) the
17507 @var{date} is ignored. If @var{zone} is a generalized time zone,
17508 the algorithms described above are used. If @var{zone} is omitted,
17509 the computation is done for the current time zone.
17511 @node Financial Functions, Binary Functions, Date Arithmetic, Arithmetic
17512 @section Financial Functions
17515 Calc's financial or business functions use the @kbd{b} prefix
17516 key followed by a shifted letter. (The @kbd{b} prefix followed by
17517 a lower-case letter is used for operations on binary numbers.)
17519 Note that the rate and the number of intervals given to these
17520 functions must be on the same time scale, e.g., both months or
17521 both years. Mixing an annual interest rate with a time expressed
17522 in months will give you very wrong answers!
17524 It is wise to compute these functions to a higher precision than
17525 you really need, just to make sure your answer is correct to the
17526 last penny; also, you may wish to check the definitions at the end
17527 of this section to make sure the functions have the meaning you expect.
17533 * Related Financial Functions::
17534 * Depreciation Functions::
17535 * Definitions of Financial Functions::
17538 @node Percentages, Future Value, Financial Functions, Financial Functions
17539 @subsection Percentages
17542 @pindex calc-percent
17545 The @kbd{M-%} (@code{calc-percent}) command takes a percentage value,
17546 say 5.4, and converts it to an equivalent actual number. For example,
17547 @kbd{5.4 M-%} enters 0.054 on the stack. (That's the @key{META} or
17548 @key{ESC} key combined with @kbd{%}.)
17550 Actually, @kbd{M-%} creates a formula of the form @samp{5.4%}.
17551 You can enter @samp{5.4%} yourself during algebraic entry. The
17552 @samp{%} operator simply means, ``the preceding value divided by
17553 100.'' The @samp{%} operator has very high precedence, so that
17554 @samp{1+8%} is interpreted as @samp{1+(8%)}, not as @samp{(1+8)%}.
17555 (The @samp{%} operator is just a postfix notation for the
17556 @code{percent} function, just like @samp{20!} is the notation for
17557 @samp{fact(20)}, or twenty-factorial.)
17559 The formula @samp{5.4%} would normally evaluate immediately to
17560 0.054, but the @kbd{M-%} command suppresses evaluation as it puts
17561 the formula onto the stack. However, the next Calc command that
17562 uses the formula @samp{5.4%} will evaluate it as its first step.
17563 The net effect is that you get to look at @samp{5.4%} on the stack,
17564 but Calc commands see it as @samp{0.054}, which is what they expect.
17566 In particular, @samp{5.4%} and @samp{0.054} are suitable values
17567 for the @var{rate} arguments of the various financial functions,
17568 but the number @samp{5.4} is probably @emph{not} suitable---it
17569 represents a rate of 540 percent!
17571 The key sequence @kbd{M-% *} effectively means ``percent-of.''
17572 For example, @kbd{68 @key{RET} 25 M-% *} computes 17, which is 25% of
17573 68 (and also 68% of 25, which comes out to the same thing).
17576 @pindex calc-convert-percent
17577 The @kbd{c %} (@code{calc-convert-percent}) command converts the
17578 value on the top of the stack from numeric to percentage form.
17579 For example, if 0.08 is on the stack, @kbd{c %} converts it to
17580 @samp{8%}. The quantity is the same, it's just represented
17581 differently. (Contrast this with @kbd{M-%}, which would convert
17582 this number to @samp{0.08%}.) The @kbd{=} key is a convenient way
17583 to convert a formula like @samp{8%} back to numeric form, 0.08.
17585 To compute what percentage one quantity is of another quantity,
17586 use @kbd{/ c %}. For example, @w{@kbd{17 @key{RET} 68 / c %}} displays
17590 @pindex calc-percent-change
17592 The @kbd{b %} (@code{calc-percent-change}) [@code{relch}] command
17593 calculates the percentage change from one number to another.
17594 For example, @kbd{40 @key{RET} 50 b %} produces the answer @samp{25%},
17595 since 50 is 25% larger than 40. A negative result represents a
17596 decrease: @kbd{50 @key{RET} 40 b %} produces @samp{-20%}, since 40 is
17597 20% smaller than 50. (The answers are different in magnitude
17598 because, in the first case, we're increasing by 25% of 40, but
17599 in the second case, we're decreasing by 20% of 50.) The effect
17600 of @kbd{40 @key{RET} 50 b %} is to compute @expr{(50-40)/40}, converting
17601 the answer to percentage form as if by @kbd{c %}.
17603 @node Future Value, Present Value, Percentages, Financial Functions
17604 @subsection Future Value
17608 @pindex calc-fin-fv
17610 The @kbd{b F} (@code{calc-fin-fv}) [@code{fv}] command computes
17611 the future value of an investment. It takes three arguments
17612 from the stack: @samp{fv(@var{rate}, @var{n}, @var{payment})}.
17613 If you give payments of @var{payment} every year for @var{n}
17614 years, and the money you have paid earns interest at @var{rate} per
17615 year, then this function tells you what your investment would be
17616 worth at the end of the period. (The actual interval doesn't
17617 have to be years, as long as @var{n} and @var{rate} are expressed
17618 in terms of the same intervals.) This function assumes payments
17619 occur at the @emph{end} of each interval.
17623 The @kbd{I b F} [@code{fvb}] command does the same computation,
17624 but assuming your payments are at the beginning of each interval.
17625 Suppose you plan to deposit $1000 per year in a savings account
17626 earning 5.4% interest, starting right now. How much will be
17627 in the account after five years? @code{fvb(5.4%, 5, 1000) = 5870.73}.
17628 Thus you will have earned $870 worth of interest over the years.
17629 Using the stack, this calculation would have been
17630 @kbd{5.4 M-% 5 @key{RET} 1000 I b F}. Note that the rate is expressed
17631 as a number between 0 and 1, @emph{not} as a percentage.
17635 The @kbd{H b F} [@code{fvl}] command computes the future value
17636 of an initial lump sum investment. Suppose you could deposit
17637 those five thousand dollars in the bank right now; how much would
17638 they be worth in five years? @code{fvl(5.4%, 5, 5000) = 6503.89}.
17640 The algebraic functions @code{fv} and @code{fvb} accept an optional
17641 fourth argument, which is used as an initial lump sum in the sense
17642 of @code{fvl}. In other words, @code{fv(@var{rate}, @var{n},
17643 @var{payment}, @var{initial}) = fv(@var{rate}, @var{n}, @var{payment})
17644 + fvl(@var{rate}, @var{n}, @var{initial})}.
17646 To illustrate the relationships between these functions, we could
17647 do the @code{fvb} calculation ``by hand'' using @code{fvl}. The
17648 final balance will be the sum of the contributions of our five
17649 deposits at various times. The first deposit earns interest for
17650 five years: @code{fvl(5.4%, 5, 1000) = 1300.78}. The second
17651 deposit only earns interest for four years: @code{fvl(5.4%, 4, 1000) =
17652 1234.13}. And so on down to the last deposit, which earns one
17653 year's interest: @code{fvl(5.4%, 1, 1000) = 1054.00}. The sum of
17654 these five values is, sure enough, $5870.73, just as was computed
17655 by @code{fvb} directly.
17657 What does @code{fv(5.4%, 5, 1000) = 5569.96} mean? The payments
17658 are now at the ends of the periods. The end of one year is the same
17659 as the beginning of the next, so what this really means is that we've
17660 lost the payment at year zero (which contributed $1300.78), but we're
17661 now counting the payment at year five (which, since it didn't have
17662 a chance to earn interest, counts as $1000). Indeed, @expr{5569.96 =
17663 5870.73 - 1300.78 + 1000} (give or take a bit of roundoff error).
17665 @node Present Value, Related Financial Functions, Future Value, Financial Functions
17666 @subsection Present Value
17670 @pindex calc-fin-pv
17672 The @kbd{b P} (@code{calc-fin-pv}) [@code{pv}] command computes
17673 the present value of an investment. Like @code{fv}, it takes
17674 three arguments: @code{pv(@var{rate}, @var{n}, @var{payment})}.
17675 It computes the present value of a series of regular payments.
17676 Suppose you have the chance to make an investment that will
17677 pay $2000 per year over the next four years; as you receive
17678 these payments you can put them in the bank at 9% interest.
17679 You want to know whether it is better to make the investment, or
17680 to keep the money in the bank where it earns 9% interest right
17681 from the start. The calculation @code{pv(9%, 4, 2000)} gives the
17682 result 6479.44. If your initial investment must be less than this,
17683 say, $6000, then the investment is worthwhile. But if you had to
17684 put up $7000, then it would be better just to leave it in the bank.
17686 Here is the interpretation of the result of @code{pv}: You are
17687 trying to compare the return from the investment you are
17688 considering, which is @code{fv(9%, 4, 2000) = 9146.26}, with
17689 the return from leaving the money in the bank, which is
17690 @code{fvl(9%, 4, @var{x})} where @var{x} is the amount of money
17691 you would have to put up in advance. The @code{pv} function
17692 finds the break-even point, @expr{x = 6479.44}, at which
17693 @code{fvl(9%, 4, 6479.44)} is also equal to 9146.26. This is
17694 the largest amount you should be willing to invest.
17698 The @kbd{I b P} [@code{pvb}] command solves the same problem,
17699 but with payments occurring at the beginning of each interval.
17700 It has the same relationship to @code{fvb} as @code{pv} has
17701 to @code{fv}. For example @code{pvb(9%, 4, 2000) = 7062.59},
17702 a larger number than @code{pv} produced because we get to start
17703 earning interest on the return from our investment sooner.
17707 The @kbd{H b P} [@code{pvl}] command computes the present value of
17708 an investment that will pay off in one lump sum at the end of the
17709 period. For example, if we get our $8000 all at the end of the
17710 four years, @code{pvl(9%, 4, 8000) = 5667.40}. This is much
17711 less than @code{pv} reported, because we don't earn any interest
17712 on the return from this investment. Note that @code{pvl} and
17713 @code{fvl} are simple inverses: @code{fvl(9%, 4, 5667.40) = 8000}.
17715 You can give an optional fourth lump-sum argument to @code{pv}
17716 and @code{pvb}; this is handled in exactly the same way as the
17717 fourth argument for @code{fv} and @code{fvb}.
17720 @pindex calc-fin-npv
17722 The @kbd{b N} (@code{calc-fin-npv}) [@code{npv}] command computes
17723 the net present value of a series of irregular investments.
17724 The first argument is the interest rate. The second argument is
17725 a vector which represents the expected return from the investment
17726 at the end of each interval. For example, if the rate represents
17727 a yearly interest rate, then the vector elements are the return
17728 from the first year, second year, and so on.
17730 Thus, @code{npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44}.
17731 Obviously this function is more interesting when the payments are
17734 The @code{npv} function can actually have two or more arguments.
17735 Multiple arguments are interpreted in the same way as for the
17736 vector statistical functions like @code{vsum}.
17737 @xref{Single-Variable Statistics}. Basically, if there are several
17738 payment arguments, each either a vector or a plain number, all these
17739 values are collected left-to-right into the complete list of payments.
17740 A numeric prefix argument on the @kbd{b N} command says how many
17741 payment values or vectors to take from the stack.
17745 The @kbd{I b N} [@code{npvb}] command computes the net present
17746 value where payments occur at the beginning of each interval
17747 rather than at the end.
17749 @node Related Financial Functions, Depreciation Functions, Present Value, Financial Functions
17750 @subsection Related Financial Functions
17753 The functions in this section are basically inverses of the
17754 present value functions with respect to the various arguments.
17757 @pindex calc-fin-pmt
17759 The @kbd{b M} (@code{calc-fin-pmt}) [@code{pmt}] command computes
17760 the amount of periodic payment necessary to amortize a loan.
17761 Thus @code{pmt(@var{rate}, @var{n}, @var{amount})} equals the
17762 value of @var{payment} such that @code{pv(@var{rate}, @var{n},
17763 @var{payment}) = @var{amount}}.
17767 The @kbd{I b M} [@code{pmtb}] command does the same computation
17768 but using @code{pvb} instead of @code{pv}. Like @code{pv} and
17769 @code{pvb}, these functions can also take a fourth argument which
17770 represents an initial lump-sum investment.
17773 The @kbd{H b M} key just invokes the @code{fvl} function, which is
17774 the inverse of @code{pvl}. There is no explicit @code{pmtl} function.
17777 @pindex calc-fin-nper
17779 The @kbd{b #} (@code{calc-fin-nper}) [@code{nper}] command computes
17780 the number of regular payments necessary to amortize a loan.
17781 Thus @code{nper(@var{rate}, @var{payment}, @var{amount})} equals
17782 the value of @var{n} such that @code{pv(@var{rate}, @var{n},
17783 @var{payment}) = @var{amount}}. If @var{payment} is too small
17784 ever to amortize a loan for @var{amount} at interest rate @var{rate},
17785 the @code{nper} function is left in symbolic form.
17789 The @kbd{I b #} [@code{nperb}] command does the same computation
17790 but using @code{pvb} instead of @code{pv}. You can give a fourth
17791 lump-sum argument to these functions, but the computation will be
17792 rather slow in the four-argument case.
17796 The @kbd{H b #} [@code{nperl}] command does the same computation
17797 using @code{pvl}. By exchanging @var{payment} and @var{amount} you
17798 can also get the solution for @code{fvl}. For example,
17799 @code{nperl(8%, 2000, 1000) = 9.006}, so if you place $1000 in a
17800 bank account earning 8%, it will take nine years to grow to $2000.
17803 @pindex calc-fin-rate
17805 The @kbd{b T} (@code{calc-fin-rate}) [@code{rate}] command computes
17806 the rate of return on an investment. This is also an inverse of @code{pv}:
17807 @code{rate(@var{n}, @var{payment}, @var{amount})} computes the value of
17808 @var{rate} such that @code{pv(@var{rate}, @var{n}, @var{payment}) =
17809 @var{amount}}. The result is expressed as a formula like @samp{6.3%}.
17815 The @kbd{I b T} [@code{rateb}] and @kbd{H b T} [@code{ratel}]
17816 commands solve the analogous equations with @code{pvb} or @code{pvl}
17817 in place of @code{pv}. Also, @code{rate} and @code{rateb} can
17818 accept an optional fourth argument just like @code{pv} and @code{pvb}.
17819 To redo the above example from a different perspective,
17820 @code{ratel(9, 2000, 1000) = 8.00597%}, which says you will need an
17821 interest rate of 8% in order to double your account in nine years.
17824 @pindex calc-fin-irr
17826 The @kbd{b I} (@code{calc-fin-irr}) [@code{irr}] command is the
17827 analogous function to @code{rate} but for net present value.
17828 Its argument is a vector of payments. Thus @code{irr(@var{payments})}
17829 computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0};
17830 this rate is known as the @dfn{internal rate of return}.
17834 The @kbd{I b I} [@code{irrb}] command computes the internal rate of
17835 return assuming payments occur at the beginning of each period.
17837 @node Depreciation Functions, Definitions of Financial Functions, Related Financial Functions, Financial Functions
17838 @subsection Depreciation Functions
17841 The functions in this section calculate @dfn{depreciation}, which is
17842 the amount of value that a possession loses over time. These functions
17843 are characterized by three parameters: @var{cost}, the original cost
17844 of the asset; @var{salvage}, the value the asset will have at the end
17845 of its expected ``useful life''; and @var{life}, the number of years
17846 (or other periods) of the expected useful life.
17848 There are several methods for calculating depreciation that differ in
17849 the way they spread the depreciation over the lifetime of the asset.
17852 @pindex calc-fin-sln
17854 The @kbd{b S} (@code{calc-fin-sln}) [@code{sln}] command computes the
17855 ``straight-line'' depreciation. In this method, the asset depreciates
17856 by the same amount every year (or period). For example,
17857 @samp{sln(12000, 2000, 5)} returns 2000. The asset costs $12000
17858 initially and will be worth $2000 after five years; it loses $2000
17862 @pindex calc-fin-syd
17864 The @kbd{b Y} (@code{calc-fin-syd}) [@code{syd}] command computes the
17865 accelerated ``sum-of-years'-digits'' depreciation. Here the depreciation
17866 is higher during the early years of the asset's life. Since the
17867 depreciation is different each year, @kbd{b Y} takes a fourth @var{period}
17868 parameter which specifies which year is requested, from 1 to @var{life}.
17869 If @var{period} is outside this range, the @code{syd} function will
17873 @pindex calc-fin-ddb
17875 The @kbd{b D} (@code{calc-fin-ddb}) [@code{ddb}] command computes an
17876 accelerated depreciation using the double-declining balance method.
17877 It also takes a fourth @var{period} parameter.
17879 For symmetry, the @code{sln} function will accept a @var{period}
17880 parameter as well, although it will ignore its value except that the
17881 return value will as usual be zero if @var{period} is out of range.
17883 For example, pushing the vector @expr{[1,2,3,4,5]} (perhaps with @kbd{v x 5})
17884 and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
17885 ddb(12000,2000,5,$)] @key{RET}} produces a matrix that allows us to compare
17886 the three depreciation methods:
17890 [ [ 2000, 3333, 4800 ]
17891 [ 2000, 2667, 2880 ]
17892 [ 2000, 2000, 1728 ]
17893 [ 2000, 1333, 592 ]
17899 (Values have been rounded to nearest integers in this figure.)
17900 We see that @code{sln} depreciates by the same amount each year,
17901 @kbd{syd} depreciates more at the beginning and less at the end,
17902 and @kbd{ddb} weights the depreciation even more toward the beginning.
17904 Summing columns with @kbd{V R : +} yields @expr{[10000, 10000, 10000]};
17905 the total depreciation in any method is (by definition) the
17906 difference between the cost and the salvage value.
17908 @node Definitions of Financial Functions, , Depreciation Functions, Financial Functions
17909 @subsection Definitions
17912 For your reference, here are the actual formulas used to compute
17913 Calc's financial functions.
17915 Calc will not evaluate a financial function unless the @var{rate} or
17916 @var{n} argument is known. However, @var{payment} or @var{amount} can
17917 be a variable. Calc expands these functions according to the
17918 formulas below for symbolic arguments only when you use the @kbd{a "}
17919 (@code{calc-expand-formula}) command, or when taking derivatives or
17920 integrals or solving equations involving the functions.
17923 These formulas are shown using the conventions of Big display
17924 mode (@kbd{d B}); for example, the formula for @code{fv} written
17925 linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}.
17930 fv(rate, n, pmt) = pmt * ---------------
17934 ((1 + rate) - 1) (1 + rate)
17935 fvb(rate, n, pmt) = pmt * ----------------------------
17939 fvl(rate, n, pmt) = pmt * (1 + rate)
17943 pv(rate, n, pmt) = pmt * ----------------
17947 (1 - (1 + rate) ) (1 + rate)
17948 pvb(rate, n, pmt) = pmt * -----------------------------
17952 pvl(rate, n, pmt) = pmt * (1 + rate)
17955 npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate)
17958 npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate)
17961 (amt - x * (1 + rate) ) * rate
17962 pmt(rate, n, amt, x) = -------------------------------
17967 (amt - x * (1 + rate) ) * rate
17968 pmtb(rate, n, amt, x) = -------------------------------
17970 (1 - (1 + rate) ) (1 + rate)
17973 nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate)
17977 nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
17981 nperl(rate, pmt, amt) = - log(---, 1 + rate)
17986 ratel(n, pmt, amt) = ------ - 1
17991 sln(cost, salv, life) = -----------
17994 (cost - salv) * (life - per + 1)
17995 syd(cost, salv, life, per) = --------------------------------
17996 life * (life + 1) / 2
17999 ddb(cost, salv, life, per) = --------, book = cost - depreciation so far
18004 $$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
18005 $$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
18006 $$ \code{fvl}(r, n, p) = p (1 + r)^n $$
18007 $$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$
18008 $$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$
18009 $$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$
18010 $$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$
18011 $$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$
18012 $$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$
18013 $$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over
18014 (1 - (1 + r)^{-n}) (1 + r) } $$
18015 $$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$
18016 $$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$
18017 $$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$
18018 $$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$
18019 $$ \code{sln}(c, s, l) = { c - s \over l } $$
18020 $$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$
18021 $$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$
18025 In @code{pmt} and @code{pmtb}, @expr{x=0} if omitted.
18027 These functions accept any numeric objects, including error forms,
18028 intervals, and even (though not very usefully) complex numbers. The
18029 above formulas specify exactly the behavior of these functions with
18030 all sorts of inputs.
18032 Note that if the first argument to the @code{log} in @code{nper} is
18033 negative, @code{nper} leaves itself in symbolic form rather than
18034 returning a (financially meaningless) complex number.
18036 @samp{rate(num, pmt, amt)} solves the equation
18037 @samp{pv(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R}
18038 (@code{calc-find-root}), with the interval @samp{[.01% .. 100%]}
18039 for an initial guess. The @code{rateb} function is the same except
18040 that it uses @code{pvb}. Note that @code{ratel} can be solved
18041 directly; its formula is shown in the above list.
18043 Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0}
18046 If you give a fourth argument to @code{nper} or @code{nperb}, Calc
18047 will also use @kbd{H a R} to solve the equation using an initial
18048 guess interval of @samp{[0 .. 100]}.
18050 A fourth argument to @code{fv} simply sums the two components
18051 calculated from the above formulas for @code{fv} and @code{fvl}.
18052 The same is true of @code{fvb}, @code{pv}, and @code{pvb}.
18054 The @kbd{ddb} function is computed iteratively; the ``book'' value
18055 starts out equal to @var{cost}, and decreases according to the above
18056 formula for the specified number of periods. If the book value
18057 would decrease below @var{salvage}, it only decreases to @var{salvage}
18058 and the depreciation is zero for all subsequent periods. The @code{ddb}
18059 function returns the amount the book value decreased in the specified
18062 @node Binary Functions, , Financial Functions, Arithmetic
18063 @section Binary Number Functions
18066 The commands in this chapter all use two-letter sequences beginning with
18067 the @kbd{b} prefix.
18069 @cindex Binary numbers
18070 The ``binary'' operations actually work regardless of the currently
18071 displayed radix, although their results make the most sense in a radix
18072 like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @w{@kbd{d 6}}
18073 commands, respectively). You may also wish to enable display of leading
18074 zeros with @kbd{d z}. @xref{Radix Modes}.
18076 @cindex Word size for binary operations
18077 The Calculator maintains a current @dfn{word size} @expr{w}, an
18078 arbitrary positive or negative integer. For a positive word size, all
18079 of the binary operations described here operate modulo @expr{2^w}. In
18080 particular, negative arguments are converted to positive integers modulo
18081 @expr{2^w} by all binary functions.
18083 If the word size is negative, binary operations produce twos-complement
18085 @texline @math{-2^{-w-1}}
18086 @infoline @expr{-(2^(-w-1))}
18088 @texline @math{2^{-w-1}-1}
18089 @infoline @expr{2^(-w-1)-1}
18090 inclusive. Either mode accepts inputs in any range; the sign of
18091 @expr{w} affects only the results produced.
18096 The @kbd{b c} (@code{calc-clip})
18097 [@code{clip}] command can be used to clip a number by reducing it modulo
18098 @expr{2^w}. The commands described in this chapter automatically clip
18099 their results to the current word size. Note that other operations like
18100 addition do not use the current word size, since integer addition
18101 generally is not ``binary.'' (However, @pxref{Simplification Modes},
18102 @code{calc-bin-simplify-mode}.) For example, with a word size of 8
18103 bits @kbd{b c} converts a number to the range 0 to 255; with a word
18104 size of @mathit{-8} @kbd{b c} converts to the range @mathit{-128} to 127.
18107 @pindex calc-word-size
18108 The default word size is 32 bits. All operations except the shifts and
18109 rotates allow you to specify a different word size for that one
18110 operation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the
18111 top of stack to the range 0 to 255 regardless of the current word size.
18112 To set the word size permanently, use @kbd{b w} (@code{calc-word-size}).
18113 This command displays a prompt with the current word size; press @key{RET}
18114 immediately to keep this word size, or type a new word size at the prompt.
18116 When the binary operations are written in symbolic form, they take an
18117 optional second (or third) word-size parameter. When a formula like
18118 @samp{and(a,b)} is finally evaluated, the word size current at that time
18119 will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of
18120 @mathit{-8} will always be used. A symbolic binary function will be left
18121 in symbolic form unless the all of its argument(s) are integers or
18122 integer-valued floats.
18124 If either or both arguments are modulo forms for which @expr{M} is a
18125 power of two, that power of two is taken as the word size unless a
18126 numeric prefix argument overrides it. The current word size is never
18127 consulted when modulo-power-of-two forms are involved.
18132 The @kbd{b a} (@code{calc-and}) [@code{and}] command computes the bitwise
18133 AND of the two numbers on the top of the stack. In other words, for each
18134 of the @expr{w} binary digits of the two numbers (pairwise), the corresponding
18135 bit of the result is 1 if and only if both input bits are 1:
18136 @samp{and(2#1100, 2#1010) = 2#1000}.
18141 The @kbd{b o} (@code{calc-or}) [@code{or}] command computes the bitwise
18142 inclusive OR of two numbers. A bit is 1 if either of the input bits, or
18143 both, are 1: @samp{or(2#1100, 2#1010) = 2#1110}.
18148 The @kbd{b x} (@code{calc-xor}) [@code{xor}] command computes the bitwise
18149 exclusive OR of two numbers. A bit is 1 if exactly one of the input bits
18150 is 1: @samp{xor(2#1100, 2#1010) = 2#0110}.
18155 The @kbd{b d} (@code{calc-diff}) [@code{diff}] command computes the bitwise
18156 difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))},
18157 so that @samp{diff(2#1100, 2#1010) = 2#0100}.
18162 The @kbd{b n} (@code{calc-not}) [@code{not}] command computes the bitwise
18163 NOT of a number. A bit is 1 if the input bit is 0 and vice-versa.
18166 @pindex calc-lshift-binary
18168 The @kbd{b l} (@code{calc-lshift-binary}) [@code{lsh}] command shifts a
18169 number left by one bit, or by the number of bits specified in the numeric
18170 prefix argument. A negative prefix argument performs a logical right shift,
18171 in which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)}
18172 is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}.
18173 Bits shifted ``off the end,'' according to the current word size, are lost.
18189 The @kbd{H b l} command also does a left shift, but it takes two arguments
18190 from the stack (the value to shift, and, at top-of-stack, the number of
18191 bits to shift). This version interprets the prefix argument just like
18192 the regular binary operations, i.e., as a word size. The Hyperbolic flag
18193 has a similar effect on the rest of the binary shift and rotate commands.
18196 @pindex calc-rshift-binary
18198 The @kbd{b r} (@code{calc-rshift-binary}) [@code{rsh}] command shifts a
18199 number right by one bit, or by the number of bits specified in the numeric
18200 prefix argument: @samp{rsh(a,n) = lsh(a,-n)}.
18203 @pindex calc-lshift-arith
18205 The @kbd{b L} (@code{calc-lshift-arith}) [@code{ash}] command shifts a
18206 number left. It is analogous to @code{lsh}, except that if the shift
18207 is rightward (the prefix argument is negative), an arithmetic shift
18208 is performed as described below.
18211 @pindex calc-rshift-arith
18213 The @kbd{b R} (@code{calc-rshift-arith}) [@code{rash}] command performs
18214 an ``arithmetic'' shift to the right, in which the leftmost bit (according
18215 to the current word size) is duplicated rather than shifting in zeros.
18216 This corresponds to dividing by a power of two where the input is interpreted
18217 as a signed, twos-complement number. (The distinction between the @samp{rsh}
18218 and @samp{rash} operations is totally independent from whether the word
18219 size is positive or negative.) With a negative prefix argument, this
18220 performs a standard left shift.
18223 @pindex calc-rotate-binary
18225 The @kbd{b t} (@code{calc-rotate-binary}) [@code{rot}] command rotates a
18226 number one bit to the left. The leftmost bit (according to the current
18227 word size) is dropped off the left and shifted in on the right. With a
18228 numeric prefix argument, the number is rotated that many bits to the left
18231 @xref{Set Operations}, for the @kbd{b p} and @kbd{b u} commands that
18232 pack and unpack binary integers into sets. (For example, @kbd{b u}
18233 unpacks the number @samp{2#11001} to the set of bit-numbers
18234 @samp{[0, 3, 4]}.) Type @kbd{b u V #} to count the number of ``1''
18235 bits in a binary integer.
18237 Another interesting use of the set representation of binary integers
18238 is to reverse the bits in, say, a 32-bit integer. Type @kbd{b u} to
18239 unpack; type @kbd{31 @key{TAB} -} to replace each bit-number in the set
18240 with 31 minus that bit-number; type @kbd{b p} to pack the set back
18241 into a binary integer.
18243 @node Scientific Functions, Matrix Functions, Arithmetic, Top
18244 @chapter Scientific Functions
18247 The functions described here perform trigonometric and other transcendental
18248 calculations. They generally produce floating-point answers correct to the
18249 full current precision. The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse)
18250 flag keys must be used to get some of these functions from the keyboard.
18254 @cindex @code{pi} variable
18257 @cindex @code{e} variable
18260 @cindex @code{gamma} variable
18262 @cindex Gamma constant, Euler's
18263 @cindex Euler's gamma constant
18265 @cindex @code{phi} variable
18266 @cindex Phi, golden ratio
18267 @cindex Golden ratio
18268 One miscellaneous command is shift-@kbd{P} (@code{calc-pi}), which pushes
18269 the value of @cpi{} (at the current precision) onto the stack. With the
18270 Hyperbolic flag, it pushes the value @expr{e}, the base of natural logarithms.
18271 With the Inverse flag, it pushes Euler's constant
18272 @texline @math{\gamma}
18273 @infoline @expr{gamma}
18274 (about 0.5772). With both Inverse and Hyperbolic, it
18275 pushes the ``golden ratio''
18276 @texline @math{\phi}
18277 @infoline @expr{phi}
18278 (about 1.618). (At present, Euler's constant is not available
18279 to unlimited precision; Calc knows only the first 100 digits.)
18280 In Symbolic mode, these commands push the
18281 actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi},
18282 respectively, instead of their values; @pxref{Symbolic Mode}.
18292 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] function is described elsewhere;
18293 @pxref{Basic Arithmetic}. With the Inverse flag [@code{sqr}], this command
18294 computes the square of the argument.
18296 @xref{Prefix Arguments}, for a discussion of the effect of numeric
18297 prefix arguments on commands in this chapter which do not otherwise
18298 interpret a prefix argument.
18301 * Logarithmic Functions::
18302 * Trigonometric and Hyperbolic Functions::
18303 * Advanced Math Functions::
18306 * Combinatorial Functions::
18307 * Probability Distribution Functions::
18310 @node Logarithmic Functions, Trigonometric and Hyperbolic Functions, Scientific Functions, Scientific Functions
18311 @section Logarithmic Functions
18321 The shift-@kbd{L} (@code{calc-ln}) [@code{ln}] command computes the natural
18322 logarithm of the real or complex number on the top of the stack. With
18323 the Inverse flag it computes the exponential function instead, although
18324 this is redundant with the @kbd{E} command.
18333 The shift-@kbd{E} (@code{calc-exp}) [@code{exp}] command computes the
18334 exponential, i.e., @expr{e} raised to the power of the number on the stack.
18335 The meanings of the Inverse and Hyperbolic flags follow from those for
18336 the @code{calc-ln} command.
18351 The @kbd{H L} (@code{calc-log10}) [@code{log10}] command computes the common
18352 (base-10) logarithm of a number. (With the Inverse flag [@code{exp10}],
18353 it raises ten to a given power.) Note that the common logarithm of a
18354 complex number is computed by taking the natural logarithm and dividing
18356 @texline @math{\ln10}.
18357 @infoline @expr{ln(10)}.
18364 The @kbd{B} (@code{calc-log}) [@code{log}] command computes a logarithm
18365 to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since
18366 @texline @math{2^{10} = 1024}.
18367 @infoline @expr{2^10 = 1024}.
18368 In certain cases like @samp{log(3,9)}, the result
18369 will be either @expr{1:2} or @expr{0.5} depending on the current Fraction
18370 mode setting. With the Inverse flag [@code{alog}], this command is
18371 similar to @kbd{^} except that the order of the arguments is reversed.
18376 The @kbd{f I} (@code{calc-ilog}) [@code{ilog}] command computes the
18377 integer logarithm of a number to any base. The number and the base must
18378 themselves be positive integers. This is the true logarithm, rounded
18379 down to an integer. Thus @kbd{ilog(x,10)} is 3 for all @expr{x} in the
18380 range from 1000 to 9999. If both arguments are positive integers, exact
18381 integer arithmetic is used; otherwise, this is equivalent to
18382 @samp{floor(log(x,b))}.
18387 The @kbd{f E} (@code{calc-expm1}) [@code{expm1}] command computes
18388 @texline @math{e^x - 1},
18389 @infoline @expr{exp(x)-1},
18390 but using an algorithm that produces a more accurate
18391 answer when the result is close to zero, i.e., when
18392 @texline @math{e^x}
18393 @infoline @expr{exp(x)}
18399 The @kbd{f L} (@code{calc-lnp1}) [@code{lnp1}] command computes
18400 @texline @math{\ln(x+1)},
18401 @infoline @expr{ln(x+1)},
18402 producing a more accurate answer when @expr{x} is close to zero.
18404 @node Trigonometric and Hyperbolic Functions, Advanced Math Functions, Logarithmic Functions, Scientific Functions
18405 @section Trigonometric/Hyperbolic Functions
18411 The shift-@kbd{S} (@code{calc-sin}) [@code{sin}] command computes the sine
18412 of an angle or complex number. If the input is an HMS form, it is interpreted
18413 as degrees-minutes-seconds; otherwise, the input is interpreted according
18414 to the current angular mode. It is best to use Radians mode when operating
18415 on complex numbers.
18417 Calc's ``units'' mechanism includes angular units like @code{deg},
18418 @code{rad}, and @code{grad}. While @samp{sin(45 deg)} is not evaluated
18419 all the time, the @kbd{u s} (@code{calc-simplify-units}) command will
18420 simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless
18421 of the current angular mode. @xref{Basic Operations on Units}.
18423 Also, the symbolic variable @code{pi} is not ordinarily recognized in
18424 arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but
18425 the default algebraic simplifications recognize many such
18426 formulas when the current angular mode is Radians @emph{and} Symbolic
18427 mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}.
18428 @xref{Symbolic Mode}. Beware, this simplification occurs even if you
18429 have stored a different value in the variable @samp{pi}; this is one
18430 reason why changing built-in variables is a bad idea. Arguments of
18431 the form @expr{x} plus a multiple of @cpiover{2} are also simplified.
18432 Calc includes similar formulas for @code{cos} and @code{tan}.
18434 Calc's algebraic simplifications know all angles which are integer multiples of
18435 @cpiover{12}, @cpiover{10}, or @cpiover{8} radians. In Degrees mode,
18436 analogous simplifications occur for integer multiples of 15 or 18
18437 degrees, and for arguments plus multiples of 90 degrees.
18440 @pindex calc-arcsin
18442 With the Inverse flag, @code{calc-sin} computes an arcsine. This is also
18443 available as the @code{calc-arcsin} command or @code{arcsin} algebraic
18444 function. The returned argument is converted to degrees, radians, or HMS
18445 notation depending on the current angular mode.
18451 @pindex calc-arcsinh
18453 With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic
18454 sine, also available as @code{calc-sinh} [@code{sinh}]. With the
18455 Hyperbolic and Inverse flags, it computes the hyperbolic arcsine
18456 (@code{calc-arcsinh}) [@code{arcsinh}].
18465 @pindex calc-arccos
18483 @pindex calc-arccosh
18501 @pindex calc-arctan
18519 @pindex calc-arctanh
18524 The shift-@kbd{C} (@code{calc-cos}) [@code{cos}] command computes the cosine
18525 of an angle or complex number, and shift-@kbd{T} (@code{calc-tan}) [@code{tan}]
18526 computes the tangent, along with all the various inverse and hyperbolic
18527 variants of these functions.
18530 @pindex calc-arctan2
18532 The @kbd{f T} (@code{calc-arctan2}) [@code{arctan2}] command takes two
18533 numbers from the stack and computes the arc tangent of their ratio. The
18534 result is in the full range from @mathit{-180} (exclusive) to @mathit{+180}
18535 (inclusive) degrees, or the analogous range in radians. A similar
18536 result would be obtained with @kbd{/} followed by @kbd{I T}, but the
18537 value would only be in the range from @mathit{-90} to @mathit{+90} degrees
18538 since the division loses information about the signs of the two
18539 components, and an error might result from an explicit division by zero
18540 which @code{arctan2} would avoid. By (arbitrary) definition,
18541 @samp{arctan2(0,0)=0}.
18543 @pindex calc-sincos
18555 The @code{calc-sincos} [@code{sincos}] command computes the sine and
18556 cosine of a number, returning them as a vector of the form
18557 @samp{[@var{cos}, @var{sin}]}.
18558 With the Inverse flag [@code{arcsincos}], this command takes a two-element
18559 vector as an argument and computes @code{arctan2} of the elements.
18560 (This command does not accept the Hyperbolic flag.)
18574 The remaining trigonometric functions, @code{calc-sec} [@code{sec}],
18575 @code{calc-csc} [@code{csc}] and @code{calc-cot} [@code{cot}], are also
18576 available. With the Hyperbolic flag, these compute their hyperbolic
18577 counterparts, which are also available separately as @code{calc-sech}
18578 [@code{sech}], @code{calc-csch} [@code{csch}] and @code{calc-coth}
18579 [@code{coth}]. (These commands do not accept the Inverse flag.)
18581 @node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions
18582 @section Advanced Mathematical Functions
18585 Calc can compute a variety of less common functions that arise in
18586 various branches of mathematics. All of the functions described in
18587 this section allow arbitrary complex arguments and, except as noted,
18588 will work to arbitrarily large precision. They can not at present
18589 handle error forms or intervals as arguments.
18591 NOTE: These functions are still experimental. In particular, their
18592 accuracy is not guaranteed in all domains. It is advisable to set the
18593 current precision comfortably higher than you actually need when
18594 using these functions. Also, these functions may be impractically
18595 slow for some values of the arguments.
18600 The @kbd{f g} (@code{calc-gamma}) [@code{gamma}] command computes the Euler
18601 gamma function. For positive integer arguments, this is related to the
18602 factorial function: @samp{gamma(n+1) = fact(n)}. For general complex
18603 arguments the gamma function can be defined by the following definite
18605 @texline @math{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt}.
18606 @infoline @expr{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}.
18607 (The actual implementation uses far more efficient computational methods.)
18623 @pindex calc-inc-gamma
18636 The @kbd{f G} (@code{calc-inc-gamma}) [@code{gammaP}] command computes
18637 the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by
18639 @texline @math{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)}.
18640 @infoline @expr{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}.
18641 This implies that @samp{gammaP(a,inf) = 1} for any @expr{a} (see the
18642 definition of the normal gamma function).
18644 Several other varieties of incomplete gamma function are defined.
18645 The complement of @expr{P(a,x)}, called @expr{Q(a,x) = 1-P(a,x)} by
18646 some authors, is computed by the @kbd{I f G} [@code{gammaQ}] command.
18647 You can think of this as taking the other half of the integral, from
18648 @expr{x} to infinity.
18651 The functions corresponding to the integrals that define @expr{P(a,x)}
18652 and @expr{Q(a,x)} but without the normalizing @expr{1/gamma(a)}
18653 factor are called @expr{g(a,x)} and @expr{G(a,x)}, respectively
18654 (where @expr{g} and @expr{G} represent the lower- and upper-case Greek
18655 letter gamma). You can obtain these using the @kbd{H f G} [@code{gammag}]
18656 and @kbd{H I f G} [@code{gammaG}] commands.
18659 The functions corresponding to the integrals that define $P(a,x)$
18660 and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
18661 factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
18662 You can obtain these using the \kbd{H f G} [\code{gammag}] and
18663 \kbd{I H f G} [\code{gammaG}] commands.
18669 The @kbd{f b} (@code{calc-beta}) [@code{beta}] command computes the
18670 Euler beta function, which is defined in terms of the gamma function as
18671 @texline @math{B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)},
18672 @infoline @expr{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)},
18674 @texline @math{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}.
18675 @infoline @expr{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}.
18679 @pindex calc-inc-beta
18682 The @kbd{f B} (@code{calc-inc-beta}) [@code{betaI}] command computes
18683 the incomplete beta function @expr{I(x,a,b)}. It is defined by
18684 @texline @math{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) / B(a,b)}.
18685 @infoline @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)}.
18686 Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding
18687 un-normalized version [@code{betaB}].
18694 The @kbd{f e} (@code{calc-erf}) [@code{erf}] command computes the
18696 @texline @math{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt}.
18697 @infoline @expr{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}.
18698 The complementary error function @kbd{I f e} (@code{calc-erfc}) [@code{erfc}]
18699 is the corresponding integral from @samp{x} to infinity; the sum
18700 @texline @math{\hbox{erf}(x) + \hbox{erfc}(x) = 1}.
18701 @infoline @expr{erf(x) + erfc(x) = 1}.
18705 @pindex calc-bessel-J
18706 @pindex calc-bessel-Y
18709 The @kbd{f j} (@code{calc-bessel-J}) [@code{besJ}] and @kbd{f y}
18710 (@code{calc-bessel-Y}) [@code{besY}] commands compute the Bessel
18711 functions of the first and second kinds, respectively.
18712 In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter
18713 @expr{n} is often an integer, but is not required to be one.
18714 Calc's implementation of the Bessel functions currently limits the
18715 precision to 8 digits, and may not be exact even to that precision.
18718 @node Branch Cuts, Random Numbers, Advanced Math Functions, Scientific Functions
18719 @section Branch Cuts and Principal Values
18722 @cindex Branch cuts
18723 @cindex Principal values
18724 All of the logarithmic, trigonometric, and other scientific functions are
18725 defined for complex numbers as well as for reals.
18726 This section describes the values
18727 returned in cases where the general result is a family of possible values.
18728 Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language},
18729 second edition, in these matters. This section will describe each
18730 function briefly; for a more detailed discussion (including some nifty
18731 diagrams), consult Steele's book.
18733 Note that the branch cuts for @code{arctan} and @code{arctanh} were
18734 changed between the first and second editions of Steele. Recent
18735 versions of Calc follow the second edition.
18737 The new branch cuts exactly match those of the HP-28/48 calculators.
18738 They also match those of Mathematica 1.2, except that Mathematica's
18739 @code{arctan} cut is always in the right half of the complex plane,
18740 and its @code{arctanh} cut is always in the top half of the plane.
18741 Calc's cuts are continuous with quadrants I and III for @code{arctan},
18742 or II and IV for @code{arctanh}.
18744 Note: The current implementations of these functions with complex arguments
18745 are designed with proper behavior around the branch cuts in mind, @emph{not}
18746 efficiency or accuracy. You may need to increase the floating precision
18747 and wait a while to get suitable answers from them.
18749 For @samp{sqrt(a+bi)}: When @expr{a<0} and @expr{b} is small but positive
18750 or zero, the result is close to the @expr{+i} axis. For @expr{b} small and
18751 negative, the result is close to the @expr{-i} axis. The result always lies
18752 in the right half of the complex plane.
18754 For @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}.
18755 The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}.
18756 Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the
18757 negative real axis.
18759 The following table describes these branch cuts in another way.
18760 If the real and imaginary parts of @expr{z} are as shown, then
18761 the real and imaginary parts of @expr{f(z)} will be as shown.
18762 Here @code{eps} stands for a small positive value; each
18763 occurrence of @code{eps} may stand for a different small value.
18767 ----------------------------------------
18770 -, +eps +eps, + +eps, +
18771 -, -eps +eps, - +eps, -
18774 For @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}.
18775 One interesting consequence of this is that @samp{(-8)^1:3} does
18776 not evaluate to @mathit{-2} as you might expect, but to the complex
18777 number @expr{(1., 1.732)}. Both of these are valid cube roots
18778 of @mathit{-8} (as is @expr{(1., -1.732)}); Calc chooses a perhaps
18779 less-obvious root for the sake of mathematical consistency.
18781 For @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}.
18782 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18784 For @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))},
18785 or equivalently by @samp{pi/2 - arcsin(z)}. The branch cuts are on
18786 the real axis, less than @mathit{-1} and greater than 1.
18788 For @samp{arctan(z)}: This is defined by
18789 @samp{(ln(1+i*z) - ln(1-i*z)) / (2*i)}. The branch cuts are on the
18790 imaginary axis, below @expr{-i} and above @expr{i}.
18792 For @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}.
18793 The branch cuts are on the imaginary axis, below @expr{-i} and
18796 For @samp{arccosh(z)}: This is defined by
18797 @samp{ln(z + (z+1)*sqrt((z-1)/(z+1)))}. The branch cut is on the
18798 real axis less than 1.
18800 For @samp{arctanh(z)}: This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}.
18801 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18803 The following tables for @code{arcsin}, @code{arccos}, and
18804 @code{arctan} assume the current angular mode is Radians. The
18805 hyperbolic functions operate independently of the angular mode.
18808 z arcsin(z) arccos(z)
18809 -------------------------------------------------------
18810 (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0
18811 (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps
18812 (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps
18813 <-1, 0 -pi/2, + pi, -
18814 <-1, +eps -pi/2 + eps, + pi - eps, -
18815 <-1, -eps -pi/2 + eps, - pi - eps, +
18817 >1, +eps pi/2 - eps, + +eps, -
18818 >1, -eps pi/2 - eps, - +eps, +
18822 z arccosh(z) arctanh(z)
18823 -----------------------------------------------------
18824 (-1..1), 0 0, (0..pi) any, 0
18825 (-1..1), +eps +eps, (0..pi) any, +eps
18826 (-1..1), -eps +eps, (-pi..0) any, -eps
18827 <-1, 0 +, pi -, pi/2
18828 <-1, +eps +, pi - eps -, pi/2 - eps
18829 <-1, -eps +, -pi + eps -, -pi/2 + eps
18830 >1, 0 +, 0 +, -pi/2
18831 >1, +eps +, +eps +, pi/2 - eps
18832 >1, -eps +, -eps +, -pi/2 + eps
18836 z arcsinh(z) arctan(z)
18837 -----------------------------------------------------
18838 0, (-1..1) 0, (-pi/2..pi/2) 0, any
18839 0, <-1 -, -pi/2 -pi/2, -
18840 +eps, <-1 +, -pi/2 + eps pi/2 - eps, -
18841 -eps, <-1 -, -pi/2 + eps -pi/2 + eps, -
18842 0, >1 +, pi/2 pi/2, +
18843 +eps, >1 +, pi/2 - eps pi/2 - eps, +
18844 -eps, >1 -, pi/2 - eps -pi/2 + eps, +
18847 Finally, the following identities help to illustrate the relationship
18848 between the complex trigonometric and hyperbolic functions. They
18849 are valid everywhere, including on the branch cuts.
18852 sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z)
18853 cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z)
18854 tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z)
18855 sinh(i*z) = i*sin(z) cosh(i*z) = cos(z)
18858 The ``advanced math'' functions (gamma, Bessel, etc.@:) are also defined
18859 for general complex arguments, but their branch cuts and principal values
18860 are not rigorously specified at present.
18862 @node Random Numbers, Combinatorial Functions, Branch Cuts, Scientific Functions
18863 @section Random Numbers
18867 @pindex calc-random
18869 The @kbd{k r} (@code{calc-random}) [@code{random}] command produces
18870 random numbers of various sorts.
18872 Given a positive numeric prefix argument @expr{M}, it produces a random
18873 integer @expr{N} in the range
18874 @texline @math{0 \le N < M}.
18875 @infoline @expr{0 <= N < M}.
18876 Each possible value @expr{N} appears with equal probability.
18878 With no numeric prefix argument, the @kbd{k r} command takes its argument
18879 from the stack instead. Once again, if this is a positive integer @expr{M}
18880 the result is a random integer less than @expr{M}. However, note that
18881 while numeric prefix arguments are limited to six digits or so, an @expr{M}
18882 taken from the stack can be arbitrarily large. If @expr{M} is negative,
18883 the result is a random integer in the range
18884 @texline @math{M < N \le 0}.
18885 @infoline @expr{M < N <= 0}.
18887 If the value on the stack is a floating-point number @expr{M}, the result
18888 is a random floating-point number @expr{N} in the range
18889 @texline @math{0 \le N < M}
18890 @infoline @expr{0 <= N < M}
18892 @texline @math{M < N \le 0},
18893 @infoline @expr{M < N <= 0},
18894 according to the sign of @expr{M}.
18896 If @expr{M} is zero, the result is a Gaussian-distributed random real
18897 number; the distribution has a mean of zero and a standard deviation
18898 of one. The algorithm used generates random numbers in pairs; thus,
18899 every other call to this function will be especially fast.
18901 If @expr{M} is an error form
18902 @texline @math{m} @code{+/-} @math{\sigma}
18903 @infoline @samp{m +/- s}
18905 @texline @math{\sigma}
18907 are both real numbers, the result uses a Gaussian distribution with mean
18908 @var{m} and standard deviation
18909 @texline @math{\sigma}.
18912 If @expr{M} is an interval form, the lower and upper bounds specify the
18913 acceptable limits of the random numbers. If both bounds are integers,
18914 the result is a random integer in the specified range. If either bound
18915 is floating-point, the result is a random real number in the specified
18916 range. If the interval is open at either end, the result will be sure
18917 not to equal that end value. (This makes a big difference for integer
18918 intervals, but for floating-point intervals it's relatively minor:
18919 with a precision of 6, @samp{random([1.0..2.0))} will return any of one
18920 million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may
18921 additionally return 2.00000, but the probability of this happening is
18924 If @expr{M} is a vector, the result is one element taken at random from
18925 the vector. All elements of the vector are given equal probabilities.
18928 The sequence of numbers produced by @kbd{k r} is completely random by
18929 default, i.e., the sequence is seeded each time you start Calc using
18930 the current time and other information. You can get a reproducible
18931 sequence by storing a particular ``seed value'' in the Calc variable
18932 @code{RandSeed}. Any integer will do for a seed; integers of from 1
18933 to 12 digits are good. If you later store a different integer into
18934 @code{RandSeed}, Calc will switch to a different pseudo-random
18935 sequence. If you ``unstore'' @code{RandSeed}, Calc will re-seed itself
18936 from the current time. If you store the same integer that you used
18937 before back into @code{RandSeed}, you will get the exact same sequence
18938 of random numbers as before.
18940 @pindex calc-rrandom
18941 The @code{calc-rrandom} command (not on any key) produces a random real
18942 number between zero and one. It is equivalent to @samp{random(1.0)}.
18945 @pindex calc-random-again
18946 The @kbd{k a} (@code{calc-random-again}) command produces another random
18947 number, re-using the most recent value of @expr{M}. With a numeric
18948 prefix argument @var{n}, it produces @var{n} more random numbers using
18949 that value of @expr{M}.
18952 @pindex calc-shuffle
18954 The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several
18955 random values with no duplicates. The value on the top of the stack
18956 specifies the set from which the random values are drawn, and may be any
18957 of the @expr{M} formats described above. The numeric prefix argument
18958 gives the length of the desired list. (If you do not provide a numeric
18959 prefix argument, the length of the list is taken from the top of the
18960 stack, and @expr{M} from second-to-top.)
18962 If @expr{M} is a floating-point number, zero, or an error form (so
18963 that the random values are being drawn from the set of real numbers)
18964 there is little practical difference between using @kbd{k h} and using
18965 @kbd{k r} several times. But if the set of possible values consists
18966 of just a few integers, or the elements of a vector, then there is
18967 a very real chance that multiple @kbd{k r}'s will produce the same
18968 number more than once. The @kbd{k h} command produces a vector whose
18969 elements are always distinct. (Actually, there is a slight exception:
18970 If @expr{M} is a vector, no given vector element will be drawn more
18971 than once, but if several elements of @expr{M} are equal, they may
18972 each make it into the result vector.)
18974 One use of @kbd{k h} is to rearrange a list at random. This happens
18975 if the prefix argument is equal to the number of values in the list:
18976 @kbd{[1, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list
18977 @samp{[2.5, 1, 1.5, 3, 2]}. As a convenient feature, if the argument
18978 @var{n} is negative it is replaced by the size of the set represented
18979 by @expr{M}. Naturally, this is allowed only when @expr{M} specifies
18980 a small discrete set of possibilities.
18982 To do the equivalent of @kbd{k h} but with duplications allowed,
18983 given @expr{M} on the stack and with @var{n} just entered as a numeric
18984 prefix, use @kbd{v b} to build a vector of copies of @expr{M}, then use
18985 @kbd{V M k r} to ``map'' the normal @kbd{k r} function over the
18986 elements of this vector. @xref{Matrix Functions}.
18989 * Random Number Generator:: (Complete description of Calc's algorithm)
18992 @node Random Number Generator, , Random Numbers, Random Numbers
18993 @subsection Random Number Generator
18995 Calc's random number generator uses several methods to ensure that
18996 the numbers it produces are highly random. Knuth's @emph{Art of
18997 Computer Programming}, Volume II, contains a thorough description
18998 of the theory of random number generators and their measurement and
19001 If @code{RandSeed} has no stored value, Calc calls Emacs's built-in
19002 @code{random} function to get a stream of random numbers, which it
19003 then treats in various ways to avoid problems inherent in the simple
19004 random number generators that many systems use to implement @code{random}.
19006 When Calc's random number generator is first invoked, it ``seeds''
19007 the low-level random sequence using the time of day, so that the
19008 random number sequence will be different every time you use Calc.
19010 Since Emacs Lisp doesn't specify the range of values that will be
19011 returned by its @code{random} function, Calc exercises the function
19012 several times to estimate the range. When Calc subsequently uses
19013 the @code{random} function, it takes only 10 bits of the result
19014 near the most-significant end. (It avoids at least the bottom
19015 four bits, preferably more, and also tries to avoid the top two
19016 bits.) This strategy works well with the linear congruential
19017 generators that are typically used to implement @code{random}.
19019 If @code{RandSeed} contains an integer, Calc uses this integer to
19020 seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A,
19022 @texline @math{X_{n-55} - X_{n-24}}.
19023 @infoline @expr{X_n-55 - X_n-24}).
19024 This method expands the seed
19025 value into a large table which is maintained internally; the variable
19026 @code{RandSeed} is changed from, e.g., 42 to the vector @expr{[42]}
19027 to indicate that the seed has been absorbed into this table. When
19028 @code{RandSeed} contains a vector, @kbd{k r} and related commands
19029 continue to use the same internal table as last time. There is no
19030 way to extract the complete state of the random number generator
19031 so that you can restart it from any point; you can only restart it
19032 from the same initial seed value. A simple way to restart from the
19033 same seed is to type @kbd{s r RandSeed} to get the seed vector,
19034 @kbd{v u} to unpack it back into a number, then @kbd{s t RandSeed}
19035 to reseed the generator with that number.
19037 Calc uses a ``shuffling'' method as described in algorithm 3.2.2B
19038 of Knuth. It fills a table with 13 random 10-bit numbers. Then,
19039 to generate a new random number, it uses the previous number to
19040 index into the table, picks the value it finds there as the new
19041 random number, then replaces that table entry with a new value
19042 obtained from a call to the base random number generator (either
19043 the additive congruential generator or the @code{random} function
19044 supplied by the system). If there are any flaws in the base
19045 generator, shuffling will tend to even them out. But if the system
19046 provides an excellent @code{random} function, shuffling will not
19047 damage its randomness.
19049 To create a random integer of a certain number of digits, Calc
19050 builds the integer three decimal digits at a time. For each group
19051 of three digits, Calc calls its 10-bit shuffling random number generator
19052 (which returns a value from 0 to 1023); if the random value is 1000
19053 or more, Calc throws it out and tries again until it gets a suitable
19056 To create a random floating-point number with precision @var{p}, Calc
19057 simply creates a random @var{p}-digit integer and multiplies by
19058 @texline @math{10^{-p}}.
19059 @infoline @expr{10^-p}.
19060 The resulting random numbers should be very clean, but note
19061 that relatively small numbers will have few significant random digits.
19062 In other words, with a precision of 12, you will occasionally get
19063 numbers on the order of
19064 @texline @math{10^{-9}}
19065 @infoline @expr{10^-9}
19067 @texline @math{10^{-10}},
19068 @infoline @expr{10^-10},
19069 but those numbers will only have two or three random digits since they
19070 correspond to small integers times
19071 @texline @math{10^{-12}}.
19072 @infoline @expr{10^-12}.
19074 To create a random integer in the interval @samp{[0 .. @var{m})}, Calc
19075 counts the digits in @var{m}, creates a random integer with three
19076 additional digits, then reduces modulo @var{m}. Unless @var{m} is a
19077 power of ten the resulting values will be very slightly biased toward
19078 the lower numbers, but this bias will be less than 0.1%. (For example,
19079 if @var{m} is 42, Calc will reduce a random integer less than 100000
19080 modulo 42 to get a result less than 42. It is easy to show that the
19081 numbers 40 and 41 will be only 2380/2381 as likely to result from this
19082 modulo operation as numbers 39 and below.) If @var{m} is a power of
19083 ten, however, the numbers should be completely unbiased.
19085 The Gaussian random numbers generated by @samp{random(0.0)} use the
19086 ``polar'' method described in Knuth section 3.4.1C@. This method
19087 generates a pair of Gaussian random numbers at a time, so only every
19088 other call to @samp{random(0.0)} will require significant calculations.
19090 @node Combinatorial Functions, Probability Distribution Functions, Random Numbers, Scientific Functions
19091 @section Combinatorial Functions
19094 Commands relating to combinatorics and number theory begin with the
19095 @kbd{k} key prefix.
19100 The @kbd{k g} (@code{calc-gcd}) [@code{gcd}] command computes the
19101 Greatest Common Divisor of two integers. It also accepts fractions;
19102 the GCD of two fractions is defined by taking the GCD of the
19103 numerators, and the LCM of the denominators. This definition is
19104 consistent with the idea that @samp{a / gcd(a,x)} should yield an
19105 integer for any @samp{a} and @samp{x}. For other types of arguments,
19106 the operation is left in symbolic form.
19111 The @kbd{k l} (@code{calc-lcm}) [@code{lcm}] command computes the
19112 Least Common Multiple of two integers or fractions. The product of
19113 the LCM and GCD of two numbers is equal to the product of the
19117 @pindex calc-extended-gcd
19119 The @kbd{k E} (@code{calc-extended-gcd}) [@code{egcd}] command computes
19120 the GCD of two integers @expr{x} and @expr{y} and returns a vector
19121 @expr{[g, a, b]} where
19122 @texline @math{g = \gcd(x,y) = a x + b y}.
19123 @infoline @expr{g = gcd(x,y) = a x + b y}.
19126 @pindex calc-factorial
19132 The @kbd{!} (@code{calc-factorial}) [@code{fact}] command computes the
19133 factorial of the number at the top of the stack. If the number is an
19134 integer, the result is an exact integer. If the number is an
19135 integer-valued float, the result is a floating-point approximation. If
19136 the number is a non-integral real number, the generalized factorial is used,
19137 as defined by the Euler Gamma function. Please note that computation of
19138 large factorials can be slow; using floating-point format will help
19139 since fewer digits must be maintained. The same is true of many of
19140 the commands in this section.
19143 @pindex calc-double-factorial
19149 The @kbd{k d} (@code{calc-double-factorial}) [@code{dfact}] command
19150 computes the ``double factorial'' of an integer. For an even integer,
19151 this is the product of even integers from 2 to @expr{N}. For an odd
19152 integer, this is the product of odd integers from 3 to @expr{N}. If
19153 the argument is an integer-valued float, the result is a floating-point
19154 approximation. This function is undefined for negative even integers.
19155 The notation @expr{N!!} is also recognized for double factorials.
19158 @pindex calc-choose
19160 The @kbd{k c} (@code{calc-choose}) [@code{choose}] command computes the
19161 binomial coefficient @expr{N}-choose-@expr{M}, where @expr{M} is the number
19162 on the top of the stack and @expr{N} is second-to-top. If both arguments
19163 are integers, the result is an exact integer. Otherwise, the result is a
19164 floating-point approximation. The binomial coefficient is defined for all
19166 @texline @math{N! \over M! (N-M)!\,}.
19167 @infoline @expr{N! / M! (N-M)!}.
19173 The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the
19174 number-of-permutations function @expr{N! / (N-M)!}.
19177 The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
19178 number-of-perm\-utations function $N! \over (N-M)!\,$.
19183 @pindex calc-bernoulli-number
19185 The @kbd{k b} (@code{calc-bernoulli-number}) [@code{bern}] command
19186 computes a given Bernoulli number. The value at the top of the stack
19187 is a nonnegative integer @expr{n} that specifies which Bernoulli number
19188 is desired. The @kbd{H k b} command computes a Bernoulli polynomial,
19189 taking @expr{n} from the second-to-top position and @expr{x} from the
19190 top of the stack. If @expr{x} is a variable or formula the result is
19191 a polynomial in @expr{x}; if @expr{x} is a number the result is a number.
19195 @pindex calc-euler-number
19197 The @kbd{k e} (@code{calc-euler-number}) [@code{euler}] command similarly
19198 computes an Euler number, and @w{@kbd{H k e}} computes an Euler polynomial.
19199 Bernoulli and Euler numbers occur in the Taylor expansions of several
19204 @pindex calc-stirling-number
19207 The @kbd{k s} (@code{calc-stirling-number}) [@code{stir1}] command
19208 computes a Stirling number of the first
19209 @texline kind@tie{}@math{n \brack m},
19211 given two integers @expr{n} and @expr{m} on the stack. The @kbd{H k s}
19212 [@code{stir2}] command computes a Stirling number of the second
19213 @texline kind@tie{}@math{n \brace m}.
19215 These are the number of @expr{m}-cycle permutations of @expr{n} objects,
19216 and the number of ways to partition @expr{n} objects into @expr{m}
19217 non-empty sets, respectively.
19220 @pindex calc-prime-test
19222 The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on
19223 the top of the stack is prime. For integers less than eight million, the
19224 answer is always exact and reasonably fast. For larger integers, a
19225 probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P).
19226 The number is first checked against small prime factors (up to 13). Then,
19227 any number of iterations of the algorithm are performed. Each step either
19228 discovers that the number is non-prime, or substantially increases the
19229 certainty that the number is prime. After a few steps, the chance that
19230 a number was mistakenly described as prime will be less than one percent.
19231 (Indeed, this is a worst-case estimate of the probability; in practice
19232 even a single iteration is quite reliable.) After the @kbd{k p} command,
19233 the number will be reported as definitely prime or non-prime if possible,
19234 or otherwise ``probably'' prime with a certain probability of error.
19240 The normal @kbd{k p} command performs one iteration of the primality
19241 test. Pressing @kbd{k p} repeatedly for the same integer will perform
19242 additional iterations. Also, @kbd{k p} with a numeric prefix performs
19243 the specified number of iterations. There is also an algebraic function
19244 @samp{prime(n)} or @samp{prime(n,iters)} which returns 1 if @expr{n}
19245 is (probably) prime and 0 if not.
19248 @pindex calc-prime-factors
19250 The @kbd{k f} (@code{calc-prime-factors}) [@code{prfac}] command
19251 attempts to decompose an integer into its prime factors. For numbers up
19252 to 25 million, the answer is exact although it may take some time. The
19253 result is a vector of the prime factors in increasing order. For larger
19254 inputs, prime factors above 5000 may not be found, in which case the
19255 last number in the vector will be an unfactored integer greater than 25
19256 million (with a warning message). For negative integers, the first
19257 element of the list will be @mathit{-1}. For inputs @mathit{-1}, @mathit{0}, and
19258 @mathit{1}, the result is a list of the same number.
19261 @pindex calc-next-prime
19263 @mindex nextpr@idots
19266 The @kbd{k n} (@code{calc-next-prime}) [@code{nextprime}] command finds
19267 the next prime above a given number. Essentially, it searches by calling
19268 @code{calc-prime-test} on successive integers until it finds one that
19269 passes the test. This is quite fast for integers less than eight million,
19270 but once the probabilistic test comes into play the search may be rather
19271 slow. Ordinarily this command stops for any prime that passes one iteration
19272 of the primality test. With a numeric prefix argument, a number must pass
19273 the specified number of iterations before the search stops. (This only
19274 matters when searching above eight million.) You can always use additional
19275 @kbd{k p} commands to increase your certainty that the number is indeed
19279 @pindex calc-prev-prime
19281 @mindex prevpr@idots
19284 The @kbd{I k n} (@code{calc-prev-prime}) [@code{prevprime}] command
19285 analogously finds the next prime less than a given number.
19288 @pindex calc-totient
19290 The @kbd{k t} (@code{calc-totient}) [@code{totient}] command computes the
19292 @texline function@tie{}@math{\phi(n)},
19293 @infoline function,
19294 the number of integers less than @expr{n} which
19295 are relatively prime to @expr{n}.
19298 @pindex calc-moebius
19300 The @kbd{k m} (@code{calc-moebius}) [@code{moebius}] command computes the
19301 @texline M@"obius @math{\mu}
19302 @infoline Moebius ``mu''
19303 function. If the input number is a product of @expr{k}
19304 distinct factors, this is @expr{(-1)^k}. If the input number has any
19305 duplicate factors (i.e., can be divided by the same prime more than once),
19306 the result is zero.
19308 @node Probability Distribution Functions, , Combinatorial Functions, Scientific Functions
19309 @section Probability Distribution Functions
19312 The functions in this section compute various probability distributions.
19313 For continuous distributions, this is the integral of the probability
19314 density function from @expr{x} to infinity. (These are the ``upper
19315 tail'' distribution functions; there are also corresponding ``lower
19316 tail'' functions which integrate from minus infinity to @expr{x}.)
19317 For discrete distributions, the upper tail function gives the sum
19318 from @expr{x} to infinity; the lower tail function gives the sum
19319 from minus infinity up to, but not including,@w{ }@expr{x}.
19321 To integrate from @expr{x} to @expr{y}, just use the distribution
19322 function twice and subtract. For example, the probability that a
19323 Gaussian random variable with mean 2 and standard deviation 1 will
19324 lie in the range from 2.5 to 2.8 is @samp{utpn(2.5,2,1) - utpn(2.8,2,1)}
19325 (``the probability that it is greater than 2.5, but not greater than 2.8''),
19326 or equivalently @samp{ltpn(2.8,2,1) - ltpn(2.5,2,1)}.
19333 The @kbd{k B} (@code{calc-utpb}) [@code{utpb}] function uses the
19334 binomial distribution. Push the parameters @var{n}, @var{p}, and
19335 then @var{x} onto the stack; the result (@samp{utpb(x,n,p)}) is the
19336 probability that an event will occur @var{x} or more times out
19337 of @var{n} trials, if its probability of occurring in any given
19338 trial is @var{p}. The @kbd{I k B} [@code{ltpb}] function is
19339 the probability that the event will occur fewer than @var{x} times.
19341 The other probability distribution functions similarly take the
19342 form @kbd{k @var{X}} (@code{calc-utp@var{x}}) [@code{utp@var{x}}]
19343 and @kbd{I k @var{X}} [@code{ltp@var{x}}], for various letters
19344 @var{x}. The arguments to the algebraic functions are the value of
19345 the random variable first, then whatever other parameters define the
19346 distribution. Note these are among the few Calc functions where the
19347 order of the arguments in algebraic form differs from the order of
19348 arguments as found on the stack. (The random variable comes last on
19349 the stack, so that you can type, e.g., @kbd{2 @key{RET} 1 @key{RET} 2.5
19350 k N M-@key{RET} @key{DEL} 2.8 k N -}, using @kbd{M-@key{RET} @key{DEL}} to
19351 recover the original arguments but substitute a new value for @expr{x}.)
19364 The @samp{utpc(x,v)} function uses the chi-square distribution with
19365 @texline @math{\nu}
19367 degrees of freedom. It is the probability that a model is
19368 correct if its chi-square statistic is @expr{x}.
19381 The @samp{utpf(F,v1,v2)} function uses the F distribution, used in
19382 various statistical tests. The parameters
19383 @texline @math{\nu_1}
19384 @infoline @expr{v1}
19386 @texline @math{\nu_2}
19387 @infoline @expr{v2}
19388 are the degrees of freedom in the numerator and denominator,
19389 respectively, used in computing the statistic @expr{F}.
19402 The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution
19403 with mean @expr{m} and standard deviation
19404 @texline @math{\sigma}.
19405 @infoline @expr{s}.
19406 It is the probability that such a normal-distributed random variable
19407 would exceed @expr{x}.
19420 The @samp{utpp(n,x)} function uses a Poisson distribution with
19421 mean @expr{x}. It is the probability that @expr{n} or more such
19422 Poisson random events will occur.
19435 The @samp{utpt(t,v)} function uses the Student's ``t'' distribution
19437 @texline @math{\nu}
19439 degrees of freedom. It is the probability that a
19440 t-distributed random variable will be greater than @expr{t}.
19441 (Note: This computes the distribution function
19442 @texline @math{A(t|\nu)}
19443 @infoline @expr{A(t|v)}
19445 @texline @math{A(0|\nu) = 1}
19446 @infoline @expr{A(0|v) = 1}
19448 @texline @math{A(\infty|\nu) \to 0}.
19449 @infoline @expr{A(inf|v) -> 0}.
19450 The @code{UTPT} operation on the HP-48 uses a different definition which
19451 returns half of Calc's value: @samp{UTPT(t,v) = .5*utpt(t,v)}.)
19453 While Calc does not provide inverses of the probability distribution
19454 functions, the @kbd{a R} command can be used to solve for the inverse.
19455 Since the distribution functions are monotonic, @kbd{a R} is guaranteed
19456 to be able to find a solution given any initial guess.
19457 @xref{Numerical Solutions}.
19459 @node Matrix Functions, Algebra, Scientific Functions, Top
19460 @chapter Vector/Matrix Functions
19463 Many of the commands described here begin with the @kbd{v} prefix.
19464 (For convenience, the shift-@kbd{V} prefix is equivalent to @kbd{v}.)
19465 The commands usually apply to both plain vectors and matrices; some
19466 apply only to matrices or only to square matrices. If the argument
19467 has the wrong dimensions the operation is left in symbolic form.
19469 Vectors are entered and displayed using @samp{[a,b,c]} notation.
19470 Matrices are vectors of which all elements are vectors of equal length.
19471 (Though none of the standard Calc commands use this concept, a
19472 three-dimensional matrix or rank-3 tensor could be defined as a
19473 vector of matrices, and so on.)
19476 * Packing and Unpacking::
19477 * Building Vectors::
19478 * Extracting Elements::
19479 * Manipulating Vectors::
19480 * Vector and Matrix Arithmetic::
19482 * Statistical Operations::
19483 * Reducing and Mapping::
19484 * Vector and Matrix Formats::
19487 @node Packing and Unpacking, Building Vectors, Matrix Functions, Matrix Functions
19488 @section Packing and Unpacking
19491 Calc's ``pack'' and ``unpack'' commands collect stack entries to build
19492 composite objects such as vectors and complex numbers. They are
19493 described in this chapter because they are most often used to build
19499 The @kbd{v p} (@code{calc-pack}) [@code{pack}] command collects several
19500 elements from the stack into a matrix, complex number, HMS form, error
19501 form, etc. It uses a numeric prefix argument to specify the kind of
19502 object to be built; this argument is referred to as the ``packing mode.''
19503 If the packing mode is a nonnegative integer, a vector of that
19504 length is created. For example, @kbd{C-u 5 v p} will pop the top
19505 five stack elements and push back a single vector of those five
19506 elements. (@kbd{C-u 0 v p} simply creates an empty vector.)
19508 The same effect can be had by pressing @kbd{[} to push an incomplete
19509 vector on the stack, using @key{TAB} (@code{calc-roll-down}) to sneak
19510 the incomplete object up past a certain number of elements, and
19511 then pressing @kbd{]} to complete the vector.
19513 Negative packing modes create other kinds of composite objects:
19517 Two values are collected to build a complex number. For example,
19518 @kbd{5 @key{RET} 7 C-u -1 v p} creates the complex number
19519 @expr{(5, 7)}. The result is always a rectangular complex
19520 number. The two input values must both be real numbers,
19521 i.e., integers, fractions, or floats. If they are not, Calc
19522 will instead build a formula like @samp{a + (0, 1) b}. (The
19523 other packing modes also create a symbolic answer if the
19524 components are not suitable.)
19527 Two values are collected to build a polar complex number.
19528 The first is the magnitude; the second is the phase expressed
19529 in either degrees or radians according to the current angular
19533 Three values are collected into an HMS form. The first
19534 two values (hours and minutes) must be integers or
19535 integer-valued floats. The third value may be any real
19539 Two values are collected into an error form. The inputs
19540 may be real numbers or formulas.
19543 Two values are collected into a modulo form. The inputs
19544 must be real numbers.
19547 Two values are collected into the interval @samp{[a .. b]}.
19548 The inputs may be real numbers, HMS or date forms, or formulas.
19551 Two values are collected into the interval @samp{[a .. b)}.
19554 Two values are collected into the interval @samp{(a .. b]}.
19557 Two values are collected into the interval @samp{(a .. b)}.
19560 Two integer values are collected into a fraction.
19563 Two values are collected into a floating-point number.
19564 The first is the mantissa; the second, which must be an
19565 integer, is the exponent. The result is the mantissa
19566 times ten to the power of the exponent.
19569 This is treated the same as @mathit{-11} by the @kbd{v p} command.
19570 When unpacking, @mathit{-12} specifies that a floating-point mantissa
19574 A real number is converted into a date form.
19577 Three numbers (year, month, day) are packed into a pure date form.
19580 Six numbers are packed into a date/time form.
19583 With any of the two-input negative packing modes, either or both
19584 of the inputs may be vectors. If both are vectors of the same
19585 length, the result is another vector made by packing corresponding
19586 elements of the input vectors. If one input is a vector and the
19587 other is a plain number, the number is packed along with each vector
19588 element to produce a new vector. For example, @kbd{C-u -4 v p}
19589 could be used to convert a vector of numbers and a vector of errors
19590 into a single vector of error forms; @kbd{C-u -5 v p} could convert
19591 a vector of numbers and a single number @var{M} into a vector of
19592 numbers modulo @var{M}.
19594 If you don't give a prefix argument to @kbd{v p}, it takes
19595 the packing mode from the top of the stack. The elements to
19596 be packed then begin at stack level 2. Thus
19597 @kbd{1 @key{RET} 2 @key{RET} 4 n v p} is another way to
19598 enter the error form @samp{1 +/- 2}.
19600 If the packing mode taken from the stack is a vector, the result is a
19601 matrix with the dimensions specified by the elements of the vector,
19602 which must each be integers. For example, if the packing mode is
19603 @samp{[2, 3]}, then six numbers will be taken from the stack and
19604 returned in the form @samp{[@w{[a, b, c]}, [d, e, f]]}.
19606 If any elements of the vector are negative, other kinds of
19607 packing are done at that level as described above. For
19608 example, @samp{[2, 3, -4]} takes 12 objects and creates a
19609 @texline @math{2\times3}
19611 matrix of error forms: @samp{[[a +/- b, c +/- d ... ]]}.
19612 Also, @samp{[-4, -10]} will convert four integers into an
19613 error form consisting of two fractions: @samp{a:b +/- c:d}.
19619 There is an equivalent algebraic function,
19620 @samp{pack(@var{mode}, @var{items})} where @var{mode} is a
19621 packing mode (an integer or a vector of integers) and @var{items}
19622 is a vector of objects to be packed (re-packed, really) according
19623 to that mode. For example, @samp{pack([3, -4], [a,b,c,d,e,f])}
19624 yields @samp{[a +/- b, @w{c +/- d}, e +/- f]}. The function is
19625 left in symbolic form if the packing mode is invalid, or if the
19626 number of data items does not match the number of items required
19631 @pindex calc-unpack
19632 The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
19633 number, HMS form, or other composite object on the top of the stack and
19634 ``unpacks'' it, pushing each of its elements onto the stack as separate
19635 objects. Thus, it is the ``inverse'' of @kbd{v p}. If the value
19636 at the top of the stack is a formula, @kbd{v u} unpacks it by pushing
19637 each of the arguments of the top-level operator onto the stack.
19639 You can optionally give a numeric prefix argument to @kbd{v u}
19640 to specify an explicit (un)packing mode. If the packing mode is
19641 negative and the input is actually a vector or matrix, the result
19642 will be two or more similar vectors or matrices of the elements.
19643 For example, given the vector @samp{[@w{a +/- b}, c^2, d +/- 7]},
19644 the result of @kbd{C-u -4 v u} will be the two vectors
19645 @samp{[a, c^2, d]} and @w{@samp{[b, 0, 7]}}.
19647 Note that the prefix argument can have an effect even when the input is
19648 not a vector. For example, if the input is the number @mathit{-5}, then
19649 @kbd{c-u -1 v u} yields @mathit{-5} and 0 (the components of @mathit{-5}
19650 when viewed as a rectangular complex number); @kbd{C-u -2 v u} yields 5
19651 and 180 (assuming Degrees mode); and @kbd{C-u -10 v u} yields @mathit{-5}
19652 and 1 (the numerator and denominator of @mathit{-5}, viewed as a rational
19653 number). Plain @kbd{v u} with this input would complain that the input
19654 is not a composite object.
19656 Unpacking mode @mathit{-11} converts a float into an integer mantissa and
19657 an integer exponent, where the mantissa is not divisible by 10
19658 (except that 0.0 is represented by a mantissa and exponent of 0).
19659 Unpacking mode @mathit{-12} converts a float into a floating-point mantissa
19660 and integer exponent, where the mantissa (for non-zero numbers)
19661 is guaranteed to lie in the range [1 .. 10). In both cases,
19662 the mantissa is shifted left or right (and the exponent adjusted
19663 to compensate) in order to satisfy these constraints.
19665 Positive unpacking modes are treated differently than for @kbd{v p}.
19666 A mode of 1 is much like plain @kbd{v u} with no prefix argument,
19667 except that in addition to the components of the input object,
19668 a suitable packing mode to re-pack the object is also pushed.
19669 Thus, @kbd{C-u 1 v u} followed by @kbd{v p} will re-build the
19672 A mode of 2 unpacks two levels of the object; the resulting
19673 re-packing mode will be a vector of length 2. This might be used
19674 to unpack a matrix, say, or a vector of error forms. Higher
19675 unpacking modes unpack the input even more deeply.
19681 There are two algebraic functions analogous to @kbd{v u}.
19682 The @samp{unpack(@var{mode}, @var{item})} function unpacks the
19683 @var{item} using the given @var{mode}, returning the result as
19684 a vector of components. Here the @var{mode} must be an
19685 integer, not a vector. For example, @samp{unpack(-4, a +/- b)}
19686 returns @samp{[a, b]}, as does @samp{unpack(1, a +/- b)}.
19692 The @code{unpackt} function is like @code{unpack} but instead
19693 of returning a simple vector of items, it returns a vector of
19694 two things: The mode, and the vector of items. For example,
19695 @samp{unpackt(1, 2:3 +/- 1:4)} returns @samp{[-4, [2:3, 1:4]]},
19696 and @samp{unpackt(2, 2:3 +/- 1:4)} returns @samp{[[-4, -10], [2, 3, 1, 4]]}.
19697 The identity for re-building the original object is
19698 @samp{apply(pack, unpackt(@var{n}, @var{x})) = @var{x}}. (The
19699 @code{apply} function builds a function call given the function
19700 name and a vector of arguments.)
19702 @cindex Numerator of a fraction, extracting
19703 Subscript notation is a useful way to extract a particular part
19704 of an object. For example, to get the numerator of a rational
19705 number, you can use @samp{unpack(-10, @var{x})_1}.
19707 @node Building Vectors, Extracting Elements, Packing and Unpacking, Matrix Functions
19708 @section Building Vectors
19711 Vectors and matrices can be added,
19712 subtracted, multiplied, and divided; @pxref{Basic Arithmetic}.
19715 @pindex calc-concat
19720 The @kbd{|} (@code{calc-concat}) [@code{vconcat}] command ``concatenates'' two vectors
19721 into one. For example, after @kbd{@w{[ 1 , 2 ]} [ 3 , 4 ] |}, the stack
19722 will contain the single vector @samp{[1, 2, 3, 4]}. If the arguments
19723 are matrices, the rows of the first matrix are concatenated with the
19724 rows of the second. (In other words, two matrices are just two vectors
19725 of row-vectors as far as @kbd{|} is concerned.)
19727 If either argument to @kbd{|} is a scalar (a non-vector), it is treated
19728 like a one-element vector for purposes of concatenation: @kbd{1 [ 2 , 3 ] |}
19729 produces the vector @samp{[1, 2, 3]}. Likewise, if one argument is a
19730 matrix and the other is a plain vector, the vector is treated as a
19735 The @kbd{H |} (@code{calc-append}) [@code{append}] command concatenates
19736 two vectors without any special cases. Both inputs must be vectors.
19737 Whether or not they are matrices is not taken into account. If either
19738 argument is a scalar, the @code{append} function is left in symbolic form.
19739 See also @code{cons} and @code{rcons} below.
19743 The @kbd{I |} and @kbd{H I |} commands are similar, but they use their
19744 two stack arguments in the opposite order. Thus @kbd{I |} is equivalent
19745 to @kbd{@key{TAB} |}, but possibly more convenient and also a bit faster.
19751 The @kbd{v d} (@code{calc-diag}) [@code{diag}] function builds a diagonal
19752 square matrix. The optional numeric prefix gives the number of rows
19753 and columns in the matrix. If the value at the top of the stack is a
19754 vector, the elements of the vector are used as the diagonal elements; the
19755 prefix, if specified, must match the size of the vector. If the value on
19756 the stack is a scalar, it is used for each element on the diagonal, and
19757 the prefix argument is required.
19759 To build a constant square matrix, e.g., a
19760 @texline @math{3\times3}
19762 matrix filled with ones, use @kbd{0 M-3 v d 1 +}, i.e., build a zero
19763 matrix first and then add a constant value to that matrix. (Another
19764 alternative would be to use @kbd{v b} and @kbd{v a}; see below.)
19770 The @kbd{v i} (@code{calc-ident}) [@code{idn}] function builds an identity
19771 matrix of the specified size. It is a convenient form of @kbd{v d}
19772 where the diagonal element is always one. If no prefix argument is given,
19773 this command prompts for one.
19775 In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)},
19776 except that @expr{a} is required to be a scalar (non-vector) quantity.
19777 If @expr{n} is omitted, @samp{idn(a)} represents @expr{a} times an
19778 identity matrix of unknown size. Calc can operate algebraically on
19779 such generic identity matrices, and if one is combined with a matrix
19780 whose size is known, it is converted automatically to an identity
19781 matrix of a suitable matching size. The @kbd{v i} command with an
19782 argument of zero creates a generic identity matrix, @samp{idn(1)}.
19783 Note that in dimensioned Matrix mode (@pxref{Matrix Mode}), generic
19784 identity matrices are immediately expanded to the current default
19791 The @kbd{v x} (@code{calc-index}) [@code{index}] function builds a vector
19792 of consecutive integers from 1 to @var{n}, where @var{n} is the numeric
19793 prefix argument. If you do not provide a prefix argument, you will be
19794 prompted to enter a suitable number. If @var{n} is negative, the result
19795 is a vector of negative integers from @var{n} to @mathit{-1}.
19797 With a prefix argument of just @kbd{C-u}, the @kbd{v x} command takes
19798 three values from the stack: @var{n}, @var{start}, and @var{incr} (with
19799 @var{incr} at top-of-stack). Counting starts at @var{start} and increases
19800 by @var{incr} for successive vector elements. If @var{start} or @var{n}
19801 is in floating-point format, the resulting vector elements will also be
19802 floats. Note that @var{start} and @var{incr} may in fact be any kind
19803 of numbers or formulas.
19805 When @var{start} and @var{incr} are specified, a negative @var{n} has a
19806 different interpretation: It causes a geometric instead of arithmetic
19807 sequence to be generated. For example, @samp{index(-3, a, b)} produces
19808 @samp{[a, a b, a b^2]}. If you omit @var{incr} in the algebraic form,
19809 @samp{index(@var{n}, @var{start})}, the default value for @var{incr}
19810 is one for positive @var{n} or two for negative @var{n}.
19814 @pindex calc-build-vector
19816 The @kbd{v b} (@code{calc-build-vector}) [@code{cvec}] function builds a
19817 vector of @var{n} copies of the value on the top of the stack, where @var{n}
19818 is the numeric prefix argument. In algebraic formulas, @samp{cvec(x,n,m)}
19819 can also be used to build an @var{n}-by-@var{m} matrix of copies of @var{x}.
19820 (Interactively, just use @kbd{v b} twice: once to build a row, then again
19821 to build a matrix of copies of that row.)
19831 The @kbd{v h} (@code{calc-head}) [@code{head}] function returns the first
19832 element of a vector. The @kbd{I v h} (@code{calc-tail}) [@code{tail}]
19833 function returns the vector with its first element removed. In both
19834 cases, the argument must be a non-empty vector.
19840 The @kbd{v k} (@code{calc-cons}) [@code{cons}] function takes a value @var{h}
19841 and a vector @var{t} from the stack, and produces the vector whose head is
19842 @var{h} and whose tail is @var{t}. This is similar to @kbd{|}, except
19843 if @var{h} is itself a vector, @kbd{|} will concatenate the two vectors
19844 whereas @code{cons} will insert @var{h} at the front of the vector @var{t}.
19867 Each of these three functions also accepts the Hyperbolic flag [@code{rhead},
19868 @code{rtail}, @code{rcons}] in which case @var{t} instead represents
19869 the @emph{last} single element of the vector, with @var{h}
19870 representing the remainder of the vector. Thus the vector
19871 @samp{[a, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)}.
19872 Also, @samp{head([a, b, c, d]) = a}, @samp{tail([a, b, c, d]) = [b, c, d]},
19873 @samp{rhead([a, b, c, d]) = [a, b, c]}, and @samp{rtail([a, b, c, d]) = d}.
19875 @node Extracting Elements, Manipulating Vectors, Building Vectors, Matrix Functions
19876 @section Extracting Vector Elements
19883 The @kbd{v r} (@code{calc-mrow}) [@code{mrow}] command extracts one row of
19884 the matrix on the top of the stack, or one element of the plain vector on
19885 the top of the stack. The row or element is specified by the numeric
19886 prefix argument; the default is to prompt for the row or element number.
19887 The matrix or vector is replaced by the specified row or element in the
19888 form of a vector or scalar, respectively.
19890 @cindex Permutations, applying
19891 With a prefix argument of @kbd{C-u} only, @kbd{v r} takes the index of
19892 the element or row from the top of the stack, and the vector or matrix
19893 from the second-to-top position. If the index is itself a vector of
19894 integers, the result is a vector of the corresponding elements of the
19895 input vector, or a matrix of the corresponding rows of the input matrix.
19896 This command can be used to obtain any permutation of a vector.
19898 With @kbd{C-u}, if the index is an interval form with integer components,
19899 it is interpreted as a range of indices and the corresponding subvector or
19900 submatrix is returned.
19902 @cindex Subscript notation
19904 @pindex calc-subscript
19907 Subscript notation in algebraic formulas (@samp{a_b}) stands for the
19908 Calc function @code{subscr}, which is synonymous with @code{mrow}.
19909 Thus, @samp{[x, y, z]_k} produces @expr{x}, @expr{y}, or @expr{z} if
19910 @expr{k} is one, two, or three, respectively. A double subscript
19911 (@samp{M_i_j}, equivalent to @samp{subscr(subscr(M, i), j)}) will
19912 access the element at row @expr{i}, column @expr{j} of a matrix.
19913 The @kbd{a _} (@code{calc-subscript}) command creates a subscript
19914 formula @samp{a_b} out of two stack entries. (It is on the @kbd{a}
19915 ``algebra'' prefix because subscripted variables are often used
19916 purely as an algebraic notation.)
19919 Given a negative prefix argument, @kbd{v r} instead deletes one row or
19920 element from the matrix or vector on the top of the stack. Thus
19921 @kbd{C-u 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v r}
19922 replaces the matrix with the same matrix with its second row removed.
19923 In algebraic form this function is called @code{mrrow}.
19926 Given a prefix argument of zero, @kbd{v r} extracts the diagonal elements
19927 of a square matrix in the form of a vector. In algebraic form this
19928 function is called @code{getdiag}.
19935 The @kbd{v c} (@code{calc-mcol}) [@code{mcol} or @code{mrcol}] command is
19936 the analogous operation on columns of a matrix. Given a plain vector
19937 it extracts (or removes) one element, just like @kbd{v r}. If the
19938 index in @kbd{C-u v c} is an interval or vector and the argument is a
19939 matrix, the result is a submatrix with only the specified columns
19940 retained (and possibly permuted in the case of a vector index).
19942 To extract a matrix element at a given row and column, use @kbd{v r} to
19943 extract the row as a vector, then @kbd{v c} to extract the column element
19944 from that vector. In algebraic formulas, it is often more convenient to
19945 use subscript notation: @samp{m_i_j} gives row @expr{i}, column @expr{j}
19946 of matrix @expr{m}.
19950 @pindex calc-subvector
19952 The @kbd{v s} (@code{calc-subvector}) [@code{subvec}] command extracts
19953 a subvector of a vector. The arguments are the vector, the starting
19954 index, and the ending index, with the ending index in the top-of-stack
19955 position. The starting index indicates the first element of the vector
19956 to take. The ending index indicates the first element @emph{past} the
19957 range to be taken. Thus, @samp{subvec([a, b, c, d, e], 2, 4)} produces
19958 the subvector @samp{[b, c]}. You could get the same result using
19959 @samp{mrow([a, b, c, d, e], @w{[2 .. 4)})}.
19961 If either the start or the end index is zero or negative, it is
19962 interpreted as relative to the end of the vector. Thus
19963 @samp{subvec([a, b, c, d, e], 2, -2)} also produces @samp{[b, c]}. In
19964 the algebraic form, the end index can be omitted in which case it
19965 is taken as zero, i.e., elements from the starting element to the
19966 end of the vector are used. The infinity symbol, @code{inf}, also
19967 has this effect when used as the ending index.
19972 With the Inverse flag, @kbd{I v s} [@code{rsubvec}] removes a subvector
19973 from a vector. The arguments are interpreted the same as for the
19974 normal @kbd{v s} command. Thus, @samp{rsubvec([a, b, c, d, e], 2, 4)}
19975 produces @samp{[a, d, e]}. It is always true that @code{subvec} and
19976 @code{rsubvec} return complementary parts of the input vector.
19978 @xref{Selecting Subformulas}, for an alternative way to operate on
19979 vectors one element at a time.
19981 @node Manipulating Vectors, Vector and Matrix Arithmetic, Extracting Elements, Matrix Functions
19982 @section Manipulating Vectors
19987 @pindex calc-vlength
19989 The @kbd{v l} (@code{calc-vlength}) [@code{vlen}] command computes the
19990 length of a vector. The length of a non-vector is considered to be zero.
19991 Note that matrices are just vectors of vectors for the purposes of this
19997 With the Hyperbolic flag, @kbd{H v l} [@code{mdims}] computes a vector
19998 of the dimensions of a vector, matrix, or higher-order object. For
19999 example, @samp{mdims([[a,b,c],[d,e,f]])} returns @samp{[2, 3]} since
20001 @texline @math{2\times3}
20007 @pindex calc-vector-find
20009 The @kbd{v f} (@code{calc-vector-find}) [@code{find}] command searches
20010 along a vector for the first element equal to a given target. The target
20011 is on the top of the stack; the vector is in the second-to-top position.
20012 If a match is found, the result is the index of the matching element.
20013 Otherwise, the result is zero. The numeric prefix argument, if given,
20014 allows you to select any starting index for the search.
20018 @pindex calc-arrange-vector
20020 @cindex Arranging a matrix
20021 @cindex Reshaping a matrix
20022 @cindex Flattening a matrix
20023 The @kbd{v a} (@code{calc-arrange-vector}) [@code{arrange}] command
20024 rearranges a vector to have a certain number of columns and rows. The
20025 numeric prefix argument specifies the number of columns; if you do not
20026 provide an argument, you will be prompted for the number of columns.
20027 The vector or matrix on the top of the stack is @dfn{flattened} into a
20028 plain vector. If the number of columns is nonzero, this vector is
20029 then formed into a matrix by taking successive groups of @var{n} elements.
20030 If the number of columns does not evenly divide the number of elements
20031 in the vector, the last row will be short and the result will not be
20032 suitable for use as a matrix. For example, with the matrix
20033 @samp{[[1, 2], @w{[3, 4]}]} on the stack, @kbd{v a 4} produces
20034 @samp{[[1, 2, 3, 4]]} (a
20035 @texline @math{1\times4}
20037 matrix), @kbd{v a 1} produces @samp{[[1], [2], [3], [4]]} (a
20038 @texline @math{4\times1}
20040 matrix), @kbd{v a 2} produces @samp{[[1, 2], [3, 4]]} (the original
20041 @texline @math{2\times2}
20043 matrix), @w{@kbd{v a 3}} produces @samp{[[1, 2, 3], [4]]} (not a
20044 matrix), and @kbd{v a 0} produces the flattened list
20045 @samp{[1, 2, @w{3, 4}]}.
20047 @cindex Sorting data
20055 The @kbd{V S} (@code{calc-sort}) [@code{sort}] command sorts the elements of
20056 a vector into increasing order. Real numbers, real infinities, and
20057 constant interval forms come first in this ordering; next come other
20058 kinds of numbers, then variables (in alphabetical order), then finally
20059 come formulas and other kinds of objects; these are sorted according
20060 to a kind of lexicographic ordering with the useful property that
20061 one vector is less or greater than another if the first corresponding
20062 unequal elements are less or greater, respectively. Since quoted strings
20063 are stored by Calc internally as vectors of ASCII character codes
20064 (@pxref{Strings}), this means vectors of strings are also sorted into
20065 alphabetical order by this command.
20067 The @kbd{I V S} [@code{rsort}] command sorts a vector into decreasing order.
20069 @cindex Permutation, inverse of
20070 @cindex Inverse of permutation
20071 @cindex Index tables
20072 @cindex Rank tables
20080 The @kbd{V G} (@code{calc-grade}) [@code{grade}, @code{rgrade}] command
20081 produces an index table or permutation vector which, if applied to the
20082 input vector (as the index of @kbd{C-u v r}, say), would sort the vector.
20083 A permutation vector is just a vector of integers from 1 to @var{n}, where
20084 each integer occurs exactly once. One application of this is to sort a
20085 matrix of data rows using one column as the sort key; extract that column,
20086 grade it with @kbd{V G}, then use the result to reorder the original matrix
20087 with @kbd{C-u v r}. Another interesting property of the @code{V G} command
20088 is that, if the input is itself a permutation vector, the result will
20089 be the inverse of the permutation. The inverse of an index table is
20090 a rank table, whose @var{k}th element says where the @var{k}th original
20091 vector element will rest when the vector is sorted. To get a rank
20092 table, just use @kbd{V G V G}.
20094 With the Inverse flag, @kbd{I V G} produces an index table that would
20095 sort the input into decreasing order. Note that @kbd{V S} and @kbd{V G}
20096 use a ``stable'' sorting algorithm, i.e., any two elements which are equal
20097 will not be moved out of their original order. Generally there is no way
20098 to tell with @kbd{V S}, since two elements which are equal look the same,
20099 but with @kbd{V G} this can be an important issue. In the matrix-of-rows
20100 example, suppose you have names and telephone numbers as two columns and
20101 you wish to sort by phone number primarily, and by name when the numbers
20102 are equal. You can sort the data matrix by names first, and then again
20103 by phone numbers. Because the sort is stable, any two rows with equal
20104 phone numbers will remain sorted by name even after the second sort.
20109 @pindex calc-histogram
20111 @mindex histo@idots
20114 The @kbd{V H} (@code{calc-histogram}) [@code{histogram}] command builds a
20115 histogram of a vector of numbers. Vector elements are assumed to be
20116 integers or real numbers in the range [0..@var{n}) for some ``number of
20117 bins'' @var{n}, which is the numeric prefix argument given to the
20118 command. The result is a vector of @var{n} counts of how many times
20119 each value appeared in the original vector. Non-integers in the input
20120 are rounded down to integers. Any vector elements outside the specified
20121 range are ignored. (You can tell if elements have been ignored by noting
20122 that the counts in the result vector don't add up to the length of the
20125 If no prefix is given, then you will be prompted for a vector which
20126 will be used to determine the bins. (If a positive integer is given at
20127 this prompt, it will be still treated as if it were given as a
20128 prefix.) Each bin will consist of the interval of numbers closest to
20129 the corresponding number of this new vector; if the vector
20130 @expr{[a, b, c, ...]} is entered at the prompt, the bins will be
20131 @expr{(-inf, (a+b)/2]}, @expr{((a+b)/2, (b+c)/2]}, etc. The result of
20132 this command will be a vector counting how many elements of the
20133 original vector are in each bin.
20135 The result will then be a vector with the same length as this new vector;
20136 each element of the new vector will be replaced by the number of
20137 elements of the original vector which are closest to it.
20141 With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack.
20142 The second-to-top vector is the list of numbers as before. The top
20143 vector is an equal-sized list of ``weights'' to attach to the elements
20144 of the data vector. For example, if the first data element is 4.2 and
20145 the first weight is 10, then 10 will be added to bin 4 of the result
20146 vector. Without the hyperbolic flag, every element has a weight of one.
20150 @pindex calc-transpose
20152 The @kbd{v t} (@code{calc-transpose}) [@code{trn}] command computes
20153 the transpose of the matrix at the top of the stack. If the argument
20154 is a plain vector, it is treated as a row vector and transposed into
20155 a one-column matrix.
20159 @pindex calc-reverse-vector
20161 The @kbd{v v} (@code{calc-reverse-vector}) [@code{rev}] command reverses
20162 a vector end-for-end. Given a matrix, it reverses the order of the rows.
20163 (To reverse the columns instead, just use @kbd{v t v v v t}. The same
20164 principle can be used to apply other vector commands to the columns of
20169 @pindex calc-mask-vector
20171 The @kbd{v m} (@code{calc-mask-vector}) [@code{vmask}] command uses
20172 one vector as a mask to extract elements of another vector. The mask
20173 is in the second-to-top position; the target vector is on the top of
20174 the stack. These vectors must have the same length. The result is
20175 the same as the target vector, but with all elements which correspond
20176 to zeros in the mask vector deleted. Thus, for example,
20177 @samp{vmask([1, 0, 1, 0, 1], [a, b, c, d, e])} produces @samp{[a, c, e]}.
20178 @xref{Logical Operations}.
20182 @pindex calc-expand-vector
20184 The @kbd{v e} (@code{calc-expand-vector}) [@code{vexp}] command
20185 expands a vector according to another mask vector. The result is a
20186 vector the same length as the mask, but with nonzero elements replaced
20187 by successive elements from the target vector. The length of the target
20188 vector is normally the number of nonzero elements in the mask. If the
20189 target vector is longer, its last few elements are lost. If the target
20190 vector is shorter, the last few nonzero mask elements are left
20191 unreplaced in the result. Thus @samp{vexp([2, 0, 3, 0, 7], [a, b])}
20192 produces @samp{[a, 0, b, 0, 7]}.
20196 With the Hyperbolic flag, @kbd{H v e} takes a filler value from the
20197 top of the stack; the mask and target vectors come from the third and
20198 second elements of the stack. This filler is used where the mask is
20199 zero: @samp{vexp([2, 0, 3, 0, 7], [a, b], z)} produces
20200 @samp{[a, z, c, z, 7]}. If the filler value is itself a vector,
20201 then successive values are taken from it, so that the effect is to
20202 interleave two vectors according to the mask:
20203 @samp{vexp([2, 0, 3, 7, 0, 0], [a, b], [x, y])} produces
20204 @samp{[a, x, b, 7, y, 0]}.
20206 Another variation on the masking idea is to combine @samp{[a, b, c, d, e]}
20207 with the mask @samp{[1, 0, 1, 0, 1]} to produce @samp{[a, 0, c, 0, e]}.
20208 You can accomplish this with @kbd{V M a &}, mapping the logical ``and''
20209 operation across the two vectors. @xref{Logical Operations}. Note that
20210 the @code{? :} operation also discussed there allows other types of
20211 masking using vectors.
20213 @node Vector and Matrix Arithmetic, Set Operations, Manipulating Vectors, Matrix Functions
20214 @section Vector and Matrix Arithmetic
20217 Basic arithmetic operations like addition and multiplication are defined
20218 for vectors and matrices as well as for numbers. Division of matrices, in
20219 the sense of multiplying by the inverse, is supported. (Division by a
20220 matrix actually uses LU-decomposition for greater accuracy and speed.)
20221 @xref{Basic Arithmetic}.
20223 The following functions are applied element-wise if their arguments are
20224 vectors or matrices: @code{change-sign}, @code{conj}, @code{arg},
20225 @code{re}, @code{im}, @code{polar}, @code{rect}, @code{clean},
20226 @code{float}, @code{frac}. @xref{Function Index}.
20230 @pindex calc-conj-transpose
20232 The @kbd{V J} (@code{calc-conj-transpose}) [@code{ctrn}] command computes
20233 the conjugate transpose of its argument, i.e., @samp{conj(trn(x))}.
20238 @kindex A (vectors)
20239 @pindex calc-abs (vectors)
20243 @tindex abs (vectors)
20244 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the
20245 Frobenius norm of a vector or matrix argument. This is the square
20246 root of the sum of the squares of the absolute values of the
20247 elements of the vector or matrix. If the vector is interpreted as
20248 a point in two- or three-dimensional space, this is the distance
20249 from that point to the origin.
20255 The @kbd{v n} (@code{calc-rnorm}) [@code{rnorm}] command computes the
20256 infinity-norm of a vector, or the row norm of a matrix. For a plain
20257 vector, this is the maximum of the absolute values of the elements. For
20258 a matrix, this is the maximum of the row-absolute-value-sums, i.e., of
20259 the sums of the absolute values of the elements along the various rows.
20265 The @kbd{V N} (@code{calc-cnorm}) [@code{cnorm}] command computes
20266 the one-norm of a vector, or column norm of a matrix. For a plain
20267 vector, this is the sum of the absolute values of the elements.
20268 For a matrix, this is the maximum of the column-absolute-value-sums.
20269 General @expr{k}-norms for @expr{k} other than one or infinity are
20270 not provided. However, the 2-norm (or Frobenius norm) is provided for
20271 vectors by the @kbd{A} (@code{calc-abs}) command.
20277 The @kbd{V C} (@code{calc-cross}) [@code{cross}] command computes the
20278 right-handed cross product of two vectors, each of which must have
20279 exactly three elements.
20284 @kindex & (matrices)
20285 @pindex calc-inv (matrices)
20289 @tindex inv (matrices)
20290 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
20291 inverse of a square matrix. If the matrix is singular, the inverse
20292 operation is left in symbolic form. Matrix inverses are recorded so
20293 that once an inverse (or determinant) of a particular matrix has been
20294 computed, the inverse and determinant of the matrix can be recomputed
20295 quickly in the future.
20297 If the argument to @kbd{&} is a plain number @expr{x}, this
20298 command simply computes @expr{1/x}. This is okay, because the
20299 @samp{/} operator also does a matrix inversion when dividing one
20306 The @kbd{V D} (@code{calc-mdet}) [@code{det}] command computes the
20307 determinant of a square matrix.
20313 The @kbd{V L} (@code{calc-mlud}) [@code{lud}] command computes the
20314 LU decomposition of a matrix. The result is a list of three matrices
20315 which, when multiplied together left-to-right, form the original matrix.
20316 The first is a permutation matrix that arises from pivoting in the
20317 algorithm, the second is lower-triangular with ones on the diagonal,
20318 and the third is upper-triangular.
20322 @pindex calc-mtrace
20324 The @kbd{V T} (@code{calc-mtrace}) [@code{tr}] command computes the
20325 trace of a square matrix. This is defined as the sum of the diagonal
20326 elements of the matrix.
20332 The @kbd{V K} (@code{calc-kron}) [@code{kron}] command computes
20333 the Kronecker product of two matrices.
20335 @node Set Operations, Statistical Operations, Vector and Matrix Arithmetic, Matrix Functions
20336 @section Set Operations using Vectors
20339 @cindex Sets, as vectors
20340 Calc includes several commands which interpret vectors as @dfn{sets} of
20341 objects. A set is a collection of objects; any given object can appear
20342 only once in the set. Calc stores sets as vectors of objects in
20343 sorted order. Objects in a Calc set can be any of the usual things,
20344 such as numbers, variables, or formulas. Two set elements are considered
20345 equal if they are identical, except that numerically equal numbers like
20346 the integer 4 and the float 4.0 are considered equal even though they
20347 are not ``identical.'' Variables are treated like plain symbols without
20348 attached values by the set operations; subtracting the set @samp{[b]}
20349 from @samp{[a, b]} always yields the set @samp{[a]} even though if
20350 the variables @samp{a} and @samp{b} both equaled 17, you might
20351 expect the answer @samp{[]}.
20353 If a set contains interval forms, then it is assumed to be a set of
20354 real numbers. In this case, all set operations require the elements
20355 of the set to be only things that are allowed in intervals: Real
20356 numbers, plus and minus infinity, HMS forms, and date forms. If
20357 there are variables or other non-real objects present in a real set,
20358 all set operations on it will be left in unevaluated form.
20360 If the input to a set operation is a plain number or interval form
20361 @var{a}, it is treated like the one-element vector @samp{[@var{a}]}.
20362 The result is always a vector, except that if the set consists of a
20363 single interval, the interval itself is returned instead.
20365 @xref{Logical Operations}, for the @code{in} function which tests if
20366 a certain value is a member of a given set. To test if the set @expr{A}
20367 is a subset of the set @expr{B}, use @samp{vdiff(A, B) = []}.
20371 @pindex calc-remove-duplicates
20373 The @kbd{V +} (@code{calc-remove-duplicates}) [@code{rdup}] command
20374 converts an arbitrary vector into set notation. It works by sorting
20375 the vector as if by @kbd{V S}, then removing duplicates. (For example,
20376 @kbd{[a, 5, 4, a, 4.0]} is sorted to @samp{[4, 4.0, 5, a, a]} and then
20377 reduced to @samp{[4, 5, a]}). Overlapping intervals are merged as
20378 necessary. You rarely need to use @kbd{V +} explicitly, since all the
20379 other set-based commands apply @kbd{V +} to their inputs before using
20384 @pindex calc-set-union
20386 The @kbd{V V} (@code{calc-set-union}) [@code{vunion}] command computes
20387 the union of two sets. An object is in the union of two sets if and
20388 only if it is in either (or both) of the input sets. (You could
20389 accomplish the same thing by concatenating the sets with @kbd{|},
20390 then using @kbd{V +}.)
20394 @pindex calc-set-intersect
20396 The @kbd{V ^} (@code{calc-set-intersect}) [@code{vint}] command computes
20397 the intersection of two sets. An object is in the intersection if
20398 and only if it is in both of the input sets. Thus if the input
20399 sets are disjoint, i.e., if they share no common elements, the result
20400 will be the empty vector @samp{[]}. Note that the characters @kbd{V}
20401 and @kbd{^} were chosen to be close to the conventional mathematical
20403 @texline union@tie{}(@math{A \cup B})
20406 @texline intersection@tie{}(@math{A \cap B}).
20407 @infoline intersection.
20411 @pindex calc-set-difference
20413 The @kbd{V -} (@code{calc-set-difference}) [@code{vdiff}] command computes
20414 the difference between two sets. An object is in the difference
20415 @expr{A - B} if and only if it is in @expr{A} but not in @expr{B}.
20416 Thus subtracting @samp{[y,z]} from a set will remove the elements
20417 @samp{y} and @samp{z} if they are present. You can also think of this
20418 as a general @dfn{set complement} operator; if @expr{A} is the set of
20419 all possible values, then @expr{A - B} is the ``complement'' of @expr{B}.
20420 Obviously this is only practical if the set of all possible values in
20421 your problem is small enough to list in a Calc vector (or simple
20422 enough to express in a few intervals).
20426 @pindex calc-set-xor
20428 The @kbd{V X} (@code{calc-set-xor}) [@code{vxor}] command computes
20429 the ``exclusive-or,'' or ``symmetric difference'' of two sets.
20430 An object is in the symmetric difference of two sets if and only
20431 if it is in one, but @emph{not} both, of the sets. Objects that
20432 occur in both sets ``cancel out.''
20436 @pindex calc-set-complement
20438 The @kbd{V ~} (@code{calc-set-complement}) [@code{vcompl}] command
20439 computes the complement of a set with respect to the real numbers.
20440 Thus @samp{vcompl(x)} is equivalent to @samp{vdiff([-inf .. inf], x)}.
20441 For example, @samp{vcompl([2, (3 .. 4]])} evaluates to
20442 @samp{[[-inf .. 2), (2 .. 3], (4 .. inf]]}.
20446 @pindex calc-set-floor
20448 The @kbd{V F} (@code{calc-set-floor}) [@code{vfloor}] command
20449 reinterprets a set as a set of integers. Any non-integer values,
20450 and intervals that do not enclose any integers, are removed. Open
20451 intervals are converted to equivalent closed intervals. Successive
20452 integers are converted into intervals of integers. For example, the
20453 complement of the set @samp{[2, 6, 7, 8]} is messy, but if you wanted
20454 the complement with respect to the set of integers you could type
20455 @kbd{V ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}.
20459 @pindex calc-set-enumerate
20461 The @kbd{V E} (@code{calc-set-enumerate}) [@code{venum}] command
20462 converts a set of integers into an explicit vector. Intervals in
20463 the set are expanded out to lists of all integers encompassed by
20464 the intervals. This only works for finite sets (i.e., sets which
20465 do not involve @samp{-inf} or @samp{inf}).
20469 @pindex calc-set-span
20471 The @kbd{V :} (@code{calc-set-span}) [@code{vspan}] command converts any
20472 set of reals into an interval form that encompasses all its elements.
20473 The lower limit will be the smallest element in the set; the upper
20474 limit will be the largest element. For an empty set, @samp{vspan([])}
20475 returns the empty interval @w{@samp{[0 .. 0)}}.
20479 @pindex calc-set-cardinality
20481 The @kbd{V #} (@code{calc-set-cardinality}) [@code{vcard}] command counts
20482 the number of integers in a set. The result is the length of the vector
20483 that would be produced by @kbd{V E}, although the computation is much
20484 more efficient than actually producing that vector.
20486 @cindex Sets, as binary numbers
20487 Another representation for sets that may be more appropriate in some
20488 cases is binary numbers. If you are dealing with sets of integers
20489 in the range 0 to 49, you can use a 50-bit binary number where a
20490 particular bit is 1 if the corresponding element is in the set.
20491 @xref{Binary Functions}, for a list of commands that operate on
20492 binary numbers. Note that many of the above set operations have
20493 direct equivalents in binary arithmetic: @kbd{b o} (@code{calc-or}),
20494 @kbd{b a} (@code{calc-and}), @kbd{b d} (@code{calc-diff}),
20495 @kbd{b x} (@code{calc-xor}), and @kbd{b n} (@code{calc-not}),
20496 respectively. You can use whatever representation for sets is most
20501 @pindex calc-pack-bits
20502 @pindex calc-unpack-bits
20505 The @kbd{b u} (@code{calc-unpack-bits}) [@code{vunpack}] command
20506 converts an integer that represents a set in binary into a set
20507 in vector/interval notation. For example, @samp{vunpack(67)}
20508 returns @samp{[[0 .. 1], 6]}. If the input is negative, the set
20509 it represents is semi-infinite: @samp{vunpack(-4) = [2 .. inf)}.
20510 Use @kbd{V E} afterwards to expand intervals to individual
20511 values if you wish. Note that this command uses the @kbd{b}
20512 (binary) prefix key.
20514 The @kbd{b p} (@code{calc-pack-bits}) [@code{vpack}] command
20515 converts the other way, from a vector or interval representing
20516 a set of nonnegative integers into a binary integer describing
20517 the same set. The set may include positive infinity, but must
20518 not include any negative numbers. The input is interpreted as a
20519 set of integers in the sense of @kbd{V F} (@code{vfloor}). Beware
20520 that a simple input like @samp{[100]} can result in a huge integer
20522 @texline (@math{2^{100}}, a 31-digit integer, in this case).
20523 @infoline (@expr{2^100}, a 31-digit integer, in this case).
20525 @node Statistical Operations, Reducing and Mapping, Set Operations, Matrix Functions
20526 @section Statistical Operations on Vectors
20529 @cindex Statistical functions
20530 The commands in this section take vectors as arguments and compute
20531 various statistical measures on the data stored in the vectors. The
20532 references used in the definitions of these functions are Bevington's
20533 @emph{Data Reduction and Error Analysis for the Physical Sciences},
20534 and @emph{Numerical Recipes} by Press, Flannery, Teukolsky and
20537 The statistical commands use the @kbd{u} prefix key followed by
20538 a shifted letter or other character.
20540 @xref{Manipulating Vectors}, for a description of @kbd{V H}
20541 (@code{calc-histogram}).
20543 @xref{Curve Fitting}, for the @kbd{a F} command for doing
20544 least-squares fits to statistical data.
20546 @xref{Probability Distribution Functions}, for several common
20547 probability distribution functions.
20550 * Single-Variable Statistics::
20551 * Paired-Sample Statistics::
20554 @node Single-Variable Statistics, Paired-Sample Statistics, Statistical Operations, Statistical Operations
20555 @subsection Single-Variable Statistics
20558 These functions do various statistical computations on single
20559 vectors. Given a numeric prefix argument, they actually pop
20560 @var{n} objects from the stack and combine them into a data
20561 vector. Each object may be either a number or a vector; if a
20562 vector, any sub-vectors inside it are ``flattened'' as if by
20563 @kbd{v a 0}; @pxref{Manipulating Vectors}. By default one object
20564 is popped, which (in order to be useful) is usually a vector.
20566 If an argument is a variable name, and the value stored in that
20567 variable is a vector, then the stored vector is used. This method
20568 has the advantage that if your data vector is large, you can avoid
20569 the slow process of manipulating it directly on the stack.
20571 These functions are left in symbolic form if any of their arguments
20572 are not numbers or vectors, e.g., if an argument is a formula, or
20573 a non-vector variable. However, formulas embedded within vector
20574 arguments are accepted; the result is a symbolic representation
20575 of the computation, based on the assumption that the formula does
20576 not itself represent a vector. All varieties of numbers such as
20577 error forms and interval forms are acceptable.
20579 Some of the functions in this section also accept a single error form
20580 or interval as an argument. They then describe a property of the
20581 normal or uniform (respectively) statistical distribution described
20582 by the argument. The arguments are interpreted in the same way as
20583 the @var{M} argument of the random number function @kbd{k r}. In
20584 particular, an interval with integer limits is considered an integer
20585 distribution, so that @samp{[2 .. 6)} is the same as @samp{[2 .. 5]}.
20586 An interval with at least one floating-point limit is a continuous
20587 distribution: @samp{[2.0 .. 6.0)} is @emph{not} the same as
20588 @samp{[2.0 .. 5.0]}!
20591 @pindex calc-vector-count
20593 The @kbd{u #} (@code{calc-vector-count}) [@code{vcount}] command
20594 computes the number of data values represented by the inputs.
20595 For example, @samp{vcount(1, [2, 3], [[4, 5], [], x, y])} returns 7.
20596 If the argument is a single vector with no sub-vectors, this
20597 simply computes the length of the vector.
20601 @pindex calc-vector-sum
20602 @pindex calc-vector-prod
20605 @cindex Summations (statistical)
20606 The @kbd{u +} (@code{calc-vector-sum}) [@code{vsum}] command
20607 computes the sum of the data values. The @kbd{u *}
20608 (@code{calc-vector-prod}) [@code{vprod}] command computes the
20609 product of the data values. If the input is a single flat vector,
20610 these are the same as @kbd{V R +} and @kbd{V R *}
20611 (@pxref{Reducing and Mapping}).
20615 @pindex calc-vector-max
20616 @pindex calc-vector-min
20619 The @kbd{u X} (@code{calc-vector-max}) [@code{vmax}] command
20620 computes the maximum of the data values, and the @kbd{u N}
20621 (@code{calc-vector-min}) [@code{vmin}] command computes the minimum.
20622 If the argument is an interval, this finds the minimum or maximum
20623 value in the interval. (Note that @samp{vmax([2..6)) = 5} as
20624 described above.) If the argument is an error form, this returns
20625 plus or minus infinity.
20628 @pindex calc-vector-mean
20630 @cindex Mean of data values
20631 The @kbd{u M} (@code{calc-vector-mean}) [@code{vmean}] command
20632 computes the average (arithmetic mean) of the data values.
20633 If the inputs are error forms
20634 @texline @math{x \pm \sigma},
20635 @infoline @samp{x +/- s},
20636 this is the weighted mean of the @expr{x} values with weights
20637 @texline @math{1 /\sigma^2}.
20638 @infoline @expr{1 / s^2}.
20640 $$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
20641 \displaystyle \sum { 1 \over \sigma_i^2 } } $$
20643 If the inputs are not error forms, this is simply the sum of the
20644 values divided by the count of the values.
20646 Note that a plain number can be considered an error form with
20648 @texline @math{\sigma = 0}.
20649 @infoline @expr{s = 0}.
20650 If the input to @kbd{u M} is a mixture of
20651 plain numbers and error forms, the result is the mean of the
20652 plain numbers, ignoring all values with non-zero errors. (By the
20653 above definitions it's clear that a plain number effectively
20654 has an infinite weight, next to which an error form with a finite
20655 weight is completely negligible.)
20657 This function also works for distributions (error forms or
20658 intervals). The mean of an error form `@var{a} @tfn{+/-} @var{b}' is simply
20659 @expr{a}. The mean of an interval is the mean of the minimum
20660 and maximum values of the interval.
20663 @pindex calc-vector-mean-error
20665 The @kbd{I u M} (@code{calc-vector-mean-error}) [@code{vmeane}]
20666 command computes the mean of the data points expressed as an
20667 error form. This includes the estimated error associated with
20668 the mean. If the inputs are error forms, the error is the square
20669 root of the reciprocal of the sum of the reciprocals of the squares
20670 of the input errors. (I.e., the variance is the reciprocal of the
20671 sum of the reciprocals of the variances.)
20673 $$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
20675 If the inputs are plain
20676 numbers, the error is equal to the standard deviation of the values
20677 divided by the square root of the number of values. (This works
20678 out to be equivalent to calculating the standard deviation and
20679 then assuming each value's error is equal to this standard
20682 $$ \sigma_\mu^2 = {\sigma^2 \over N} $$
20686 @pindex calc-vector-median
20688 @cindex Median of data values
20689 The @kbd{H u M} (@code{calc-vector-median}) [@code{vmedian}]
20690 command computes the median of the data values. The values are
20691 first sorted into numerical order; the median is the middle
20692 value after sorting. (If the number of data values is even,
20693 the median is taken to be the average of the two middle values.)
20694 The median function is different from the other functions in
20695 this section in that the arguments must all be real numbers;
20696 variables are not accepted even when nested inside vectors.
20697 (Otherwise it is not possible to sort the data values.) If
20698 any of the input values are error forms, their error parts are
20701 The median function also accepts distributions. For both normal
20702 (error form) and uniform (interval) distributions, the median is
20703 the same as the mean.
20706 @pindex calc-vector-harmonic-mean
20708 @cindex Harmonic mean
20709 The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) [@code{vhmean}]
20710 command computes the harmonic mean of the data values. This is
20711 defined as the reciprocal of the arithmetic mean of the reciprocals
20714 $$ { N \over \displaystyle \sum {1 \over x_i} } $$
20718 @pindex calc-vector-geometric-mean
20720 @cindex Geometric mean
20721 The @kbd{u G} (@code{calc-vector-geometric-mean}) [@code{vgmean}]
20722 command computes the geometric mean of the data values. This
20723 is the @var{n}th root of the product of the values. This is also
20724 equal to the @code{exp} of the arithmetic mean of the logarithms
20725 of the data values.
20727 $$ \exp \left ( \sum { \ln x_i } \right ) =
20728 \left ( \prod { x_i } \right)^{1 / N} $$
20733 The @kbd{H u G} [@code{agmean}] command computes the ``arithmetic-geometric
20734 mean'' of two numbers taken from the stack. This is computed by
20735 replacing the two numbers with their arithmetic mean and geometric
20736 mean, then repeating until the two values converge.
20738 $$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
20741 @cindex Root-mean-square
20742 Another commonly used mean, the RMS (root-mean-square), can be computed
20743 for a vector of numbers simply by using the @kbd{A} command.
20746 @pindex calc-vector-sdev
20748 @cindex Standard deviation
20749 @cindex Sample statistics
20750 The @kbd{u S} (@code{calc-vector-sdev}) [@code{vsdev}] command
20751 computes the standard
20752 @texline deviation@tie{}@math{\sigma}
20753 @infoline deviation
20754 of the data values. If the values are error forms, the errors are used
20755 as weights just as for @kbd{u M}. This is the @emph{sample} standard
20756 deviation, whose value is the square root of the sum of the squares of
20757 the differences between the values and the mean of the @expr{N} values,
20758 divided by @expr{N-1}.
20760 $$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
20763 This function also applies to distributions. The standard deviation
20764 of a single error form is simply the error part. The standard deviation
20765 of a continuous interval happens to equal the difference between the
20767 @texline @math{\sqrt{12}}.
20768 @infoline @expr{sqrt(12)}.
20769 The standard deviation of an integer interval is the same as the
20770 standard deviation of a vector of those integers.
20773 @pindex calc-vector-pop-sdev
20775 @cindex Population statistics
20776 The @kbd{I u S} (@code{calc-vector-pop-sdev}) [@code{vpsdev}]
20777 command computes the @emph{population} standard deviation.
20778 It is defined by the same formula as above but dividing
20779 by @expr{N} instead of by @expr{N-1}. The population standard
20780 deviation is used when the input represents the entire set of
20781 data values in the distribution; the sample standard deviation
20782 is used when the input represents a sample of the set of all
20783 data values, so that the mean computed from the input is itself
20784 only an estimate of the true mean.
20786 $$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
20789 For error forms and continuous intervals, @code{vpsdev} works
20790 exactly like @code{vsdev}. For integer intervals, it computes the
20791 population standard deviation of the equivalent vector of integers.
20795 @pindex calc-vector-variance
20796 @pindex calc-vector-pop-variance
20799 @cindex Variance of data values
20800 The @kbd{H u S} (@code{calc-vector-variance}) [@code{vvar}] and
20801 @kbd{H I u S} (@code{calc-vector-pop-variance}) [@code{vpvar}]
20802 commands compute the variance of the data values. The variance
20804 @texline square@tie{}@math{\sigma^2}
20806 of the standard deviation, i.e., the sum of the
20807 squares of the deviations of the data values from the mean.
20808 (This definition also applies when the argument is a distribution.)
20814 The @code{vflat} algebraic function returns a vector of its
20815 arguments, interpreted in the same way as the other functions
20816 in this section. For example, @samp{vflat(1, [2, [3, 4]], 5)}
20817 returns @samp{[1, 2, 3, 4, 5]}.
20819 @node Paired-Sample Statistics, , Single-Variable Statistics, Statistical Operations
20820 @subsection Paired-Sample Statistics
20823 The functions in this section take two arguments, which must be
20824 vectors of equal size. The vectors are each flattened in the same
20825 way as by the single-variable statistical functions. Given a numeric
20826 prefix argument of 1, these functions instead take one object from
20827 the stack, which must be an
20828 @texline @math{N\times2}
20830 matrix of data values. Once again, variable names can be used in place
20831 of actual vectors and matrices.
20834 @pindex calc-vector-covariance
20837 The @kbd{u C} (@code{calc-vector-covariance}) [@code{vcov}] command
20838 computes the sample covariance of two vectors. The covariance
20839 of vectors @var{x} and @var{y} is the sum of the products of the
20840 differences between the elements of @var{x} and the mean of @var{x}
20841 times the differences between the corresponding elements of @var{y}
20842 and the mean of @var{y}, all divided by @expr{N-1}. Note that
20843 the variance of a vector is just the covariance of the vector
20844 with itself. Once again, if the inputs are error forms the
20845 errors are used as weight factors. If both @var{x} and @var{y}
20846 are composed of error forms, the error for a given data point
20847 is taken as the square root of the sum of the squares of the two
20850 $$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
20851 $$ \sigma_{x\!y}^2 =
20852 {\displaystyle {1 \over N-1}
20853 \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2}
20854 \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}}
20859 @pindex calc-vector-pop-covariance
20861 The @kbd{I u C} (@code{calc-vector-pop-covariance}) [@code{vpcov}]
20862 command computes the population covariance, which is the same as the
20863 sample covariance computed by @kbd{u C} except dividing by @expr{N}
20864 instead of @expr{N-1}.
20867 @pindex calc-vector-correlation
20869 @cindex Correlation coefficient
20870 @cindex Linear correlation
20871 The @kbd{H u C} (@code{calc-vector-correlation}) [@code{vcorr}]
20872 command computes the linear correlation coefficient of two vectors.
20873 This is defined by the covariance of the vectors divided by the
20874 product of their standard deviations. (There is no difference
20875 between sample or population statistics here.)
20877 $$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
20880 @node Reducing and Mapping, Vector and Matrix Formats, Statistical Operations, Matrix Functions
20881 @section Reducing and Mapping Vectors
20884 The commands in this section allow for more general operations on the
20885 elements of vectors.
20891 The simplest of these operations is @kbd{V A} (@code{calc-apply})
20892 [@code{apply}], which applies a given operator to the elements of a vector.
20893 For example, applying the hypothetical function @code{f} to the vector
20894 @w{@samp{[1, 2, 3]}} would produce the function call @samp{f(1, 2, 3)}.
20895 Applying the @code{+} function to the vector @samp{[a, b]} gives
20896 @samp{a + b}. Applying @code{+} to the vector @samp{[a, b, c]} is an
20897 error, since the @code{+} function expects exactly two arguments.
20899 While @kbd{V A} is useful in some cases, you will usually find that either
20900 @kbd{V R} or @kbd{V M}, described below, is closer to what you want.
20903 * Specifying Operators::
20906 * Nesting and Fixed Points::
20907 * Generalized Products::
20910 @node Specifying Operators, Mapping, Reducing and Mapping, Reducing and Mapping
20911 @subsection Specifying Operators
20914 Commands in this section (like @kbd{V A}) prompt you to press the key
20915 corresponding to the desired operator. Press @kbd{?} for a partial
20916 list of the available operators. Generally, an operator is any key or
20917 sequence of keys that would normally take one or more arguments from
20918 the stack and replace them with a result. For example, @kbd{V A H C}
20919 uses the hyperbolic cosine operator, @code{cosh}. (Since @code{cosh}
20920 expects one argument, @kbd{V A H C} requires a vector with a single
20921 element as its argument.)
20923 You can press @kbd{x} at the operator prompt to select any algebraic
20924 function by name to use as the operator. This includes functions you
20925 have defined yourself using the @kbd{Z F} command. (@xref{Algebraic
20926 Definitions}.) If you give a name for which no function has been
20927 defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}.
20928 Calc will prompt for the number of arguments the function takes if it
20929 can't figure it out on its own (say, because you named a function that
20930 is currently undefined). It is also possible to type a digit key before
20931 the function name to specify the number of arguments, e.g.,
20932 @kbd{V M 3 x f @key{RET}} calls @code{f} with three arguments even if it
20933 looks like it ought to have only two. This technique may be necessary
20934 if the function allows a variable number of arguments. For example,
20935 the @kbd{v e} [@code{vexp}] function accepts two or three arguments;
20936 if you want to map with the three-argument version, you will have to
20937 type @kbd{V M 3 v e}.
20939 It is also possible to apply any formula to a vector by treating that
20940 formula as a function. When prompted for the operator to use, press
20941 @kbd{'} (the apostrophe) and type your formula as an algebraic entry.
20942 You will then be prompted for the argument list, which defaults to a
20943 list of all variables that appear in the formula, sorted into alphabetic
20944 order. For example, suppose you enter the formula @w{@samp{x + 2y^x}}.
20945 The default argument list would be @samp{(x y)}, which means that if
20946 this function is applied to the arguments @samp{[3, 10]} the result will
20947 be @samp{3 + 2*10^3}. (If you plan to use a certain formula in this
20948 way often, you might consider defining it as a function with @kbd{Z F}.)
20950 Another way to specify the arguments to the formula you enter is with
20951 @kbd{$}, @kbd{$$}, and so on. For example, @kbd{V A ' $$ + 2$^$$}
20952 has the same effect as the previous example. The argument list is
20953 automatically taken to be @samp{($$ $)}. (The order of the arguments
20954 may seem backwards, but it is analogous to the way normal algebraic
20955 entry interacts with the stack.)
20957 If you press @kbd{$} at the operator prompt, the effect is similar to
20958 the apostrophe except that the relevant formula is taken from top-of-stack
20959 instead. The actual vector arguments of the @kbd{V A $} or related command
20960 then start at the second-to-top stack position. You will still be
20961 prompted for an argument list.
20963 @cindex Nameless functions
20964 @cindex Generic functions
20965 A function can be written without a name using the notation @samp{<#1 - #2>},
20966 which means ``a function of two arguments that computes the first
20967 argument minus the second argument.'' The symbols @samp{#1} and @samp{#2}
20968 are placeholders for the arguments. You can use any names for these
20969 placeholders if you wish, by including an argument list followed by a
20970 colon: @samp{<x, y : x - y>}. When you type @kbd{V A ' $$ + 2$^$$ @key{RET}},
20971 Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function
20972 to map across the vectors. When you type @kbd{V A ' x + 2y^x @key{RET} @key{RET}},
20973 Calc builds the nameless function @w{@samp{<x, y : x + 2 y^x>}}. In both
20974 cases, Calc also writes the nameless function to the Trail so that you
20975 can get it back later if you wish.
20977 If there is only one argument, you can write @samp{#} in place of @samp{#1}.
20978 (Note that @samp{< >} notation is also used for date forms. Calc tells
20979 that @samp{<@var{stuff}>} is a nameless function by the presence of
20980 @samp{#} signs inside @var{stuff}, or by the fact that @var{stuff}
20981 begins with a list of variables followed by a colon.)
20983 You can type a nameless function directly to @kbd{V A '}, or put one on
20984 the stack and use it with @w{@kbd{V A $}}. Calc will not prompt for an
20985 argument list in this case, since the nameless function specifies the
20986 argument list as well as the function itself. In @kbd{V A '}, you can
20987 omit the @samp{< >} marks if you use @samp{#} notation for the arguments,
20988 so that @kbd{V A ' #1+#2 @key{RET}} is the same as @kbd{V A ' <#1+#2> @key{RET}},
20989 which in turn is the same as @kbd{V A ' $$+$ @key{RET}}.
20991 @cindex Lambda expressions
20996 The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}.
20997 (The word @code{lambda} derives from Lisp notation and the theory of
20998 functions.) The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA,
20999 ArgB, ArgA + ArgB)}. Note that there is no actual Calc function called
21000 @code{lambda}; the whole point is that the @code{lambda} expression is
21001 used in its symbolic form, not evaluated for an answer until it is applied
21002 to specific arguments by a command like @kbd{V A} or @kbd{V M}.
21004 (Actually, @code{lambda} does have one special property: Its arguments
21005 are never evaluated; for example, putting @samp{<(2/3) #>} on the stack
21006 will not simplify the @samp{2/3} until the nameless function is actually
21035 As usual, commands like @kbd{V A} have algebraic function name equivalents.
21036 For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to
21037 @samp{apply(gcd, v)}. The first argument specifies the operator name,
21038 and is either a variable whose name is the same as the function name,
21039 or a nameless function like @samp{<#^3+1>}. Operators that are normally
21040 written as algebraic symbols have the names @code{add}, @code{sub},
21041 @code{mul}, @code{div}, @code{pow}, @code{neg}, @code{mod}, and
21048 The @code{call} function builds a function call out of several arguments:
21049 @samp{call(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which
21050 in turn is the same as @samp{gcd(x, y)}. The first argument of @code{call},
21051 like the other functions described here, may be either a variable naming a
21052 function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same
21055 (Experts will notice that it's not quite proper to use a variable to name
21056 a function, since the name @code{gcd} corresponds to the Lisp variable
21057 @code{var-gcd} but to the Lisp function @code{calcFunc-gcd}. Calc
21058 automatically makes this translation, so you don't have to worry
21061 @node Mapping, Reducing, Specifying Operators, Reducing and Mapping
21062 @subsection Mapping
21069 The @kbd{V M} (@code{calc-map}) [@code{map}] command applies a given
21070 operator elementwise to one or more vectors. For example, mapping
21071 @code{A} [@code{abs}] produces a vector of the absolute values of the
21072 elements in the input vector. Mapping @code{+} pops two vectors from
21073 the stack, which must be of equal length, and produces a vector of the
21074 pairwise sums of the elements. If either argument is a non-vector, it
21075 is duplicated for each element of the other vector. For example,
21076 @kbd{[1,2,3] 2 V M ^} squares the elements of the specified vector.
21077 With the 2 listed first, it would have computed a vector of powers of
21078 two. Mapping a user-defined function pops as many arguments from the
21079 stack as the function requires. If you give an undefined name, you will
21080 be prompted for the number of arguments to use.
21082 If any argument to @kbd{V M} is a matrix, the operator is normally mapped
21083 across all elements of the matrix. For example, given the matrix
21084 @expr{[[1, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to
21086 @texline @math{3\times2}
21088 matrix, @expr{[[1, 2, 3], [4, 5, 6]]}.
21091 The command @kbd{V M _} [@code{mapr}] (i.e., type an underscore at the
21092 operator prompt) maps by rows instead. For example, @kbd{V M _ A} views
21093 the above matrix as a vector of two 3-element row vectors. It produces
21094 a new vector which contains the absolute values of those row vectors,
21095 namely @expr{[3.74, 8.77]}. (Recall, the absolute value of a vector is
21096 defined as the square root of the sum of the squares of the elements.)
21097 Some operators accept vectors and return new vectors; for example,
21098 @kbd{v v} reverses a vector, so @kbd{V M _ v v} would reverse each row
21099 of the matrix to get a new matrix, @expr{[[3, -2, 1], [-6, 5, -4]]}.
21101 Sometimes a vector of vectors (representing, say, strings, sets, or lists)
21102 happens to look like a matrix. If so, remember to use @kbd{V M _} if you
21103 want to map a function across the whole strings or sets rather than across
21104 their individual elements.
21107 The command @kbd{V M :} [@code{mapc}] maps by columns. Basically, it
21108 transposes the input matrix, maps by rows, and then, if the result is a
21109 matrix, transposes again. For example, @kbd{V M : A} takes the absolute
21110 values of the three columns of the matrix, treating each as a 2-vector,
21111 and @kbd{V M : v v} reverses the columns to get the matrix
21112 @expr{[[-4, 5, -6], [1, -2, 3]]}.
21114 (The symbols @kbd{_} and @kbd{:} were chosen because they had row-like
21115 and column-like appearances, and were not already taken by useful
21116 operators. Also, they appear shifted on most keyboards so they are easy
21117 to type after @kbd{V M}.)
21119 The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are
21120 not matrices (so if none of the arguments are matrices, they have no
21121 effect at all). If some of the arguments are matrices and others are
21122 plain numbers, the plain numbers are held constant for all rows of the
21123 matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring
21124 a vector takes a dot product of the vector with itself).
21126 If some of the arguments are vectors with the same lengths as the
21127 rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix
21128 arguments, those vectors are also held constant for every row or
21131 Sometimes it is useful to specify another mapping command as the operator
21132 to use with @kbd{V M}. For example, @kbd{V M _ V A +} applies @kbd{V A +}
21133 to each row of the input matrix, which in turn adds the two values on that
21134 row. If you give another vector-operator command as the operator for
21135 @kbd{V M}, it automatically uses map-by-rows mode if you don't specify
21136 otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}. (If
21137 you really want to map-by-elements another mapping command, you can use
21138 a triple-nested mapping command: @kbd{V M V M V A +} means to map
21139 @kbd{V M V A +} over the rows of the matrix; in turn, @kbd{V A +} is
21140 mapped over the elements of each row.)
21144 Previous versions of Calc had ``map across'' and ``map down'' modes
21145 that are now considered obsolete; the old ``map across'' is now simply
21146 @kbd{V M V A}, and ``map down'' is now @kbd{V M : V A}. The algebraic
21147 functions @code{mapa} and @code{mapd} are still supported, though.
21148 Note also that, while the old mapping modes were persistent (once you
21149 set the mode, it would apply to later mapping commands until you reset
21150 it), the new @kbd{:} and @kbd{_} modifiers apply only to the current
21151 mapping command. The default @kbd{V M} always means map-by-elements.
21153 @xref{Algebraic Manipulation}, for the @kbd{a M} command, which is like
21154 @kbd{V M} but for equations and inequalities instead of vectors.
21155 @xref{Storing Variables}, for the @kbd{s m} command which modifies a
21156 variable's stored value using a @kbd{V M}-like operator.
21158 @node Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping
21159 @subsection Reducing
21164 @pindex calc-reduce
21166 The @kbd{V R} (@code{calc-reduce}) [@code{reduce}] command applies a given
21167 binary operator across all the elements of a vector. A binary operator is
21168 a function such as @code{+} or @code{max} which takes two arguments. For
21169 example, reducing @code{+} over a vector computes the sum of the elements
21170 of the vector. Reducing @code{-} computes the first element minus each of
21171 the remaining elements. Reducing @code{max} computes the maximum element
21172 and so on. In general, reducing @code{f} over the vector @samp{[a, b, c, d]}
21173 produces @samp{f(f(f(a, b), c), d)}.
21178 The @kbd{I V R} [@code{rreduce}] command is similar to @kbd{V R} except
21179 that works from right to left through the vector. For example, plain
21180 @kbd{V R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c - d}
21181 but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))},
21182 or @samp{a - b + c - d}. This ``alternating sum'' occurs frequently
21183 in power series expansions.
21188 The @kbd{V U} (@code{calc-accumulate}) [@code{accum}] command does an
21189 accumulation operation. Here Calc does the corresponding reduction
21190 operation, but instead of producing only the final result, it produces
21191 a vector of all the intermediate results. Accumulating @code{+} over
21192 the vector @samp{[a, b, c, d]} produces the vector
21193 @samp{[a, a + b, a + b + c, a + b + c + d]}.
21198 The @kbd{I V U} [@code{raccum}] command does a right-to-left accumulation.
21199 For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the
21200 vector @samp{[a - b + c - d, b - c + d, c - d, d]}.
21206 As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise. For
21207 example, given the matrix @expr{[[a, b, c], [d, e, f]]}, @kbd{V R +} will
21208 compute @expr{a + b + c + d + e + f}. You can type @kbd{V R _} or
21209 @kbd{V R :} to modify this behavior. The @kbd{V R _} [@code{reducea}]
21210 command reduces ``across'' the matrix; it reduces each row of the matrix
21211 as a vector, then collects the results. Thus @kbd{V R _ +} of this
21212 matrix would produce @expr{[a + b + c, d + e + f]}. Similarly, @kbd{V R :}
21213 [@code{reduced}] reduces down; @kbd{V R : +} would produce @expr{[a + d,
21218 There is a third ``by rows'' mode for reduction that is occasionally
21219 useful; @kbd{V R =} [@code{reducer}] simply reduces the operator over
21220 the rows of the matrix themselves. Thus @kbd{V R = +} on the above
21221 matrix would get the same result as @kbd{V R : +}, since adding two
21222 row vectors is equivalent to adding their elements. But @kbd{V R = *}
21223 would multiply the two rows (to get a single number, their dot product),
21224 while @kbd{V R : *} would produce a vector of the products of the columns.
21226 These three matrix reduction modes work with @kbd{V R} and @kbd{I V R},
21227 but they are not currently supported with @kbd{V U} or @kbd{I V U}.
21231 The obsolete reduce-by-columns function, @code{reducec}, is still
21232 supported but there is no way to get it through the @kbd{V R} command.
21234 The commands @kbd{C-x * :} and @kbd{C-x * _} are equivalent to typing
21235 @kbd{C-x * r} to grab a rectangle of data into Calc, and then typing
21236 @kbd{V R : +} or @kbd{V R _ +}, respectively, to sum the columns or
21237 rows of the matrix. @xref{Grabbing From Buffers}.
21239 @node Nesting and Fixed Points, Generalized Products, Reducing, Reducing and Mapping
21240 @subsection Nesting and Fixed Points
21246 The @kbd{H V R} [@code{nest}] command applies a function to a given
21247 argument repeatedly. It takes two values, @samp{a} and @samp{n}, from
21248 the stack, where @samp{n} must be an integer. It then applies the
21249 function nested @samp{n} times; if the function is @samp{f} and @samp{n}
21250 is 3, the result is @samp{f(f(f(a)))}. The number @samp{n} may be
21251 negative if Calc knows an inverse for the function @samp{f}; for
21252 example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.
21257 The @kbd{H V U} [@code{anest}] command is an accumulating version of
21258 @code{nest}: It returns a vector of @samp{n+1} values, e.g.,
21259 @samp{[a, f(a), f(f(a)), f(f(f(a)))]}. If @samp{n} is negative and
21260 @samp{F} is the inverse of @samp{f}, then the result is of the
21261 form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}.
21266 @cindex Fixed points
21267 The @kbd{H I V R} [@code{fixp}] command is like @kbd{H V R}, except
21268 that it takes only an @samp{a} value from the stack; the function is
21269 applied until it reaches a ``fixed point,'' i.e., until the result
21275 The @kbd{H I V U} [@code{afixp}] command is an accumulating @code{fixp}.
21276 The first element of the return vector will be the initial value @samp{a};
21277 the last element will be the final result that would have been returned
21280 For example, 0.739085 is a fixed point of the cosine function (in radians):
21281 @samp{cos(0.739085) = 0.739085}. You can find this value by putting, say,
21282 1.0 on the stack and typing @kbd{H I V U C}. (We use the accumulating
21283 version so we can see the intermediate results: @samp{[1, 0.540302, 0.857553,
21284 0.65329, ...]}. With a precision of six, this command will take 36 steps
21285 to converge to 0.739085.)
21287 Newton's method for finding roots is a classic example of iteration
21288 to a fixed point. To find the square root of five starting with an
21289 initial guess, Newton's method would look for a fixed point of the
21290 function @samp{(x + 5/x) / 2}. Putting a guess of 1 on the stack
21291 and typing @kbd{H I V R ' ($ + 5/$)/2 @key{RET}} quickly yields the result
21292 2.23607. This is equivalent to using the @kbd{a R} (@code{calc-find-root})
21293 command to find a root of the equation @samp{x^2 = 5}.
21295 These examples used numbers for @samp{a} values. Calc keeps applying
21296 the function until two successive results are equal to within the
21297 current precision. For complex numbers, both the real parts and the
21298 imaginary parts must be equal to within the current precision. If
21299 @samp{a} is a formula (say, a variable name), then the function is
21300 applied until two successive results are exactly the same formula.
21301 It is up to you to ensure that the function will eventually converge;
21302 if it doesn't, you may have to press @kbd{C-g} to stop the Calculator.
21304 The algebraic @code{fixp} function takes two optional arguments, @samp{n}
21305 and @samp{tol}. The first is the maximum number of steps to be allowed,
21306 and must be either an integer or the symbol @samp{inf} (infinity, the
21307 default). The second is a convergence tolerance. If a tolerance is
21308 specified, all results during the calculation must be numbers, not
21309 formulas, and the iteration stops when the magnitude of the difference
21310 between two successive results is less than or equal to the tolerance.
21311 (This implies that a tolerance of zero iterates until the results are
21314 Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)}
21315 computes the square root of @samp{A} given the initial guess @samp{B},
21316 stopping when the result is correct within the specified tolerance, or
21317 when 20 steps have been taken, whichever is sooner.
21319 @node Generalized Products, , Nesting and Fixed Points, Reducing and Mapping
21320 @subsection Generalized Products
21324 @pindex calc-outer-product
21326 The @kbd{V O} (@code{calc-outer-product}) [@code{outer}] command applies
21327 a given binary operator to all possible pairs of elements from two
21328 vectors, to produce a matrix. For example, @kbd{V O *} with @samp{[a, b]}
21329 and @samp{[x, y, z]} on the stack produces a multiplication table:
21330 @samp{[[a x, a y, a z], [b x, b y, b z]]}. Element @var{r},@var{c} of
21331 the result matrix is obtained by applying the operator to element @var{r}
21332 of the lefthand vector and element @var{c} of the righthand vector.
21336 @pindex calc-inner-product
21338 The @kbd{V I} (@code{calc-inner-product}) [@code{inner}] command computes
21339 the generalized inner product of two vectors or matrices, given a
21340 ``multiplicative'' operator and an ``additive'' operator. These can each
21341 actually be any binary operators; if they are @samp{*} and @samp{+},
21342 respectively, the result is a standard matrix multiplication. Element
21343 @var{r},@var{c} of the result matrix is obtained by mapping the
21344 multiplicative operator across row @var{r} of the lefthand matrix and
21345 column @var{c} of the righthand matrix, and then reducing with the additive
21346 operator. Just as for the standard @kbd{*} command, this can also do a
21347 vector-matrix or matrix-vector inner product, or a vector-vector
21348 generalized dot product.
21350 Since @kbd{V I} requires two operators, it prompts twice. In each case,
21351 you can use any of the usual methods for entering the operator. If you
21352 use @kbd{$} twice to take both operator formulas from the stack, the
21353 first (multiplicative) operator is taken from the top of the stack
21354 and the second (additive) operator is taken from second-to-top.
21356 @node Vector and Matrix Formats, , Reducing and Mapping, Matrix Functions
21357 @section Vector and Matrix Display Formats
21360 Commands for controlling vector and matrix display use the @kbd{v} prefix
21361 instead of the usual @kbd{d} prefix. But they are display modes; in
21362 particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys
21363 in the same way (@pxref{Display Modes}). Matrix display is also
21364 influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
21365 @pxref{Normal Language Modes}.
21369 @pindex calc-matrix-left-justify
21372 @pindex calc-matrix-center-justify
21375 @pindex calc-matrix-right-justify
21376 The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
21377 (@code{calc-matrix-right-justify}), and @w{@kbd{v =}}
21378 (@code{calc-matrix-center-justify}) control whether matrix elements
21379 are justified to the left, right, or center of their columns.
21383 @pindex calc-vector-brackets
21386 @pindex calc-vector-braces
21389 @pindex calc-vector-parens
21390 The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
21391 brackets that surround vectors and matrices displayed in the stack on
21392 and off. The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
21393 (@code{calc-vector-parens}) commands use curly braces or parentheses,
21394 respectively, instead of square brackets. For example, @kbd{v @{} might
21395 be used in preparation for yanking a matrix into a buffer running
21396 Mathematica. (In fact, the Mathematica language mode uses this mode;
21397 @pxref{Mathematica Language Mode}.) Note that, regardless of the
21398 display mode, either brackets or braces may be used to enter vectors,
21399 and parentheses may never be used for this purpose.
21407 @pindex calc-matrix-brackets
21408 The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the
21409 ``big'' style display of matrices, for matrices which have more than
21410 one row. It prompts for a string of code letters; currently
21411 implemented letters are @code{R}, which enables brackets on each row
21412 of the matrix; @code{O}, which enables outer brackets in opposite
21413 corners of the matrix; and @code{C}, which enables commas or
21414 semicolons at the ends of all rows but the last. The default format
21415 is @samp{RO}. (Before Calc 2.00, the format was fixed at @samp{ROC}.)
21416 Here are some example matrices:
21420 [ [ 123, 0, 0 ] [ [ 123, 0, 0 ],
21421 [ 0, 123, 0 ] [ 0, 123, 0 ],
21422 [ 0, 0, 123 ] ] [ 0, 0, 123 ] ]
21431 [ 123, 0, 0 [ 123, 0, 0 ;
21432 0, 123, 0 0, 123, 0 ;
21433 0, 0, 123 ] 0, 0, 123 ]
21442 [ 123, 0, 0 ] 123, 0, 0
21443 [ 0, 123, 0 ] 0, 123, 0
21444 [ 0, 0, 123 ] 0, 0, 123
21451 Note that of the formats shown here, @samp{RO}, @samp{ROC}, and
21452 @samp{OC} are all recognized as matrices during reading, while
21453 the others are useful for display only.
21457 @pindex calc-vector-commas
21458 The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
21459 off in vector and matrix display.
21461 In vectors of length one, and in all vectors when commas have been
21462 turned off, Calc adds extra parentheses around formulas that might
21463 otherwise be ambiguous. For example, @samp{[a b]} could be a vector
21464 of the one formula @samp{a b}, or it could be a vector of two
21465 variables with commas turned off. Calc will display the former
21466 case as @samp{[(a b)]}. You can disable these extra parentheses
21467 (to make the output less cluttered at the expense of allowing some
21468 ambiguity) by adding the letter @code{P} to the control string you
21469 give to @kbd{v ]} (as described above).
21473 @pindex calc-full-vectors
21474 The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
21475 display of long vectors on and off. In this mode, vectors of six
21476 or more elements, or matrices of six or more rows or columns, will
21477 be displayed in an abbreviated form that displays only the first
21478 three elements and the last element: @samp{[a, b, c, ..., z]}.
21479 When very large vectors are involved this will substantially
21480 improve Calc's display speed.
21483 @pindex calc-full-trail-vectors
21484 The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a
21485 similar mode for recording vectors in the Trail. If you turn on
21486 this mode, vectors of six or more elements and matrices of six or
21487 more rows or columns will be abbreviated when they are put in the
21488 Trail. The @kbd{t y} (@code{calc-trail-yank}) command will be
21489 unable to recover those vectors. If you are working with very
21490 large vectors, this mode will improve the speed of all operations
21491 that involve the trail.
21495 @pindex calc-break-vectors
21496 The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
21497 vector display on and off. Normally, matrices are displayed with one
21498 row per line but all other types of vectors are displayed in a single
21499 line. This mode causes all vectors, whether matrices or not, to be
21500 displayed with a single element per line. Sub-vectors within the
21501 vectors will still use the normal linear form.
21503 @node Algebra, Units, Matrix Functions, Top
21507 This section covers the Calc features that help you work with
21508 algebraic formulas. First, the general sub-formula selection
21509 mechanism is described; this works in conjunction with any Calc
21510 commands. Then, commands for specific algebraic operations are
21511 described. Finally, the flexible @dfn{rewrite rule} mechanism
21514 The algebraic commands use the @kbd{a} key prefix; selection
21515 commands use the @kbd{j} (for ``just a letter that wasn't used
21516 for anything else'') prefix.
21518 @xref{Editing Stack Entries}, to see how to manipulate formulas
21519 using regular Emacs editing commands.
21521 When doing algebraic work, you may find several of the Calculator's
21522 modes to be helpful, including Algebraic Simplification mode (@kbd{m A})
21523 or No-Simplification mode (@kbd{m O}),
21524 Algebraic entry mode (@kbd{m a}), Fraction mode (@kbd{m f}), and
21525 Symbolic mode (@kbd{m s}). @xref{Mode Settings}, for discussions
21526 of these modes. You may also wish to select Big display mode (@kbd{d B}).
21527 @xref{Normal Language Modes}.
21530 * Selecting Subformulas::
21531 * Algebraic Manipulation::
21532 * Simplifying Formulas::
21535 * Solving Equations::
21536 * Numerical Solutions::
21539 * Logical Operations::
21543 @node Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra
21544 @section Selecting Sub-Formulas
21548 @cindex Sub-formulas
21549 @cindex Parts of formulas
21550 When working with an algebraic formula it is often necessary to
21551 manipulate a portion of the formula rather than the formula as a
21552 whole. Calc allows you to ``select'' a portion of any formula on
21553 the stack. Commands which would normally operate on that stack
21554 entry will now operate only on the sub-formula, leaving the
21555 surrounding part of the stack entry alone.
21557 One common non-algebraic use for selection involves vectors. To work
21558 on one element of a vector in-place, simply select that element as a
21559 ``sub-formula'' of the vector.
21562 * Making Selections::
21563 * Changing Selections::
21564 * Displaying Selections::
21565 * Operating on Selections::
21566 * Rearranging with Selections::
21569 @node Making Selections, Changing Selections, Selecting Subformulas, Selecting Subformulas
21570 @subsection Making Selections
21574 @pindex calc-select-here
21575 To select a sub-formula, move the Emacs cursor to any character in that
21576 sub-formula, and press @w{@kbd{j s}} (@code{calc-select-here}). Calc will
21577 highlight the smallest portion of the formula that contains that
21578 character. By default the sub-formula is highlighted by blanking out
21579 all of the rest of the formula with dots. Selection works in any
21580 display mode but is perhaps easiest in Big mode (@kbd{d B}).
21581 Suppose you enter the following formula:
21593 (by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}). If you move the
21594 cursor to the letter @samp{b} and press @w{@kbd{j s}}, the display changes
21607 Every character not part of the sub-formula @samp{b} has been changed
21608 to a dot. (If the customizable variable
21609 @code{calc-highlight-selections-with-faces} is non-nil, then the characters
21610 not part of the sub-formula are de-emphasized by using a less
21611 noticeable face instead of using dots. @pxref{Displaying Selections}.)
21612 The @samp{*} next to the line number is to remind you that
21613 the formula has a portion of it selected. (In this case, it's very
21614 obvious, but it might not always be. If Embedded mode is enabled,
21615 the word @samp{Sel} also appears in the mode line because the stack
21616 may not be visible. @pxref{Embedded Mode}.)
21618 If you had instead placed the cursor on the parenthesis immediately to
21619 the right of the @samp{b}, the selection would have been:
21631 The portion selected is always large enough to be considered a complete
21632 formula all by itself, so selecting the parenthesis selects the whole
21633 formula that it encloses. Putting the cursor on the @samp{+} sign
21634 would have had the same effect.
21636 (Strictly speaking, the Emacs cursor is really the manifestation of
21637 the Emacs ``point,'' which is a position @emph{between} two characters
21638 in the buffer. So purists would say that Calc selects the smallest
21639 sub-formula which contains the character to the right of ``point.'')
21641 If you supply a numeric prefix argument @var{n}, the selection is
21642 expanded to the @var{n}th enclosing sub-formula. Thus, positioning
21643 the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select
21644 @samp{a + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3},
21647 If the cursor is not on any part of the formula, or if you give a
21648 numeric prefix that is too large, the entire formula is selected.
21650 If the cursor is on the @samp{.} line that marks the top of the stack
21651 (i.e., its normal ``rest position''), this command selects the entire
21652 formula at stack level 1. Most selection commands similarly operate
21653 on the formula at the top of the stack if you haven't positioned the
21654 cursor on any stack entry.
21657 @pindex calc-select-additional
21658 The @kbd{j a} (@code{calc-select-additional}) command enlarges the
21659 current selection to encompass the cursor. To select the smallest
21660 sub-formula defined by two different points, move to the first and
21661 press @kbd{j s}, then move to the other and press @kbd{j a}. This
21662 is roughly analogous to using @kbd{C-@@} (@code{set-mark-command}) to
21663 select the two ends of a region of text during normal Emacs editing.
21666 @pindex calc-select-once
21667 The @kbd{j o} (@code{calc-select-once}) command selects a formula in
21668 exactly the same way as @kbd{j s}, except that the selection will
21669 last only as long as the next command that uses it. For example,
21670 @kbd{j o 1 +} is a handy way to add one to the sub-formula indicated
21673 (A somewhat more precise definition: The @kbd{j o} command sets a flag
21674 such that the next command involving selected stack entries will clear
21675 the selections on those stack entries afterwards. All other selection
21676 commands except @kbd{j a} and @kbd{j O} clear this flag.)
21680 @pindex calc-select-here-maybe
21681 @pindex calc-select-once-maybe
21682 The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O}
21683 (@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s}
21684 and @kbd{j o}, respectively, except that if the formula already
21685 has a selection they have no effect. This is analogous to the
21686 behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection};
21687 @pxref{Selections with Rewrite Rules}) and is mainly intended to be
21688 used in keyboard macros that implement your own selection-oriented
21691 Selection of sub-formulas normally treats associative terms like
21692 @samp{a + b - c + d} and @samp{x * y * z} as single levels of the formula.
21693 If you place the cursor anywhere inside @samp{a + b - c + d} except
21694 on one of the variable names and use @kbd{j s}, you will select the
21695 entire four-term sum.
21698 @pindex calc-break-selections
21699 The @kbd{j b} (@code{calc-break-selections}) command controls a mode
21700 in which the ``deep structure'' of these associative formulas shows
21701 through. Calc actually stores the above formulas as
21702 @samp{((a + b) - c) + d} and @samp{x * (y * z)}. (Note that for certain
21703 obscure reasons, by default Calc treats multiplication as
21704 right-associative.) Once you have enabled @kbd{j b} mode, selecting
21705 with the cursor on the @samp{-} sign would only select the @samp{a + b -
21706 c} portion, which makes sense when the deep structure of the sum is
21707 considered. There is no way to select the @samp{b - c + d} portion;
21708 although this might initially look like just as legitimate a sub-formula
21709 as @samp{a + b - c}, the deep structure shows that it isn't. The @kbd{d
21710 U} command can be used to view the deep structure of any formula
21711 (@pxref{Normal Language Modes}).
21713 When @kbd{j b} mode has not been enabled, the deep structure is
21714 generally hidden by the selection commands---what you see is what
21718 @pindex calc-unselect
21719 The @kbd{j u} (@code{calc-unselect}) command unselects the formula
21720 that the cursor is on. If there was no selection in the formula,
21721 this command has no effect. With a numeric prefix argument, it
21722 unselects the @var{n}th stack element rather than using the cursor
21726 @pindex calc-clear-selections
21727 The @kbd{j c} (@code{calc-clear-selections}) command unselects all
21730 @node Changing Selections, Displaying Selections, Making Selections, Selecting Subformulas
21731 @subsection Changing Selections
21735 @pindex calc-select-more
21736 Once you have selected a sub-formula, you can expand it using the
21737 @w{@kbd{j m}} (@code{calc-select-more}) command. If @samp{a + b} is
21738 selected, pressing @w{@kbd{j m}} repeatedly works as follows:
21743 (a + b) . . . (a + b) + V c (a + b) + V c
21744 1* ............... 1* ............... 1* ---------------
21745 . . . . . . . . 2 x + 1
21750 In the last example, the entire formula is selected. This is roughly
21751 the same as having no selection at all, but because there are subtle
21752 differences the @samp{*} character is still there on the line number.
21754 With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n}
21755 times (or until the entire formula is selected). Note that @kbd{j s}
21756 with argument @var{n} is equivalent to plain @kbd{j s} followed by
21757 @kbd{j m} with argument @var{n}. If @w{@kbd{j m}} is used when there
21758 is no current selection, it is equivalent to @w{@kbd{j s}}.
21760 Even though @kbd{j m} does not explicitly use the location of the
21761 cursor within the formula, it nevertheless uses the cursor to determine
21762 which stack element to operate on. As usual, @kbd{j m} when the cursor
21763 is not on any stack element operates on the top stack element.
21766 @pindex calc-select-less
21767 The @kbd{j l} (@code{calc-select-less}) command reduces the current
21768 selection around the cursor position. That is, it selects the
21769 immediate sub-formula of the current selection which contains the
21770 cursor, the opposite of @kbd{j m}. If the cursor is not inside the
21771 current selection, the command de-selects the formula.
21774 @pindex calc-select-part
21775 The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands
21776 select the @var{n}th sub-formula of the current selection. They are
21777 like @kbd{j l} (@code{calc-select-less}) except they use counting
21778 rather than the cursor position to decide which sub-formula to select.
21779 For example, if the current selection is @kbd{a + b + c} or
21780 @kbd{f(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a},
21781 @kbd{j 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of
21782 these cases, @kbd{j 4} through @kbd{j 9} would be errors.
21784 If there is no current selection, @kbd{j 1} through @kbd{j 9} select
21785 the @var{n}th top-level sub-formula. (In other words, they act as if
21786 the entire stack entry were selected first.) To select the @var{n}th
21787 sub-formula where @var{n} is greater than nine, you must instead invoke
21788 @w{@kbd{j 1}} with @var{n} as a numeric prefix argument.
21792 @pindex calc-select-next
21793 @pindex calc-select-previous
21794 The @kbd{j n} (@code{calc-select-next}) and @kbd{j p}
21795 (@code{calc-select-previous}) commands change the current selection
21796 to the next or previous sub-formula at the same level. For example,
21797 if @samp{b} is selected in @w{@samp{2 + a*b*c + x}}, then @kbd{j n}
21798 selects @samp{c}. Further @kbd{j n} commands would be in error because,
21799 even though there is something to the right of @samp{c} (namely, @samp{x}),
21800 it is not at the same level; in this case, it is not a term of the
21801 same product as @samp{b} and @samp{c}. However, @kbd{j m} (to select
21802 the whole product @samp{a*b*c} as a term of the sum) followed by
21803 @w{@kbd{j n}} would successfully select the @samp{x}.
21805 Similarly, @kbd{j p} moves the selection from the @samp{b} in this
21806 sample formula to the @samp{a}. Both commands accept numeric prefix
21807 arguments to move several steps at a time.
21809 It is interesting to compare Calc's selection commands with the
21810 Emacs Info system's commands for navigating through hierarchically
21811 organized documentation. Calc's @kbd{j n} command is completely
21812 analogous to Info's @kbd{n} command. Likewise, @kbd{j p} maps to
21813 @kbd{p}, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}.
21814 (Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.)
21815 The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and
21816 @kbd{j l}; in each case, you can jump directly to a sub-component
21817 of the hierarchy simply by pointing to it with the cursor.
21819 @node Displaying Selections, Operating on Selections, Changing Selections, Selecting Subformulas
21820 @subsection Displaying Selections
21824 @pindex calc-show-selections
21825 @vindex calc-highlight-selections-with-faces
21826 @vindex calc-selected-face
21827 @vindex calc-nonselected-face
21828 The @kbd{j d} (@code{calc-show-selections}) command controls how
21829 selected sub-formulas are displayed. One of the alternatives is
21830 illustrated in the above examples; if we press @kbd{j d} we switch
21831 to the other style in which the selected portion itself is obscured
21837 (a + b) . . . ## # ## + V c
21838 1* ............... 1* ---------------
21842 If the customizable variable
21843 @code{calc-highlight-selections-with-faces} is non-nil, then the
21844 non-selected portion of the formula will be de-emphasized by using a
21845 less noticeable face (@code{calc-nonselected-face}) instead of dots
21846 and the selected sub-formula will be highlighted by using a more
21847 noticeable face (@code{calc-selected-face}) instead of @samp{#}
21848 signs. (@pxref{Customizing Calc}.)
21850 @node Operating on Selections, Rearranging with Selections, Displaying Selections, Selecting Subformulas
21851 @subsection Operating on Selections
21854 Once a selection is made, all Calc commands that manipulate items
21855 on the stack will operate on the selected portions of the items
21856 instead. (Note that several stack elements may have selections
21857 at once, though there can be only one selection at a time in any
21858 given stack element.)
21861 @pindex calc-enable-selections
21862 The @kbd{j e} (@code{calc-enable-selections}) command disables the
21863 effect that selections have on Calc commands. The current selections
21864 still exist, but Calc commands operate on whole stack elements anyway.
21865 This mode can be identified by the fact that the @samp{*} markers on
21866 the line numbers are gone, even though selections are visible. To
21867 reactivate the selections, press @kbd{j e} again.
21869 To extract a sub-formula as a new formula, simply select the
21870 sub-formula and press @key{RET}. This normally duplicates the top
21871 stack element; here it duplicates only the selected portion of that
21874 To replace a sub-formula with something different, you can enter the
21875 new value onto the stack and press @key{TAB}. This normally exchanges
21876 the top two stack elements; here it swaps the value you entered into
21877 the selected portion of the formula, returning the old selected
21878 portion to the top of the stack.
21883 (a + b) . . . 17 x y . . . 17 x y + V c
21884 2* ............... 2* ............. 2: -------------
21885 . . . . . . . . 2 x + 1
21888 1: 17 x y 1: (a + b) 1: (a + b)
21892 In this example we select a sub-formula of our original example,
21893 enter a new formula, @key{TAB} it into place, then deselect to see
21894 the complete, edited formula.
21896 If you want to swap whole formulas around even though they contain
21897 selections, just use @kbd{j e} before and after.
21900 @pindex calc-enter-selection
21901 The @kbd{j '} (@code{calc-enter-selection}) command is another way
21902 to replace a selected sub-formula. This command does an algebraic
21903 entry just like the regular @kbd{'} key. When you press @key{RET},
21904 the formula you type replaces the original selection. You can use
21905 the @samp{$} symbol in the formula to refer to the original
21906 selection. If there is no selection in the formula under the cursor,
21907 the cursor is used to make a temporary selection for the purposes of
21908 the command. Thus, to change a term of a formula, all you have to
21909 do is move the Emacs cursor to that term and press @kbd{j '}.
21912 @pindex calc-edit-selection
21913 The @kbd{j `} (@code{calc-edit-selection}) command is a similar
21914 analogue of the @kbd{`} (@code{calc-edit}) command. It edits the
21915 selected sub-formula in a separate buffer. If there is no
21916 selection, it edits the sub-formula indicated by the cursor.
21918 To delete a sub-formula, press @key{DEL}. This generally replaces
21919 the sub-formula with the constant zero, but in a few suitable contexts
21920 it uses the constant one instead. The @key{DEL} key automatically
21921 deselects and re-simplifies the entire formula afterwards. Thus:
21926 17 x y + # # 17 x y 17 # y 17 y
21927 1* ------------- 1: ------- 1* ------- 1: -------
21928 2 x + 1 2 x + 1 2 x + 1 2 x + 1
21932 In this example, we first delete the @samp{sqrt(c)} term; Calc
21933 accomplishes this by replacing @samp{sqrt(c)} with zero and
21934 resimplifying. We then delete the @kbd{x} in the numerator;
21935 since this is part of a product, Calc replaces it with @samp{1}
21938 If you select an element of a vector and press @key{DEL}, that
21939 element is deleted from the vector. If you delete one side of
21940 an equation or inequality, only the opposite side remains.
21942 @kindex j @key{DEL}
21943 @pindex calc-del-selection
21944 The @kbd{j @key{DEL}} (@code{calc-del-selection}) command is like
21945 @key{DEL} but with the auto-selecting behavior of @kbd{j '} and
21946 @kbd{j `}. It deletes the selected portion of the formula
21947 indicated by the cursor, or, in the absence of a selection, it
21948 deletes the sub-formula indicated by the cursor position.
21950 @kindex j @key{RET}
21951 @pindex calc-grab-selection
21952 (There is also an auto-selecting @kbd{j @key{RET}} (@code{calc-copy-selection})
21955 Normal arithmetic operations also apply to sub-formulas. Here we
21956 select the denominator, press @kbd{5 -} to subtract five from the
21957 denominator, press @kbd{n} to negate the denominator, then
21958 press @kbd{Q} to take the square root.
21962 .. . .. . .. . .. .
21963 1* ....... 1* ....... 1* ....... 1* ..........
21964 2 x + 1 2 x - 4 4 - 2 x _________
21969 Certain types of operations on selections are not allowed. For
21970 example, for an arithmetic function like @kbd{-} no more than one of
21971 the arguments may be a selected sub-formula. (As the above example
21972 shows, the result of the subtraction is spliced back into the argument
21973 which had the selection; if there were more than one selection involved,
21974 this would not be well-defined.) If you try to subtract two selections,
21975 the command will abort with an error message.
21977 Operations on sub-formulas sometimes leave the formula as a whole
21978 in an ``un-natural'' state. Consider negating the @samp{2 x} term
21979 of our sample formula by selecting it and pressing @kbd{n}
21980 (@code{calc-change-sign}).
21985 1* .......... 1* ...........
21986 ......... ..........
21987 . . . 2 x . . . -2 x
21991 Unselecting the sub-formula reveals that the minus sign, which would
21992 normally have canceled out with the subtraction automatically, has
21993 not been able to do so because the subtraction was not part of the
21994 selected portion. Pressing @kbd{=} (@code{calc-evaluate}) or doing
21995 any other mathematical operation on the whole formula will cause it
22001 1: ----------- 1: ----------
22002 __________ _________
22003 V 4 - -2 x V 4 + 2 x
22007 @node Rearranging with Selections, , Operating on Selections, Selecting Subformulas
22008 @subsection Rearranging Formulas using Selections
22012 @pindex calc-commute-right
22013 The @kbd{j R} (@code{calc-commute-right}) command moves the selected
22014 sub-formula to the right in its surrounding formula. Generally the
22015 selection is one term of a sum or product; the sum or product is
22016 rearranged according to the commutative laws of algebra.
22018 As with @kbd{j '} and @kbd{j @key{DEL}}, the term under the cursor is used
22019 if there is no selection in the current formula. All commands described
22020 in this section share this property. In this example, we place the
22021 cursor on the @samp{a} and type @kbd{j R}, then repeat.
22024 1: a + b - c 1: b + a - c 1: b - c + a
22028 Note that in the final step above, the @samp{a} is switched with
22029 the @samp{c} but the signs are adjusted accordingly. When moving
22030 terms of sums and products, @kbd{j R} will never change the
22031 mathematical meaning of the formula.
22033 The selected term may also be an element of a vector or an argument
22034 of a function. The term is exchanged with the one to its right.
22035 In this case, the ``meaning'' of the vector or function may of
22036 course be drastically changed.
22039 1: [a, b, c] 1: [b, a, c] 1: [b, c, a]
22041 1: f(a, b, c) 1: f(b, a, c) 1: f(b, c, a)
22045 @pindex calc-commute-left
22046 The @kbd{j L} (@code{calc-commute-left}) command is like @kbd{j R}
22047 except that it swaps the selected term with the one to its left.
22049 With numeric prefix arguments, these commands move the selected
22050 term several steps at a time. It is an error to try to move a
22051 term left or right past the end of its enclosing formula.
22052 With numeric prefix arguments of zero, these commands move the
22053 selected term as far as possible in the given direction.
22056 @pindex calc-sel-distribute
22057 The @kbd{j D} (@code{calc-sel-distribute}) command mixes the selected
22058 sum or product into the surrounding formula using the distributive
22059 law. For example, in @samp{a * (b - c)} with the @samp{b - c}
22060 selected, the result is @samp{a b - a c}. This also distributes
22061 products or quotients into surrounding powers, and can also do
22062 transformations like @samp{exp(a + b)} to @samp{exp(a) exp(b)},
22063 where @samp{a + b} is the selected term, and @samp{ln(a ^ b)}
22064 to @samp{ln(a) b}, where @samp{a ^ b} is the selected term.
22066 For multiple-term sums or products, @kbd{j D} takes off one term
22067 at a time: @samp{a * (b + c - d)} goes to @samp{a * (c - d) + a b}
22068 with the @samp{c - d} selected so that you can type @kbd{j D}
22069 repeatedly to expand completely. The @kbd{j D} command allows a
22070 numeric prefix argument which specifies the maximum number of
22071 times to expand at once; the default is one time only.
22073 @vindex DistribRules
22074 The @kbd{j D} command is implemented using rewrite rules.
22075 @xref{Selections with Rewrite Rules}. The rules are stored in
22076 the Calc variable @code{DistribRules}. A convenient way to view
22077 these rules is to use @kbd{s e} (@code{calc-edit-variable}) which
22078 displays and edits the stored value of a variable. Press @kbd{C-c C-c}
22079 to return from editing mode; be careful not to make any actual changes
22080 or else you will affect the behavior of future @kbd{j D} commands!
22082 To extend @kbd{j D} to handle new cases, just edit @code{DistribRules}
22083 as described above. You can then use the @kbd{s p} command to save
22084 this variable's value permanently for future Calc sessions.
22085 @xref{Operations on Variables}.
22088 @pindex calc-sel-merge
22090 The @kbd{j M} (@code{calc-sel-merge}) command is the complement
22091 of @kbd{j D}; given @samp{a b - a c} with either @samp{a b} or
22092 @samp{a c} selected, the result is @samp{a * (b - c)}. Once
22093 again, @kbd{j M} can also merge calls to functions like @code{exp}
22094 and @code{ln}; examine the variable @code{MergeRules} to see all
22095 the relevant rules.
22098 @pindex calc-sel-commute
22099 @vindex CommuteRules
22100 The @kbd{j C} (@code{calc-sel-commute}) command swaps the arguments
22101 of the selected sum, product, or equation. It always behaves as
22102 if @kbd{j b} mode were in effect, i.e., the sum @samp{a + b + c} is
22103 treated as the nested sums @samp{(a + b) + c} by this command.
22104 If you put the cursor on the first @samp{+}, the result is
22105 @samp{(b + a) + c}; if you put the cursor on the second @samp{+}, the
22106 result is @samp{c + (a + b)} (which the default simplifications
22107 will rearrange to @samp{(c + a) + b}). The relevant rules are stored
22108 in the variable @code{CommuteRules}.
22110 You may need to turn default simplifications off (with the @kbd{m O}
22111 command) in order to get the full benefit of @kbd{j C}. For example,
22112 commuting @samp{a - b} produces @samp{-b + a}, but the default
22113 simplifications will ``simplify'' this right back to @samp{a - b} if
22114 you don't turn them off. The same is true of some of the other
22115 manipulations described in this section.
22118 @pindex calc-sel-negate
22119 @vindex NegateRules
22120 The @kbd{j N} (@code{calc-sel-negate}) command replaces the selected
22121 term with the negative of that term, then adjusts the surrounding
22122 formula in order to preserve the meaning. For example, given
22123 @samp{exp(a - b)} where @samp{a - b} is selected, the result is
22124 @samp{1 / exp(b - a)}. By contrast, selecting a term and using the
22125 regular @kbd{n} (@code{calc-change-sign}) command negates the
22126 term without adjusting the surroundings, thus changing the meaning
22127 of the formula as a whole. The rules variable is @code{NegateRules}.
22130 @pindex calc-sel-invert
22131 @vindex InvertRules
22132 The @kbd{j &} (@code{calc-sel-invert}) command is similar to @kbd{j N}
22133 except it takes the reciprocal of the selected term. For example,
22134 given @samp{a - ln(b)} with @samp{b} selected, the result is
22135 @samp{a + ln(1/b)}. The rules variable is @code{InvertRules}.
22138 @pindex calc-sel-jump-equals
22140 The @kbd{j E} (@code{calc-sel-jump-equals}) command moves the
22141 selected term from one side of an equation to the other. Given
22142 @samp{a + b = c + d} with @samp{c} selected, the result is
22143 @samp{a + b - c = d}. This command also works if the selected
22144 term is part of a @samp{*}, @samp{/}, or @samp{^} formula. The
22145 relevant rules variable is @code{JumpRules}.
22149 @pindex calc-sel-isolate
22150 The @kbd{j I} (@code{calc-sel-isolate}) command isolates the
22151 selected term on its side of an equation. It uses the @kbd{a S}
22152 (@code{calc-solve-for}) command to solve the equation, and the
22153 Hyperbolic flag affects it in the same way. @xref{Solving Equations}.
22154 When it applies, @kbd{j I} is often easier to use than @kbd{j E}.
22155 It understands more rules of algebra, and works for inequalities
22156 as well as equations.
22160 @pindex calc-sel-mult-both-sides
22161 @pindex calc-sel-div-both-sides
22162 The @kbd{j *} (@code{calc-sel-mult-both-sides}) command prompts for a
22163 formula using algebraic entry, then multiplies both sides of the
22164 selected quotient or equation by that formula. It performs the
22165 default algebraic simplifications before re-forming the
22166 quotient or equation. You can suppress this simplification by
22167 providing a prefix argument: @kbd{C-u j *}. There is also a @kbd{j /}
22168 (@code{calc-sel-div-both-sides}) which is similar to @kbd{j *} but
22169 dividing instead of multiplying by the factor you enter.
22171 If the selection is a quotient with numerator 1, then Calc's default
22172 simplifications would normally cancel the new factors. To prevent
22173 this, when the @kbd{j *} command is used on a selection whose numerator is
22174 1 or -1, the denominator is expanded at the top level using the
22175 distributive law (as if using the @kbd{C-u 1 a x} command). Suppose the
22176 formula on the stack is @samp{1 / (a + 1)} and you wish to multiplying the
22177 top and bottom by @samp{a - 1}. Calc's default simplifications would
22178 normally change the result @samp{(a - 1) /(a + 1) (a - 1)} back
22179 to the original form by cancellation; when @kbd{j *} is used, Calc
22180 expands the denominator to @samp{a (a - 1) + a - 1} to prevent this.
22182 If you wish the @kbd{j *} command to completely expand the denominator
22183 of a quotient you can call it with a zero prefix: @kbd{C-u 0 j *}. For
22184 example, if the formula on the stack is @samp{1 / (sqrt(a) + 1)}, you may
22185 wish to eliminate the square root in the denominator by multiplying
22186 the top and bottom by @samp{sqrt(a) - 1}. If you did this simply by using
22187 a simple @kbd{j *} command, you would get
22188 @samp{(sqrt(a)-1)/ (sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1)}. Instead,
22189 you would probably want to use @kbd{C-u 0 j *}, which would expand the
22190 bottom and give you the desired result @samp{(sqrt(a)-1)/(a-1)}. More
22191 generally, if @kbd{j *} is called with an argument of a positive
22192 integer @var{n}, then the denominator of the expression will be
22193 expanded @var{n} times (as if with the @kbd{C-u @var{n} a x} command).
22195 If the selection is an inequality, @kbd{j *} and @kbd{j /} will
22196 accept any factor, but will warn unless they can prove the factor
22197 is either positive or negative. (In the latter case the direction
22198 of the inequality will be switched appropriately.) @xref{Declarations},
22199 for ways to inform Calc that a given variable is positive or
22200 negative. If Calc can't tell for sure what the sign of the factor
22201 will be, it will assume it is positive and display a warning
22204 For selections that are not quotients, equations, or inequalities,
22205 these commands pull out a multiplicative factor: They divide (or
22206 multiply) by the entered formula, simplify, then multiply (or divide)
22207 back by the formula.
22211 @pindex calc-sel-add-both-sides
22212 @pindex calc-sel-sub-both-sides
22213 The @kbd{j +} (@code{calc-sel-add-both-sides}) and @kbd{j -}
22214 (@code{calc-sel-sub-both-sides}) commands analogously add to or
22215 subtract from both sides of an equation or inequality. For other
22216 types of selections, they extract an additive factor. A numeric
22217 prefix argument suppresses simplification of the intermediate
22221 @pindex calc-sel-unpack
22222 The @kbd{j U} (@code{calc-sel-unpack}) command replaces the
22223 selected function call with its argument. For example, given
22224 @samp{a + sin(x^2)} with @samp{sin(x^2)} selected, the result
22225 is @samp{a + x^2}. (The @samp{x^2} will remain selected; if you
22226 wanted to change the @code{sin} to @code{cos}, just press @kbd{C}
22227 now to take the cosine of the selected part.)
22230 @pindex calc-sel-evaluate
22231 The @kbd{j v} (@code{calc-sel-evaluate}) command performs the
22232 basic simplifications on the selected sub-formula.
22233 These simplifications would normally be done automatically
22234 on all results, but may have been partially inhibited by
22235 previous selection-related operations, or turned off altogether
22236 by the @kbd{m O} command. This command is just an auto-selecting
22237 version of the @w{@kbd{a v}} command (@pxref{Algebraic Manipulation}).
22239 With a numeric prefix argument of 2, @kbd{C-u 2 j v} applies
22240 the default algebraic simplifications to the selected
22241 sub-formula. With a prefix argument of 3 or more, e.g., @kbd{C-u j v}
22242 applies the @kbd{a e} (@code{calc-simplify-extended}) command.
22243 @xref{Simplifying Formulas}. With a negative prefix argument
22244 it simplifies at the top level only, just as with @kbd{a v}.
22245 Here the ``top'' level refers to the top level of the selected
22249 @pindex calc-sel-expand-formula
22250 The @kbd{j "} (@code{calc-sel-expand-formula}) command is to @kbd{a "}
22251 (@pxref{Algebraic Manipulation}) what @kbd{j v} is to @kbd{a v}.
22253 You can use the @kbd{j r} (@code{calc-rewrite-selection}) command
22254 to define other algebraic operations on sub-formulas. @xref{Rewrite Rules}.
22256 @node Algebraic Manipulation, Simplifying Formulas, Selecting Subformulas, Algebra
22257 @section Algebraic Manipulation
22260 The commands in this section perform general-purpose algebraic
22261 manipulations. They work on the whole formula at the top of the
22262 stack (unless, of course, you have made a selection in that
22265 Many algebra commands prompt for a variable name or formula. If you
22266 answer the prompt with a blank line, the variable or formula is taken
22267 from top-of-stack, and the normal argument for the command is taken
22268 from the second-to-top stack level.
22271 @pindex calc-alg-evaluate
22272 The @kbd{a v} (@code{calc-alg-evaluate}) command performs the normal
22273 default simplifications on a formula; for example, @samp{a - -b} is
22274 changed to @samp{a + b}. These simplifications are normally done
22275 automatically on all Calc results, so this command is useful only if
22276 you have turned default simplifications off with an @kbd{m O}
22277 command. @xref{Simplification Modes}.
22279 It is often more convenient to type @kbd{=}, which is like @kbd{a v}
22280 but which also substitutes stored values for variables in the formula.
22281 Use @kbd{a v} if you want the variables to ignore their stored values.
22283 If you give a numeric prefix argument of 2 to @kbd{a v}, it simplifies
22284 using Calc's algebraic simplifications; @pxref{Simplifying Formulas}.
22285 If you give a numeric prefix of 3 or more, it uses Extended
22286 Simplification mode (@kbd{a e}).
22288 If you give a negative prefix argument @mathit{-1}, @mathit{-2}, or @mathit{-3},
22289 it simplifies in the corresponding mode but only works on the top-level
22290 function call of the formula. For example, @samp{(2 + 3) * (2 + 3)} will
22291 simplify to @samp{(2 + 3)^2}, without simplifying the sub-formulas
22292 @samp{2 + 3}. As another example, typing @kbd{V R +} to sum the vector
22293 @samp{[1, 2, 3, 4]} produces the formula @samp{reduce(add, [1, 2, 3, 4])}
22294 in No-Simplify mode. Using @kbd{a v} will evaluate this all the way to
22295 10; using @kbd{C-u - a v} will evaluate it only to @samp{1 + 2 + 3 + 4}.
22296 (@xref{Reducing and Mapping}.)
22300 The @kbd{=} command corresponds to the @code{evalv} function, and
22301 the related @kbd{N} command, which is like @kbd{=} but temporarily
22302 disables Symbolic mode (@kbd{m s}) during the evaluation, corresponds
22303 to the @code{evalvn} function. (These commands interpret their prefix
22304 arguments differently than @kbd{a v}; @kbd{=} treats the prefix as
22305 the number of stack elements to evaluate at once, and @kbd{N} treats
22306 it as a temporary different working precision.)
22308 The @code{evalvn} function can take an alternate working precision
22309 as an optional second argument. This argument can be either an
22310 integer, to set the precision absolutely, or a vector containing
22311 a single integer, to adjust the precision relative to the current
22312 precision. Note that @code{evalvn} with a larger than current
22313 precision will do the calculation at this higher precision, but the
22314 result will as usual be rounded back down to the current precision
22315 afterward. For example, @samp{evalvn(pi - 3.1415)} at a precision
22316 of 12 will return @samp{9.265359e-5}; @samp{evalvn(pi - 3.1415, 30)}
22317 will return @samp{9.26535897932e-5} (computing a 25-digit result which
22318 is then rounded down to 12); and @samp{evalvn(pi - 3.1415, [-2])}
22319 will return @samp{9.2654e-5}.
22322 @pindex calc-expand-formula
22323 The @kbd{a "} (@code{calc-expand-formula}) command expands functions
22324 into their defining formulas wherever possible. For example,
22325 @samp{deg(x^2)} is changed to @samp{180 x^2 / pi}. Most functions,
22326 like @code{sin} and @code{gcd}, are not defined by simple formulas
22327 and so are unaffected by this command. One important class of
22328 functions which @emph{can} be expanded is the user-defined functions
22329 created by the @kbd{Z F} command. @xref{Algebraic Definitions}.
22330 Other functions which @kbd{a "} can expand include the probability
22331 distribution functions, most of the financial functions, and the
22332 hyperbolic and inverse hyperbolic functions. A numeric prefix argument
22333 affects @kbd{a "} in the same way as it does @kbd{a v}: A positive
22334 argument expands all functions in the formula and then simplifies in
22335 various ways; a negative argument expands and simplifies only the
22336 top-level function call.
22339 @pindex calc-map-equation
22341 The @kbd{a M} (@code{calc-map-equation}) [@code{mapeq}] command applies
22342 a given function or operator to one or more equations. It is analogous
22343 to @kbd{V M}, which operates on vectors instead of equations.
22344 @pxref{Reducing and Mapping}. For example, @kbd{a M S} changes
22345 @samp{x = y+1} to @samp{sin(x) = sin(y+1)}, and @kbd{a M +} with
22346 @samp{x = y+1} and @expr{6} on the stack produces @samp{x+6 = y+7}.
22347 With two equations on the stack, @kbd{a M +} would add the lefthand
22348 sides together and the righthand sides together to get the two
22349 respective sides of a new equation.
22351 Mapping also works on inequalities. Mapping two similar inequalities
22352 produces another inequality of the same type. Mapping an inequality
22353 with an equation produces an inequality of the same type. Mapping a
22354 @samp{<=} with a @samp{<} or @samp{!=} (not-equal) produces a @samp{<}.
22355 If inequalities with opposite direction (e.g., @samp{<} and @samp{>})
22356 are mapped, the direction of the second inequality is reversed to
22357 match the first: Using @kbd{a M +} on @samp{a < b} and @samp{a > 2}
22358 reverses the latter to get @samp{2 < a}, which then allows the
22359 combination @samp{a + 2 < b + a}, which the algebraic simplifications
22360 can reduce to @samp{2 < b}.
22362 Using @kbd{a M *}, @kbd{a M /}, @kbd{a M n}, or @kbd{a M &} to negate
22363 or invert an inequality will reverse the direction of the inequality.
22364 Other adjustments to inequalities are @emph{not} done automatically;
22365 @kbd{a M S} will change @w{@samp{x < y}} to @samp{sin(x) < sin(y)} even
22366 though this is not true for all values of the variables.
22370 With the Hyperbolic flag, @kbd{H a M} [@code{mapeqp}] does a plain
22371 mapping operation without reversing the direction of any inequalities.
22372 Thus, @kbd{H a M &} would change @kbd{x > 2} to @kbd{1/x > 0.5}.
22373 (This change is mathematically incorrect, but perhaps you were
22374 fixing an inequality which was already incorrect.)
22378 With the Inverse flag, @kbd{I a M} [@code{mapeqr}] always reverses
22379 the direction of the inequality. You might use @kbd{I a M C} to
22380 change @samp{x < y} to @samp{cos(x) > cos(y)} if you know you are
22381 working with small positive angles.
22384 @pindex calc-substitute
22386 The @kbd{a b} (@code{calc-substitute}) [@code{subst}] command substitutes
22388 of some variable or sub-expression of an expression with a new
22389 sub-expression. For example, substituting @samp{sin(x)} with @samp{cos(y)}
22390 in @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces
22391 @samp{2 cos(y)^2 + x cos(y) + @w{sin(2 x)}}.
22392 Note that this is a purely structural substitution; the lone @samp{x} and
22393 the @samp{sin(2 x)} stayed the same because they did not look like
22394 @samp{sin(x)}. @xref{Rewrite Rules}, for a more general method for
22395 doing substitutions.
22397 The @kbd{a b} command normally prompts for two formulas, the old
22398 one and the new one. If you enter a blank line for the first
22399 prompt, all three arguments are taken from the stack (new, then old,
22400 then target expression). If you type an old formula but then enter a
22401 blank line for the new one, the new formula is taken from top-of-stack
22402 and the target from second-to-top. If you answer both prompts, the
22403 target is taken from top-of-stack as usual.
22405 Note that @kbd{a b} has no understanding of commutativity or
22406 associativity. The pattern @samp{x+y} will not match the formula
22407 @samp{y+x}. Also, @samp{y+z} will not match inside the formula @samp{x+y+z}
22408 because the @samp{+} operator is left-associative, so the ``deep
22409 structure'' of that formula is @samp{(x+y) + z}. Use @kbd{d U}
22410 (@code{calc-unformatted-language}) mode to see the true structure of
22411 a formula. The rewrite rule mechanism, discussed later, does not have
22414 As an algebraic function, @code{subst} takes three arguments:
22415 Target expression, old, new. Note that @code{subst} is always
22416 evaluated immediately, even if its arguments are variables, so if
22417 you wish to put a call to @code{subst} onto the stack you must
22418 turn the default simplifications off first (with @kbd{m O}).
22420 @node Simplifying Formulas, Polynomials, Algebraic Manipulation, Algebra
22421 @section Simplifying Formulas
22427 @pindex calc-simplify
22430 The sections below describe all the various kinds of
22431 simplifications Calc provides in full detail. None of Calc's
22432 simplification commands are designed to pull rabbits out of hats;
22433 they simply apply certain specific rules to put formulas into
22434 less redundant or more pleasing forms. Serious algebra in Calc
22435 must be done manually, usually with a combination of selections
22436 and rewrite rules. @xref{Rearranging with Selections}.
22437 @xref{Rewrite Rules}.
22439 @xref{Simplification Modes}, for commands to control what level of
22440 simplification occurs automatically. Normally the algebraic
22441 simplifications described below occur. If you have turned on a
22442 simplification mode which does not do these algebraic simplifications,
22443 you can still apply them to a formula with the @kbd{a s}
22444 (@code{calc-simplify}) [@code{simplify}] command.
22446 There are some simplifications that, while sometimes useful, are never
22447 done automatically. For example, the @kbd{I} prefix can be given to
22448 @kbd{a s}; the @kbd{I a s} command will change any trigonometric
22449 function to the appropriate combination of @samp{sin}s and @samp{cos}s
22450 before simplifying. This can be useful in simplifying even mildly
22451 complicated trigonometric expressions. For example, while the algebraic
22452 simplifications can reduce @samp{sin(x) csc(x)} to @samp{1}, they will not
22453 simplify @samp{sin(x)^2 csc(x)}. The command @kbd{I a s} can be used to
22454 simplify this latter expression; it will transform @samp{sin(x)^2
22455 csc(x)} into @samp{sin(x)}. However, @kbd{I a s} will also perform
22456 some ``simplifications'' which may not be desired; for example, it
22457 will transform @samp{tan(x)^2} into @samp{sin(x)^2 / cos(x)^2}. The
22458 Hyperbolic prefix @kbd{H} can be used similarly; the @kbd{H a s} will
22459 replace any hyperbolic functions in the formula with the appropriate
22460 combinations of @samp{sinh}s and @samp{cosh}s before simplifying.
22464 * Basic Simplifications::
22465 * Algebraic Simplifications::
22466 * Unsafe Simplifications::
22467 * Simplification of Units::
22470 @node Basic Simplifications, Algebraic Simplifications, Simplifying Formulas, Simplifying Formulas
22471 @subsection Basic Simplifications
22474 @cindex Basic simplifications
22475 This section describes basic simplifications which Calc performs in many
22476 situations. For example, both binary simplifications and algebraic
22477 simplifications begin by performing these basic simplifications. You
22478 can type @kbd{m I} to restrict the simplifications done on the stack to
22479 these simplifications.
22481 The most basic simplification is the evaluation of functions.
22482 For example, @expr{2 + 3} is evaluated to @expr{5}, and @expr{@tfn{sqrt}(9)}
22483 is evaluated to @expr{3}. Evaluation does not occur if the arguments
22484 to a function are somehow of the wrong type @expr{@tfn{tan}([2,3,4])}),
22485 range (@expr{@tfn{tan}(90)}), or number (@expr{@tfn{tan}(3,5)}),
22486 or if the function name is not recognized (@expr{@tfn{f}(5)}), or if
22487 Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation
22488 (@expr{@tfn{sqrt}(2)}).
22490 Calc simplifies (evaluates) the arguments to a function before it
22491 simplifies the function itself. Thus @expr{@tfn{sqrt}(5+4)} is
22492 simplified to @expr{@tfn{sqrt}(9)} before the @code{sqrt} function
22493 itself is applied. There are very few exceptions to this rule:
22494 @code{quote}, @code{lambda}, and @code{condition} (the @code{::}
22495 operator) do not evaluate their arguments, @code{if} (the @code{? :}
22496 operator) does not evaluate all of its arguments, and @code{evalto}
22497 does not evaluate its lefthand argument.
22499 Most commands apply at least these basic simplifications to all
22500 arguments they take from the stack, perform a particular operation,
22501 then simplify the result before pushing it back on the stack. In the
22502 common special case of regular arithmetic commands like @kbd{+} and
22503 @kbd{Q} [@code{sqrt}], the arguments are simply popped from the stack
22504 and collected into a suitable function call, which is then simplified
22505 (the arguments being simplified first as part of the process, as
22508 Even the basic set of simplifications are too numerous to describe
22509 completely here, but this section will describe the ones that apply to the
22510 major arithmetic operators. This list will be rather technical in
22511 nature, and will probably be interesting to you only if you are
22512 a serious user of Calc's algebra facilities.
22518 As well as the simplifications described here, if you have stored
22519 any rewrite rules in the variable @code{EvalRules} then these rules
22520 will also be applied before any of the basic simplifications.
22521 @xref{Automatic Rewrites}, for details.
22527 And now, on with the basic simplifications:
22529 Arithmetic operators like @kbd{+} and @kbd{*} always take two
22530 arguments in Calc's internal form. Sums and products of three or
22531 more terms are arranged by the associative law of algebra into
22532 a left-associative form for sums, @expr{((a + b) + c) + d}, and
22533 (by default) a right-associative form for products,
22534 @expr{a * (b * (c * d))}. Formulas like @expr{(a + b) + (c + d)} are
22535 rearranged to left-associative form, though this rarely matters since
22536 Calc's algebra commands are designed to hide the inner structure of sums
22537 and products as much as possible. Sums and products in their proper
22538 associative form will be written without parentheses in the examples
22541 Sums and products are @emph{not} rearranged according to the
22542 commutative law (@expr{a + b} to @expr{b + a}) except in a few
22543 special cases described below. Some algebra programs always
22544 rearrange terms into a canonical order, which enables them to
22545 see that @expr{a b + b a} can be simplified to @expr{2 a b}.
22546 If you are using Basic Simplification mode, Calc assumes you have put
22547 the terms into the order you want and generally leaves that order alone,
22548 with the consequence that formulas like the above will only be
22549 simplified if you explicitly give the @kbd{a s} command.
22550 @xref{Algebraic Simplifications}.
22552 Differences @expr{a - b} are treated like sums @expr{a + (-b)}
22553 for purposes of simplification; one of the default simplifications
22554 is to rewrite @expr{a + (-b)} or @expr{(-b) + a}, where @expr{-b}
22555 represents a ``negative-looking'' term, into @expr{a - b} form.
22556 ``Negative-looking'' means negative numbers, negated formulas like
22557 @expr{-x}, and products or quotients in which either term is
22560 Other simplifications involving negation are @expr{-(-x)} to @expr{x};
22561 @expr{-(a b)} or @expr{-(a/b)} where either @expr{a} or @expr{b} is
22562 negative-looking, simplified by negating that term, or else where
22563 @expr{a} or @expr{b} is any number, by negating that number;
22564 @expr{-(a + b)} to @expr{-a - b}, and @expr{-(b - a)} to @expr{a - b}.
22565 (This, and rewriting @expr{(-b) + a} to @expr{a - b}, are the only
22566 cases where the order of terms in a sum is changed by the default
22569 The distributive law is used to simplify sums in some cases:
22570 @expr{a x + b x} to @expr{(a + b) x}, where @expr{a} represents
22571 a number or an implicit 1 or @mathit{-1} (as in @expr{x} or @expr{-x})
22572 and similarly for @expr{b}. Use the @kbd{a c}, @w{@kbd{a f}}, or
22573 @kbd{j M} commands to merge sums with non-numeric coefficients
22574 using the distributive law.
22576 The distributive law is only used for sums of two terms, or
22577 for adjacent terms in a larger sum. Thus @expr{a + b + b + c}
22578 is simplified to @expr{a + 2 b + c}, but @expr{a + b + c + b}
22579 is not simplified. The reason is that comparing all terms of a
22580 sum with one another would require time proportional to the
22581 square of the number of terms; Calc omits potentially slow
22582 operations like this in basic simplification mode.
22584 Finally, @expr{a + 0} and @expr{0 + a} are simplified to @expr{a}.
22585 A consequence of the above rules is that @expr{0 - a} is simplified
22592 The products @expr{1 a} and @expr{a 1} are simplified to @expr{a};
22593 @expr{(-1) a} and @expr{a (-1)} are simplified to @expr{-a};
22594 @expr{0 a} and @expr{a 0} are simplified to @expr{0}, except that
22595 in Matrix mode where @expr{a} is not provably scalar the result
22596 is the generic zero matrix @samp{idn(0)}, and that if @expr{a} is
22597 infinite the result is @samp{nan}.
22599 Also, @expr{(-a) b} and @expr{a (-b)} are simplified to @expr{-(a b)},
22600 where this occurs for negated formulas but not for regular negative
22603 Products are commuted only to move numbers to the front:
22604 @expr{a b 2} is commuted to @expr{2 a b}.
22606 The product @expr{a (b + c)} is distributed over the sum only if
22607 @expr{a} and at least one of @expr{b} and @expr{c} are numbers:
22608 @expr{2 (x + 3)} goes to @expr{2 x + 6}. The formula
22609 @expr{(-a) (b - c)}, where @expr{-a} is a negative number, is
22610 rewritten to @expr{a (c - b)}.
22612 The distributive law of products and powers is used for adjacent
22613 terms of the product: @expr{x^a x^b} goes to
22614 @texline @math{x^{a+b}}
22615 @infoline @expr{x^(a+b)}
22616 where @expr{a} is a number, or an implicit 1 (as in @expr{x}),
22617 or the implicit one-half of @expr{@tfn{sqrt}(x)}, and similarly for
22618 @expr{b}. The result is written using @samp{sqrt} or @samp{1/sqrt}
22619 if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively.
22620 If the sum of the powers is zero, the product is simplified to
22621 @expr{1} or to @samp{idn(1)} if Matrix mode is enabled.
22623 The product of a negative power times anything but another negative
22624 power is changed to use division:
22625 @texline @math{x^{-2} y}
22626 @infoline @expr{x^(-2) y}
22627 goes to @expr{y / x^2} unless Matrix mode is
22628 in effect and neither @expr{x} nor @expr{y} are scalar (in which
22629 case it is considered unsafe to rearrange the order of the terms).
22631 Finally, @expr{a (b/c)} is rewritten to @expr{(a b)/c}, and also
22632 @expr{(a/b) c} is changed to @expr{(a c)/b} unless in Matrix mode.
22638 Simplifications for quotients are analogous to those for products.
22639 The quotient @expr{0 / x} is simplified to @expr{0}, with the same
22640 exceptions that were noted for @expr{0 x}. Likewise, @expr{x / 1}
22641 and @expr{x / (-1)} are simplified to @expr{x} and @expr{-x},
22644 The quotient @expr{x / 0} is left unsimplified or changed to an
22645 infinite quantity, as directed by the current infinite mode.
22646 @xref{Infinite Mode}.
22649 @texline @math{a / b^{-c}}
22650 @infoline @expr{a / b^(-c)}
22651 is changed to @expr{a b^c}, where @expr{-c} is any negative-looking
22652 power. Also, @expr{1 / b^c} is changed to
22653 @texline @math{b^{-c}}
22654 @infoline @expr{b^(-c)}
22655 for any power @expr{c}.
22657 Also, @expr{(-a) / b} and @expr{a / (-b)} go to @expr{-(a/b)};
22658 @expr{(a/b) / c} goes to @expr{a / (b c)}; and @expr{a / (b/c)}
22659 goes to @expr{(a c) / b} unless Matrix mode prevents this
22660 rearrangement. Similarly, @expr{a / (b:c)} is simplified to
22661 @expr{(c:b) a} for any fraction @expr{b:c}.
22663 The distributive law is applied to @expr{(a + b) / c} only if
22664 @expr{c} and at least one of @expr{a} and @expr{b} are numbers.
22665 Quotients of powers and square roots are distributed just as
22666 described for multiplication.
22668 Quotients of products cancel only in the leading terms of the
22669 numerator and denominator. In other words, @expr{a x b / a y b}
22670 is canceled to @expr{x b / y b} but not to @expr{x / y}. Once
22671 again this is because full cancellation can be slow; use @kbd{a s}
22672 to cancel all terms of the quotient.
22674 Quotients of negative-looking values are simplified according
22675 to @expr{(-a) / (-b)} to @expr{a / b}, @expr{(-a) / (b - c)}
22676 to @expr{a / (c - b)}, and @expr{(a - b) / (-c)} to @expr{(b - a) / c}.
22682 The formula @expr{x^0} is simplified to @expr{1}, or to @samp{idn(1)}
22683 in Matrix mode. The formula @expr{0^x} is simplified to @expr{0}
22684 unless @expr{x} is a negative number, complex number or zero.
22685 If @expr{x} is negative, complex or @expr{0.0}, @expr{0^x} is an
22686 infinity or an unsimplified formula according to the current infinite
22687 mode. The expression @expr{0^0} is simplified to @expr{1}.
22689 Powers of products or quotients @expr{(a b)^c}, @expr{(a/b)^c}
22690 are distributed to @expr{a^c b^c}, @expr{a^c / b^c} only if @expr{c}
22691 is an integer, or if either @expr{a} or @expr{b} are nonnegative
22692 real numbers. Powers of powers @expr{(a^b)^c} are simplified to
22693 @texline @math{a^{b c}}
22694 @infoline @expr{a^(b c)}
22695 only when @expr{c} is an integer and @expr{b c} also
22696 evaluates to an integer. Without these restrictions these simplifications
22697 would not be safe because of problems with principal values.
22699 @texline @math{((-3)^{1/2})^2}
22700 @infoline @expr{((-3)^1:2)^2}
22701 is safe to simplify, but
22702 @texline @math{((-3)^2)^{1/2}}
22703 @infoline @expr{((-3)^2)^1:2}
22704 is not.) @xref{Declarations}, for ways to inform Calc that your
22705 variables satisfy these requirements.
22707 As a special case of this rule, @expr{@tfn{sqrt}(x)^n} is simplified to
22708 @texline @math{x^{n/2}}
22709 @infoline @expr{x^(n/2)}
22710 only for even integers @expr{n}.
22712 If @expr{a} is known to be real, @expr{b} is an even integer, and
22713 @expr{c} is a half- or quarter-integer, then @expr{(a^b)^c} is
22714 simplified to @expr{@tfn{abs}(a^(b c))}.
22716 Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an
22717 even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer,
22718 for any negative-looking expression @expr{-a}.
22720 Square roots @expr{@tfn{sqrt}(x)} generally act like one-half powers
22721 @texline @math{x^{1:2}}
22722 @infoline @expr{x^1:2}
22723 for the purposes of the above-listed simplifications.
22726 @texline @math{1 / x^{1:2}}
22727 @infoline @expr{1 / x^1:2}
22729 @texline @math{x^{-1:2}},
22730 @infoline @expr{x^(-1:2)},
22731 but @expr{1 / @tfn{sqrt}(x)} is left alone.
22737 Generic identity matrices (@pxref{Matrix Mode}) are simplified by the
22738 following rules: @expr{@tfn{idn}(a) + b} to @expr{a + b} if @expr{b}
22739 is provably scalar, or expanded out if @expr{b} is a matrix;
22740 @expr{@tfn{idn}(a) + @tfn{idn}(b)} to @expr{@tfn{idn}(a + b)};
22741 @expr{-@tfn{idn}(a)} to @expr{@tfn{idn}(-a)}; @expr{a @tfn{idn}(b)} to
22742 @expr{@tfn{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b}
22743 if @expr{a} is provably non-scalar; @expr{@tfn{idn}(a) @tfn{idn}(b)} to
22744 @expr{@tfn{idn}(a b)}; analogous simplifications for quotients involving
22745 @code{idn}; and @expr{@tfn{idn}(a)^n} to @expr{@tfn{idn}(a^n)} where
22746 @expr{n} is an integer.
22752 The @code{floor} function and other integer truncation functions
22753 vanish if the argument is provably integer-valued, so that
22754 @expr{@tfn{floor}(@tfn{round}(x))} simplifies to @expr{@tfn{round}(x)}.
22755 Also, combinations of @code{float}, @code{floor} and its friends,
22756 and @code{ffloor} and its friends, are simplified in appropriate
22757 ways. @xref{Integer Truncation}.
22759 The expression @expr{@tfn{abs}(-x)} changes to @expr{@tfn{abs}(x)}.
22760 The expression @expr{@tfn{abs}(@tfn{abs}(x))} changes to
22761 @expr{@tfn{abs}(x)}; in fact, @expr{@tfn{abs}(x)} changes to @expr{x} or
22762 @expr{-x} if @expr{x} is provably nonnegative or nonpositive
22763 (@pxref{Declarations}).
22765 While most functions do not recognize the variable @code{i} as an
22766 imaginary number, the @code{arg} function does handle the two cases
22767 @expr{@tfn{arg}(@tfn{i})} and @expr{@tfn{arg}(-@tfn{i})} just for convenience.
22769 The expression @expr{@tfn{conj}(@tfn{conj}(x))} simplifies to @expr{x}.
22770 Various other expressions involving @code{conj}, @code{re}, and
22771 @code{im} are simplified, especially if some of the arguments are
22772 provably real or involve the constant @code{i}. For example,
22773 @expr{@tfn{conj}(a + b i)} is changed to
22774 @expr{@tfn{conj}(a) - @tfn{conj}(b) i}, or to @expr{a - b i} if @expr{a}
22775 and @expr{b} are known to be real.
22777 Functions like @code{sin} and @code{arctan} generally don't have
22778 any default simplifications beyond simply evaluating the functions
22779 for suitable numeric arguments and infinity. The algebraic
22780 simplifications described in the next section do provide some
22781 simplifications for these functions, though.
22783 One important simplification that does occur is that
22784 @expr{@tfn{ln}(@tfn{e})} is simplified to 1, and @expr{@tfn{ln}(@tfn{e}^x)} is
22785 simplified to @expr{x} for any @expr{x}. This occurs even if you have
22786 stored a different value in the Calc variable @samp{e}; but this would
22787 be a bad idea in any case if you were also using natural logarithms!
22789 Among the logical functions, @tfn{!(@var{a} <= @var{b})} changes to
22790 @tfn{@var{a} > @var{b}} and so on. Equations and inequalities where both sides
22791 are either negative-looking or zero are simplified by negating both sides
22792 and reversing the inequality. While it might seem reasonable to simplify
22793 @expr{!!x} to @expr{x}, this would not be valid in general because
22794 @expr{!!2} is 1, not 2.
22796 Most other Calc functions have few if any basic simplifications
22797 defined, aside of course from evaluation when the arguments are
22800 @node Algebraic Simplifications, Unsafe Simplifications, Basic Simplifications, Simplifying Formulas
22801 @subsection Algebraic Simplifications
22804 @cindex Algebraic simplifications
22807 This section describes all simplifications that are performed by
22808 the algebraic simplification mode, which is the default simplification
22809 mode. If you have switched to a different simplification mode, you can
22810 switch back with the @kbd{m A} command. Even in other simplification
22811 modes, the @kbd{a s} command will use these algebraic simplifications to
22812 simplify the formula.
22814 There is a variable, @code{AlgSimpRules}, in which you can put rewrites
22815 to be applied. Its use is analogous to @code{EvalRules},
22816 but without the special restrictions. Basically, the simplifier does
22817 @samp{@w{a r} AlgSimpRules} with an infinite repeat count on the whole
22818 expression being simplified, then it traverses the expression applying
22819 the built-in rules described below. If the result is different from
22820 the original expression, the process repeats with the basic
22821 simplifications (including @code{EvalRules}), then @code{AlgSimpRules},
22822 then the built-in simplifications, and so on.
22828 Sums are simplified in two ways. Constant terms are commuted to the
22829 end of the sum, so that @expr{a + 2 + b} changes to @expr{a + b + 2}.
22830 The only exception is that a constant will not be commuted away
22831 from the first position of a difference, i.e., @expr{2 - x} is not
22832 commuted to @expr{-x + 2}.
22834 Also, terms of sums are combined by the distributive law, as in
22835 @expr{x + y + 2 x} to @expr{y + 3 x}. This always occurs for
22836 adjacent terms, but Calc's algebraic simplifications compare all pairs
22837 of terms including non-adjacent ones.
22843 Products are sorted into a canonical order using the commutative
22844 law. For example, @expr{b c a} is commuted to @expr{a b c}.
22845 This allows easier comparison of products; for example, the basic
22846 simplifications will not change @expr{x y + y x} to @expr{2 x y},
22847 but the algebraic simplifications; it first rewrites the sum to
22848 @expr{x y + x y} which can then be recognized as a sum of identical
22851 The canonical ordering used to sort terms of products has the
22852 property that real-valued numbers, interval forms and infinities
22853 come first, and are sorted into increasing order. The @kbd{V S}
22854 command uses the same ordering when sorting a vector.
22856 Sorting of terms of products is inhibited when Matrix mode is
22857 turned on; in this case, Calc will never exchange the order of
22858 two terms unless it knows at least one of the terms is a scalar.
22860 Products of powers are distributed by comparing all pairs of
22861 terms, using the same method that the default simplifications
22862 use for adjacent terms of products.
22864 Even though sums are not sorted, the commutative law is still
22865 taken into account when terms of a product are being compared.
22866 Thus @expr{(x + y) (y + x)} will be simplified to @expr{(x + y)^2}.
22867 A subtle point is that @expr{(x - y) (y - x)} will @emph{not}
22868 be simplified to @expr{-(x - y)^2}; Calc does not notice that
22869 one term can be written as a constant times the other, even if
22870 that constant is @mathit{-1}.
22872 A fraction times any expression, @expr{(a:b) x}, is changed to
22873 a quotient involving integers: @expr{a x / b}. This is not
22874 done for floating-point numbers like @expr{0.5}, however. This
22875 is one reason why you may find it convenient to turn Fraction mode
22876 on while doing algebra; @pxref{Fraction Mode}.
22882 Quotients are simplified by comparing all terms in the numerator
22883 with all terms in the denominator for possible cancellation using
22884 the distributive law. For example, @expr{a x^2 b / c x^3 d} will
22885 cancel @expr{x^2} from the top and bottom to get @expr{a b / c x d}.
22886 (The terms in the denominator will then be rearranged to @expr{c d x}
22887 as described above.) If there is any common integer or fractional
22888 factor in the numerator and denominator, it is canceled out;
22889 for example, @expr{(4 x + 6) / 8 x} simplifies to @expr{(2 x + 3) / 4 x}.
22891 Non-constant common factors are not found even by algebraic
22892 simplifications. To cancel the factor @expr{a} in
22893 @expr{(a x + a) / a^2} you could first use @kbd{j M} on the product
22894 @expr{a x} to Merge the numerator to @expr{a (1+x)}, which can then be
22895 simplified successfully.
22901 Integer powers of the variable @code{i} are simplified according
22902 to the identity @expr{i^2 = -1}. If you store a new value other
22903 than the complex number @expr{(0,1)} in @code{i}, this simplification
22904 will no longer occur. This is not done by the basic
22905 simplifications; in case someone (unwisely) wants to use the name
22906 @code{i} for a variable unrelated to complex numbers, they can use
22907 basic simplification mode.
22909 Square roots of integer or rational arguments are simplified in
22910 several ways. (Note that these will be left unevaluated only in
22911 Symbolic mode.) First, square integer or rational factors are
22912 pulled out so that @expr{@tfn{sqrt}(8)} is rewritten as
22913 @texline @math{2\,@tfn{sqrt}(2)}.
22914 @infoline @expr{2 sqrt(2)}.
22915 Conceptually speaking this implies factoring the argument into primes
22916 and moving pairs of primes out of the square root, but for reasons of
22917 efficiency Calc only looks for primes up to 29.
22919 Square roots in the denominator of a quotient are moved to the
22920 numerator: @expr{1 / @tfn{sqrt}(3)} changes to @expr{@tfn{sqrt}(3) / 3}.
22921 The same effect occurs for the square root of a fraction:
22922 @expr{@tfn{sqrt}(2:3)} changes to @expr{@tfn{sqrt}(6) / 3}.
22928 The @code{%} (modulo) operator is simplified in several ways
22929 when the modulus @expr{M} is a positive real number. First, if
22930 the argument is of the form @expr{x + n} for some real number
22931 @expr{n}, then @expr{n} is itself reduced modulo @expr{M}. For
22932 example, @samp{(x - 23) % 10} is simplified to @samp{(x + 7) % 10}.
22934 If the argument is multiplied by a constant, and this constant
22935 has a common integer divisor with the modulus, then this factor is
22936 canceled out. For example, @samp{12 x % 15} is changed to
22937 @samp{3 (4 x % 5)} by factoring out 3. Also, @samp{(12 x + 1) % 15}
22938 is changed to @samp{3 ((4 x + 1:3) % 5)}. While these forms may
22939 not seem ``simpler,'' they allow Calc to discover useful information
22940 about modulo forms in the presence of declarations.
22942 If the modulus is 1, then Calc can use @code{int} declarations to
22943 evaluate the expression. For example, the idiom @samp{x % 2} is
22944 often used to check whether a number is odd or even. As described
22945 above, @w{@samp{2 n % 2}} and @samp{(2 n + 1) % 2} are simplified to
22946 @samp{2 (n % 1)} and @samp{2 ((n + 1:2) % 1)}, respectively; Calc
22947 can simplify these to 0 and 1 (respectively) if @code{n} has been
22948 declared to be an integer.
22954 Trigonometric functions are simplified in several ways. Whenever a
22955 products of two trigonometric functions can be replaced by a single
22956 function, the replacement is made; for example,
22957 @expr{@tfn{tan}(x) @tfn{cos}(x)} is simplified to @expr{@tfn{sin}(x)}.
22958 Reciprocals of trigonometric functions are replaced by their reciprocal
22959 function; for example, @expr{1/@tfn{sec}(x)} is simplified to
22960 @expr{@tfn{cos}(x)}. The corresponding simplifications for the
22961 hyperbolic functions are also handled.
22963 Trigonometric functions of their inverse functions are
22964 simplified. The expression @expr{@tfn{sin}(@tfn{arcsin}(x))} is
22965 simplified to @expr{x}, and similarly for @code{cos} and @code{tan}.
22966 Trigonometric functions of inverses of different trigonometric
22967 functions can also be simplified, as in @expr{@tfn{sin}(@tfn{arccos}(x))}
22968 to @expr{@tfn{sqrt}(1 - x^2)}.
22970 If the argument to @code{sin} is negative-looking, it is simplified to
22971 @expr{-@tfn{sin}(x)}, and similarly for @code{cos} and @code{tan}.
22972 Finally, certain special values of the argument are recognized;
22973 @pxref{Trigonometric and Hyperbolic Functions}.
22975 Hyperbolic functions of their inverses and of negative-looking
22976 arguments are also handled, as are exponentials of inverse
22977 hyperbolic functions.
22979 No simplifications for inverse trigonometric and hyperbolic
22980 functions are known, except for negative arguments of @code{arcsin},
22981 @code{arctan}, @code{arcsinh}, and @code{arctanh}. Note that
22982 @expr{@tfn{arcsin}(@tfn{sin}(x))} can @emph{not} safely change to
22983 @expr{x}, since this only correct within an integer multiple of
22984 @texline @math{2 \pi}
22985 @infoline @expr{2 pi}
22986 radians or 360 degrees. However, @expr{@tfn{arcsinh}(@tfn{sinh}(x))} is
22987 simplified to @expr{x} if @expr{x} is known to be real.
22989 Several simplifications that apply to logarithms and exponentials
22990 are that @expr{@tfn{exp}(@tfn{ln}(x))},
22991 @texline @tfn{e}@math{^{\ln(x)}},
22992 @infoline @expr{e^@tfn{ln}(x)},
22994 @texline @math{10^{{\rm log10}(x)}}
22995 @infoline @expr{10^@tfn{log10}(x)}
22996 all reduce to @expr{x}. Also, @expr{@tfn{ln}(@tfn{exp}(x))}, etc., can
22997 reduce to @expr{x} if @expr{x} is provably real. The form
22998 @expr{@tfn{exp}(x)^y} is simplified to @expr{@tfn{exp}(x y)}. If @expr{x}
22999 is a suitable multiple of
23000 @texline @math{\pi i}
23001 @infoline @expr{pi i}
23002 (as described above for the trigonometric functions), then
23003 @expr{@tfn{exp}(x)} or @expr{e^x} will be expanded. Finally,
23004 @expr{@tfn{ln}(x)} is simplified to a form involving @code{pi} and
23005 @code{i} where @expr{x} is provably negative, positive imaginary, or
23006 negative imaginary.
23008 The error functions @code{erf} and @code{erfc} are simplified when
23009 their arguments are negative-looking or are calls to the @code{conj}
23016 Equations and inequalities are simplified by canceling factors
23017 of products, quotients, or sums on both sides. Inequalities
23018 change sign if a negative multiplicative factor is canceled.
23019 Non-constant multiplicative factors as in @expr{a b = a c} are
23020 canceled from equations only if they are provably nonzero (generally
23021 because they were declared so; @pxref{Declarations}). Factors
23022 are canceled from inequalities only if they are nonzero and their
23025 Simplification also replaces an equation or inequality with
23026 1 or 0 (``true'' or ``false'') if it can through the use of
23027 declarations. If @expr{x} is declared to be an integer greater
23028 than 5, then @expr{x < 3}, @expr{x = 3}, and @expr{x = 7.5} are
23029 all simplified to 0, but @expr{x > 3} is simplified to 1.
23030 By a similar analysis, @expr{abs(x) >= 0} is simplified to 1,
23031 as is @expr{x^2 >= 0} if @expr{x} is known to be real.
23033 @node Unsafe Simplifications, Simplification of Units, Algebraic Simplifications, Simplifying Formulas
23034 @subsection ``Unsafe'' Simplifications
23037 @cindex Unsafe simplifications
23038 @cindex Extended simplification
23041 @pindex calc-simplify-extended
23043 @mindex esimpl@idots
23046 Calc is capable of performing some simplifications which may sometimes
23047 be desired but which are not ``safe'' in all cases. The @kbd{a e}
23048 (@code{calc-simplify-extended}) [@code{esimplify}] command
23049 applies the algebraic simplifications as well as these extended, or
23050 ``unsafe'', simplifications. Use this only if you know the values in
23051 your formula lie in the restricted ranges for which these
23052 simplifications are valid. You can use Extended Simplification mode
23053 (@kbd{m E}) to have these simplifications done automatically.
23055 The symbolic integrator uses these extended simplifications; one effect
23056 of this is that the integrator's results must be used with caution.
23057 Where an integral table will often attach conditions like ``for positive
23058 @expr{a} only,'' Calc (like most other symbolic integration programs)
23059 will simply produce an unqualified result.
23061 Because @kbd{a e}'s simplifications are unsafe, it is sometimes better
23062 to type @kbd{C-u -3 a v}, which does extended simplification only
23063 on the top level of the formula without affecting the sub-formulas.
23064 In fact, @kbd{C-u -3 j v} allows you to target extended simplification
23065 to any specific part of a formula.
23067 The variable @code{ExtSimpRules} contains rewrites to be applied when
23068 the extended simplifications are used. These are applied in addition to
23069 @code{EvalRules} and @code{AlgSimpRules}. (The @kbd{a r AlgSimpRules}
23070 step described above is simply followed by an @kbd{a r ExtSimpRules} step.)
23072 Following is a complete list of the ``unsafe'' simplifications.
23078 Inverse trigonometric or hyperbolic functions, called with their
23079 corresponding non-inverse functions as arguments, are simplified.
23080 For example, @expr{@tfn{arcsin}(@tfn{sin}(x))} changes
23081 to @expr{x}. Also, @expr{@tfn{arcsin}(@tfn{cos}(x))} and
23082 @expr{@tfn{arccos}(@tfn{sin}(x))} both change to @expr{@tfn{pi}/2 - x}.
23083 These simplifications are unsafe because they are valid only for
23084 values of @expr{x} in a certain range; outside that range, values
23085 are folded down to the 360-degree range that the inverse trigonometric
23086 functions always produce.
23088 Powers of powers @expr{(x^a)^b} are simplified to
23089 @texline @math{x^{a b}}
23090 @infoline @expr{x^(a b)}
23091 for all @expr{a} and @expr{b}. These results will be valid only
23092 in a restricted range of @expr{x}; for example, in
23093 @texline @math{(x^2)^{1:2}}
23094 @infoline @expr{(x^2)^1:2}
23095 the powers cancel to get @expr{x}, which is valid for positive values
23096 of @expr{x} but not for negative or complex values.
23098 Similarly, @expr{@tfn{sqrt}(x^a)} and @expr{@tfn{sqrt}(x)^a} are both
23099 simplified (possibly unsafely) to
23100 @texline @math{x^{a/2}}.
23101 @infoline @expr{x^(a/2)}.
23103 Forms like @expr{@tfn{sqrt}(1 - sin(x)^2)} are simplified to, e.g.,
23104 @expr{@tfn{cos}(x)}. Calc has identities of this sort for @code{sin},
23105 @code{cos}, @code{tan}, @code{sinh}, and @code{cosh}.
23107 Arguments of square roots are partially factored to look for
23108 squared terms that can be extracted. For example,
23109 @expr{@tfn{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to
23110 @expr{a b @tfn{sqrt}(a+b)}.
23112 The simplifications of @expr{@tfn{ln}(@tfn{exp}(x))},
23113 @expr{@tfn{ln}(@tfn{e}^x)}, and @expr{@tfn{log10}(10^x)} to @expr{x} are also
23114 unsafe because of problems with principal values (although these
23115 simplifications are safe if @expr{x} is known to be real).
23117 Common factors are canceled from products on both sides of an
23118 equation, even if those factors may be zero: @expr{a x / b x}
23119 to @expr{a / b}. Such factors are never canceled from
23120 inequalities: Even the extended simplifications are not bold enough to
23121 reduce @expr{a x < b x} to @expr{a < b} (or @expr{a > b}, depending
23122 on whether you believe @expr{x} is positive or negative).
23123 The @kbd{a M /} command can be used to divide a factor out of
23124 both sides of an inequality.
23126 @node Simplification of Units, , Unsafe Simplifications, Simplifying Formulas
23127 @subsection Simplification of Units
23130 The simplifications described in this section (as well as the algebraic
23131 simplifications) are applied when units need to be simplified. They can
23132 be applied using the @kbd{u s} (@code{calc-simplify-units}) command, or
23133 will be done automatically in Units Simplification mode (@kbd{m U}).
23134 @xref{Basic Operations on Units}.
23136 The variable @code{UnitSimpRules} contains rewrites to be applied by
23137 units simplifications. These are applied in addition to @code{EvalRules}
23138 and @code{AlgSimpRules}.
23140 Scalar mode is automatically put into effect when simplifying units.
23141 @xref{Matrix Mode}.
23143 Sums @expr{a + b} involving units are simplified by extracting the
23144 units of @expr{a} as if by the @kbd{u x} command (call the result
23145 @expr{u_a}), then simplifying the expression @expr{b / u_a}
23146 using @kbd{u b} and @kbd{u s}. If the result has units then the sum
23147 is inconsistent and is left alone. Otherwise, it is rewritten
23148 in terms of the units @expr{u_a}.
23150 If units auto-ranging mode is enabled, products or quotients in
23151 which the first argument is a number which is out of range for the
23152 leading unit are modified accordingly.
23154 When canceling and combining units in products and quotients,
23155 Calc accounts for unit names that differ only in the prefix letter.
23156 For example, @samp{2 km m} is simplified to @samp{2000 m^2}.
23157 However, compatible but different units like @code{ft} and @code{in}
23158 are not combined in this way.
23160 Quotients @expr{a / b} are simplified in three additional ways. First,
23161 if @expr{b} is a number or a product beginning with a number, Calc
23162 computes the reciprocal of this number and moves it to the numerator.
23164 Second, for each pair of unit names from the numerator and denominator
23165 of a quotient, if the units are compatible (e.g., they are both
23166 units of area) then they are replaced by the ratio between those
23167 units. For example, in @samp{3 s in N / kg cm} the units
23168 @samp{in / cm} will be replaced by @expr{2.54}.
23170 Third, if the units in the quotient exactly cancel out, so that
23171 a @kbd{u b} command on the quotient would produce a dimensionless
23172 number for an answer, then the quotient simplifies to that number.
23174 For powers and square roots, the ``unsafe'' simplifications
23175 @expr{(a b)^c} to @expr{a^c b^c}, @expr{(a/b)^c} to @expr{a^c / b^c},
23176 and @expr{(a^b)^c} to
23177 @texline @math{a^{b c}}
23178 @infoline @expr{a^(b c)}
23179 are done if the powers are real numbers. (These are safe in the context
23180 of units because all numbers involved can reasonably be assumed to be
23183 Also, if a unit name is raised to a fractional power, and the
23184 base units in that unit name all occur to powers which are a
23185 multiple of the denominator of the power, then the unit name
23186 is expanded out into its base units, which can then be simplified
23187 according to the previous paragraph. For example, @samp{acre^1.5}
23188 is simplified by noting that @expr{1.5 = 3:2}, that @samp{acre}
23189 is defined in terms of @samp{m^2}, and that the 2 in the power of
23190 @code{m} is a multiple of 2 in @expr{3:2}. Thus, @code{acre^1.5} is
23191 replaced by approximately
23192 @texline @math{(4046 m^2)^{1.5}}
23193 @infoline @expr{(4046 m^2)^1.5},
23194 which is then changed to
23195 @texline @math{4046^{1.5} \, (m^2)^{1.5}},
23196 @infoline @expr{4046^1.5 (m^2)^1.5},
23197 then to @expr{257440 m^3}.
23199 The functions @code{float}, @code{frac}, @code{clean}, @code{abs},
23200 as well as @code{floor} and the other integer truncation functions,
23201 applied to unit names or products or quotients involving units, are
23202 simplified. For example, @samp{round(1.6 in)} is changed to
23203 @samp{round(1.6) round(in)}; the lefthand term evaluates to 2,
23204 and the righthand term simplifies to @code{in}.
23206 The functions @code{sin}, @code{cos}, and @code{tan} with arguments
23207 that have angular units like @code{rad} or @code{arcmin} are
23208 simplified by converting to base units (radians), then evaluating
23209 with the angular mode temporarily set to radians.
23211 @node Polynomials, Calculus, Simplifying Formulas, Algebra
23212 @section Polynomials
23214 A @dfn{polynomial} is a sum of terms which are coefficients times
23215 various powers of a ``base'' variable. For example, @expr{2 x^2 + 3 x - 4}
23216 is a polynomial in @expr{x}. Some formulas can be considered
23217 polynomials in several different variables: @expr{1 + 2 x + 3 y + 4 x y^2}
23218 is a polynomial in both @expr{x} and @expr{y}. Polynomial coefficients
23219 are often numbers, but they may in general be any formulas not
23220 involving the base variable.
23223 @pindex calc-factor
23225 The @kbd{a f} (@code{calc-factor}) [@code{factor}] command factors a
23226 polynomial into a product of terms. For example, the polynomial
23227 @expr{x^3 + 2 x^2 + x} is factored into @samp{x*(x+1)^2}. As another
23228 example, @expr{a c + b d + b c + a d} is factored into the product
23229 @expr{(a + b) (c + d)}.
23231 Calc currently has three algorithms for factoring. Formulas which are
23232 linear in several variables, such as the second example above, are
23233 merged according to the distributive law. Formulas which are
23234 polynomials in a single variable, with constant integer or fractional
23235 coefficients, are factored into irreducible linear and/or quadratic
23236 terms. The first example above factors into three linear terms
23237 (@expr{x}, @expr{x+1}, and @expr{x+1} again). Finally, formulas
23238 which do not fit the above criteria are handled by the algebraic
23241 Calc's polynomial factorization algorithm works by using the general
23242 root-finding command (@w{@kbd{a P}}) to solve for the roots of the
23243 polynomial. It then looks for roots which are rational numbers
23244 or complex-conjugate pairs, and converts these into linear and
23245 quadratic terms, respectively. Because it uses floating-point
23246 arithmetic, it may be unable to find terms that involve large
23247 integers (whose number of digits approaches the current precision).
23248 Also, irreducible factors of degree higher than quadratic are not
23249 found, and polynomials in more than one variable are not treated.
23250 (A more robust factorization algorithm may be included in a future
23253 @vindex FactorRules
23265 The rewrite-based factorization method uses rules stored in the variable
23266 @code{FactorRules}. @xref{Rewrite Rules}, for a discussion of the
23267 operation of rewrite rules. The default @code{FactorRules} are able
23268 to factor quadratic forms symbolically into two linear terms,
23269 @expr{(a x + b) (c x + d)}. You can edit these rules to include other
23270 cases if you wish. To use the rules, Calc builds the formula
23271 @samp{thecoefs(x, [a, b, c, ...])} where @code{x} is the polynomial
23272 base variable and @code{a}, @code{b}, etc., are polynomial coefficients
23273 (which may be numbers or formulas). The constant term is written first,
23274 i.e., in the @code{a} position. When the rules complete, they should have
23275 changed the formula into the form @samp{thefactors(x, [f1, f2, f3, ...])}
23276 where each @code{fi} should be a factored term, e.g., @samp{x - ai}.
23277 Calc then multiplies these terms together to get the complete
23278 factored form of the polynomial. If the rules do not change the
23279 @code{thecoefs} call to a @code{thefactors} call, @kbd{a f} leaves the
23280 polynomial alone on the assumption that it is unfactorable. (Note that
23281 the function names @code{thecoefs} and @code{thefactors} are used only
23282 as placeholders; there are no actual Calc functions by those names.)
23286 The @kbd{H a f} [@code{factors}] command also factors a polynomial,
23287 but it returns a list of factors instead of an expression which is the
23288 product of the factors. Each factor is represented by a sub-vector
23289 of the factor, and the power with which it appears. For example,
23290 @expr{x^5 + x^4 - 33 x^3 + 63 x^2} factors to @expr{(x + 7) x^2 (x - 3)^2}
23291 in @kbd{a f}, or to @expr{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}.
23292 If there is an overall numeric factor, it always comes first in the list.
23293 The functions @code{factor} and @code{factors} allow a second argument
23294 when written in algebraic form; @samp{factor(x,v)} factors @expr{x} with
23295 respect to the specific variable @expr{v}. The default is to factor with
23296 respect to all the variables that appear in @expr{x}.
23299 @pindex calc-collect
23301 The @kbd{a c} (@code{calc-collect}) [@code{collect}] command rearranges a
23303 polynomial in a given variable, ordered in decreasing powers of that
23304 variable. For example, given @expr{1 + 2 x + 3 y + 4 x y^2} on
23305 the stack, @kbd{a c x} would produce @expr{(2 + 4 y^2) x + (1 + 3 y)},
23306 and @kbd{a c y} would produce @expr{(4 x) y^2 + 3 y + (1 + 2 x)}.
23307 The polynomial will be expanded out using the distributive law as
23308 necessary: Collecting @expr{x} in @expr{(x - 1)^3} produces
23309 @expr{x^3 - 3 x^2 + 3 x - 1}. Terms not involving @expr{x} will
23312 The ``variable'' you specify at the prompt can actually be any
23313 expression: @kbd{a c ln(x+1)} will collect together all terms multiplied
23314 by @samp{ln(x+1)} or integer powers thereof. If @samp{x} also appears
23315 in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will
23316 treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants.
23319 @pindex calc-expand
23321 The @kbd{a x} (@code{calc-expand}) [@code{expand}] command expands an
23322 expression by applying the distributive law everywhere. It applies to
23323 products, quotients, and powers involving sums. By default, it fully
23324 distributes all parts of the expression. With a numeric prefix argument,
23325 the distributive law is applied only the specified number of times, then
23326 the partially expanded expression is left on the stack.
23328 The @kbd{a x} and @kbd{j D} commands are somewhat redundant. Use
23329 @kbd{a x} if you want to expand all products of sums in your formula.
23330 Use @kbd{j D} if you want to expand a particular specified term of
23331 the formula. There is an exactly analogous correspondence between
23332 @kbd{a f} and @kbd{j M}. (The @kbd{j D} and @kbd{j M} commands
23333 also know many other kinds of expansions, such as
23334 @samp{exp(a + b) = exp(a) exp(b)}, which @kbd{a x} and @kbd{a f}
23337 Calc's automatic simplifications will sometimes reverse a partial
23338 expansion. For example, the first step in expanding @expr{(x+1)^3} is
23339 to write @expr{(x+1) (x+1)^2}. If @kbd{a x} stops there and tries
23340 to put this formula onto the stack, though, Calc will automatically
23341 simplify it back to @expr{(x+1)^3} form. The solution is to turn
23342 simplification off first (@pxref{Simplification Modes}), or to run
23343 @kbd{a x} without a numeric prefix argument so that it expands all
23344 the way in one step.
23349 The @kbd{a a} (@code{calc-apart}) [@code{apart}] command expands a
23350 rational function by partial fractions. A rational function is the
23351 quotient of two polynomials; @code{apart} pulls this apart into a
23352 sum of rational functions with simple denominators. In algebraic
23353 notation, the @code{apart} function allows a second argument that
23354 specifies which variable to use as the ``base''; by default, Calc
23355 chooses the base variable automatically.
23358 @pindex calc-normalize-rat
23360 The @kbd{a n} (@code{calc-normalize-rat}) [@code{nrat}] command
23361 attempts to arrange a formula into a quotient of two polynomials.
23362 For example, given @expr{1 + (a + b/c) / d}, the result would be
23363 @expr{(b + a c + c d) / c d}. The quotient is reduced, so that
23364 @kbd{a n} will simplify @expr{(x^2 + 2x + 1) / (x^2 - 1)} by dividing
23365 out the common factor @expr{x + 1}, yielding @expr{(x + 1) / (x - 1)}.
23368 @pindex calc-poly-div
23370 The @kbd{a \} (@code{calc-poly-div}) [@code{pdiv}] command divides
23371 two polynomials @expr{u} and @expr{v}, yielding a new polynomial
23372 @expr{q}. If several variables occur in the inputs, the inputs are
23373 considered multivariate polynomials. (Calc divides by the variable
23374 with the largest power in @expr{u} first, or, in the case of equal
23375 powers, chooses the variables in alphabetical order.) For example,
23376 dividing @expr{x^2 + 3 x + 2} by @expr{x + 2} yields @expr{x + 1}.
23377 The remainder from the division, if any, is reported at the bottom
23378 of the screen and is also placed in the Trail along with the quotient.
23380 Using @code{pdiv} in algebraic notation, you can specify the particular
23381 variable to be used as the base: @code{pdiv(@var{a},@var{b},@var{x})}.
23382 If @code{pdiv} is given only two arguments (as is always the case with
23383 the @kbd{a \} command), then it does a multivariate division as outlined
23387 @pindex calc-poly-rem
23389 The @kbd{a %} (@code{calc-poly-rem}) [@code{prem}] command divides
23390 two polynomials and keeps the remainder @expr{r}. The quotient
23391 @expr{q} is discarded. For any formulas @expr{a} and @expr{b}, the
23392 results of @kbd{a \} and @kbd{a %} satisfy @expr{a = q b + r}.
23393 (This is analogous to plain @kbd{\} and @kbd{%}, which compute the
23394 integer quotient and remainder from dividing two numbers.)
23398 @pindex calc-poly-div-rem
23401 The @kbd{a /} (@code{calc-poly-div-rem}) [@code{pdivrem}] command
23402 divides two polynomials and reports both the quotient and the
23403 remainder as a vector @expr{[q, r]}. The @kbd{H a /} [@code{pdivide}]
23404 command divides two polynomials and constructs the formula
23405 @expr{q + r/b} on the stack. (Naturally if the remainder is zero,
23406 this will immediately simplify to @expr{q}.)
23409 @pindex calc-poly-gcd
23411 The @kbd{a g} (@code{calc-poly-gcd}) [@code{pgcd}] command computes
23412 the greatest common divisor of two polynomials. (The GCD actually
23413 is unique only to within a constant multiplier; Calc attempts to
23414 choose a GCD which will be unsurprising.) For example, the @kbd{a n}
23415 command uses @kbd{a g} to take the GCD of the numerator and denominator
23416 of a quotient, then divides each by the result using @kbd{a \}. (The
23417 definition of GCD ensures that this division can take place without
23418 leaving a remainder.)
23420 While the polynomials used in operations like @kbd{a /} and @kbd{a g}
23421 often have integer coefficients, this is not required. Calc can also
23422 deal with polynomials over the rationals or floating-point reals.
23423 Polynomials with modulo-form coefficients are also useful in many
23424 applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc
23425 automatically transforms this into a polynomial over the field of
23426 integers mod 5: @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}.
23428 Congratulations and thanks go to Ove Ewerlid
23429 (@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the
23430 polynomial routines used in the above commands.
23432 @xref{Decomposing Polynomials}, for several useful functions for
23433 extracting the individual coefficients of a polynomial.
23435 @node Calculus, Solving Equations, Polynomials, Algebra
23439 The following calculus commands do not automatically simplify their
23440 inputs or outputs using @code{calc-simplify}. You may find it helps
23441 to do this by hand by typing @kbd{a s} or @kbd{a e}. It may also help
23442 to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most
23446 * Differentiation::
23448 * Customizing the Integrator::
23449 * Numerical Integration::
23453 @node Differentiation, Integration, Calculus, Calculus
23454 @subsection Differentiation
23459 @pindex calc-derivative
23462 The @kbd{a d} (@code{calc-derivative}) [@code{deriv}] command computes
23463 the derivative of the expression on the top of the stack with respect to
23464 some variable, which it will prompt you to enter. Normally, variables
23465 in the formula other than the specified differentiation variable are
23466 considered constant, i.e., @samp{deriv(y,x)} is reduced to zero. With
23467 the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used
23468 instead, in which derivatives of variables are not reduced to zero
23469 unless those variables are known to be ``constant,'' i.e., independent
23470 of any other variables. (The built-in special variables like @code{pi}
23471 are considered constant, as are variables that have been declared
23472 @code{const}; @pxref{Declarations}.)
23474 With a numeric prefix argument @var{n}, this command computes the
23475 @var{n}th derivative.
23477 When working with trigonometric functions, it is best to switch to
23478 Radians mode first (with @w{@kbd{m r}}). The derivative of @samp{sin(x)}
23479 in degrees is @samp{(pi/180) cos(x)}, probably not the expected
23482 If you use the @code{deriv} function directly in an algebraic formula,
23483 you can write @samp{deriv(f,x,x0)} which represents the derivative
23484 of @expr{f} with respect to @expr{x}, evaluated at the point
23485 @texline @math{x=x_0}.
23486 @infoline @expr{x=x0}.
23488 If the formula being differentiated contains functions which Calc does
23489 not know, the derivatives of those functions are produced by adding
23490 primes (apostrophe characters). For example, @samp{deriv(f(2x), x)}
23491 produces @samp{2 f'(2 x)}, where the function @code{f'} represents the
23492 derivative of @code{f}.
23494 For functions you have defined with the @kbd{Z F} command, Calc expands
23495 the functions according to their defining formulas unless you have
23496 also defined @code{f'} suitably. For example, suppose we define
23497 @samp{sinc(x) = sin(x)/x} using @kbd{Z F}. If we then differentiate
23498 the formula @samp{sinc(2 x)}, the formula will be expanded to
23499 @samp{sin(2 x) / (2 x)} and differentiated. However, if we also
23500 define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the
23501 result as @samp{2 dsinc(2 x)}. @xref{Algebraic Definitions}.
23503 For multi-argument functions @samp{f(x,y,z)}, the derivative with respect
23504 to the first argument is written @samp{f'(x,y,z)}; derivatives with
23505 respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}.
23506 Various higher-order derivatives can be formed in the obvious way, e.g.,
23507 @samp{f'@var{}'(x)} (the second derivative of @code{f}) or
23508 @samp{f'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each
23511 @node Integration, Customizing the Integrator, Differentiation, Calculus
23512 @subsection Integration
23516 @pindex calc-integral
23518 The @kbd{a i} (@code{calc-integral}) [@code{integ}] command computes the
23519 indefinite integral of the expression on the top of the stack with
23520 respect to a prompted-for variable. The integrator is not guaranteed to
23521 work for all integrable functions, but it is able to integrate several
23522 large classes of formulas. In particular, any polynomial or rational
23523 function (a polynomial divided by a polynomial) is acceptable.
23524 (Rational functions don't have to be in explicit quotient form, however;
23525 @texline @math{x/(1+x^{-2})}
23526 @infoline @expr{x/(1+x^-2)}
23527 is not strictly a quotient of polynomials, but it is equivalent to
23528 @expr{x^3/(x^2+1)}, which is.) Also, square roots of terms involving
23529 @expr{x} and @expr{x^2} may appear in rational functions being
23530 integrated. Finally, rational functions involving trigonometric or
23531 hyperbolic functions can be integrated.
23533 With an argument (@kbd{C-u a i}), this command will compute the definite
23534 integral of the expression on top of the stack. In this case, the
23535 command will again prompt for an integration variable, then prompt for a
23536 lower limit and an upper limit.
23539 If you use the @code{integ} function directly in an algebraic formula,
23540 you can also write @samp{integ(f,x,v)} which expresses the resulting
23541 indefinite integral in terms of variable @code{v} instead of @code{x}.
23542 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23543 integral from @code{a} to @code{b}.
23546 If you use the @code{integ} function directly in an algebraic formula,
23547 you can also write @samp{integ(f,x,v)} which expresses the resulting
23548 indefinite integral in terms of variable @code{v} instead of @code{x}.
23549 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23550 integral $\int_a^b f(x) \, dx$.
23553 Please note that the current implementation of Calc's integrator sometimes
23554 produces results that are significantly more complex than they need to
23555 be. For example, the integral Calc finds for
23556 @texline @math{1/(x+\sqrt{x^2+1})}
23557 @infoline @expr{1/(x+sqrt(x^2+1))}
23558 is several times more complicated than the answer Mathematica
23559 returns for the same input, although the two forms are numerically
23560 equivalent. Also, any indefinite integral should be considered to have
23561 an arbitrary constant of integration added to it, although Calc does not
23562 write an explicit constant of integration in its result. For example,
23563 Calc's solution for
23564 @texline @math{1/(1+\tan x)}
23565 @infoline @expr{1/(1+tan(x))}
23566 differs from the solution given in the @emph{CRC Math Tables} by a
23568 @texline @math{\pi i / 2}
23569 @infoline @expr{pi i / 2},
23570 due to a different choice of constant of integration.
23572 The Calculator remembers all the integrals it has done. If conditions
23573 change in a way that would invalidate the old integrals, say, a switch
23574 from Degrees to Radians mode, then they will be thrown out. If you
23575 suspect this is not happening when it should, use the
23576 @code{calc-flush-caches} command; @pxref{Caches}.
23579 Calc normally will pursue integration by substitution or integration by
23580 parts up to 3 nested times before abandoning an approach as fruitless.
23581 If the integrator is taking too long, you can lower this limit by storing
23582 a number (like 2) in the variable @code{IntegLimit}. (The @kbd{s I}
23583 command is a convenient way to edit @code{IntegLimit}.) If this variable
23584 has no stored value or does not contain a nonnegative integer, a limit
23585 of 3 is used. The lower this limit is, the greater the chance that Calc
23586 will be unable to integrate a function it could otherwise handle. Raising
23587 this limit allows the Calculator to solve more integrals, though the time
23588 it takes may grow exponentially. You can monitor the integrator's actions
23589 by creating an Emacs buffer called @code{*Trace*}. If such a buffer
23590 exists, the @kbd{a i} command will write a log of its actions there.
23592 If you want to manipulate integrals in a purely symbolic way, you can
23593 set the integration nesting limit to 0 to prevent all but fast
23594 table-lookup solutions of integrals. You might then wish to define
23595 rewrite rules for integration by parts, various kinds of substitutions,
23596 and so on. @xref{Rewrite Rules}.
23598 @node Customizing the Integrator, Numerical Integration, Integration, Calculus
23599 @subsection Customizing the Integrator
23603 Calc has two built-in rewrite rules called @code{IntegRules} and
23604 @code{IntegAfterRules} which you can edit to define new integration
23605 methods. @xref{Rewrite Rules}. At each step of the integration process,
23606 Calc wraps the current integrand in a call to the fictitious function
23607 @samp{integtry(@var{expr},@var{var})}, where @var{expr} is the
23608 integrand and @var{var} is the integration variable. If your rules
23609 rewrite this to be a plain formula (not a call to @code{integtry}), then
23610 Calc will use this formula as the integral of @var{expr}. For example,
23611 the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to
23612 integrate a function @code{mysin} that acts like the sine function.
23613 Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y}
23614 will produce the integral @samp{-2 mycos(2y+1)}. Note that Calc has
23615 automatically made various transformations on the integral to allow it
23616 to use your rule; integral tables generally give rules for
23617 @samp{mysin(a x + b)}, but you don't need to use this much generality
23618 in your @code{IntegRules}.
23620 @cindex Exponential integral Ei(x)
23625 As a more serious example, the expression @samp{exp(x)/x} cannot be
23626 integrated in terms of the standard functions, so the ``exponential
23627 integral'' function
23628 @texline @math{{\rm Ei}(x)}
23629 @infoline @expr{Ei(x)}
23630 was invented to describe it.
23631 We can get Calc to do this integral in terms of a made-up @code{Ei}
23632 function by adding the rule @samp{[integtry(exp(x)/x, x) := Ei(x)]}
23633 to @code{IntegRules}. Now entering @samp{exp(2x)/x} on the stack
23634 and typing @kbd{a i x} yields @samp{Ei(2 x)}. This new rule will
23635 work with Calc's various built-in integration methods (such as
23636 integration by substitution) to solve a variety of other problems
23637 involving @code{Ei}: For example, now Calc will also be able to
23638 integrate @samp{exp(exp(x))} and @samp{ln(ln(x))} (to get @samp{Ei(exp(x))}
23639 and @samp{x ln(ln(x)) - Ei(ln(x))}, respectively).
23641 Your rule may do further integration by calling @code{integ}. For
23642 example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc
23643 to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}.
23644 Note that @code{integ} was called with only one argument. This notation
23645 is allowed only within @code{IntegRules}; it means ``integrate this
23646 with respect to the same integration variable.'' If Calc is unable
23647 to integrate @code{u}, the integration that invoked @code{IntegRules}
23648 also fails. Thus integrating @samp{twice(f(x))} fails, returning the
23649 unevaluated integral @samp{integ(twice(f(x)), x)}. It is still valid
23650 to call @code{integ} with two or more arguments, however; in this case,
23651 if @code{u} is not integrable, @code{twice} itself will still be
23652 integrated: If the above rule is changed to @samp{... := twice(integ(u,x))},
23653 then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}.
23655 If a rule instead produces the formula @samp{integsubst(@var{sexpr},
23656 @var{svar})}, either replacing the top-level @code{integtry} call or
23657 nested anywhere inside the expression, then Calc will apply the
23658 substitution @samp{@var{u} = @var{sexpr}(@var{svar})} to try to
23659 integrate the original @var{expr}. For example, the rule
23660 @samp{sqrt(a) := integsubst(sqrt(x),x)} says that if Calc ever finds
23661 a square root in the integrand, it should attempt the substitution
23662 @samp{u = sqrt(x)}. (This particular rule is unnecessary because
23663 Calc always tries ``obvious'' substitutions where @var{sexpr} actually
23664 appears in the integrand.) The variable @var{svar} may be the same
23665 as the @var{var} that appeared in the call to @code{integtry}, but
23668 When integrating according to an @code{integsubst}, Calc uses the
23669 equation solver to find the inverse of @var{sexpr} (if the integrand
23670 refers to @var{var} anywhere except in subexpressions that exactly
23671 match @var{sexpr}). It uses the differentiator to find the derivative
23672 of @var{sexpr} and/or its inverse (it has two methods that use one
23673 derivative or the other). You can also specify these items by adding
23674 extra arguments to the @code{integsubst} your rules construct; the
23675 general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv},
23676 @var{sprime})}, where @var{sinv} is the inverse of @var{sexpr} (still
23677 written as a function of @var{svar}), and @var{sprime} is the
23678 derivative of @var{sexpr} with respect to @var{svar}. If you don't
23679 specify these things, and Calc is not able to work them out on its
23680 own with the information it knows, then your substitution rule will
23681 work only in very specific, simple cases.
23683 Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules};
23684 in other words, Calc stops rewriting as soon as any rule in your rule
23685 set succeeds. (If it weren't for this, the @samp{integsubst(sqrt(x),x)}
23686 example above would keep on adding layers of @code{integsubst} calls
23689 @vindex IntegSimpRules
23690 Another set of rules, stored in @code{IntegSimpRules}, are applied
23691 every time the integrator uses algebraic simplifications to simplify an
23692 intermediate result. For example, putting the rule
23693 @samp{twice(x) := 2 x} into @code{IntegSimpRules} would tell Calc to
23694 convert the @code{twice} function into a form it knows whenever
23695 integration is attempted.
23697 One more way to influence the integrator is to define a function with
23698 the @kbd{Z F} command (@pxref{Algebraic Definitions}). Calc's
23699 integrator automatically expands such functions according to their
23700 defining formulas, even if you originally asked for the function to
23701 be left unevaluated for symbolic arguments. (Certain other Calc
23702 systems, such as the differentiator and the equation solver, also
23705 @vindex IntegAfterRules
23706 Sometimes Calc is able to find a solution to your integral, but it
23707 expresses the result in a way that is unnecessarily complicated. If
23708 this happens, you can either use @code{integsubst} as described
23709 above to try to hint at a more direct path to the desired result, or
23710 you can use @code{IntegAfterRules}. This is an extra rule set that
23711 runs after the main integrator returns its result; basically, Calc does
23712 an @kbd{a r IntegAfterRules} on the result before showing it to you.
23713 (It also does algebraic simplifications, without @code{IntegSimpRules},
23714 after that to further simplify the result.) For example, Calc's integrator
23715 sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)};
23716 the default @code{IntegAfterRules} rewrite this into the more readable
23717 form @samp{2 arctanh(x)}. Note that, unlike @code{IntegRules},
23718 @code{IntegSimpRules} and @code{IntegAfterRules} are applied any number
23719 of times until no further changes are possible. Rewriting by
23720 @code{IntegAfterRules} occurs only after the main integrator has
23721 finished, not at every step as for @code{IntegRules} and
23722 @code{IntegSimpRules}.
23724 @node Numerical Integration, Taylor Series, Customizing the Integrator, Calculus
23725 @subsection Numerical Integration
23729 @pindex calc-num-integral
23731 If you want a purely numerical answer to an integration problem, you can
23732 use the @kbd{a I} (@code{calc-num-integral}) [@code{ninteg}] command. This
23733 command prompts for an integration variable, a lower limit, and an
23734 upper limit. Except for the integration variable, all other variables
23735 that appear in the integrand formula must have stored values. (A stored
23736 value, if any, for the integration variable itself is ignored.)
23738 Numerical integration works by evaluating your formula at many points in
23739 the specified interval. Calc uses an ``open Romberg'' method; this means
23740 that it does not evaluate the formula actually at the endpoints (so that
23741 it is safe to integrate @samp{sin(x)/x} from zero, for example). Also,
23742 the Romberg method works especially well when the function being
23743 integrated is fairly smooth. If the function is not smooth, Calc will
23744 have to evaluate it at quite a few points before it can accurately
23745 determine the value of the integral.
23747 Integration is much faster when the current precision is small. It is
23748 best to set the precision to the smallest acceptable number of digits
23749 before you use @kbd{a I}. If Calc appears to be taking too long, press
23750 @kbd{C-g} to halt it and try a lower precision. If Calc still appears
23751 to need hundreds of evaluations, check to make sure your function is
23752 well-behaved in the specified interval.
23754 It is possible for the lower integration limit to be @samp{-inf} (minus
23755 infinity). Likewise, the upper limit may be plus infinity. Calc
23756 internally transforms the integral into an equivalent one with finite
23757 limits. However, integration to or across singularities is not supported:
23758 The integral of @samp{1/sqrt(x)} from 0 to 1 exists (it can be found
23759 by Calc's symbolic integrator, for example), but @kbd{a I} will fail
23760 because the integrand goes to infinity at one of the endpoints.
23762 @node Taylor Series, , Numerical Integration, Calculus
23763 @subsection Taylor Series
23767 @pindex calc-taylor
23769 The @kbd{a t} (@code{calc-taylor}) [@code{taylor}] command computes a
23770 power series expansion or Taylor series of a function. You specify the
23771 variable and the desired number of terms. You may give an expression of
23772 the form @samp{@var{var} = @var{a}} or @samp{@var{var} - @var{a}} instead
23773 of just a variable to produce a Taylor expansion about the point @var{a}.
23774 You may specify the number of terms with a numeric prefix argument;
23775 otherwise the command will prompt you for the number of terms. Note that
23776 many series expansions have coefficients of zero for some terms, so you
23777 may appear to get fewer terms than you asked for.
23779 If the @kbd{a i} command is unable to find a symbolic integral for a
23780 function, you can get an approximation by integrating the function's
23783 @node Solving Equations, Numerical Solutions, Calculus, Algebra
23784 @section Solving Equations
23788 @pindex calc-solve-for
23790 @cindex Equations, solving
23791 @cindex Solving equations
23792 The @kbd{a S} (@code{calc-solve-for}) [@code{solve}] command rearranges
23793 an equation to solve for a specific variable. An equation is an
23794 expression of the form @expr{L = R}. For example, the command @kbd{a S x}
23795 will rearrange @expr{y = 3x + 6} to the form, @expr{x = y/3 - 2}. If the
23796 input is not an equation, it is treated like an equation of the
23799 This command also works for inequalities, as in @expr{y < 3x + 6}.
23800 Some inequalities cannot be solved where the analogous equation could
23801 be; for example, solving
23802 @texline @math{a < b \, c}
23803 @infoline @expr{a < b c}
23804 for @expr{b} is impossible
23805 without knowing the sign of @expr{c}. In this case, @kbd{a S} will
23807 @texline @math{b \mathbin{\hbox{\code{!=}}} a/c}
23808 @infoline @expr{b != a/c}
23809 (using the not-equal-to operator) to signify that the direction of the
23810 inequality is now unknown. The inequality
23811 @texline @math{a \le b \, c}
23812 @infoline @expr{a <= b c}
23813 is not even partially solved. @xref{Declarations}, for a way to tell
23814 Calc that the signs of the variables in a formula are in fact known.
23816 Two useful commands for working with the result of @kbd{a S} are
23817 @kbd{a .} (@pxref{Logical Operations}), which converts @expr{x = y/3 - 2}
23818 to @expr{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates
23819 another formula with @expr{x} set equal to @expr{y/3 - 2}.
23822 * Multiple Solutions::
23823 * Solving Systems of Equations::
23824 * Decomposing Polynomials::
23827 @node Multiple Solutions, Solving Systems of Equations, Solving Equations, Solving Equations
23828 @subsection Multiple Solutions
23833 Some equations have more than one solution. The Hyperbolic flag
23834 (@code{H a S}) [@code{fsolve}] tells the solver to report the fully
23835 general family of solutions. It will invent variables @code{n1},
23836 @code{n2}, @dots{}, which represent independent arbitrary integers, and
23837 @code{s1}, @code{s2}, @dots{}, which represent independent arbitrary
23838 signs (either @mathit{+1} or @mathit{-1}). If you don't use the Hyperbolic
23839 flag, Calc will use zero in place of all arbitrary integers, and plus
23840 one in place of all arbitrary signs. Note that variables like @code{n1}
23841 and @code{s1} are not given any special interpretation in Calc except by
23842 the equation solver itself. As usual, you can use the @w{@kbd{s l}}
23843 (@code{calc-let}) command to obtain solutions for various actual values
23844 of these variables.
23846 For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to
23847 get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the
23848 equation are @samp{sqrt(y)} and @samp{-sqrt(y)}. Another way to
23849 think about it is that the square-root operation is really a
23850 two-valued function; since every Calc function must return a
23851 single result, @code{sqrt} chooses to return the positive result.
23852 Then @kbd{H a S} doctors this result using @code{s1} to indicate
23853 the full set of possible values of the mathematical square-root.
23855 There is a similar phenomenon going the other direction: Suppose
23856 we solve @samp{sqrt(y) = x} for @code{y}. Calc squares both sides
23857 to get @samp{y = x^2}. This is correct, except that it introduces
23858 some dubious solutions. Consider solving @samp{sqrt(y) = -3}:
23859 Calc will report @expr{y = 9} as a valid solution, which is true
23860 in the mathematical sense of square-root, but false (there is no
23861 solution) for the actual Calc positive-valued @code{sqrt}. This
23862 happens for both @kbd{a S} and @kbd{H a S}.
23864 @cindex @code{GenCount} variable
23874 If you store a positive integer in the Calc variable @code{GenCount},
23875 then Calc will generate formulas of the form @samp{as(@var{n})} for
23876 arbitrary signs, and @samp{an(@var{n})} for arbitrary integers,
23877 where @var{n} represents successive values taken by incrementing
23878 @code{GenCount} by one. While the normal arbitrary sign and
23879 integer symbols start over at @code{s1} and @code{n1} with each
23880 new Calc command, the @code{GenCount} approach will give each
23881 arbitrary value a name that is unique throughout the entire Calc
23882 session. Also, the arbitrary values are function calls instead
23883 of variables, which is advantageous in some cases. For example,
23884 you can make a rewrite rule that recognizes all arbitrary signs
23885 using a pattern like @samp{as(n)}. The @kbd{s l} command only works
23886 on variables, but you can use the @kbd{a b} (@code{calc-substitute})
23887 command to substitute actual values for function calls like @samp{as(3)}.
23889 The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient
23890 way to create or edit this variable. Press @kbd{C-c C-c} to finish.
23892 If you have not stored a value in @code{GenCount}, or if the value
23893 in that variable is not a positive integer, the regular
23894 @code{s1}/@code{n1} notation is used.
23900 With the Inverse flag, @kbd{I a S} [@code{finv}] treats the expression
23901 on top of the stack as a function of the specified variable and solves
23902 to find the inverse function, written in terms of the same variable.
23903 For example, @kbd{I a S x} inverts @expr{2x + 6} to @expr{x/2 - 3}.
23904 You can use both Inverse and Hyperbolic [@code{ffinv}] to obtain a
23905 fully general inverse, as described above.
23908 @pindex calc-poly-roots
23910 Some equations, specifically polynomials, have a known, finite number
23911 of solutions. The @kbd{a P} (@code{calc-poly-roots}) [@code{roots}]
23912 command uses @kbd{H a S} to solve an equation in general form, then, for
23913 all arbitrary-sign variables like @code{s1}, and all arbitrary-integer
23914 variables like @code{n1} for which @code{n1} only usefully varies over
23915 a finite range, it expands these variables out to all their possible
23916 values. The results are collected into a vector, which is returned.
23917 For example, @samp{roots(x^4 = 1, x)} returns the four solutions
23918 @samp{[1, -1, (0, 1), (0, -1)]}. Generally an @var{n}th degree
23919 polynomial will always have @var{n} roots on the complex plane.
23920 (If you have given a @code{real} declaration for the solution
23921 variable, then only the real-valued solutions, if any, will be
23922 reported; @pxref{Declarations}.)
23924 Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver
23925 symbolic solutions if the polynomial has symbolic coefficients. Also
23926 note that Calc's solver is not able to get exact symbolic solutions
23927 to all polynomials. Polynomials containing powers up to @expr{x^4}
23928 can always be solved exactly; polynomials of higher degree sometimes
23929 can be: @expr{x^6 + x^3 + 1} is converted to @expr{(x^3)^2 + (x^3) + 1},
23930 which can be solved for @expr{x^3} using the quadratic equation, and then
23931 for @expr{x} by taking cube roots. But in many cases, like
23932 @expr{x^6 + x + 1}, Calc does not know how to rewrite the polynomial
23933 into a form it can solve. The @kbd{a P} command can still deliver a
23934 list of numerical roots, however, provided that Symbolic mode (@kbd{m s})
23935 is not turned on. (If you work with Symbolic mode on, recall that the
23936 @kbd{N} (@code{calc-eval-num}) key is a handy way to reevaluate the
23937 formula on the stack with Symbolic mode temporarily off.) Naturally,
23938 @kbd{a P} can only provide numerical roots if the polynomial coefficients
23939 are all numbers (real or complex).
23941 @node Solving Systems of Equations, Decomposing Polynomials, Multiple Solutions, Solving Equations
23942 @subsection Solving Systems of Equations
23945 @cindex Systems of equations, symbolic
23946 You can also use the commands described above to solve systems of
23947 simultaneous equations. Just create a vector of equations, then
23948 specify a vector of variables for which to solve. (You can omit
23949 the surrounding brackets when entering the vector of variables
23952 For example, putting @samp{[x + y = a, x - y = b]} on the stack
23953 and typing @kbd{a S x,y @key{RET}} produces the vector of solutions
23954 @samp{[x = a - (a-b)/2, y = (a-b)/2]}. The result vector will
23955 have the same length as the variables vector, and the variables
23956 will be listed in the same order there. Note that the solutions
23957 are not always simplified as far as possible; the solution for
23958 @expr{x} here could be improved by an application of the @kbd{a n}
23961 Calc's algorithm works by trying to eliminate one variable at a
23962 time by solving one of the equations for that variable and then
23963 substituting into the other equations. Calc will try all the
23964 possibilities, but you can speed things up by noting that Calc
23965 first tries to eliminate the first variable with the first
23966 equation, then the second variable with the second equation,
23967 and so on. It also helps to put the simpler (e.g., more linear)
23968 equations toward the front of the list. Calc's algorithm will
23969 solve any system of linear equations, and also many kinds of
23976 Normally there will be as many variables as equations. If you
23977 give fewer variables than equations (an ``over-determined'' system
23978 of equations), Calc will find a partial solution. For example,
23979 typing @kbd{a S y @key{RET}} with the above system of equations
23980 would produce @samp{[y = a - x]}. There are now several ways to
23981 express this solution in terms of the original variables; Calc uses
23982 the first one that it finds. You can control the choice by adding
23983 variable specifiers of the form @samp{elim(@var{v})} to the
23984 variables list. This says that @var{v} should be eliminated from
23985 the equations; the variable will not appear at all in the solution.
23986 For example, typing @kbd{a S y,elim(x)} would yield
23987 @samp{[y = a - (b+a)/2]}.
23989 If the variables list contains only @code{elim} specifiers,
23990 Calc simply eliminates those variables from the equations
23991 and then returns the resulting set of equations. For example,
23992 @kbd{a S elim(x)} produces @samp{[a - 2 y = b]}. Every variable
23993 eliminated will reduce the number of equations in the system
23996 Again, @kbd{a S} gives you one solution to the system of
23997 equations. If there are several solutions, you can use @kbd{H a S}
23998 to get a general family of solutions, or, if there is a finite
23999 number of solutions, you can use @kbd{a P} to get a list. (In
24000 the latter case, the result will take the form of a matrix where
24001 the rows are different solutions and the columns correspond to the
24002 variables you requested.)
24004 Another way to deal with certain kinds of overdetermined systems of
24005 equations is the @kbd{a F} command, which does least-squares fitting
24006 to satisfy the equations. @xref{Curve Fitting}.
24008 @node Decomposing Polynomials, , Solving Systems of Equations, Solving Equations
24009 @subsection Decomposing Polynomials
24016 The @code{poly} function takes a polynomial and a variable as
24017 arguments, and returns a vector of polynomial coefficients (constant
24018 coefficient first). For example, @samp{poly(x^3 + 2 x, x)} returns
24019 @expr{[0, 2, 0, 1]}. If the input is not a polynomial in @expr{x},
24020 the call to @code{poly} is left in symbolic form. If the input does
24021 not involve the variable @expr{x}, the input is returned in a list
24022 of length one, representing a polynomial with only a constant
24023 coefficient. The call @samp{poly(x, x)} returns the vector @expr{[0, 1]}.
24024 The last element of the returned vector is guaranteed to be nonzero;
24025 note that @samp{poly(0, x)} returns the empty vector @expr{[]}.
24026 Note also that @expr{x} may actually be any formula; for example,
24027 @samp{poly(sin(x)^2 - sin(x) + 3, sin(x))} returns @expr{[3, -1, 1]}.
24029 @cindex Coefficients of polynomial
24030 @cindex Degree of polynomial
24031 To get the @expr{x^k} coefficient of polynomial @expr{p}, use
24032 @samp{poly(p, x)_(k+1)}. To get the degree of polynomial @expr{p},
24033 use @samp{vlen(poly(p, x)) - 1}. For example, @samp{poly((x+1)^4, x)}
24034 returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)}
24035 gives the @expr{x^2} coefficient of this polynomial, 6.
24041 One important feature of the solver is its ability to recognize
24042 formulas which are ``essentially'' polynomials. This ability is
24043 made available to the user through the @code{gpoly} function, which
24044 is used just like @code{poly}: @samp{gpoly(@var{expr}, @var{var})}.
24045 If @var{expr} is a polynomial in some term which includes @var{var}, then
24046 this function will return a vector @samp{[@var{x}, @var{c}, @var{a}]}
24047 where @var{x} is the term that depends on @var{var}, @var{c} is a
24048 vector of polynomial coefficients (like the one returned by @code{poly}),
24049 and @var{a} is a multiplier which is usually 1. Basically,
24050 @samp{@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} +
24051 @var{c}_3 @var{x}^2 + ...)}. The last element of @var{c} is
24052 guaranteed to be non-zero, and @var{c} will not equal @samp{[1]}
24053 (i.e., the trivial decomposition @var{expr} = @var{x} is not
24054 considered a polynomial). One side effect is that @samp{gpoly(x, x)}
24055 and @samp{gpoly(6, x)}, both of which might be expected to recognize
24056 their arguments as polynomials, will not because the decomposition
24057 is considered trivial.
24059 For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]},
24060 since the expanded form of this polynomial is @expr{4 - 4 x + x^2}.
24062 The term @var{x} may itself be a polynomial in @var{var}. This is
24063 done to reduce the size of the @var{c} vector. For example,
24064 @samp{gpoly(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]},
24065 since a quadratic polynomial in @expr{x^2} is easier to solve than
24066 a quartic polynomial in @expr{x}.
24068 A few more examples of the kinds of polynomials @code{gpoly} can
24072 sin(x) - 1 [sin(x), [-1, 1], 1]
24073 x + 1/x - 1 [x, [1, -1, 1], 1/x]
24074 x + 1/x [x^2, [1, 1], 1/x]
24075 x^3 + 2 x [x^2, [2, 1], x]
24076 x + x^2:3 + sqrt(x) [x^1:6, [1, 1, 0, 1], x^1:2]
24077 x^(2a) + 2 x^a + 5 [x^a, [5, 2, 1], 1]
24078 (exp(-x) + exp(x)) / 2 [e^(2 x), [0.5, 0.5], e^-x]
24081 The @code{poly} and @code{gpoly} functions accept a third integer argument
24082 which specifies the largest degree of polynomial that is acceptable.
24083 If this is @expr{n}, then only @var{c} vectors of length @expr{n+1}
24084 or less will be returned. Otherwise, the @code{poly} or @code{gpoly}
24085 call will remain in symbolic form. For example, the equation solver
24086 can handle quartics and smaller polynomials, so it calls
24087 @samp{gpoly(@var{expr}, @var{var}, 4)} to discover whether @var{expr}
24088 can be treated by its linear, quadratic, cubic, or quartic formulas.
24094 The @code{pdeg} function computes the degree of a polynomial;
24095 @samp{pdeg(p,x)} is the highest power of @code{x} that appears in
24096 @code{p}. This is the same as @samp{vlen(poly(p,x))-1}, but is
24097 much more efficient. If @code{p} is constant with respect to @code{x},
24098 then @samp{pdeg(p,x) = 0}. If @code{p} is not a polynomial in @code{x}
24099 (e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated.
24100 It is possible to omit the second argument @code{x}, in which case
24101 @samp{pdeg(p)} returns the highest total degree of any term of the
24102 polynomial, counting all variables that appear in @code{p}. Note
24103 that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c};
24104 the degree of the constant zero is considered to be @code{-inf}
24111 The @code{plead} function finds the leading term of a polynomial.
24112 Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))},
24113 though again more efficient. In particular, @samp{plead((2x+1)^10, x)}
24114 returns 1024 without expanding out the list of coefficients. The
24115 value of @code{plead(p,x)} will be zero only if @expr{p = 0}.
24121 The @code{pcont} function finds the @dfn{content} of a polynomial. This
24122 is the greatest common divisor of all the coefficients of the polynomial.
24123 With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)}
24124 to get a list of coefficients, then uses @code{pgcd} (the polynomial
24125 GCD function) to combine these into an answer. For example,
24126 @samp{pcont(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}. The content is
24127 basically the ``biggest'' polynomial that can be divided into @code{p}
24128 exactly. The sign of the content is the same as the sign of the leading
24131 With only one argument, @samp{pcont(p)} computes the numerical
24132 content of the polynomial, i.e., the @code{gcd} of the numerical
24133 coefficients of all the terms in the formula. Note that @code{gcd}
24134 is defined on rational numbers as well as integers; it computes
24135 the @code{gcd} of the numerators and the @code{lcm} of the
24136 denominators. Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3.
24137 Dividing the polynomial by this number will clear all the
24138 denominators, as well as dividing by any common content in the
24139 numerators. The numerical content of a polynomial is negative only
24140 if all the coefficients in the polynomial are negative.
24146 The @code{pprim} function finds the @dfn{primitive part} of a
24147 polynomial, which is simply the polynomial divided (using @code{pdiv}
24148 if necessary) by its content. If the input polynomial has rational
24149 coefficients, the result will have integer coefficients in simplest
24152 @node Numerical Solutions, Curve Fitting, Solving Equations, Algebra
24153 @section Numerical Solutions
24156 Not all equations can be solved symbolically. The commands in this
24157 section use numerical algorithms that can find a solution to a specific
24158 instance of an equation to any desired accuracy. Note that the
24159 numerical commands are slower than their algebraic cousins; it is a
24160 good idea to try @kbd{a S} before resorting to these commands.
24162 (@xref{Curve Fitting}, for some other, more specialized, operations
24163 on numerical data.)
24168 * Numerical Systems of Equations::
24171 @node Root Finding, Minimization, Numerical Solutions, Numerical Solutions
24172 @subsection Root Finding
24176 @pindex calc-find-root
24178 @cindex Newton's method
24179 @cindex Roots of equations
24180 @cindex Numerical root-finding
24181 The @kbd{a R} (@code{calc-find-root}) [@code{root}] command finds a
24182 numerical solution (or @dfn{root}) of an equation. (This command treats
24183 inequalities the same as equations. If the input is any other kind
24184 of formula, it is interpreted as an equation of the form @expr{X = 0}.)
24186 The @kbd{a R} command requires an initial guess on the top of the
24187 stack, and a formula in the second-to-top position. It prompts for a
24188 solution variable, which must appear in the formula. All other variables
24189 that appear in the formula must have assigned values, i.e., when
24190 a value is assigned to the solution variable and the formula is
24191 evaluated with @kbd{=}, it should evaluate to a number. Any assigned
24192 value for the solution variable itself is ignored and unaffected by
24195 When the command completes, the initial guess is replaced on the stack
24196 by a vector of two numbers: The value of the solution variable that
24197 solves the equation, and the difference between the lefthand and
24198 righthand sides of the equation at that value. Ordinarily, the second
24199 number will be zero or very nearly zero. (Note that Calc uses a
24200 slightly higher precision while finding the root, and thus the second
24201 number may be slightly different from the value you would compute from
24202 the equation yourself.)
24204 The @kbd{v h} (@code{calc-head}) command is a handy way to extract
24205 the first element of the result vector, discarding the error term.
24207 The initial guess can be a real number, in which case Calc searches
24208 for a real solution near that number, or a complex number, in which
24209 case Calc searches the whole complex plane near that number for a
24210 solution, or it can be an interval form which restricts the search
24211 to real numbers inside that interval.
24213 Calc tries to use @kbd{a d} to take the derivative of the equation.
24214 If this succeeds, it uses Newton's method. If the equation is not
24215 differentiable Calc uses a bisection method. (If Newton's method
24216 appears to be going astray, Calc switches over to bisection if it
24217 can, or otherwise gives up. In this case it may help to try again
24218 with a slightly different initial guess.) If the initial guess is a
24219 complex number, the function must be differentiable.
24221 If the formula (or the difference between the sides of an equation)
24222 is negative at one end of the interval you specify and positive at
24223 the other end, the root finder is guaranteed to find a root.
24224 Otherwise, Calc subdivides the interval into small parts looking for
24225 positive and negative values to bracket the root. When your guess is
24226 an interval, Calc will not look outside that interval for a root.
24230 The @kbd{H a R} [@code{wroot}] command is similar to @kbd{a R}, except
24231 that if the initial guess is an interval for which the function has
24232 the same sign at both ends, then rather than subdividing the interval
24233 Calc attempts to widen it to enclose a root. Use this mode if
24234 you are not sure if the function has a root in your interval.
24236 If the function is not differentiable, and you give a simple number
24237 instead of an interval as your initial guess, Calc uses this widening
24238 process even if you did not type the Hyperbolic flag. (If the function
24239 @emph{is} differentiable, Calc uses Newton's method which does not
24240 require a bounding interval in order to work.)
24242 If Calc leaves the @code{root} or @code{wroot} function in symbolic
24243 form on the stack, it will normally display an explanation for why
24244 no root was found. If you miss this explanation, press @kbd{w}
24245 (@code{calc-why}) to get it back.
24247 @node Minimization, Numerical Systems of Equations, Root Finding, Numerical Solutions
24248 @subsection Minimization
24255 @pindex calc-find-minimum
24256 @pindex calc-find-maximum
24259 @cindex Minimization, numerical
24260 The @kbd{a N} (@code{calc-find-minimum}) [@code{minimize}] command
24261 finds a minimum value for a formula. It is very similar in operation
24262 to @kbd{a R} (@code{calc-find-root}): You give the formula and an initial
24263 guess on the stack, and are prompted for the name of a variable. The guess
24264 may be either a number near the desired minimum, or an interval enclosing
24265 the desired minimum. The function returns a vector containing the
24266 value of the variable which minimizes the formula's value, along
24267 with the minimum value itself.
24269 Note that this command looks for a @emph{local} minimum. Many functions
24270 have more than one minimum; some, like
24271 @texline @math{x \sin x},
24272 @infoline @expr{x sin(x)},
24273 have infinitely many. In fact, there is no easy way to define the
24274 ``global'' minimum of
24275 @texline @math{x \sin x}
24276 @infoline @expr{x sin(x)}
24277 but Calc can still locate any particular local minimum
24278 for you. Calc basically goes downhill from the initial guess until it
24279 finds a point at which the function's value is greater both to the left
24280 and to the right. Calc does not use derivatives when minimizing a function.
24282 If your initial guess is an interval and it looks like the minimum
24283 occurs at one or the other endpoint of the interval, Calc will return
24284 that endpoint only if that endpoint is closed; thus, minimizing @expr{17 x}
24285 over @expr{[2..3]} will return @expr{[2, 38]}, but minimizing over
24286 @expr{(2..3]} would report no minimum found. In general, you should
24287 use closed intervals to find literally the minimum value in that
24288 range of @expr{x}, or open intervals to find the local minimum, if
24289 any, that happens to lie in that range.
24291 Most functions are smooth and flat near their minimum values. Because
24292 of this flatness, if the current precision is, say, 12 digits, the
24293 variable can only be determined meaningfully to about six digits. Thus
24294 you should set the precision to twice as many digits as you need in your
24305 The @kbd{H a N} [@code{wminimize}] command, analogously to @kbd{H a R},
24306 expands the guess interval to enclose a minimum rather than requiring
24307 that the minimum lie inside the interval you supply.
24309 The @kbd{a X} (@code{calc-find-maximum}) [@code{maximize}] and
24310 @kbd{H a X} [@code{wmaximize}] commands effectively minimize the
24311 negative of the formula you supply.
24313 The formula must evaluate to a real number at all points inside the
24314 interval (or near the initial guess if the guess is a number). If
24315 the initial guess is a complex number the variable will be minimized
24316 over the complex numbers; if it is real or an interval it will
24317 be minimized over the reals.
24319 @node Numerical Systems of Equations, , Minimization, Numerical Solutions
24320 @subsection Systems of Equations
24323 @cindex Systems of equations, numerical
24324 The @kbd{a R} command can also solve systems of equations. In this
24325 case, the equation should instead be a vector of equations, the
24326 guess should instead be a vector of numbers (intervals are not
24327 supported), and the variable should be a vector of variables. You
24328 can omit the brackets while entering the list of variables. Each
24329 equation must be differentiable by each variable for this mode to
24330 work. The result will be a vector of two vectors: The variable
24331 values that solved the system of equations, and the differences
24332 between the sides of the equations with those variable values.
24333 There must be the same number of equations as variables. Since
24334 only plain numbers are allowed as guesses, the Hyperbolic flag has
24335 no effect when solving a system of equations.
24337 It is also possible to minimize over many variables with @kbd{a N}
24338 (or maximize with @kbd{a X}). Once again the variable name should
24339 be replaced by a vector of variables, and the initial guess should
24340 be an equal-sized vector of initial guesses. But, unlike the case of
24341 multidimensional @kbd{a R}, the formula being minimized should
24342 still be a single formula, @emph{not} a vector. Beware that
24343 multidimensional minimization is currently @emph{very} slow.
24345 @node Curve Fitting, Summations, Numerical Solutions, Algebra
24346 @section Curve Fitting
24349 The @kbd{a F} command fits a set of data to a @dfn{model formula},
24350 such as @expr{y = m x + b} where @expr{m} and @expr{b} are parameters
24351 to be determined. For a typical set of measured data there will be
24352 no single @expr{m} and @expr{b} that exactly fit the data; in this
24353 case, Calc chooses values of the parameters that provide the closest
24354 possible fit. The model formula can be entered in various ways after
24355 the key sequence @kbd{a F} is pressed.
24357 If the letter @kbd{P} is pressed after @kbd{a F} but before the model
24358 description is entered, the data as well as the model formula will be
24359 plotted after the formula is determined. This will be indicated by a
24360 ``P'' in the minibuffer after the help message.
24364 * Polynomial and Multilinear Fits::
24365 * Error Estimates for Fits::
24366 * Standard Nonlinear Models::
24367 * Curve Fitting Details::
24371 @node Linear Fits, Polynomial and Multilinear Fits, Curve Fitting, Curve Fitting
24372 @subsection Linear Fits
24376 @pindex calc-curve-fit
24378 @cindex Linear regression
24379 @cindex Least-squares fits
24380 The @kbd{a F} (@code{calc-curve-fit}) [@code{fit}] command attempts
24381 to fit a set of data (@expr{x} and @expr{y} vectors of numbers) to a
24382 straight line, polynomial, or other function of @expr{x}. For the
24383 moment we will consider only the case of fitting to a line, and we
24384 will ignore the issue of whether or not the model was in fact a good
24387 In a standard linear least-squares fit, we have a set of @expr{(x,y)}
24388 data points that we wish to fit to the model @expr{y = m x + b}
24389 by adjusting the parameters @expr{m} and @expr{b} to make the @expr{y}
24390 values calculated from the formula be as close as possible to the actual
24391 @expr{y} values in the data set. (In a polynomial fit, the model is
24392 instead, say, @expr{y = a x^3 + b x^2 + c x + d}. In a multilinear fit,
24393 we have data points of the form @expr{(x_1,x_2,x_3,y)} and our model is
24394 @expr{y = a x_1 + b x_2 + c x_3 + d}. These will be discussed later.)
24396 In the model formula, variables like @expr{x} and @expr{x_2} are called
24397 the @dfn{independent variables}, and @expr{y} is the @dfn{dependent
24398 variable}. Variables like @expr{m}, @expr{a}, and @expr{b} are called
24399 the @dfn{parameters} of the model.
24401 The @kbd{a F} command takes the data set to be fitted from the stack.
24402 By default, it expects the data in the form of a matrix. For example,
24403 for a linear or polynomial fit, this would be a
24404 @texline @math{2\times N}
24406 matrix where the first row is a list of @expr{x} values and the second
24407 row has the corresponding @expr{y} values. For the multilinear fit
24408 shown above, the matrix would have four rows (@expr{x_1}, @expr{x_2},
24409 @expr{x_3}, and @expr{y}, respectively).
24411 If you happen to have an
24412 @texline @math{N\times2}
24414 matrix instead of a
24415 @texline @math{2\times N}
24417 matrix, just press @kbd{v t} first to transpose the matrix.
24419 After you type @kbd{a F}, Calc prompts you to select a model. For a
24420 linear fit, press the digit @kbd{1}.
24422 Calc then prompts for you to name the variables. By default it chooses
24423 high letters like @expr{x} and @expr{y} for independent variables and
24424 low letters like @expr{a} and @expr{b} for parameters. (The dependent
24425 variable doesn't need a name.) The two kinds of variables are separated
24426 by a semicolon. Since you generally care more about the names of the
24427 independent variables than of the parameters, Calc also allows you to
24428 name only those and let the parameters use default names.
24430 For example, suppose the data matrix
24435 [ [ 1, 2, 3, 4, 5 ]
24436 [ 5, 7, 9, 11, 13 ] ]
24442 $$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr
24443 5 & 7 & 9 & 11 & 13 }
24449 is on the stack and we wish to do a simple linear fit. Type
24450 @kbd{a F}, then @kbd{1} for the model, then @key{RET} to use
24451 the default names. The result will be the formula @expr{3. + 2. x}
24452 on the stack. Calc has created the model expression @kbd{a + b x},
24453 then found the optimal values of @expr{a} and @expr{b} to fit the
24454 data. (In this case, it was able to find an exact fit.) Calc then
24455 substituted those values for @expr{a} and @expr{b} in the model
24458 The @kbd{a F} command puts two entries in the trail. One is, as
24459 always, a copy of the result that went to the stack; the other is
24460 a vector of the actual parameter values, written as equations:
24461 @expr{[a = 3, b = 2]}, in case you'd rather read them in a list
24462 than pick them out of the formula. (You can type @kbd{t y}
24463 to move this vector to the stack; see @ref{Trail Commands}.
24465 Specifying a different independent variable name will affect the
24466 resulting formula: @kbd{a F 1 k @key{RET}} produces @kbd{3 + 2 k}.
24467 Changing the parameter names (say, @kbd{a F 1 k;b,m @key{RET}}) will affect
24468 the equations that go into the trail.
24474 To see what happens when the fit is not exact, we could change
24475 the number 13 in the data matrix to 14 and try the fit again.
24482 Evaluating this formula, say with @kbd{v x 5 @key{RET} @key{TAB} V M $ @key{RET}}, shows
24483 a reasonably close match to the y-values in the data.
24486 [4.8, 7., 9.2, 11.4, 13.6]
24489 Since there is no line which passes through all the @var{n} data points,
24490 Calc has chosen a line that best approximates the data points using
24491 the method of least squares. The idea is to define the @dfn{chi-square}
24496 chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
24501 $$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
24506 which is clearly zero if @expr{a + b x} exactly fits all data points,
24507 and increases as various @expr{a + b x_i} values fail to match the
24508 corresponding @expr{y_i} values. There are several reasons why the
24509 summand is squared, one of them being to ensure that
24510 @texline @math{\chi^2 \ge 0}.
24511 @infoline @expr{chi^2 >= 0}.
24512 Least-squares fitting simply chooses the values of @expr{a} and @expr{b}
24513 for which the error
24514 @texline @math{\chi^2}
24515 @infoline @expr{chi^2}
24516 is as small as possible.
24518 Other kinds of models do the same thing but with a different model
24519 formula in place of @expr{a + b x_i}.
24525 A numeric prefix argument causes the @kbd{a F} command to take the
24526 data in some other form than one big matrix. A positive argument @var{n}
24527 will take @var{N} items from the stack, corresponding to the @var{n} rows
24528 of a data matrix. In the linear case, @var{n} must be 2 since there
24529 is always one independent variable and one dependent variable.
24531 A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two
24532 items from the stack, an @var{n}-row matrix of @expr{x} values, and a
24533 vector of @expr{y} values. If there is only one independent variable,
24534 the @expr{x} values can be either a one-row matrix or a plain vector,
24535 in which case the @kbd{C-u} prefix is the same as a @w{@kbd{C-u 2}} prefix.
24537 @node Polynomial and Multilinear Fits, Error Estimates for Fits, Linear Fits, Curve Fitting
24538 @subsection Polynomial and Multilinear Fits
24541 To fit the data to higher-order polynomials, just type one of the
24542 digits @kbd{2} through @kbd{9} when prompted for a model. For example,
24543 we could fit the original data matrix from the previous section
24544 (with 13, not 14) to a parabola instead of a line by typing
24545 @kbd{a F 2 @key{RET}}.
24548 2.00000000001 x - 1.5e-12 x^2 + 2.99999999999
24551 Note that since the constant and linear terms are enough to fit the
24552 data exactly, it's no surprise that Calc chose a tiny contribution
24553 for @expr{x^2}. (The fact that it's not exactly zero is due only
24554 to roundoff error. Since our data are exact integers, we could get
24555 an exact answer by typing @kbd{m f} first to get Fraction mode.
24556 Then the @expr{x^2} term would vanish altogether. Usually, though,
24557 the data being fitted will be approximate floats so Fraction mode
24560 Doing the @kbd{a F 2} fit on the data set with 14 instead of 13
24561 gives a much larger @expr{x^2} contribution, as Calc bends the
24562 line slightly to improve the fit.
24565 0.142857142855 x^2 + 1.34285714287 x + 3.59999999998
24568 An important result from the theory of polynomial fitting is that it
24569 is always possible to fit @var{n} data points exactly using a polynomial
24570 of degree @mathit{@var{n}-1}, sometimes called an @dfn{interpolating polynomial}.
24571 Using the modified (14) data matrix, a model number of 4 gives
24572 a polynomial that exactly matches all five data points:
24575 0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.
24578 The actual coefficients we get with a precision of 12, like
24579 @expr{0.0416666663588}, clearly suffer from loss of precision.
24580 It is a good idea to increase the working precision to several
24581 digits beyond what you need when you do a fitting operation.
24582 Or, if your data are exact, use Fraction mode to get exact
24585 You can type @kbd{i} instead of a digit at the model prompt to fit
24586 the data exactly to a polynomial. This just counts the number of
24587 columns of the data matrix to choose the degree of the polynomial
24590 Fitting data ``exactly'' to high-degree polynomials is not always
24591 a good idea, though. High-degree polynomials have a tendency to
24592 wiggle uncontrollably in between the fitting data points. Also,
24593 if the exact-fit polynomial is going to be used to interpolate or
24594 extrapolate the data, it is numerically better to use the @kbd{a p}
24595 command described below. @xref{Interpolation}.
24601 Another generalization of the linear model is to assume the
24602 @expr{y} values are a sum of linear contributions from several
24603 @expr{x} values. This is a @dfn{multilinear} fit, and it is also
24604 selected by the @kbd{1} digit key. (Calc decides whether the fit
24605 is linear or multilinear by counting the rows in the data matrix.)
24607 Given the data matrix,
24611 [ [ 1, 2, 3, 4, 5 ]
24613 [ 14.5, 15, 18.5, 22.5, 24 ] ]
24618 the command @kbd{a F 1 @key{RET}} will call the first row @expr{x} and the
24619 second row @expr{y}, and will fit the values in the third row to the
24620 model @expr{a + b x + c y}.
24626 Calc can do multilinear fits with any number of independent variables
24627 (i.e., with any number of data rows).
24633 Yet another variation is @dfn{homogeneous} linear models, in which
24634 the constant term is known to be zero. In the linear case, this
24635 means the model formula is simply @expr{a x}; in the multilinear
24636 case, the model might be @expr{a x + b y + c z}; and in the polynomial
24637 case, the model could be @expr{a x + b x^2 + c x^3}. You can get
24638 a homogeneous linear or multilinear model by pressing the letter
24639 @kbd{h} followed by a regular model key, like @kbd{1} or @kbd{2}.
24640 This will be indicated by an ``h'' in the minibuffer after the help
24643 It is certainly possible to have other constrained linear models,
24644 like @expr{2.3 + a x} or @expr{a - 4 x}. While there is no single
24645 key to select models like these, a later section shows how to enter
24646 any desired model by hand. In the first case, for example, you
24647 would enter @kbd{a F ' 2.3 + a x}.
24649 Another class of models that will work but must be entered by hand
24650 are multinomial fits, e.g., @expr{a + b x + c y + d x^2 + e y^2 + f x y}.
24652 @node Error Estimates for Fits, Standard Nonlinear Models, Polynomial and Multilinear Fits, Curve Fitting
24653 @subsection Error Estimates for Fits
24658 With the Hyperbolic flag, @kbd{H a F} [@code{efit}] performs the same
24659 fitting operation as @kbd{a F}, but reports the coefficients as error
24660 forms instead of plain numbers. Fitting our two data matrices (first
24661 with 13, then with 14) to a line with @kbd{H a F} gives the results,
24665 2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x
24668 In the first case the estimated errors are zero because the linear
24669 fit is perfect. In the second case, the errors are nonzero but
24670 moderately small, because the data are still very close to linear.
24672 It is also possible for the @emph{input} to a fitting operation to
24673 contain error forms. The data values must either all include errors
24674 or all be plain numbers. Error forms can go anywhere but generally
24675 go on the numbers in the last row of the data matrix. If the last
24676 row contains error forms
24677 @texline `@var{y_i}@w{ @tfn{+/-} }@math{\sigma_i}',
24678 @infoline `@var{y_i}@w{ @tfn{+/-} }@var{sigma_i}',
24680 @texline @math{\chi^2}
24681 @infoline @expr{chi^2}
24686 chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
24691 $$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
24696 so that data points with larger error estimates contribute less to
24697 the fitting operation.
24699 If there are error forms on other rows of the data matrix, all the
24700 errors for a given data point are combined; the square root of the
24701 sum of the squares of the errors forms the
24702 @texline @math{\sigma_i}
24703 @infoline @expr{sigma_i}
24704 used for the data point.
24706 Both @kbd{a F} and @kbd{H a F} can accept error forms in the input
24707 matrix, although if you are concerned about error analysis you will
24708 probably use @kbd{H a F} so that the output also contains error
24711 If the input contains error forms but all the
24712 @texline @math{\sigma_i}
24713 @infoline @expr{sigma_i}
24714 values are the same, it is easy to see that the resulting fitted model
24715 will be the same as if the input did not have error forms at all
24716 @texline (@math{\chi^2}
24717 @infoline (@expr{chi^2}
24718 is simply scaled uniformly by
24719 @texline @math{1 / \sigma^2},
24720 @infoline @expr{1 / sigma^2},
24721 which doesn't affect where it has a minimum). But there @emph{will} be
24722 a difference in the estimated errors of the coefficients reported by
24725 Consult any text on statistical modeling of data for a discussion
24726 of where these error estimates come from and how they should be
24735 With the Inverse flag, @kbd{I a F} [@code{xfit}] produces even more
24736 information. The result is a vector of six items:
24740 The model formula with error forms for its coefficients or
24741 parameters. This is the result that @kbd{H a F} would have
24745 A vector of ``raw'' parameter values for the model. These are the
24746 polynomial coefficients or other parameters as plain numbers, in the
24747 same order as the parameters appeared in the final prompt of the
24748 @kbd{I a F} command. For polynomials of degree @expr{d}, this vector
24749 will have length @expr{M = d+1} with the constant term first.
24752 The covariance matrix @expr{C} computed from the fit. This is
24753 an @var{m}x@var{m} symmetric matrix; the diagonal elements
24754 @texline @math{C_{jj}}
24755 @infoline @expr{C_j_j}
24757 @texline @math{\sigma_j^2}
24758 @infoline @expr{sigma_j^2}
24759 of the parameters. The other elements are covariances
24760 @texline @math{\sigma_{ij}^2}
24761 @infoline @expr{sigma_i_j^2}
24762 that describe the correlation between pairs of parameters. (A related
24763 set of numbers, the @dfn{linear correlation coefficients}
24764 @texline @math{r_{ij}},
24765 @infoline @expr{r_i_j},
24767 @texline @math{\sigma_{ij}^2 / \sigma_i \, \sigma_j}.)
24768 @infoline @expr{sigma_i_j^2 / sigma_i sigma_j}.)
24771 A vector of @expr{M} ``parameter filter'' functions whose
24772 meanings are described below. If no filters are necessary this
24773 will instead be an empty vector; this is always the case for the
24774 polynomial and multilinear fits described so far.
24778 @texline @math{\chi^2}
24779 @infoline @expr{chi^2}
24780 for the fit, calculated by the formulas shown above. This gives a
24781 measure of the quality of the fit; statisticians consider
24782 @texline @math{\chi^2 \approx N - M}
24783 @infoline @expr{chi^2 = N - M}
24784 to indicate a moderately good fit (where again @expr{N} is the number of
24785 data points and @expr{M} is the number of parameters).
24788 A measure of goodness of fit expressed as a probability @expr{Q}.
24789 This is computed from the @code{utpc} probability distribution
24791 @texline @math{\chi^2}
24792 @infoline @expr{chi^2}
24793 with @expr{N - M} degrees of freedom. A
24794 value of 0.5 implies a good fit; some texts recommend that often
24795 @expr{Q = 0.1} or even 0.001 can signify an acceptable fit. In
24797 @texline @math{\chi^2}
24798 @infoline @expr{chi^2}
24799 statistics assume the errors in your inputs
24800 follow a normal (Gaussian) distribution; if they don't, you may
24801 have to accept smaller values of @expr{Q}.
24803 The @expr{Q} value is computed only if the input included error
24804 estimates. Otherwise, Calc will report the symbol @code{nan}
24805 for @expr{Q}. The reason is that in this case the
24806 @texline @math{\chi^2}
24807 @infoline @expr{chi^2}
24808 value has effectively been used to estimate the original errors
24809 in the input, and thus there is no redundant information left
24810 over to use for a confidence test.
24813 @node Standard Nonlinear Models, Curve Fitting Details, Error Estimates for Fits, Curve Fitting
24814 @subsection Standard Nonlinear Models
24817 The @kbd{a F} command also accepts other kinds of models besides
24818 lines and polynomials. Some common models have quick single-key
24819 abbreviations; others must be entered by hand as algebraic formulas.
24821 Here is a complete list of the standard models recognized by @kbd{a F}:
24825 Linear or multilinear. @mathit{a + b x + c y + d z}.
24827 Polynomials. @mathit{a + b x + c x^2 + d x^3}.
24829 Exponential. @mathit{a} @tfn{exp}@mathit{(b x)} @tfn{exp}@mathit{(c y)}.
24831 Base-10 exponential. @mathit{a} @tfn{10^}@mathit{(b x)} @tfn{10^}@mathit{(c y)}.
24833 Exponential (alternate notation). @tfn{exp}@mathit{(a + b x + c y)}.
24835 Base-10 exponential (alternate). @tfn{10^}@mathit{(a + b x + c y)}.
24837 Logarithmic. @mathit{a + b} @tfn{ln}@mathit{(x) + c} @tfn{ln}@mathit{(y)}.
24839 Base-10 logarithmic. @mathit{a + b} @tfn{log10}@mathit{(x) + c} @tfn{log10}@mathit{(y)}.
24841 General exponential. @mathit{a b^x c^y}.
24843 Power law. @mathit{a x^b y^c}.
24845 Quadratic. @mathit{a + b (x-c)^2 + d (x-e)^2}.
24848 @texline @math{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)}.
24849 @infoline @mathit{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}.
24851 Logistic @emph{s} curve.
24852 @texline @math{a/(1+e^{b(x-c)})}.
24853 @infoline @mathit{a/(1 + exp(b (x - c)))}.
24855 Logistic bell curve.
24856 @texline @math{ae^{b(x-c)}/(1+e^{b(x-c)})^2}.
24857 @infoline @mathit{a exp(b (x - c))/(1 + exp(b (x - c)))^2}.
24859 Hubbert linearization.
24860 @texline @math{{y \over x} = a(1-x/b)}.
24861 @infoline @mathit{(y/x) = a (1 - x/b)}.
24864 All of these models are used in the usual way; just press the appropriate
24865 letter at the model prompt, and choose variable names if you wish. The
24866 result will be a formula as shown in the above table, with the best-fit
24867 values of the parameters substituted. (You may find it easier to read
24868 the parameter values from the vector that is placed in the trail.)
24870 All models except Gaussian, logistics, Hubbert and polynomials can
24871 generalize as shown to any number of independent variables. Also, all
24872 the built-in models except for the logistic and Hubbert curves have an
24873 additive or multiplicative parameter shown as @expr{a} in the above table
24874 which can be replaced by zero or one, as appropriate, by typing @kbd{h}
24875 before the model key.
24877 Note that many of these models are essentially equivalent, but express
24878 the parameters slightly differently. For example, @expr{a b^x} and
24879 the other two exponential models are all algebraic rearrangements of
24880 each other. Also, the ``quadratic'' model is just a degree-2 polynomial
24881 with the parameters expressed differently. Use whichever form best
24882 matches the problem.
24884 The HP-28/48 calculators support four different models for curve
24885 fitting, called @code{LIN}, @code{LOG}, @code{EXP}, and @code{PWR}.
24886 These correspond to Calc models @samp{a + b x}, @samp{a + b ln(x)},
24887 @samp{a exp(b x)}, and @samp{a x^b}, respectively. In each case,
24888 @expr{a} is what the HP-48 identifies as the ``intercept,'' and
24889 @expr{b} is what it calls the ``slope.''
24895 If the model you want doesn't appear on this list, press @kbd{'}
24896 (the apostrophe key) at the model prompt to enter any algebraic
24897 formula, such as @kbd{m x - b}, as the model. (Not all models
24898 will work, though---see the next section for details.)
24900 The model can also be an equation like @expr{y = m x + b}.
24901 In this case, Calc thinks of all the rows of the data matrix on
24902 equal terms; this model effectively has two parameters
24903 (@expr{m} and @expr{b}) and two independent variables (@expr{x}
24904 and @expr{y}), with no ``dependent'' variables. Model equations
24905 do not need to take this @expr{y =} form. For example, the
24906 implicit line equation @expr{a x + b y = 1} works fine as a
24909 When you enter a model, Calc makes an alphabetical list of all
24910 the variables that appear in the model. These are used for the
24911 default parameters, independent variables, and dependent variable
24912 (in that order). If you enter a plain formula (not an equation),
24913 Calc assumes the dependent variable does not appear in the formula
24914 and thus does not need a name.
24916 For example, if the model formula has the variables @expr{a,mu,sigma,t,x},
24917 and the data matrix has three rows (meaning two independent variables),
24918 Calc will use @expr{a,mu,sigma} as the default parameters, and the
24919 data rows will be named @expr{t} and @expr{x}, respectively. If you
24920 enter an equation instead of a plain formula, Calc will use @expr{a,mu}
24921 as the parameters, and @expr{sigma,t,x} as the three independent
24924 You can, of course, override these choices by entering something
24925 different at the prompt. If you leave some variables out of the list,
24926 those variables must have stored values and those stored values will
24927 be used as constants in the model. (Stored values for the parameters
24928 and independent variables are ignored by the @kbd{a F} command.)
24929 If you list only independent variables, all the remaining variables
24930 in the model formula will become parameters.
24932 If there are @kbd{$} signs in the model you type, they will stand
24933 for parameters and all other variables (in alphabetical order)
24934 will be independent. Use @kbd{$} for one parameter, @kbd{$$} for
24935 another, and so on. Thus @kbd{$ x + $$} is another way to describe
24938 If you type a @kbd{$} instead of @kbd{'} at the model prompt itself,
24939 Calc will take the model formula from the stack. (The data must then
24940 appear at the second stack level.) The same conventions are used to
24941 choose which variables in the formula are independent by default and
24942 which are parameters.
24944 Models taken from the stack can also be expressed as vectors of
24945 two or three elements, @expr{[@var{model}, @var{vars}]} or
24946 @expr{[@var{model}, @var{vars}, @var{params}]}. Each of @var{vars}
24947 and @var{params} may be either a variable or a vector of variables.
24948 (If @var{params} is omitted, all variables in @var{model} except
24949 those listed as @var{vars} are parameters.)
24951 When you enter a model manually with @kbd{'}, Calc puts a 3-vector
24952 describing the model in the trail so you can get it back if you wish.
24960 Finally, you can store a model in one of the Calc variables
24961 @code{Model1} or @code{Model2}, then use this model by typing
24962 @kbd{a F u} or @kbd{a F U} (respectively). The value stored in
24963 the variable can be any of the formats that @kbd{a F $} would
24964 accept for a model on the stack.
24970 Calc uses the principal values of inverse functions like @code{ln}
24971 and @code{arcsin} when doing fits. For example, when you enter
24972 the model @samp{y = sin(a t + b)} Calc actually uses the easier
24973 form @samp{arcsin(y) = a t + b}. The @code{arcsin} function always
24974 returns results in the range from @mathit{-90} to 90 degrees (or the
24975 equivalent range in radians). Suppose you had data that you
24976 believed to represent roughly three oscillations of a sine wave,
24977 so that the argument of the sine might go from zero to
24978 @texline @math{3\times360}
24979 @infoline @mathit{3*360}
24981 The above model would appear to be a good way to determine the
24982 true frequency and phase of the sine wave, but in practice it
24983 would fail utterly. The righthand side of the actual model
24984 @samp{arcsin(y) = a t + b} will grow smoothly with @expr{t}, but
24985 the lefthand side will bounce back and forth between @mathit{-90} and 90.
24986 No values of @expr{a} and @expr{b} can make the two sides match,
24987 even approximately.
24989 There is no good solution to this problem at present. You could
24990 restrict your data to small enough ranges so that the above problem
24991 doesn't occur (i.e., not straddling any peaks in the sine wave).
24992 Or, in this case, you could use a totally different method such as
24993 Fourier analysis, which is beyond the scope of the @kbd{a F} command.
24994 (Unfortunately, Calc does not currently have any facilities for
24995 taking Fourier and related transforms.)
24997 @node Curve Fitting Details, Interpolation, Standard Nonlinear Models, Curve Fitting
24998 @subsection Curve Fitting Details
25001 Calc's internal least-squares fitter can only handle multilinear
25002 models. More precisely, it can handle any model of the form
25003 @expr{a f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @expr{a,b,c}
25004 are the parameters and @expr{x,y,z} are the independent variables
25005 (of course there can be any number of each, not just three).
25007 In a simple multilinear or polynomial fit, it is easy to see how
25008 to convert the model into this form. For example, if the model
25009 is @expr{a + b x + c x^2}, then @expr{f(x) = 1}, @expr{g(x) = x},
25010 and @expr{h(x) = x^2} are suitable functions.
25012 For most other models, Calc uses a variety of algebraic manipulations
25013 to try to put the problem into the form
25016 Y(x,y,z) = A(a,b,c) F(x,y,z) + B(a,b,c) G(x,y,z) + C(a,b,c) H(x,y,z)
25020 where @expr{Y,A,B,C,F,G,H} are arbitrary functions. It computes
25021 @expr{Y}, @expr{F}, @expr{G}, and @expr{H} for all the data points,
25022 does a standard linear fit to find the values of @expr{A}, @expr{B},
25023 and @expr{C}, then uses the equation solver to solve for @expr{a,b,c}
25024 in terms of @expr{A,B,C}.
25026 A remarkable number of models can be cast into this general form.
25027 We'll look at two examples here to see how it works. The power-law
25028 model @expr{y = a x^b} with two independent variables and two parameters
25029 can be rewritten as follows:
25034 y = exp(ln(a) + b ln(x))
25035 ln(y) = ln(a) + b ln(x)
25039 which matches the desired form with
25040 @texline @math{Y = \ln(y)},
25041 @infoline @expr{Y = ln(y)},
25042 @texline @math{A = \ln(a)},
25043 @infoline @expr{A = ln(a)},
25044 @expr{F = 1}, @expr{B = b}, and
25045 @texline @math{G = \ln(x)}.
25046 @infoline @expr{G = ln(x)}.
25047 Calc thus computes the logarithms of your @expr{y} and @expr{x} values,
25048 does a linear fit for @expr{A} and @expr{B}, then solves to get
25049 @texline @math{a = \exp(A)}
25050 @infoline @expr{a = exp(A)}
25053 Another interesting example is the ``quadratic'' model, which can
25054 be handled by expanding according to the distributive law.
25057 y = a + b*(x - c)^2
25058 y = a + b c^2 - 2 b c x + b x^2
25062 which matches with @expr{Y = y}, @expr{A = a + b c^2}, @expr{F = 1},
25063 @expr{B = -2 b c}, @expr{G = x} (the @mathit{-2} factor could just as easily
25064 have been put into @expr{G} instead of @expr{B}), @expr{C = b}, and
25067 The Gaussian model looks quite complicated, but a closer examination
25068 shows that it's actually similar to the quadratic model but with an
25069 exponential that can be brought to the top and moved into @expr{Y}.
25071 The logistic models cannot be put into general linear form. For these
25072 models, and the Hubbert linearization, Calc computes a rough
25073 approximation for the parameters, then uses the Levenberg-Marquardt
25074 iterative method to refine the approximations.
25076 Another model that cannot be put into general linear
25077 form is a Gaussian with a constant background added on, i.e.,
25078 @expr{d} + the regular Gaussian formula. If you have a model like
25079 this, your best bet is to replace enough of your parameters with
25080 constants to make the model linearizable, then adjust the constants
25081 manually by doing a series of fits. You can compare the fits by
25082 graphing them, by examining the goodness-of-fit measures returned by
25083 @kbd{I a F}, or by some other method suitable to your application.
25084 Note that some models can be linearized in several ways. The
25085 Gaussian-plus-@var{d} model can be linearized by setting @expr{d}
25086 (the background) to a constant, or by setting @expr{b} (the standard
25087 deviation) and @expr{c} (the mean) to constants.
25089 To fit a model with constants substituted for some parameters, just
25090 store suitable values in those parameter variables, then omit them
25091 from the list of parameters when you answer the variables prompt.
25097 A last desperate step would be to use the general-purpose
25098 @code{minimize} function rather than @code{fit}. After all, both
25099 functions solve the problem of minimizing an expression (the
25100 @texline @math{\chi^2}
25101 @infoline @expr{chi^2}
25102 sum) by adjusting certain parameters in the expression. The @kbd{a F}
25103 command is able to use a vastly more efficient algorithm due to its
25104 special knowledge about linear chi-square sums, but the @kbd{a N}
25105 command can do the same thing by brute force.
25107 A compromise would be to pick out a few parameters without which the
25108 fit is linearizable, and use @code{minimize} on a call to @code{fit}
25109 which efficiently takes care of the rest of the parameters. The thing
25110 to be minimized would be the value of
25111 @texline @math{\chi^2}
25112 @infoline @expr{chi^2}
25113 returned as the fifth result of the @code{xfit} function:
25116 minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)
25120 where @code{gaus} represents the Gaussian model with background,
25121 @code{data} represents the data matrix, and @code{guess} represents
25122 the initial guess for @expr{d} that @code{minimize} requires.
25123 This operation will only be, shall we say, extraordinarily slow
25124 rather than astronomically slow (as would be the case if @code{minimize}
25125 were used by itself to solve the problem).
25131 The @kbd{I a F} [@code{xfit}] command is somewhat trickier when
25132 nonlinear models are used. The second item in the result is the
25133 vector of ``raw'' parameters @expr{A}, @expr{B}, @expr{C}. The
25134 covariance matrix is written in terms of those raw parameters.
25135 The fifth item is a vector of @dfn{filter} expressions. This
25136 is the empty vector @samp{[]} if the raw parameters were the same
25137 as the requested parameters, i.e., if @expr{A = a}, @expr{B = b},
25138 and so on (which is always true if the model is already linear
25139 in the parameters as written, e.g., for polynomial fits). If the
25140 parameters had to be rearranged, the fifth item is instead a vector
25141 of one formula per parameter in the original model. The raw
25142 parameters are expressed in these ``filter'' formulas as
25143 @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)} for @expr{B},
25146 When Calc needs to modify the model to return the result, it replaces
25147 @samp{fitdummy(1)} in all the filters with the first item in the raw
25148 parameters list, and so on for the other raw parameters, then
25149 evaluates the resulting filter formulas to get the actual parameter
25150 values to be substituted into the original model. In the case of
25151 @kbd{H a F} and @kbd{I a F} where the parameters must be error forms,
25152 Calc uses the square roots of the diagonal entries of the covariance
25153 matrix as error values for the raw parameters, then lets Calc's
25154 standard error-form arithmetic take it from there.
25156 If you use @kbd{I a F} with a nonlinear model, be sure to remember
25157 that the covariance matrix is in terms of the raw parameters,
25158 @emph{not} the actual requested parameters. It's up to you to
25159 figure out how to interpret the covariances in the presence of
25160 nontrivial filter functions.
25162 Things are also complicated when the input contains error forms.
25163 Suppose there are three independent and dependent variables, @expr{x},
25164 @expr{y}, and @expr{z}, one or more of which are error forms in the
25165 data. Calc combines all the error values by taking the square root
25166 of the sum of the squares of the errors. It then changes @expr{x}
25167 and @expr{y} to be plain numbers, and makes @expr{z} into an error
25168 form with this combined error. The @expr{Y(x,y,z)} part of the
25169 linearized model is evaluated, and the result should be an error
25170 form. The error part of that result is used for
25171 @texline @math{\sigma_i}
25172 @infoline @expr{sigma_i}
25173 for the data point. If for some reason @expr{Y(x,y,z)} does not return
25174 an error form, the combined error from @expr{z} is used directly for
25175 @texline @math{\sigma_i}.
25176 @infoline @expr{sigma_i}.
25177 Finally, @expr{z} is also stripped of its error
25178 for use in computing @expr{F(x,y,z)}, @expr{G(x,y,z)} and so on;
25179 the righthand side of the linearized model is computed in regular
25180 arithmetic with no error forms.
25182 (While these rules may seem complicated, they are designed to do
25183 the most reasonable thing in the typical case that @expr{Y(x,y,z)}
25184 depends only on the dependent variable @expr{z}, and in fact is
25185 often simply equal to @expr{z}. For common cases like polynomials
25186 and multilinear models, the combined error is simply used as the
25187 @texline @math{\sigma}
25188 @infoline @expr{sigma}
25189 for the data point with no further ado.)
25196 It may be the case that the model you wish to use is linearizable,
25197 but Calc's built-in rules are unable to figure it out. Calc uses
25198 its algebraic rewrite mechanism to linearize a model. The rewrite
25199 rules are kept in the variable @code{FitRules}. You can edit this
25200 variable using the @kbd{s e FitRules} command; in fact, there is
25201 a special @kbd{s F} command just for editing @code{FitRules}.
25202 @xref{Operations on Variables}.
25204 @xref{Rewrite Rules}, for a discussion of rewrite rules.
25238 Calc uses @code{FitRules} as follows. First, it converts the model
25239 to an equation if necessary and encloses the model equation in a
25240 call to the function @code{fitmodel} (which is not actually a defined
25241 function in Calc; it is only used as a placeholder by the rewrite rules).
25242 Parameter variables are renamed to function calls @samp{fitparam(1)},
25243 @samp{fitparam(2)}, and so on, and independent variables are renamed
25244 to @samp{fitvar(1)}, @samp{fitvar(2)}, etc. The dependent variable
25245 is the highest-numbered @code{fitvar}. For example, the power law
25246 model @expr{a x^b} is converted to @expr{y = a x^b}, then to
25250 fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))
25254 Calc then applies the rewrites as if by @samp{C-u 0 a r FitRules}.
25255 (The zero prefix means that rewriting should continue until no further
25256 changes are possible.)
25258 When rewriting is complete, the @code{fitmodel} call should have
25259 been replaced by a @code{fitsystem} call that looks like this:
25262 fitsystem(@var{Y}, @var{FGH}, @var{abc})
25266 where @var{Y} is a formula that describes the function @expr{Y(x,y,z)},
25267 @var{FGH} is the vector of formulas @expr{[F(x,y,z), G(x,y,z), H(x,y,z)]},
25268 and @var{abc} is the vector of parameter filters which refer to the
25269 raw parameters as @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)}
25270 for @expr{B}, etc. While the number of raw parameters (the length of
25271 the @var{FGH} vector) is usually the same as the number of original
25272 parameters (the length of the @var{abc} vector), this is not required.
25274 The power law model eventually boils down to
25278 fitsystem(ln(fitvar(2)),
25279 [1, ln(fitvar(1))],
25280 [exp(fitdummy(1)), fitdummy(2)])
25284 The actual implementation of @code{FitRules} is complicated; it
25285 proceeds in four phases. First, common rearrangements are done
25286 to try to bring linear terms together and to isolate functions like
25287 @code{exp} and @code{ln} either all the way ``out'' (so that they
25288 can be put into @var{Y}) or all the way ``in'' (so that they can
25289 be put into @var{abc} or @var{FGH}). In particular, all
25290 non-constant powers are converted to logs-and-exponentials form,
25291 and the distributive law is used to expand products of sums.
25292 Quotients are rewritten to use the @samp{fitinv} function, where
25293 @samp{fitinv(x)} represents @expr{1/x} while the @code{FitRules}
25294 are operating. (The use of @code{fitinv} makes recognition of
25295 linear-looking forms easier.) If you modify @code{FitRules}, you
25296 will probably only need to modify the rules for this phase.
25298 Phase two, whose rules can actually also apply during phases one
25299 and three, first rewrites @code{fitmodel} to a two-argument
25300 form @samp{fitmodel(@var{Y}, @var{model})}, where @var{Y} is
25301 initially zero and @var{model} has been changed from @expr{a=b}
25302 to @expr{a-b} form. It then tries to peel off invertible functions
25303 from the outside of @var{model} and put them into @var{Y} instead,
25304 calling the equation solver to invert the functions. Finally, when
25305 this is no longer possible, the @code{fitmodel} is changed to a
25306 four-argument @code{fitsystem}, where the fourth argument is
25307 @var{model} and the @var{FGH} and @var{abc} vectors are initially
25308 empty. (The last vector is really @var{ABC}, corresponding to
25309 raw parameters, for now.)
25311 Phase three converts a sum of items in the @var{model} to a sum
25312 of @samp{fitpart(@var{a}, @var{b}, @var{c})} terms which represent
25313 terms @samp{@var{a}*@var{b}*@var{c}} of the sum, where @var{a}
25314 is all factors that do not involve any variables, @var{b} is all
25315 factors that involve only parameters, and @var{c} is the factors
25316 that involve only independent variables. (If this decomposition
25317 is not possible, the rule set will not complete and Calc will
25318 complain that the model is too complex.) Then @code{fitpart}s
25319 with equal @var{b} or @var{c} components are merged back together
25320 using the distributive law in order to minimize the number of
25321 raw parameters needed.
25323 Phase four moves the @code{fitpart} terms into the @var{FGH} and
25324 @var{ABC} vectors. Also, some of the algebraic expansions that
25325 were done in phase 1 are undone now to make the formulas more
25326 computationally efficient. Finally, it calls the solver one more
25327 time to convert the @var{ABC} vector to an @var{abc} vector, and
25328 removes the fourth @var{model} argument (which by now will be zero)
25329 to obtain the three-argument @code{fitsystem} that the linear
25330 least-squares solver wants to see.
25336 @mindex hasfit@idots
25338 @tindex hasfitparams
25346 Two functions which are useful in connection with @code{FitRules}
25347 are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check
25348 whether @expr{x} refers to any parameters or independent variables,
25349 respectively. Specifically, these functions return ``true'' if the
25350 argument contains any @code{fitparam} (or @code{fitvar}) function
25351 calls, and ``false'' otherwise. (Recall that ``true'' means a
25352 nonzero number, and ``false'' means zero. The actual nonzero number
25353 returned is the largest @var{n} from all the @samp{fitparam(@var{n})}s
25354 or @samp{fitvar(@var{n})}s, respectively, that appear in the formula.)
25360 The @code{fit} function in algebraic notation normally takes four
25361 arguments, @samp{fit(@var{model}, @var{vars}, @var{params}, @var{data})},
25362 where @var{model} is the model formula as it would be typed after
25363 @kbd{a F '}, @var{vars} is the independent variable or a vector of
25364 independent variables, @var{params} likewise gives the parameter(s),
25365 and @var{data} is the data matrix. Note that the length of @var{vars}
25366 must be equal to the number of rows in @var{data} if @var{model} is
25367 an equation, or one less than the number of rows if @var{model} is
25368 a plain formula. (Actually, a name for the dependent variable is
25369 allowed but will be ignored in the plain-formula case.)
25371 If @var{params} is omitted, the parameters are all variables in
25372 @var{model} except those that appear in @var{vars}. If @var{vars}
25373 is also omitted, Calc sorts all the variables that appear in
25374 @var{model} alphabetically and uses the higher ones for @var{vars}
25375 and the lower ones for @var{params}.
25377 Alternatively, @samp{fit(@var{modelvec}, @var{data})} is allowed
25378 where @var{modelvec} is a 2- or 3-vector describing the model
25379 and variables, as discussed previously.
25381 If Calc is unable to do the fit, the @code{fit} function is left
25382 in symbolic form, ordinarily with an explanatory message. The
25383 message will be ``Model expression is too complex'' if the
25384 linearizer was unable to put the model into the required form.
25386 The @code{efit} (corresponding to @kbd{H a F}) and @code{xfit}
25387 (for @kbd{I a F}) functions are completely analogous.
25389 @node Interpolation, , Curve Fitting Details, Curve Fitting
25390 @subsection Polynomial Interpolation
25393 @pindex calc-poly-interp
25395 The @kbd{a p} (@code{calc-poly-interp}) [@code{polint}] command does
25396 a polynomial interpolation at a particular @expr{x} value. It takes
25397 two arguments from the stack: A data matrix of the sort used by
25398 @kbd{a F}, and a single number which represents the desired @expr{x}
25399 value. Calc effectively does an exact polynomial fit as if by @kbd{a F i},
25400 then substitutes the @expr{x} value into the result in order to get an
25401 approximate @expr{y} value based on the fit. (Calc does not actually
25402 use @kbd{a F i}, however; it uses a direct method which is both more
25403 efficient and more numerically stable.)
25405 The result of @kbd{a p} is actually a vector of two values: The @expr{y}
25406 value approximation, and an error measure @expr{dy} that reflects Calc's
25407 estimation of the probable error of the approximation at that value of
25408 @expr{x}. If the input @expr{x} is equal to any of the @expr{x} values
25409 in the data matrix, the output @expr{y} will be the corresponding @expr{y}
25410 value from the matrix, and the output @expr{dy} will be exactly zero.
25412 A prefix argument of 2 causes @kbd{a p} to take separate x- and
25413 y-vectors from the stack instead of one data matrix.
25415 If @expr{x} is a vector of numbers, @kbd{a p} will return a matrix of
25416 interpolated results for each of those @expr{x} values. (The matrix will
25417 have two columns, the @expr{y} values and the @expr{dy} values.)
25418 If @expr{x} is a formula instead of a number, the @code{polint} function
25419 remains in symbolic form; use the @kbd{a "} command to expand it out to
25420 a formula that describes the fit in symbolic terms.
25422 In all cases, the @kbd{a p} command leaves the data vectors or matrix
25423 on the stack. Only the @expr{x} value is replaced by the result.
25427 The @kbd{H a p} [@code{ratint}] command does a rational function
25428 interpolation. It is used exactly like @kbd{a p}, except that it
25429 uses as its model the quotient of two polynomials. If there are
25430 @expr{N} data points, the numerator and denominator polynomials will
25431 each have degree @expr{N/2} (if @expr{N} is odd, the denominator will
25432 have degree one higher than the numerator).
25434 Rational approximations have the advantage that they can accurately
25435 describe functions that have poles (points at which the function's value
25436 goes to infinity, so that the denominator polynomial of the approximation
25437 goes to zero). If @expr{x} corresponds to a pole of the fitted rational
25438 function, then the result will be a division by zero. If Infinite mode
25439 is enabled, the result will be @samp{[uinf, uinf]}.
25441 There is no way to get the actual coefficients of the rational function
25442 used by @kbd{H a p}. (The algorithm never generates these coefficients
25443 explicitly, and quotients of polynomials are beyond @w{@kbd{a F}}'s
25444 capabilities to fit.)
25446 @node Summations, Logical Operations, Curve Fitting, Algebra
25447 @section Summations
25450 @cindex Summation of a series
25452 @pindex calc-summation
25454 The @kbd{a +} (@code{calc-summation}) [@code{sum}] command computes
25455 the sum of a formula over a certain range of index values. The formula
25456 is taken from the top of the stack; the command prompts for the
25457 name of the summation index variable, the lower limit of the
25458 sum (any formula), and the upper limit of the sum. If you
25459 enter a blank line at any of these prompts, that prompt and
25460 any later ones are answered by reading additional elements from
25461 the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a + @key{RET}}
25462 produces the result 55.
25464 $$ \sum_{k=1}^5 k^2 = 55 $$
25467 The choice of index variable is arbitrary, but it's best not to
25468 use a variable with a stored value. In particular, while
25469 @code{i} is often a favorite index variable, it should be avoided
25470 in Calc because @code{i} has the imaginary constant @expr{(0, 1)}
25471 as a value. If you pressed @kbd{=} on a sum over @code{i}, it would
25472 be changed to a nonsensical sum over the ``variable'' @expr{(0, 1)}!
25473 If you really want to use @code{i} as an index variable, use
25474 @w{@kbd{s u i @key{RET}}} first to ``unstore'' this variable.
25475 (@xref{Storing Variables}.)
25477 A numeric prefix argument steps the index by that amount rather
25478 than by one. Thus @kbd{' a_k @key{RET} C-u -2 a + k @key{RET} 10 @key{RET} 0 @key{RET}}
25479 yields @samp{a_10 + a_8 + a_6 + a_4 + a_2 + a_0}. A prefix
25480 argument of plain @kbd{C-u} causes @kbd{a +} to prompt for the
25481 step value, in which case you can enter any formula or enter
25482 a blank line to take the step value from the stack. With the
25483 @kbd{C-u} prefix, @kbd{a +} can take up to five arguments from
25484 the stack: The formula, the variable, the lower limit, the
25485 upper limit, and (at the top of the stack), the step value.
25487 Calc knows how to do certain sums in closed form. For example,
25488 @samp{sum(6 k^2, k, 1, n) = @w{2 n^3} + 3 n^2 + n}. In particular,
25489 this is possible if the formula being summed is polynomial or
25490 exponential in the index variable. Sums of logarithms are
25491 transformed into logarithms of products. Sums of trigonometric
25492 and hyperbolic functions are transformed to sums of exponentials
25493 and then done in closed form. Also, of course, sums in which the
25494 lower and upper limits are both numbers can always be evaluated
25495 just by grinding them out, although Calc will use closed forms
25496 whenever it can for the sake of efficiency.
25498 The notation for sums in algebraic formulas is
25499 @samp{sum(@var{expr}, @var{var}, @var{low}, @var{high}, @var{step})}.
25500 If @var{step} is omitted, it defaults to one. If @var{high} is
25501 omitted, @var{low} is actually the upper limit and the lower limit
25502 is one. If @var{low} is also omitted, the limits are @samp{-inf}
25503 and @samp{inf}, respectively.
25505 Infinite sums can sometimes be evaluated: @samp{sum(.5^k, k, 1, inf)}
25506 returns @expr{1}. This is done by evaluating the sum in closed
25507 form (to @samp{1. - 0.5^n} in this case), then evaluating this
25508 formula with @code{n} set to @code{inf}. Calc's usual rules
25509 for ``infinite'' arithmetic can find the answer from there. If
25510 infinite arithmetic yields a @samp{nan}, or if the sum cannot be
25511 solved in closed form, Calc leaves the @code{sum} function in
25512 symbolic form. @xref{Infinities}.
25514 As a special feature, if the limits are infinite (or omitted, as
25515 described above) but the formula includes vectors subscripted by
25516 expressions that involve the iteration variable, Calc narrows
25517 the limits to include only the range of integers which result in
25518 valid subscripts for the vector. For example, the sum
25519 @samp{sum(k [a,b,c,d,e,f,g]_(2k),k)} evaluates to @samp{b + 2 d + 3 f}.
25521 The limits of a sum do not need to be integers. For example,
25522 @samp{sum(a_k, k, 0, 2 n, n)} produces @samp{a_0 + a_n + a_(2 n)}.
25523 Calc computes the number of iterations using the formula
25524 @samp{1 + (@var{high} - @var{low}) / @var{step}}, which must,
25525 after algebraic simplification, evaluate to an integer.
25527 If the number of iterations according to the above formula does
25528 not come out to an integer, the sum is invalid and will be left
25529 in symbolic form. However, closed forms are still supplied, and
25530 you are on your honor not to misuse the resulting formulas by
25531 substituting mismatched bounds into them. For example,
25532 @samp{sum(k, k, 1, 10, 2)} is invalid, but Calc will go ahead and
25533 evaluate the closed form solution for the limits 1 and 10 to get
25534 the rather dubious answer, 29.25.
25536 If the lower limit is greater than the upper limit (assuming a
25537 positive step size), the result is generally zero. However,
25538 Calc only guarantees a zero result when the upper limit is
25539 exactly one step less than the lower limit, i.e., if the number
25540 of iterations is @mathit{-1}. Thus @samp{sum(f(k), k, n, n-1)} is zero
25541 but the sum from @samp{n} to @samp{n-2} may report a nonzero value
25542 if Calc used a closed form solution.
25544 Calc's logical predicates like @expr{a < b} return 1 for ``true''
25545 and 0 for ``false.'' @xref{Logical Operations}. This can be
25546 used to advantage for building conditional sums. For example,
25547 @samp{sum(prime(k)*k^2, k, 1, 20)} is the sum of the squares of all
25548 prime numbers from 1 to 20; the @code{prime} predicate returns 1 if
25549 its argument is prime and 0 otherwise. You can read this expression
25550 as ``the sum of @expr{k^2}, where @expr{k} is prime.'' Indeed,
25551 @samp{sum(prime(k)*k^2, k)} would represent the sum of @emph{all} primes
25552 squared, since the limits default to plus and minus infinity, but
25553 there are no such sums that Calc's built-in rules can do in
25556 As another example, @samp{sum((k != k_0) * f(k), k, 1, n)} is the
25557 sum of @expr{f(k)} for all @expr{k} from 1 to @expr{n}, excluding
25558 one value @expr{k_0}. Slightly more tricky is the summand
25559 @samp{(k != k_0) / (k - k_0)}, which is an attempt to describe
25560 the sum of all @expr{1/(k-k_0)} except at @expr{k = k_0}, where
25561 this would be a division by zero. But at @expr{k = k_0}, this
25562 formula works out to the indeterminate form @expr{0 / 0}, which
25563 Calc will not assume is zero. Better would be to use
25564 @samp{(k != k_0) ? 1/(k-k_0) : 0}; the @samp{? :} operator does
25565 an ``if-then-else'' test: This expression says, ``if
25566 @texline @math{k \ne k_0},
25567 @infoline @expr{k != k_0},
25568 then @expr{1/(k-k_0)}, else zero.'' Now the formula @expr{1/(k-k_0)}
25569 will not even be evaluated by Calc when @expr{k = k_0}.
25571 @cindex Alternating sums
25573 @pindex calc-alt-summation
25575 The @kbd{a -} (@code{calc-alt-summation}) [@code{asum}] command
25576 computes an alternating sum. Successive terms of the sequence
25577 are given alternating signs, with the first term (corresponding
25578 to the lower index value) being positive. Alternating sums
25579 are converted to normal sums with an extra term of the form
25580 @samp{(-1)^(k-@var{low})}. This formula is adjusted appropriately
25581 if the step value is other than one. For example, the Taylor
25582 series for the sine function is @samp{asum(x^k / k!, k, 1, inf, 2)}.
25583 (Calc cannot evaluate this infinite series, but it can approximate
25584 it if you replace @code{inf} with any particular odd number.)
25585 Calc converts this series to a regular sum with a step of one,
25586 namely @samp{sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)}.
25588 @cindex Product of a sequence
25590 @pindex calc-product
25592 The @kbd{a *} (@code{calc-product}) [@code{prod}] command is
25593 the analogous way to take a product of many terms. Calc also knows
25594 some closed forms for products, such as @samp{prod(k, k, 1, n) = n!}.
25595 Conditional products can be written @samp{prod(k^prime(k), k, 1, n)}
25596 or @samp{prod(prime(k) ? k : 1, k, 1, n)}.
25599 @pindex calc-tabulate
25601 The @kbd{a T} (@code{calc-tabulate}) [@code{table}] command
25602 evaluates a formula at a series of iterated index values, just
25603 like @code{sum} and @code{prod}, but its result is simply a
25604 vector of the results. For example, @samp{table(a_i, i, 1, 7, 2)}
25605 produces @samp{[a_1, a_3, a_5, a_7]}.
25607 @node Logical Operations, Rewrite Rules, Summations, Algebra
25608 @section Logical Operations
25611 The following commands and algebraic functions return true/false values,
25612 where 1 represents ``true'' and 0 represents ``false.'' In cases where
25613 a truth value is required (such as for the condition part of a rewrite
25614 rule, or as the condition for a @w{@kbd{Z [ Z ]}} control structure), any
25615 nonzero value is accepted to mean ``true.'' (Specifically, anything
25616 for which @code{dnonzero} returns 1 is ``true,'' and anything for
25617 which @code{dnonzero} returns 0 or cannot decide is assumed ``false.''
25618 Note that this means that @w{@kbd{Z [ Z ]}} will execute the ``then''
25619 portion if its condition is provably true, but it will execute the
25620 ``else'' portion for any condition like @expr{a = b} that is not
25621 provably true, even if it might be true. Algebraic functions that
25622 have conditions as arguments, like @code{? :} and @code{&&}, remain
25623 unevaluated if the condition is neither provably true nor provably
25624 false. @xref{Declarations}.)
25627 @pindex calc-equal-to
25631 The @kbd{a =} (@code{calc-equal-to}) command, or @samp{eq(a,b)} function
25632 (which can also be written @samp{a = b} or @samp{a == b} in an algebraic
25633 formula) is true if @expr{a} and @expr{b} are equal, either because they
25634 are identical expressions, or because they are numbers which are
25635 numerically equal. (Thus the integer 1 is considered equal to the float
25636 1.0.) If the equality of @expr{a} and @expr{b} cannot be determined,
25637 the comparison is left in symbolic form. Note that as a command, this
25638 operation pops two values from the stack and pushes back either a 1 or
25639 a 0, or a formula @samp{a = b} if the values' equality cannot be determined.
25641 Many Calc commands use @samp{=} formulas to represent @dfn{equations}.
25642 For example, the @kbd{a S} (@code{calc-solve-for}) command rearranges
25643 an equation to solve for a given variable. The @kbd{a M}
25644 (@code{calc-map-equation}) command can be used to apply any
25645 function to both sides of an equation; for example, @kbd{2 a M *}
25646 multiplies both sides of the equation by two. Note that just
25647 @kbd{2 *} would not do the same thing; it would produce the formula
25648 @samp{2 (a = b)} which represents 2 if the equality is true or
25651 The @code{eq} function with more than two arguments (e.g., @kbd{C-u 3 a =}
25652 or @samp{a = b = c}) tests if all of its arguments are equal. In
25653 algebraic notation, the @samp{=} operator is unusual in that it is
25654 neither left- nor right-associative: @samp{a = b = c} is not the
25655 same as @samp{(a = b) = c} or @samp{a = (b = c)} (which each compare
25656 one variable with the 1 or 0 that results from comparing two other
25660 @pindex calc-not-equal-to
25663 The @kbd{a #} (@code{calc-not-equal-to}) command, or @samp{neq(a,b)} or
25664 @samp{a != b} function, is true if @expr{a} and @expr{b} are not equal.
25665 This also works with more than two arguments; @samp{a != b != c != d}
25666 tests that all four of @expr{a}, @expr{b}, @expr{c}, and @expr{d} are
25683 @pindex calc-less-than
25684 @pindex calc-greater-than
25685 @pindex calc-less-equal
25686 @pindex calc-greater-equal
25715 The @kbd{a <} (@code{calc-less-than}) [@samp{lt(a,b)} or @samp{a < b}]
25716 operation is true if @expr{a} is less than @expr{b}. Similar functions
25717 are @kbd{a >} (@code{calc-greater-than}) [@samp{gt(a,b)} or @samp{a > b}],
25718 @kbd{a [} (@code{calc-less-equal}) [@samp{leq(a,b)} or @samp{a <= b}], and
25719 @kbd{a ]} (@code{calc-greater-equal}) [@samp{geq(a,b)} or @samp{a >= b}].
25721 While the inequality functions like @code{lt} do not accept more
25722 than two arguments, the syntax @w{@samp{a <= b < c}} is translated to an
25723 equivalent expression involving intervals: @samp{b in [a .. c)}.
25724 (See the description of @code{in} below.) All four combinations
25725 of @samp{<} and @samp{<=} are allowed, or any of the four combinations
25726 of @samp{>} and @samp{>=}. Four-argument constructions like
25727 @samp{a < b < c < d}, and mixtures like @w{@samp{a < b = c}} that
25728 involve both equations and inequalities, are not allowed.
25731 @pindex calc-remove-equal
25733 The @kbd{a .} (@code{calc-remove-equal}) [@code{rmeq}] command extracts
25734 the righthand side of the equation or inequality on the top of the
25735 stack. It also works elementwise on vectors. For example, if
25736 @samp{[x = 2.34, y = z / 2]} is on the stack, then @kbd{a .} produces
25737 @samp{[2.34, z / 2]}. As a special case, if the righthand side is a
25738 variable and the lefthand side is a number (as in @samp{2.34 = x}), then
25739 Calc keeps the lefthand side instead. Finally, this command works with
25740 assignments @samp{x := 2.34} as well as equations, always taking the
25741 righthand side, and for @samp{=>} (evaluates-to) operators, always
25742 taking the lefthand side.
25745 @pindex calc-logical-and
25748 The @kbd{a &} (@code{calc-logical-and}) [@samp{land(a,b)} or @samp{a && b}]
25749 function is true if both of its arguments are true, i.e., are
25750 non-zero numbers. In this case, the result will be either @expr{a} or
25751 @expr{b}, chosen arbitrarily. If either argument is zero, the result is
25752 zero. Otherwise, the formula is left in symbolic form.
25755 @pindex calc-logical-or
25758 The @kbd{a |} (@code{calc-logical-or}) [@samp{lor(a,b)} or @samp{a || b}]
25759 function is true if either or both of its arguments are true (nonzero).
25760 The result is whichever argument was nonzero, choosing arbitrarily if both
25761 are nonzero. If both @expr{a} and @expr{b} are zero, the result is
25765 @pindex calc-logical-not
25768 The @kbd{a !} (@code{calc-logical-not}) [@samp{lnot(a)} or @samp{!@: a}]
25769 function is true if @expr{a} is false (zero), or false if @expr{a} is
25770 true (nonzero). It is left in symbolic form if @expr{a} is not a
25774 @pindex calc-logical-if
25784 @cindex Arguments, not evaluated
25785 The @kbd{a :} (@code{calc-logical-if}) [@samp{if(a,b,c)} or @samp{a ? b :@: c}]
25786 function is equal to either @expr{b} or @expr{c} if @expr{a} is a nonzero
25787 number or zero, respectively. If @expr{a} is not a number, the test is
25788 left in symbolic form and neither @expr{b} nor @expr{c} is evaluated in
25789 any way. In algebraic formulas, this is one of the few Calc functions
25790 whose arguments are not automatically evaluated when the function itself
25791 is evaluated. The others are @code{lambda}, @code{quote}, and
25794 One minor surprise to watch out for is that the formula @samp{a?3:4}
25795 will not work because the @samp{3:4} is parsed as a fraction instead of
25796 as three separate symbols. Type something like @samp{a ? 3 : 4} or
25797 @samp{a?(3):4} instead.
25799 As a special case, if @expr{a} evaluates to a vector, then both @expr{b}
25800 and @expr{c} are evaluated; the result is a vector of the same length
25801 as @expr{a} whose elements are chosen from corresponding elements of
25802 @expr{b} and @expr{c} according to whether each element of @expr{a}
25803 is zero or nonzero. Each of @expr{b} and @expr{c} must be either a
25804 vector of the same length as @expr{a}, or a non-vector which is matched
25805 with all elements of @expr{a}.
25808 @pindex calc-in-set
25810 The @kbd{a @{} (@code{calc-in-set}) [@samp{in(a,b)}] function is true if
25811 the number @expr{a} is in the set of numbers represented by @expr{b}.
25812 If @expr{b} is an interval form, @expr{a} must be one of the values
25813 encompassed by the interval. If @expr{b} is a vector, @expr{a} must be
25814 equal to one of the elements of the vector. (If any vector elements are
25815 intervals, @expr{a} must be in any of the intervals.) If @expr{b} is a
25816 plain number, @expr{a} must be numerically equal to @expr{b}.
25817 @xref{Set Operations}, for a group of commands that manipulate sets
25824 The @samp{typeof(a)} function produces an integer or variable which
25825 characterizes @expr{a}. If @expr{a} is a number, vector, or variable,
25826 the result will be one of the following numbers:
25831 3 Floating-point number
25833 5 Rectangular complex number
25834 6 Polar complex number
25840 12 Infinity (inf, uinf, or nan)
25842 101 Vector (but not a matrix)
25846 Otherwise, @expr{a} is a formula, and the result is a variable which
25847 represents the name of the top-level function call.
25861 The @samp{integer(a)} function returns true if @expr{a} is an integer.
25862 The @samp{real(a)} function
25863 is true if @expr{a} is a real number, either integer, fraction, or
25864 float. The @samp{constant(a)} function returns true if @expr{a} is
25865 any of the objects for which @code{typeof} would produce an integer
25866 code result except for variables, and provided that the components of
25867 an object like a vector or error form are themselves constant.
25868 Note that infinities do not satisfy any of these tests, nor do
25869 special constants like @code{pi} and @code{e}.
25871 @xref{Declarations}, for a set of similar functions that recognize
25872 formulas as well as actual numbers. For example, @samp{dint(floor(x))}
25873 is true because @samp{floor(x)} is provably integer-valued, but
25874 @samp{integer(floor(x))} does not because @samp{floor(x)} is not
25875 literally an integer constant.
25881 The @samp{refers(a,b)} function is true if the variable (or sub-expression)
25882 @expr{b} appears in @expr{a}, or false otherwise. Unlike the other
25883 tests described here, this function returns a definite ``no'' answer
25884 even if its arguments are still in symbolic form. The only case where
25885 @code{refers} will be left unevaluated is if @expr{a} is a plain
25886 variable (different from @expr{b}).
25892 The @samp{negative(a)} function returns true if @expr{a} ``looks'' negative,
25893 because it is a negative number, because it is of the form @expr{-x},
25894 or because it is a product or quotient with a term that looks negative.
25895 This is most useful in rewrite rules. Beware that @samp{negative(a)}
25896 evaluates to 1 or 0 for @emph{any} argument @expr{a}, so it can only
25897 be stored in a formula if the default simplifications are turned off
25898 first with @kbd{m O} (or if it appears in an unevaluated context such
25899 as a rewrite rule condition).
25905 The @samp{variable(a)} function is true if @expr{a} is a variable,
25906 or false if not. If @expr{a} is a function call, this test is left
25907 in symbolic form. Built-in variables like @code{pi} and @code{inf}
25908 are considered variables like any others by this test.
25914 The @samp{nonvar(a)} function is true if @expr{a} is a non-variable.
25915 If its argument is a variable it is left unsimplified; it never
25916 actually returns zero. However, since Calc's condition-testing
25917 commands consider ``false'' anything not provably true, this is
25936 @cindex Linearity testing
25937 The functions @code{lin}, @code{linnt}, @code{islin}, and @code{islinnt}
25938 check if an expression is ``linear,'' i.e., can be written in the form
25939 @expr{a + b x} for some constants @expr{a} and @expr{b}, and some
25940 variable or subformula @expr{x}. The function @samp{islin(f,x)} checks
25941 if formula @expr{f} is linear in @expr{x}, returning 1 if so. For
25942 example, @samp{islin(x,x)}, @samp{islin(-x,x)}, @samp{islin(3,x)}, and
25943 @samp{islin(x y / 3 - 2, x)} all return 1. The @samp{lin(f,x)} function
25944 is similar, except that instead of returning 1 it returns the vector
25945 @expr{[a, b, x]}. For the above examples, this vector would be
25946 @expr{[0, 1, x]}, @expr{[0, -1, x]}, @expr{[3, 0, x]}, and
25947 @expr{[-2, y/3, x]}, respectively. Both @code{lin} and @code{islin}
25948 generally remain unevaluated for expressions which are not linear,
25949 e.g., @samp{lin(2 x^2, x)} and @samp{lin(sin(x), x)}. The second
25950 argument can also be a formula; @samp{islin(2 + 3 sin(x), sin(x))}
25953 The @code{linnt} and @code{islinnt} functions perform a similar check,
25954 but require a ``non-trivial'' linear form, which means that the
25955 @expr{b} coefficient must be non-zero. For example, @samp{lin(2,x)}
25956 returns @expr{[2, 0, x]} and @samp{lin(y,x)} returns @expr{[y, 0, x]},
25957 but @samp{linnt(2,x)} and @samp{linnt(y,x)} are left unevaluated
25958 (in other words, these formulas are considered to be only ``trivially''
25959 linear in @expr{x}).
25961 All four linearity-testing functions allow you to omit the second
25962 argument, in which case the input may be linear in any non-constant
25963 formula. Here, the @expr{a=0}, @expr{b=1} case is also considered
25964 trivial, and only constant values for @expr{a} and @expr{b} are
25965 recognized. Thus, @samp{lin(2 x y)} returns @expr{[0, 2, x y]},
25966 @samp{lin(2 - x y)} returns @expr{[2, -1, x y]}, and @samp{lin(x y)}
25967 returns @expr{[0, 1, x y]}. The @code{linnt} function would allow the
25968 first two cases but not the third. Also, neither @code{lin} nor
25969 @code{linnt} accept plain constants as linear in the one-argument
25970 case: @samp{islin(2,x)} is true, but @samp{islin(2)} is false.
25976 The @samp{istrue(a)} function returns 1 if @expr{a} is a nonzero
25977 number or provably nonzero formula, or 0 if @expr{a} is anything else.
25978 Calls to @code{istrue} can only be manipulated if @kbd{m O} mode is
25979 used to make sure they are not evaluated prematurely. (Note that
25980 declarations are used when deciding whether a formula is true;
25981 @code{istrue} returns 1 when @code{dnonzero} would return 1, and
25982 it returns 0 when @code{dnonzero} would return 0 or leave itself
25985 @node Rewrite Rules, , Logical Operations, Algebra
25986 @section Rewrite Rules
25989 @cindex Rewrite rules
25990 @cindex Transformations
25991 @cindex Pattern matching
25993 @pindex calc-rewrite
25995 The @kbd{a r} (@code{calc-rewrite}) [@code{rewrite}] command makes
25996 substitutions in a formula according to a specified pattern or patterns
25997 known as @dfn{rewrite rules}. Whereas @kbd{a b} (@code{calc-substitute})
25998 matches literally, so that substituting @samp{sin(x)} with @samp{cos(x)}
25999 matches only the @code{sin} function applied to the variable @code{x},
26000 rewrite rules match general kinds of formulas; rewriting using the rule
26001 @samp{sin(x) := cos(x)} matches @code{sin} of any argument and replaces
26002 it with @code{cos} of that same argument. The only significance of the
26003 name @code{x} is that the same name is used on both sides of the rule.
26005 Rewrite rules rearrange formulas already in Calc's memory.
26006 @xref{Syntax Tables}, to read about @dfn{syntax rules}, which are
26007 similar to algebraic rewrite rules but operate when new algebraic
26008 entries are being parsed, converting strings of characters into
26012 * Entering Rewrite Rules::
26013 * Basic Rewrite Rules::
26014 * Conditional Rewrite Rules::
26015 * Algebraic Properties of Rewrite Rules::
26016 * Other Features of Rewrite Rules::
26017 * Composing Patterns in Rewrite Rules::
26018 * Nested Formulas with Rewrite Rules::
26019 * Multi-Phase Rewrite Rules::
26020 * Selections with Rewrite Rules::
26021 * Matching Commands::
26022 * Automatic Rewrites::
26023 * Debugging Rewrites::
26024 * Examples of Rewrite Rules::
26027 @node Entering Rewrite Rules, Basic Rewrite Rules, Rewrite Rules, Rewrite Rules
26028 @subsection Entering Rewrite Rules
26031 Rewrite rules normally use the ``assignment'' operator
26032 @samp{@var{old} := @var{new}}.
26033 This operator is equivalent to the function call @samp{assign(old, new)}.
26034 The @code{assign} function is undefined by itself in Calc, so an
26035 assignment formula such as a rewrite rule will be left alone by ordinary
26036 Calc commands. But certain commands, like the rewrite system, interpret
26037 assignments in special ways.
26039 For example, the rule @samp{sin(x)^2 := 1-cos(x)^2} says to replace
26040 every occurrence of the sine of something, squared, with one minus the
26041 square of the cosine of that same thing. All by itself as a formula
26042 on the stack it does nothing, but when given to the @kbd{a r} command
26043 it turns that command into a sine-squared-to-cosine-squared converter.
26045 To specify a set of rules to be applied all at once, make a vector of
26048 When @kbd{a r} prompts you to enter the rewrite rules, you can answer
26053 With a rule: @kbd{f(x) := g(x) @key{RET}}.
26055 With a vector of rules: @kbd{[f1(x) := g1(x), f2(x) := g2(x)] @key{RET}}.
26056 (You can omit the enclosing square brackets if you wish.)
26058 With the name of a variable that contains the rule or rules vector:
26059 @kbd{myrules @key{RET}}.
26061 With any formula except a rule, a vector, or a variable name; this
26062 will be interpreted as the @var{old} half of a rewrite rule,
26063 and you will be prompted a second time for the @var{new} half:
26064 @kbd{f(x) @key{RET} g(x) @key{RET}}.
26066 With a blank line, in which case the rule, rules vector, or variable
26067 will be taken from the top of the stack (and the formula to be
26068 rewritten will come from the second-to-top position).
26071 If you enter the rules directly (as opposed to using rules stored
26072 in a variable), those rules will be put into the Trail so that you
26073 can retrieve them later. @xref{Trail Commands}.
26075 It is most convenient to store rules you use often in a variable and
26076 invoke them by giving the variable name. The @kbd{s e}
26077 (@code{calc-edit-variable}) command is an easy way to create or edit a
26078 rule set stored in a variable. You may also wish to use @kbd{s p}
26079 (@code{calc-permanent-variable}) to save your rules permanently;
26080 @pxref{Operations on Variables}.
26082 Rewrite rules are compiled into a special internal form for faster
26083 matching. If you enter a rule set directly it must be recompiled
26084 every time. If you store the rules in a variable and refer to them
26085 through that variable, they will be compiled once and saved away
26086 along with the variable for later reference. This is another good
26087 reason to store your rules in a variable.
26089 Calc also accepts an obsolete notation for rules, as vectors
26090 @samp{[@var{old}, @var{new}]}. But because it is easily confused with a
26091 vector of two rules, the use of this notation is no longer recommended.
26093 @node Basic Rewrite Rules, Conditional Rewrite Rules, Entering Rewrite Rules, Rewrite Rules
26094 @subsection Basic Rewrite Rules
26097 To match a particular formula @expr{x} with a particular rewrite rule
26098 @samp{@var{old} := @var{new}}, Calc compares the structure of @expr{x} with
26099 the structure of @var{old}. Variables that appear in @var{old} are
26100 treated as @dfn{meta-variables}; the corresponding positions in @expr{x}
26101 may contain any sub-formulas. For example, the pattern @samp{f(x,y)}
26102 would match the expression @samp{f(12, a+1)} with the meta-variable
26103 @samp{x} corresponding to 12 and with @samp{y} corresponding to
26104 @samp{a+1}. However, this pattern would not match @samp{f(12)} or
26105 @samp{g(12, a+1)}, since there is no assignment of the meta-variables
26106 that will make the pattern match these expressions. Notice that if
26107 the pattern is a single meta-variable, it will match any expression.
26109 If a given meta-variable appears more than once in @var{old}, the
26110 corresponding sub-formulas of @expr{x} must be identical. Thus
26111 the pattern @samp{f(x,x)} would match @samp{f(12, 12)} and
26112 @samp{f(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(a+b, b+a)}.
26113 (@xref{Conditional Rewrite Rules}, for a way to match the latter.)
26115 Things other than variables must match exactly between the pattern
26116 and the target formula. To match a particular variable exactly, use
26117 the pseudo-function @samp{quote(v)} in the pattern. For example, the
26118 pattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or
26121 The special variable names @samp{e}, @samp{pi}, @samp{i}, @samp{phi},
26122 @samp{gamma}, @samp{inf}, @samp{uinf}, and @samp{nan} always match
26123 literally. Thus the pattern @samp{sin(d + e + f)} acts exactly like
26124 @samp{sin(d + quote(e) + f)}.
26126 If the @var{old} pattern is found to match a given formula, that
26127 formula is replaced by @var{new}, where any occurrences in @var{new}
26128 of meta-variables from the pattern are replaced with the sub-formulas
26129 that they matched. Thus, applying the rule @samp{f(x,y) := g(y+x,x)}
26130 to @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}.
26132 The normal @kbd{a r} command applies rewrite rules over and over
26133 throughout the target formula until no further changes are possible
26134 (up to a limit of 100 times). Use @kbd{C-u 1 a r} to make only one
26137 @node Conditional Rewrite Rules, Algebraic Properties of Rewrite Rules, Basic Rewrite Rules, Rewrite Rules
26138 @subsection Conditional Rewrite Rules
26141 A rewrite rule can also be @dfn{conditional}, written in the form
26142 @samp{@var{old} := @var{new} :: @var{cond}}. (There is also the obsolete
26143 form @samp{[@var{old}, @var{new}, @var{cond}]}.) If a @var{cond} part
26145 rule, this is an additional condition that must be satisfied before
26146 the rule is accepted. Once @var{old} has been successfully matched
26147 to the target expression, @var{cond} is evaluated (with all the
26148 meta-variables substituted for the values they matched) and simplified
26149 with Calc's algebraic simplifications. If the result is a nonzero
26150 number or any other object known to be nonzero (@pxref{Declarations}),
26151 the rule is accepted. If the result is zero or if it is a symbolic
26152 formula that is not known to be nonzero, the rule is rejected.
26153 @xref{Logical Operations}, for a number of functions that return
26154 1 or 0 according to the results of various tests.
26156 For example, the formula @samp{n > 0} simplifies to 1 or 0 if @expr{n}
26157 is replaced by a positive or nonpositive number, respectively (or if
26158 @expr{n} has been declared to be positive or nonpositive). Thus,
26159 the rule @samp{f(x,y) := g(y+x,x) :: x+y > 0} would apply to
26160 @samp{f(0, 4)} but not to @samp{f(-3, 2)} or @samp{f(12, a+1)}
26161 (assuming no outstanding declarations for @expr{a}). In the case of
26162 @samp{f(-3, 2)}, the condition can be shown not to be satisfied; in
26163 the case of @samp{f(12, a+1)}, the condition merely cannot be shown
26164 to be satisfied, but that is enough to reject the rule.
26166 While Calc will use declarations to reason about variables in the
26167 formula being rewritten, declarations do not apply to meta-variables.
26168 For example, the rule @samp{f(a) := g(a+1)} will match for any values
26169 of @samp{a}, such as complex numbers, vectors, or formulas, even if
26170 @samp{a} has been declared to be real or scalar. If you want the
26171 meta-variable @samp{a} to match only literal real numbers, use
26172 @samp{f(a) := g(a+1) :: real(a)}. If you want @samp{a} to match only
26173 reals and formulas which are provably real, use @samp{dreal(a)} as
26176 The @samp{::} operator is a shorthand for the @code{condition}
26177 function; @samp{@var{old} := @var{new} :: @var{cond}} is equivalent to
26178 the formula @samp{condition(assign(@var{old}, @var{new}), @var{cond})}.
26180 If you have several conditions, you can use @samp{... :: c1 :: c2 :: c3}
26181 or @samp{... :: c1 && c2 && c3}. The two are entirely equivalent.
26183 It is also possible to embed conditions inside the pattern:
26184 @samp{f(x :: x>0, y) := g(y+x, x)}. This is purely a notational
26185 convenience, though; where a condition appears in a rule has no
26186 effect on when it is tested. The rewrite-rule compiler automatically
26187 decides when it is best to test each condition while a rule is being
26190 Certain conditions are handled as special cases by the rewrite rule
26191 system and are tested very efficiently: Where @expr{x} is any
26192 meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)},
26193 @samp{constant(x)}, @samp{negative(x)}, @samp{x >= y} where @expr{y}
26194 is either a constant or another meta-variable and @samp{>=} may be
26195 replaced by any of the six relational operators, and @samp{x % a = b}
26196 where @expr{a} and @expr{b} are constants. Other conditions, like
26197 @samp{x >= y+1} or @samp{dreal(x)}, will be less efficient to check
26198 since Calc must bring the whole evaluator and simplifier into play.
26200 An interesting property of @samp{::} is that neither of its arguments
26201 will be touched by Calc's default simplifications. This is important
26202 because conditions often are expressions that cannot safely be
26203 evaluated early. For example, the @code{typeof} function never
26204 remains in symbolic form; entering @samp{typeof(a)} will put the
26205 number 100 (the type code for variables like @samp{a}) on the stack.
26206 But putting the condition @samp{... :: typeof(a) = 6} on the stack
26207 is safe since @samp{::} prevents the @code{typeof} from being
26208 evaluated until the condition is actually used by the rewrite system.
26210 Since @samp{::} protects its lefthand side, too, you can use a dummy
26211 condition to protect a rule that must itself not evaluate early.
26212 For example, it's not safe to put @samp{a(f,x) := apply(f, [x])} on
26213 the stack because it will immediately evaluate to @samp{a(f,x) := f(x)},
26214 where the meta-variable-ness of @code{f} on the righthand side has been
26215 lost. But @samp{a(f,x) := apply(f, [x]) :: 1} is safe, and of course
26216 the condition @samp{1} is always true (nonzero) so it has no effect on
26217 the functioning of the rule. (The rewrite compiler will ensure that
26218 it doesn't even impact the speed of matching the rule.)
26220 @node Algebraic Properties of Rewrite Rules, Other Features of Rewrite Rules, Conditional Rewrite Rules, Rewrite Rules
26221 @subsection Algebraic Properties of Rewrite Rules
26224 The rewrite mechanism understands the algebraic properties of functions
26225 like @samp{+} and @samp{*}. In particular, pattern matching takes
26226 the associativity and commutativity of the following functions into
26230 + - * = != && || and or xor vint vunion vxor gcd lcm max min beta
26233 For example, the rewrite rule:
26236 a x + b x := (a + b) x
26240 will match formulas of the form,
26243 a x + b x, x a + x b, a x + x b, x a + b x
26246 Rewrites also understand the relationship between the @samp{+} and @samp{-}
26247 operators. The above rewrite rule will also match the formulas,
26250 a x - b x, x a - x b, a x - x b, x a - b x
26254 by matching @samp{b} in the pattern to @samp{-b} from the formula.
26256 Applied to a sum of many terms like @samp{r + a x + s + b x + t}, this
26257 pattern will check all pairs of terms for possible matches. The rewrite
26258 will take whichever suitable pair it discovers first.
26260 In general, a pattern using an associative operator like @samp{a + b}
26261 will try @var{2 n} different ways to match a sum of @var{n} terms
26262 like @samp{x + y + z - w}. First, @samp{a} is matched against each
26263 of @samp{x}, @samp{y}, @samp{z}, and @samp{-w} in turn, with @samp{b}
26264 being matched to the remainders @samp{y + z - w}, @samp{x + z - w}, etc.
26265 If none of these succeed, then @samp{b} is matched against each of the
26266 four terms with @samp{a} matching the remainder. Half-and-half matches,
26267 like @samp{(x + y) + (z - w)}, are not tried.
26269 Note that @samp{*} is not commutative when applied to matrices, but
26270 rewrite rules pretend that it is. If you type @kbd{m v} to enable
26271 Matrix mode (@pxref{Matrix Mode}), rewrite rules will match @samp{*}
26272 literally, ignoring its usual commutativity property. (In the
26273 current implementation, the associativity also vanishes---it is as
26274 if the pattern had been enclosed in a @code{plain} marker; see below.)
26275 If you are applying rewrites to formulas with matrices, it's best to
26276 enable Matrix mode first to prevent algebraically incorrect rewrites
26279 The pattern @samp{-x} will actually match any expression. For example,
26287 will rewrite @samp{f(a)} to @samp{-f(-a)}. To avoid this, either use
26288 a @code{plain} marker as described below, or add a @samp{negative(x)}
26289 condition. The @code{negative} function is true if its argument
26290 ``looks'' negative, for example, because it is a negative number or
26291 because it is a formula like @samp{-x}. The new rule using this
26295 f(x) := -f(-x) :: negative(x) @r{or, equivalently,}
26296 f(-x) := -f(x) :: negative(-x)
26299 In the same way, the pattern @samp{x - y} will match the sum @samp{a + b}
26300 by matching @samp{y} to @samp{-b}.
26302 The pattern @samp{a b} will also match the formula @samp{x/y} if
26303 @samp{y} is a number. Thus the rule @samp{a x + @w{b x} := (a+b) x}
26304 will also convert @samp{a x + x / 2} to @samp{(a + 0.5) x} (or
26305 @samp{(a + 1:2) x}, depending on the current fraction mode).
26307 Calc will @emph{not} take other liberties with @samp{*}, @samp{/}, and
26308 @samp{^}. For example, the pattern @samp{f(a b)} will not match
26309 @samp{f(x^2)}, and @samp{f(a + b)} will not match @samp{f(2 x)}, even
26310 though conceivably these patterns could match with @samp{a = b = x}.
26311 Nor will @samp{f(a b)} match @samp{f(x / y)} if @samp{y} is not a
26312 constant, even though it could be considered to match with @samp{a = x}
26313 and @samp{b = 1/y}. The reasons are partly for efficiency, and partly
26314 because while few mathematical operations are substantively different
26315 for addition and subtraction, often it is preferable to treat the cases
26316 of multiplication, division, and integer powers separately.
26318 Even more subtle is the rule set
26321 [ f(a) + f(b) := f(a + b), -f(a) := f(-a) ]
26325 attempting to match @samp{f(x) - f(y)}. You might think that Calc
26326 will view this subtraction as @samp{f(x) + (-f(y))} and then apply
26327 the above two rules in turn, but actually this will not work because
26328 Calc only does this when considering rules for @samp{+} (like the
26329 first rule in this set). So it will see first that @samp{f(x) + (-f(y))}
26330 does not match @samp{f(a) + f(b)} for any assignments of the
26331 meta-variables, and then it will see that @samp{f(x) - f(y)} does
26332 not match @samp{-f(a)} for any assignment of @samp{a}. Because Calc
26333 tries only one rule at a time, it will not be able to rewrite
26334 @samp{f(x) - f(y)} with this rule set. An explicit @samp{f(a) - f(b)}
26335 rule will have to be added.
26337 Another thing patterns will @emph{not} do is break up complex numbers.
26338 The pattern @samp{myconj(a + @w{b i)} := a - b i} will work for formulas
26339 involving the special constant @samp{i} (such as @samp{3 - 4 i}), but
26340 it will not match actual complex numbers like @samp{(3, -4)}. A version
26341 of the above rule for complex numbers would be
26344 myconj(a) := re(a) - im(a) (0,1) :: im(a) != 0
26348 (Because the @code{re} and @code{im} functions understand the properties
26349 of the special constant @samp{i}, this rule will also work for
26350 @samp{3 - 4 i}. In fact, this particular rule would probably be better
26351 without the @samp{im(a) != 0} condition, since if @samp{im(a) = 0} the
26352 righthand side of the rule will still give the correct answer for the
26353 conjugate of a real number.)
26355 It is also possible to specify optional arguments in patterns. The rule
26358 opt(a) x + opt(b) (x^opt(c) + opt(d)) := f(a, b, c, d)
26362 will match the formula
26369 in a fairly straightforward manner, but it will also match reduced
26373 x + x^2, 2(x + 1) - x, x + x
26377 producing, respectively,
26380 f(1, 1, 2, 0), f(-1, 2, 1, 1), f(1, 1, 1, 0)
26383 (The latter two formulas can be entered only if default simplifications
26384 have been turned off with @kbd{m O}.)
26386 The default value for a term of a sum is zero. The default value
26387 for a part of a product, for a power, or for the denominator of a
26388 quotient, is one. Also, @samp{-x} matches the pattern @samp{opt(a) b}
26389 with @samp{a = -1}.
26391 In particular, the distributive-law rule can be refined to
26394 opt(a) x + opt(b) x := (a + b) x
26398 so that it will convert, e.g., @samp{a x - x}, to @samp{(a - 1) x}.
26400 The pattern @samp{opt(a) + opt(b) x} matches almost any formulas which
26401 are linear in @samp{x}. You can also use the @code{lin} and @code{islin}
26402 functions with rewrite conditions to test for this; @pxref{Logical
26403 Operations}. These functions are not as convenient to use in rewrite
26404 rules, but they recognize more kinds of formulas as linear:
26405 @samp{x/z} is considered linear with @expr{b = 1/z} by @code{lin},
26406 but it will not match the above pattern because that pattern calls
26407 for a multiplication, not a division.
26409 As another example, the obvious rule to replace @samp{sin(x)^2 + cos(x)^2}
26413 sin(x)^2 + cos(x)^2 := 1
26417 misses many cases because the sine and cosine may both be multiplied by
26418 an equal factor. Here's a more successful rule:
26421 opt(a) sin(x)^2 + opt(a) cos(x)^2 := a
26424 Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2}
26425 because one @expr{a} would have ``matched'' 1 while the other matched 6.
26427 Calc automatically converts a rule like
26437 f(temp, x) := g(x) :: temp = x-1
26441 (where @code{temp} stands for a new, invented meta-variable that
26442 doesn't actually have a name). This modified rule will successfully
26443 match @samp{f(6, 7)}, binding @samp{temp} and @samp{x} to 6 and 7,
26444 respectively, then verifying that they differ by one even though
26445 @samp{6} does not superficially look like @samp{x-1}.
26447 However, Calc does not solve equations to interpret a rule. The
26451 f(x-1, x+1) := g(x)
26455 will not work. That is, it will match @samp{f(a - 1 + b, a + 1 + b)}
26456 but not @samp{f(6, 8)}. Calc always interprets at least one occurrence
26457 of a variable by literal matching. If the variable appears ``isolated''
26458 then Calc is smart enough to use it for literal matching. But in this
26459 last example, Calc is forced to rewrite the rule to @samp{f(x-1, temp)
26460 := g(x) :: temp = x+1} where the @samp{x-1} term must correspond to an
26461 actual ``something-minus-one'' in the target formula.
26463 A successful way to write this would be @samp{f(x, x+2) := g(x+1)}.
26464 You could make this resemble the original form more closely by using
26465 @code{let} notation, which is described in the next section:
26468 f(xm1, x+1) := g(x) :: let(x := xm1+1)
26471 Calc does this rewriting or ``conditionalizing'' for any sub-pattern
26472 which involves only the functions in the following list, operating
26473 only on constants and meta-variables which have already been matched
26474 elsewhere in the pattern. When matching a function call, Calc is
26475 careful to match arguments which are plain variables before arguments
26476 which are calls to any of the functions below, so that a pattern like
26477 @samp{f(x-1, x)} can be conditionalized even though the isolated
26478 @samp{x} comes after the @samp{x-1}.
26481 + - * / \ % ^ abs sign round rounde roundu trunc floor ceil
26482 max min re im conj arg
26485 You can suppress all of the special treatments described in this
26486 section by surrounding a function call with a @code{plain} marker.
26487 This marker causes the function call which is its argument to be
26488 matched literally, without regard to commutativity, associativity,
26489 negation, or conditionalization. When you use @code{plain}, the
26490 ``deep structure'' of the formula being matched can show through.
26494 plain(a - a b) := f(a, b)
26498 will match only literal subtractions. However, the @code{plain}
26499 marker does not affect its arguments' arguments. In this case,
26500 commutativity and associativity is still considered while matching
26501 the @w{@samp{a b}} sub-pattern, so the whole pattern will match
26502 @samp{x - y x} as well as @samp{x - x y}. We could go still
26506 plain(a - plain(a b)) := f(a, b)
26510 which would do a completely strict match for the pattern.
26512 By contrast, the @code{quote} marker means that not only the
26513 function name but also the arguments must be literally the same.
26514 The above pattern will match @samp{x - x y} but
26517 quote(a - a b) := f(a, b)
26521 will match only the single formula @samp{a - a b}. Also,
26524 quote(a - quote(a b)) := f(a, b)
26528 will match only @samp{a - quote(a b)}---probably not the desired
26531 A certain amount of algebra is also done when substituting the
26532 meta-variables on the righthand side of a rule. For example,
26536 a + f(b) := f(a + b)
26540 matching @samp{f(x) - y} would produce @samp{f((-y) + x)} if
26541 taken literally, but the rewrite mechanism will simplify the
26542 righthand side to @samp{f(x - y)} automatically. (Of course,
26543 the default simplifications would do this anyway, so this
26544 special simplification is only noticeable if you have turned the
26545 default simplifications off.) This rewriting is done only when
26546 a meta-variable expands to a ``negative-looking'' expression.
26547 If this simplification is not desirable, you can use a @code{plain}
26548 marker on the righthand side:
26551 a + f(b) := f(plain(a + b))
26555 In this example, we are still allowing the pattern-matcher to
26556 use all the algebra it can muster, but the righthand side will
26557 always simplify to a literal addition like @samp{f((-y) + x)}.
26559 @node Other Features of Rewrite Rules, Composing Patterns in Rewrite Rules, Algebraic Properties of Rewrite Rules, Rewrite Rules
26560 @subsection Other Features of Rewrite Rules
26563 Certain ``function names'' serve as markers in rewrite rules.
26564 Here is a complete list of these markers. First are listed the
26565 markers that work inside a pattern; then come the markers that
26566 work in the righthand side of a rule.
26572 One kind of marker, @samp{import(x)}, takes the place of a whole
26573 rule. Here @expr{x} is the name of a variable containing another
26574 rule set; those rules are ``spliced into'' the rule set that
26575 imports them. For example, if @samp{[f(a+b) := f(a) + f(b),
26576 f(a b) := a f(b) :: real(a)]} is stored in variable @samp{linearF},
26577 then the rule set @samp{[f(0) := 0, import(linearF)]} will apply
26578 all three rules. It is possible to modify the imported rules
26579 slightly: @samp{import(x, v1, x1, v2, x2, @dots{})} imports
26580 the rule set @expr{x} with all occurrences of
26581 @texline @math{v_1},
26582 @infoline @expr{v1},
26583 as either a variable name or a function name, replaced with
26584 @texline @math{x_1}
26585 @infoline @expr{x1}
26587 @texline @math{v_1}
26588 @infoline @expr{v1}
26589 is used as a function name, then
26590 @texline @math{x_1}
26591 @infoline @expr{x1}
26592 must be either a function name itself or a @w{@samp{< >}} nameless
26593 function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0,
26594 import(linearF, f, g)]} applies the linearity rules to the function
26595 @samp{g} instead of @samp{f}. Imports can be nested, but the
26596 import-with-renaming feature may fail to rename sub-imports properly.
26598 The special functions allowed in patterns are:
26606 This pattern matches exactly @expr{x}; variable names in @expr{x} are
26607 not interpreted as meta-variables. The only flexibility is that
26608 numbers are compared for numeric equality, so that the pattern
26609 @samp{f(quote(12))} will match both @samp{f(12)} and @samp{f(12.0)}.
26610 (Numbers are always treated this way by the rewrite mechanism:
26611 The rule @samp{f(x,x) := g(x)} will match @samp{f(12, 12.0)}.
26612 The rewrite may produce either @samp{g(12)} or @samp{g(12.0)}
26613 as a result in this case.)
26620 Here @expr{x} must be a function call @samp{f(x1,x2,@dots{})}. This
26621 pattern matches a call to function @expr{f} with the specified
26622 argument patterns. No special knowledge of the properties of the
26623 function @expr{f} is used in this case; @samp{+} is not commutative or
26624 associative. Unlike @code{quote}, the arguments @samp{x1,x2,@dots{}}
26625 are treated as patterns. If you wish them to be treated ``plainly''
26626 as well, you must enclose them with more @code{plain} markers:
26627 @samp{plain(plain(@w{-a}) + plain(b c))}.
26634 Here @expr{x} must be a variable name. This must appear as an
26635 argument to a function or an element of a vector; it specifies that
26636 the argument or element is optional.
26637 As an argument to @samp{+}, @samp{-}, @samp{*}, @samp{&&}, or @samp{||},
26638 or as the second argument to @samp{/} or @samp{^}, the value @var{def}
26639 may be omitted. The pattern @samp{x + opt(y)} matches a sum by
26640 binding one summand to @expr{x} and the other to @expr{y}, and it
26641 matches anything else by binding the whole expression to @expr{x} and
26642 zero to @expr{y}. The other operators above work similarly.
26644 For general miscellaneous functions, the default value @code{def}
26645 must be specified. Optional arguments are dropped starting with
26646 the rightmost one during matching. For example, the pattern
26647 @samp{f(opt(a,0), b, opt(c,b))} will match @samp{f(b)}, @samp{f(a,b)},
26648 or @samp{f(a,b,c)}. Default values of zero and @expr{b} are
26649 supplied in this example for the omitted arguments. Note that
26650 the literal variable @expr{b} will be the default in the latter
26651 case, @emph{not} the value that matched the meta-variable @expr{b}.
26652 In other words, the default @var{def} is effectively quoted.
26654 @item condition(x,c)
26660 This matches the pattern @expr{x}, with the attached condition
26661 @expr{c}. It is the same as @samp{x :: c}.
26669 This matches anything that matches both pattern @expr{x} and
26670 pattern @expr{y}. It is the same as @samp{x &&& y}.
26671 @pxref{Composing Patterns in Rewrite Rules}.
26679 This matches anything that matches either pattern @expr{x} or
26680 pattern @expr{y}. It is the same as @w{@samp{x ||| y}}.
26688 This matches anything that does not match pattern @expr{x}.
26689 It is the same as @samp{!!! x}.
26695 @tindex cons (rewrites)
26696 This matches any vector of one or more elements. The first
26697 element is matched to @expr{h}; a vector of the remaining
26698 elements is matched to @expr{t}. Note that vectors of fixed
26699 length can also be matched as actual vectors: The rule
26700 @samp{cons(a,cons(b,[])) := cons(a+b,[])} is equivalent
26701 to the rule @samp{[a,b] := [a+b]}.
26707 @tindex rcons (rewrites)
26708 This is like @code{cons}, except that the @emph{last} element
26709 is matched to @expr{h}, with the remaining elements matched
26712 @item apply(f,args)
26716 @tindex apply (rewrites)
26717 This matches any function call. The name of the function, in
26718 the form of a variable, is matched to @expr{f}. The arguments
26719 of the function, as a vector of zero or more objects, are
26720 matched to @samp{args}. Constants, variables, and vectors
26721 do @emph{not} match an @code{apply} pattern. For example,
26722 @samp{apply(f,x)} matches any function call, @samp{apply(quote(f),x)}
26723 matches any call to the function @samp{f}, @samp{apply(f,[a,b])}
26724 matches any function call with exactly two arguments, and
26725 @samp{apply(quote(f), cons(a,cons(b,x)))} matches any call
26726 to the function @samp{f} with two or more arguments. Another
26727 way to implement the latter, if the rest of the rule does not
26728 need to refer to the first two arguments of @samp{f} by name,
26729 would be @samp{apply(quote(f), x :: vlen(x) >= 2)}.
26730 Here's a more interesting sample use of @code{apply}:
26733 apply(f,[x+n]) := n + apply(f,[x])
26734 :: in(f, [floor,ceil,round,trunc]) :: integer(n)
26737 Note, however, that this will be slower to match than a rule
26738 set with four separate rules. The reason is that Calc sorts
26739 the rules of a rule set according to top-level function name;
26740 if the top-level function is @code{apply}, Calc must try the
26741 rule for every single formula and sub-formula. If the top-level
26742 function in the pattern is, say, @code{floor}, then Calc invokes
26743 the rule only for sub-formulas which are calls to @code{floor}.
26745 Formulas normally written with operators like @code{+} are still
26746 considered function calls: @code{apply(f,x)} matches @samp{a+b}
26747 with @samp{f = add}, @samp{x = [a,b]}.
26749 You must use @code{apply} for meta-variables with function names
26750 on both sides of a rewrite rule: @samp{apply(f, [x]) := f(x+1)}
26751 is @emph{not} correct, because it rewrites @samp{spam(6)} into
26752 @samp{f(7)}. The righthand side should be @samp{apply(f, [x+1])}.
26753 Also note that you will have to use No-Simplify mode (@kbd{m O})
26754 when entering this rule so that the @code{apply} isn't
26755 evaluated immediately to get the new rule @samp{f(x) := f(x+1)}.
26756 Or, use @kbd{s e} to enter the rule without going through the stack,
26757 or enter the rule as @samp{apply(f, [x]) := apply(f, [x+1]) @w{:: 1}}.
26758 @xref{Conditional Rewrite Rules}.
26765 This is used for applying rules to formulas with selections;
26766 @pxref{Selections with Rewrite Rules}.
26769 Special functions for the righthand sides of rules are:
26773 The notation @samp{quote(x)} is changed to @samp{x} when the
26774 righthand side is used. As far as the rewrite rule is concerned,
26775 @code{quote} is invisible. However, @code{quote} has the special
26776 property in Calc that its argument is not evaluated. Thus,
26777 while it will not work to put the rule @samp{t(a) := typeof(a)}
26778 on the stack because @samp{typeof(a)} is evaluated immediately
26779 to produce @samp{t(a) := 100}, you can use @code{quote} to
26780 protect the righthand side: @samp{t(a) := quote(typeof(a))}.
26781 (@xref{Conditional Rewrite Rules}, for another trick for
26782 protecting rules from evaluation.)
26785 Special properties of and simplifications for the function call
26786 @expr{x} are not used. One interesting case where @code{plain}
26787 is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a
26788 shorthand notation for the @code{quote} function. This rule will
26789 not work as shown; instead of replacing @samp{q(foo)} with
26790 @samp{quote(foo)}, it will replace it with @samp{foo}! The correct
26791 rule would be @samp{q(x) := plain(quote(x))}.
26794 Where @expr{t} is a vector, this is converted into an expanded
26795 vector during rewrite processing. Note that @code{cons} is a regular
26796 Calc function which normally does this anyway; the only way @code{cons}
26797 is treated specially by rewrites is that @code{cons} on the righthand
26798 side of a rule will be evaluated even if default simplifications
26799 have been turned off.
26802 Analogous to @code{cons} except putting @expr{h} at the @emph{end} of
26803 the vector @expr{t}.
26805 @item apply(f,args)
26806 Where @expr{f} is a variable and @var{args} is a vector, this
26807 is converted to a function call. Once again, note that @code{apply}
26808 is also a regular Calc function.
26815 The formula @expr{x} is handled in the usual way, then the
26816 default simplifications are applied to it even if they have
26817 been turned off normally. This allows you to treat any function
26818 similarly to the way @code{cons} and @code{apply} are always
26819 treated. However, there is a slight difference: @samp{cons(2+3, [])}
26820 with default simplifications off will be converted to @samp{[2+3]},
26821 whereas @samp{eval(cons(2+3, []))} will be converted to @samp{[5]}.
26828 The formula @expr{x} has meta-variables substituted in the usual
26829 way, then algebraically simplified.
26831 @item evalextsimp(x)
26835 @tindex evalextsimp
26836 The formula @expr{x} has meta-variables substituted in the normal
26837 way, then ``extendedly'' simplified as if by the @kbd{a e} command.
26840 @xref{Selections with Rewrite Rules}.
26843 There are also some special functions you can use in conditions.
26851 The expression @expr{x} is evaluated with meta-variables substituted.
26852 The algebraic simplifications are @emph{not} applied by
26853 default, but @expr{x} can include calls to @code{evalsimp} or
26854 @code{evalextsimp} as described above to invoke higher levels
26855 of simplification. The result of @expr{x} is then bound to the
26856 meta-variable @expr{v}. As usual, if this meta-variable has already
26857 been matched to something else the two values must be equal; if the
26858 meta-variable is new then it is bound to the result of the expression.
26859 This variable can then appear in later conditions, and on the righthand
26861 In fact, @expr{v} may be any pattern in which case the result of
26862 evaluating @expr{x} is matched to that pattern, binding any
26863 meta-variables that appear in that pattern. Note that @code{let}
26864 can only appear by itself as a condition, or as one term of an
26865 @samp{&&} which is a whole condition: It cannot be inside
26866 an @samp{||} term or otherwise buried.
26868 The alternate, equivalent form @samp{let(v, x)} is also recognized.
26869 Note that the use of @samp{:=} by @code{let}, while still being
26870 assignment-like in character, is unrelated to the use of @samp{:=}
26871 in the main part of a rewrite rule.
26873 As an example, @samp{f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)}
26874 replaces @samp{f(a)} with @samp{g} of the inverse of @samp{a}, if
26875 that inverse exists and is constant. For example, if @samp{a} is a
26876 singular matrix the operation @samp{1/a} is left unsimplified and
26877 @samp{constant(ia)} fails, but if @samp{a} is an invertible matrix
26878 then the rule succeeds. Without @code{let} there would be no way
26879 to express this rule that didn't have to invert the matrix twice.
26880 Note that, because the meta-variable @samp{ia} is otherwise unbound
26881 in this rule, the @code{let} condition itself always ``succeeds''
26882 because no matter what @samp{1/a} evaluates to, it can successfully
26883 be bound to @code{ia}.
26885 Here's another example, for integrating cosines of linear
26886 terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}.
26887 The @code{lin} function returns a 3-vector if its argument is linear,
26888 or leaves itself unevaluated if not. But an unevaluated @code{lin}
26889 call will not match the 3-vector on the lefthand side of the @code{let},
26890 so this @code{let} both verifies that @code{y} is linear, and binds
26891 the coefficients @code{a} and @code{b} for use elsewhere in the rule.
26892 (It would have been possible to use @samp{sin(a x + b)/b} for the
26893 righthand side instead, but using @samp{sin(y)/b} avoids gratuitous
26894 rearrangement of the argument of the sine.)
26900 Similarly, here is a rule that implements an inverse-@code{erf}
26901 function. It uses @code{root} to search for a solution. If
26902 @code{root} succeeds, it will return a vector of two numbers
26903 where the first number is the desired solution. If no solution
26904 is found, @code{root} remains in symbolic form. So we use
26905 @code{let} to check that the result was indeed a vector.
26908 ierf(x) := y :: let([y,z] := root(erf(a) = x, a, .5))
26912 The meta-variable @var{v}, which must already have been matched
26913 to something elsewhere in the rule, is compared against pattern
26914 @var{p}. Since @code{matches} is a standard Calc function, it
26915 can appear anywhere in a condition. But if it appears alone or
26916 as a term of a top-level @samp{&&}, then you get the special
26917 extra feature that meta-variables which are bound to things
26918 inside @var{p} can be used elsewhere in the surrounding rewrite
26921 The only real difference between @samp{let(p := v)} and
26922 @samp{matches(v, p)} is that the former evaluates @samp{v} using
26923 the default simplifications, while the latter does not.
26927 This is actually a variable, not a function. If @code{remember}
26928 appears as a condition in a rule, then when that rule succeeds
26929 the original expression and rewritten expression are added to the
26930 front of the rule set that contained the rule. If the rule set
26931 was not stored in a variable, @code{remember} is ignored. The
26932 lefthand side is enclosed in @code{quote} in the added rule if it
26933 contains any variables.
26935 For example, the rule @samp{f(n) := n f(n-1) :: remember} applied
26936 to @samp{f(7)} will add the rule @samp{f(7) := 7 f(6)} to the front
26937 of the rule set. The rule set @code{EvalRules} works slightly
26938 differently: There, the evaluation of @samp{f(6)} will complete before
26939 the result is added to the rule set, in this case as @samp{f(7) := 5040}.
26940 Thus @code{remember} is most useful inside @code{EvalRules}.
26942 It is up to you to ensure that the optimization performed by
26943 @code{remember} is safe. For example, the rule @samp{foo(n) := n
26944 :: evalv(eatfoo) > 0 :: remember} is a bad idea (@code{evalv} is
26945 the function equivalent of the @kbd{=} command); if the variable
26946 @code{eatfoo} ever contains 1, rules like @samp{foo(7) := 7} will
26947 be added to the rule set and will continue to operate even if
26948 @code{eatfoo} is later changed to 0.
26955 Remember the match as described above, but only if condition @expr{c}
26956 is true. For example, @samp{remember(n % 4 = 0)} in the above factorial
26957 rule remembers only every fourth result. Note that @samp{remember(1)}
26958 is equivalent to @samp{remember}, and @samp{remember(0)} has no effect.
26961 @node Composing Patterns in Rewrite Rules, Nested Formulas with Rewrite Rules, Other Features of Rewrite Rules, Rewrite Rules
26962 @subsection Composing Patterns in Rewrite Rules
26965 There are three operators, @samp{&&&}, @samp{|||}, and @samp{!!!},
26966 that combine rewrite patterns to make larger patterns. The
26967 combinations are ``and,'' ``or,'' and ``not,'' respectively, and
26968 these operators are the pattern equivalents of @samp{&&}, @samp{||}
26969 and @samp{!} (which operate on zero-or-nonzero logical values).
26971 Note that @samp{&&&}, @samp{|||}, and @samp{!!!} are left in symbolic
26972 form by all regular Calc features; they have special meaning only in
26973 the context of rewrite rule patterns.
26975 The pattern @samp{@var{p1} &&& @var{p2}} matches anything that
26976 matches both @var{p1} and @var{p2}. One especially useful case is
26977 when one of @var{p1} or @var{p2} is a meta-variable. For example,
26978 here is a rule that operates on error forms:
26981 f(x &&& a +/- b, x) := g(x)
26984 This does the same thing, but is arguably simpler than, the rule
26987 f(a +/- b, a +/- b) := g(a +/- b)
26994 Here's another interesting example:
26997 ends(cons(a, x) &&& rcons(y, b)) := [a, b]
27001 which effectively clips out the middle of a vector leaving just
27002 the first and last elements. This rule will change a one-element
27003 vector @samp{[a]} to @samp{[a, a]}. The similar rule
27006 ends(cons(a, rcons(y, b))) := [a, b]
27010 would do the same thing except that it would fail to match a
27011 one-element vector.
27017 The pattern @samp{@var{p1} ||| @var{p2}} matches anything that
27018 matches either @var{p1} or @var{p2}. Calc first tries matching
27019 against @var{p1}; if that fails, it goes on to try @var{p2}.
27025 A simple example of @samp{|||} is
27028 curve(inf ||| -inf) := 0
27032 which converts both @samp{curve(inf)} and @samp{curve(-inf)} to zero.
27034 Here is a larger example:
27037 log(a, b) ||| (ln(a) :: let(b := e)) := mylog(a, b)
27040 This matches both generalized and natural logarithms in a single rule.
27041 Note that the @samp{::} term must be enclosed in parentheses because
27042 that operator has lower precedence than @samp{|||} or @samp{:=}.
27044 (In practice this rule would probably include a third alternative,
27045 omitted here for brevity, to take care of @code{log10}.)
27047 While Calc generally treats interior conditions exactly the same as
27048 conditions on the outside of a rule, it does guarantee that if all the
27049 variables in the condition are special names like @code{e}, or already
27050 bound in the pattern to which the condition is attached (say, if
27051 @samp{a} had appeared in this condition), then Calc will process this
27052 condition right after matching the pattern to the left of the @samp{::}.
27053 Thus, we know that @samp{b} will be bound to @samp{e} only if the
27054 @code{ln} branch of the @samp{|||} was taken.
27056 Note that this rule was careful to bind the same set of meta-variables
27057 on both sides of the @samp{|||}. Calc does not check this, but if
27058 you bind a certain meta-variable only in one branch and then use that
27059 meta-variable elsewhere in the rule, results are unpredictable:
27062 f(a,b) ||| g(b) := h(a,b)
27065 Here if the pattern matches @samp{g(17)}, Calc makes no promises about
27066 the value that will be substituted for @samp{a} on the righthand side.
27072 The pattern @samp{!!! @var{pat}} matches anything that does not
27073 match @var{pat}. Any meta-variables that are bound while matching
27074 @var{pat} remain unbound outside of @var{pat}.
27079 f(x &&& !!! a +/- b, !!![]) := g(x)
27083 converts @code{f} whose first argument is anything @emph{except} an
27084 error form, and whose second argument is not the empty vector, into
27085 a similar call to @code{g} (but without the second argument).
27087 If we know that the second argument will be a vector (empty or not),
27088 then an equivalent rule would be:
27091 f(x, y) := g(x) :: typeof(x) != 7 :: vlen(y) > 0
27095 where of course 7 is the @code{typeof} code for error forms.
27096 Another final condition, that works for any kind of @samp{y},
27097 would be @samp{!istrue(y == [])}. (The @code{istrue} function
27098 returns an explicit 0 if its argument was left in symbolic form;
27099 plain @samp{!(y == [])} or @samp{y != []} would not work to replace
27100 @samp{!!![]} since these would be left unsimplified, and thus cause
27101 the rule to fail, if @samp{y} was something like a variable name.)
27103 It is possible for a @samp{!!!} to refer to meta-variables bound
27104 elsewhere in the pattern. For example,
27111 matches any call to @code{f} with different arguments, changing
27112 this to @code{g} with only the first argument.
27114 If a function call is to be matched and one of the argument patterns
27115 contains a @samp{!!!} somewhere inside it, that argument will be
27123 will be careful to bind @samp{a} to the second argument of @code{f}
27124 before testing the first argument. If Calc had tried to match the
27125 first argument of @code{f} first, the results would have been
27126 disastrous: since @code{a} was unbound so far, the pattern @samp{a}
27127 would have matched anything at all, and the pattern @samp{!!!a}
27128 therefore would @emph{not} have matched anything at all!
27130 @node Nested Formulas with Rewrite Rules, Multi-Phase Rewrite Rules, Composing Patterns in Rewrite Rules, Rewrite Rules
27131 @subsection Nested Formulas with Rewrite Rules
27134 When @kbd{a r} (@code{calc-rewrite}) is used, it takes an expression from
27135 the top of the stack and attempts to match any of the specified rules
27136 to any part of the expression, starting with the whole expression
27137 and then, if that fails, trying deeper and deeper sub-expressions.
27138 For each part of the expression, the rules are tried in the order
27139 they appear in the rules vector. The first rule to match the first
27140 sub-expression wins; it replaces the matched sub-expression according
27141 to the @var{new} part of the rule.
27143 Often, the rule set will match and change the formula several times.
27144 The top-level formula is first matched and substituted repeatedly until
27145 it no longer matches the pattern; then, sub-formulas are tried, and
27146 so on. Once every part of the formula has gotten its chance, the
27147 rewrite mechanism starts over again with the top-level formula
27148 (in case a substitution of one of its arguments has caused it again
27149 to match). This continues until no further matches can be made
27150 anywhere in the formula.
27152 It is possible for a rule set to get into an infinite loop. The
27153 most obvious case, replacing a formula with itself, is not a problem
27154 because a rule is not considered to ``succeed'' unless the righthand
27155 side actually comes out to something different than the original
27156 formula or sub-formula that was matched. But if you accidentally
27157 had both @samp{ln(a b) := ln(a) + ln(b)} and the reverse
27158 @samp{ln(a) + ln(b) := ln(a b)} in your rule set, Calc would
27159 run forever switching a formula back and forth between the two
27162 To avoid disaster, Calc normally stops after 100 changes have been
27163 made to the formula. This will be enough for most multiple rewrites,
27164 but it will keep an endless loop of rewrites from locking up the
27165 computer forever. (On most systems, you can also type @kbd{C-g} to
27166 halt any Emacs command prematurely.)
27168 To change this limit, give a positive numeric prefix argument.
27169 In particular, @kbd{M-1 a r} applies only one rewrite at a time,
27170 useful when you are first testing your rule (or just if repeated
27171 rewriting is not what is called for by your application).
27180 You can also put a ``function call'' @samp{iterations(@var{n})}
27181 in place of a rule anywhere in your rules vector (but usually at
27182 the top). Then, @var{n} will be used instead of 100 as the default
27183 number of iterations for this rule set. You can use
27184 @samp{iterations(inf)} if you want no iteration limit by default.
27185 A prefix argument will override the @code{iterations} limit in the
27193 More precisely, the limit controls the number of ``iterations,''
27194 where each iteration is a successful matching of a rule pattern whose
27195 righthand side, after substituting meta-variables and applying the
27196 default simplifications, is different from the original sub-formula
27199 A prefix argument of zero sets the limit to infinity. Use with caution!
27201 Given a negative numeric prefix argument, @kbd{a r} will match and
27202 substitute the top-level expression up to that many times, but
27203 will not attempt to match the rules to any sub-expressions.
27205 In a formula, @code{rewrite(@var{expr}, @var{rules}, @var{n})}
27206 does a rewriting operation. Here @var{expr} is the expression
27207 being rewritten, @var{rules} is the rule, vector of rules, or
27208 variable containing the rules, and @var{n} is the optional
27209 iteration limit, which may be a positive integer, a negative
27210 integer, or @samp{inf} or @samp{-inf}. If @var{n} is omitted
27211 the @code{iterations} value from the rule set is used; if both
27212 are omitted, 100 is used.
27214 @node Multi-Phase Rewrite Rules, Selections with Rewrite Rules, Nested Formulas with Rewrite Rules, Rewrite Rules
27215 @subsection Multi-Phase Rewrite Rules
27218 It is possible to separate a rewrite rule set into several @dfn{phases}.
27219 During each phase, certain rules will be enabled while certain others
27220 will be disabled. A @dfn{phase schedule} controls the order in which
27221 phases occur during the rewriting process.
27228 If a call to the marker function @code{phase} appears in the rules
27229 vector in place of a rule, all rules following that point will be
27230 members of the phase(s) identified in the arguments to @code{phase}.
27231 Phases are given integer numbers. The markers @samp{phase()} and
27232 @samp{phase(all)} both mean the following rules belong to all phases;
27233 this is the default at the start of the rule set.
27235 If you do not explicitly schedule the phases, Calc sorts all phase
27236 numbers that appear in the rule set and executes the phases in
27237 ascending order. For example, the rule set
27254 has three phases, 1 through 3. Phase 1 consists of the @code{f0},
27255 @code{f1}, and @code{f4} rules (in that order). Phase 2 consists of
27256 @code{f0}, @code{f2}, and @code{f4}. Phase 3 consists of @code{f0}
27259 When Calc rewrites a formula using this rule set, it first rewrites
27260 the formula using only the phase 1 rules until no further changes are
27261 possible. Then it switches to the phase 2 rule set and continues
27262 until no further changes occur, then finally rewrites with phase 3.
27263 When no more phase 3 rules apply, rewriting finishes. (This is
27264 assuming @kbd{a r} with a large enough prefix argument to allow the
27265 rewriting to run to completion; the sequence just described stops
27266 early if the number of iterations specified in the prefix argument,
27267 100 by default, is reached.)
27269 During each phase, Calc descends through the nested levels of the
27270 formula as described previously. (@xref{Nested Formulas with Rewrite
27271 Rules}.) Rewriting starts at the top of the formula, then works its
27272 way down to the parts, then goes back to the top and works down again.
27273 The phase 2 rules do not begin until no phase 1 rules apply anywhere
27280 A @code{schedule} marker appearing in the rule set (anywhere, but
27281 conventionally at the top) changes the default schedule of phases.
27282 In the simplest case, @code{schedule} has a sequence of phase numbers
27283 for arguments; each phase number is invoked in turn until the
27284 arguments to @code{schedule} are exhausted. Thus adding
27285 @samp{schedule(3,2,1)} at the top of the above rule set would
27286 reverse the order of the phases; @samp{schedule(1,2,3)} would have
27287 no effect since this is the default schedule; and @samp{schedule(1,2,1,3)}
27288 would give phase 1 a second chance after phase 2 has completed, before
27289 moving on to phase 3.
27291 Any argument to @code{schedule} can instead be a vector of phase
27292 numbers (or even of sub-vectors). Then the sub-sequence of phases
27293 described by the vector are tried repeatedly until no change occurs
27294 in any phase in the sequence. For example, @samp{schedule([1, 2], 3)}
27295 tries phase 1, then phase 2, then, if either phase made any changes
27296 to the formula, repeats these two phases until they can make no
27297 further progress. Finally, it goes on to phase 3 for finishing
27300 Also, items in @code{schedule} can be variable names as well as
27301 numbers. A variable name is interpreted as the name of a function
27302 to call on the whole formula. For example, @samp{schedule(1, simplify)}
27303 says to apply the phase-1 rules (presumably, all of them), then to
27304 call @code{simplify} which is the function name equivalent of @kbd{a s}.
27305 Likewise, @samp{schedule([1, simplify])} says to alternate between
27306 phase 1 and @kbd{a s} until no further changes occur.
27308 Phases can be used purely to improve efficiency; if it is known that
27309 a certain group of rules will apply only at the beginning of rewriting,
27310 and a certain other group will apply only at the end, then rewriting
27311 will be faster if these groups are identified as separate phases.
27312 Once the phase 1 rules are done, Calc can put them aside and no longer
27313 spend any time on them while it works on phase 2.
27315 There are also some problems that can only be solved with several
27316 rewrite phases. For a real-world example of a multi-phase rule set,
27317 examine the set @code{FitRules}, which is used by the curve-fitting
27318 command to convert a model expression to linear form.
27319 @xref{Curve Fitting Details}. This set is divided into four phases.
27320 The first phase rewrites certain kinds of expressions to be more
27321 easily linearizable, but less computationally efficient. After the
27322 linear components have been picked out, the final phase includes the
27323 opposite rewrites to put each component back into an efficient form.
27324 If both sets of rules were included in one big phase, Calc could get
27325 into an infinite loop going back and forth between the two forms.
27327 Elsewhere in @code{FitRules}, the components are first isolated,
27328 then recombined where possible to reduce the complexity of the linear
27329 fit, then finally packaged one component at a time into vectors.
27330 If the packaging rules were allowed to begin before the recombining
27331 rules were finished, some components might be put away into vectors
27332 before they had a chance to recombine. By putting these rules in
27333 two separate phases, this problem is neatly avoided.
27335 @node Selections with Rewrite Rules, Matching Commands, Multi-Phase Rewrite Rules, Rewrite Rules
27336 @subsection Selections with Rewrite Rules
27339 If a sub-formula of the current formula is selected (as by @kbd{j s};
27340 @pxref{Selecting Subformulas}), the @kbd{a r} (@code{calc-rewrite})
27341 command applies only to that sub-formula. Together with a negative
27342 prefix argument, you can use this fact to apply a rewrite to one
27343 specific part of a formula without affecting any other parts.
27346 @pindex calc-rewrite-selection
27347 The @kbd{j r} (@code{calc-rewrite-selection}) command allows more
27348 sophisticated operations on selections. This command prompts for
27349 the rules in the same way as @kbd{a r}, but it then applies those
27350 rules to the whole formula in question even though a sub-formula
27351 of it has been selected. However, the selected sub-formula will
27352 first have been surrounded by a @samp{select( )} function call.
27353 (Calc's evaluator does not understand the function name @code{select};
27354 this is only a tag used by the @kbd{j r} command.)
27356 For example, suppose the formula on the stack is @samp{2 (a + b)^2}
27357 and the sub-formula @samp{a + b} is selected. This formula will
27358 be rewritten to @samp{2 select(a + b)^2} and then the rewrite
27359 rules will be applied in the usual way. The rewrite rules can
27360 include references to @code{select} to tell where in the pattern
27361 the selected sub-formula should appear.
27363 If there is still exactly one @samp{select( )} function call in
27364 the formula after rewriting is done, it indicates which part of
27365 the formula should be selected afterwards. Otherwise, the
27366 formula will be unselected.
27368 You can make @kbd{j r} act much like @kbd{a r} by enclosing both parts
27369 of the rewrite rule with @samp{select()}. However, @kbd{j r}
27370 allows you to use the current selection in more flexible ways.
27371 Suppose you wished to make a rule which removed the exponent from
27372 the selected term; the rule @samp{select(a)^x := select(a)} would
27373 work. In the above example, it would rewrite @samp{2 select(a + b)^2}
27374 to @samp{2 select(a + b)}. This would then be returned to the
27375 stack as @samp{2 (a + b)} with the @samp{a + b} selected.
27377 The @kbd{j r} command uses one iteration by default, unlike
27378 @kbd{a r} which defaults to 100 iterations. A numeric prefix
27379 argument affects @kbd{j r} in the same way as @kbd{a r}.
27380 @xref{Nested Formulas with Rewrite Rules}.
27382 As with other selection commands, @kbd{j r} operates on the stack
27383 entry that contains the cursor. (If the cursor is on the top-of-stack
27384 @samp{.} marker, it works as if the cursor were on the formula
27387 If you don't specify a set of rules, the rules are taken from the
27388 top of the stack, just as with @kbd{a r}. In this case, the
27389 cursor must indicate stack entry 2 or above as the formula to be
27390 rewritten (otherwise the same formula would be used as both the
27391 target and the rewrite rules).
27393 If the indicated formula has no selection, the cursor position within
27394 the formula temporarily selects a sub-formula for the purposes of this
27395 command. If the cursor is not on any sub-formula (e.g., it is in
27396 the line-number area to the left of the formula), the @samp{select( )}
27397 markers are ignored by the rewrite mechanism and the rules are allowed
27398 to apply anywhere in the formula.
27400 As a special feature, the normal @kbd{a r} command also ignores
27401 @samp{select( )} calls in rewrite rules. For example, if you used the
27402 above rule @samp{select(a)^x := select(a)} with @kbd{a r}, it would apply
27403 the rule as if it were @samp{a^x := a}. Thus, you can write general
27404 purpose rules with @samp{select( )} hints inside them so that they
27405 will ``do the right thing'' in both @kbd{a r} and @kbd{j r},
27406 both with and without selections.
27408 @node Matching Commands, Automatic Rewrites, Selections with Rewrite Rules, Rewrite Rules
27409 @subsection Matching Commands
27415 The @kbd{a m} (@code{calc-match}) [@code{match}] function takes a
27416 vector of formulas and a rewrite-rule-style pattern, and produces
27417 a vector of all formulas which match the pattern. The command
27418 prompts you to enter the pattern; as for @kbd{a r}, you can enter
27419 a single pattern (i.e., a formula with meta-variables), or a
27420 vector of patterns, or a variable which contains patterns, or
27421 you can give a blank response in which case the patterns are taken
27422 from the top of the stack. The pattern set will be compiled once
27423 and saved if it is stored in a variable. If there are several
27424 patterns in the set, vector elements are kept if they match any
27427 For example, @samp{match(a+b, [x, x+y, x-y, 7, x+y+z])}
27428 will return @samp{[x+y, x-y, x+y+z]}.
27430 The @code{import} mechanism is not available for pattern sets.
27432 The @kbd{a m} command can also be used to extract all vector elements
27433 which satisfy any condition: The pattern @samp{x :: x>0} will select
27434 all the positive vector elements.
27438 With the Inverse flag [@code{matchnot}], this command extracts all
27439 vector elements which do @emph{not} match the given pattern.
27445 There is also a function @samp{matches(@var{x}, @var{p})} which
27446 evaluates to 1 if expression @var{x} matches pattern @var{p}, or
27447 to 0 otherwise. This is sometimes useful for including into the
27448 conditional clauses of other rewrite rules.
27454 The function @code{vmatches} is just like @code{matches}, except
27455 that if the match succeeds it returns a vector of assignments to
27456 the meta-variables instead of the number 1. For example,
27457 @samp{vmatches(f(1,2), f(a,b))} returns @samp{[a := 1, b := 2]}.
27458 If the match fails, the function returns the number 0.
27460 @node Automatic Rewrites, Debugging Rewrites, Matching Commands, Rewrite Rules
27461 @subsection Automatic Rewrites
27464 @cindex @code{EvalRules} variable
27466 It is possible to get Calc to apply a set of rewrite rules on all
27467 results, effectively adding to the built-in set of default
27468 simplifications. To do this, simply store your rule set in the
27469 variable @code{EvalRules}. There is a convenient @kbd{s E} command
27470 for editing @code{EvalRules}; @pxref{Operations on Variables}.
27472 For example, suppose you want @samp{sin(a + b)} to be expanded out
27473 to @samp{sin(b) cos(a) + cos(b) sin(a)} wherever it appears, and
27474 similarly for @samp{cos(a + b)}. The corresponding rewrite rule
27479 [ sin(a + b) := cos(a) sin(b) + sin(a) cos(b),
27480 cos(a + b) := cos(a) cos(b) - sin(a) sin(b) ]
27484 To apply these manually, you could put them in a variable called
27485 @code{trigexp} and then use @kbd{a r trigexp} every time you wanted
27486 to expand trig functions. But if instead you store them in the
27487 variable @code{EvalRules}, they will automatically be applied to all
27488 sines and cosines of sums. Then, with @samp{2 x} and @samp{45} on
27489 the stack, typing @kbd{+ S} will (assuming Degrees mode) result in
27490 @samp{0.7071 sin(2 x) + 0.7071 cos(2 x)} automatically.
27492 As each level of a formula is evaluated, the rules from
27493 @code{EvalRules} are applied before the default simplifications.
27494 Rewriting continues until no further @code{EvalRules} apply.
27495 Note that this is different from the usual order of application of
27496 rewrite rules: @code{EvalRules} works from the bottom up, simplifying
27497 the arguments to a function before the function itself, while @kbd{a r}
27498 applies rules from the top down.
27500 Because the @code{EvalRules} are tried first, you can use them to
27501 override the normal behavior of any built-in Calc function.
27503 It is important not to write a rule that will get into an infinite
27504 loop. For example, the rule set @samp{[f(0) := 1, f(n) := n f(n-1)]}
27505 appears to be a good definition of a factorial function, but it is
27506 unsafe. Imagine what happens if @samp{f(2.5)} is simplified. Calc
27507 will continue to subtract 1 from this argument forever without reaching
27508 zero. A safer second rule would be @samp{f(n) := n f(n-1) :: n>0}.
27509 Another dangerous rule is @samp{g(x, y) := g(y, x)}. Rewriting
27510 @samp{g(2, 4)}, this would bounce back and forth between that and
27511 @samp{g(4, 2)} forever. If an infinite loop in @code{EvalRules}
27512 occurs, Emacs will eventually stop with a ``Computation got stuck
27513 or ran too long'' message.
27515 Another subtle difference between @code{EvalRules} and regular rewrites
27516 concerns rules that rewrite a formula into an identical formula. For
27517 example, @samp{f(n) := f(floor(n))} ``fails to match'' when @expr{n} is
27518 already an integer. But in @code{EvalRules} this case is detected only
27519 if the righthand side literally becomes the original formula before any
27520 further simplification. This means that @samp{f(n) := f(floor(n))} will
27521 get into an infinite loop if it occurs in @code{EvalRules}. Calc will
27522 replace @samp{f(6)} with @samp{f(floor(6))}, which is different from
27523 @samp{f(6)}, so it will consider the rule to have matched and will
27524 continue simplifying that formula; first the argument is simplified
27525 to get @samp{f(6)}, then the rule matches again to get @samp{f(floor(6))}
27526 again, ad infinitum. A much safer rule would check its argument first,
27527 say, with @samp{f(n) := f(floor(n)) :: !dint(n)}.
27529 (What really happens is that the rewrite mechanism substitutes the
27530 meta-variables in the righthand side of a rule, compares to see if the
27531 result is the same as the original formula and fails if so, then uses
27532 the default simplifications to simplify the result and compares again
27533 (and again fails if the formula has simplified back to its original
27534 form). The only special wrinkle for the @code{EvalRules} is that the
27535 same rules will come back into play when the default simplifications
27536 are used. What Calc wants to do is build @samp{f(floor(6))}, see that
27537 this is different from the original formula, simplify to @samp{f(6)},
27538 see that this is the same as the original formula, and thus halt the
27539 rewriting. But while simplifying, @samp{f(6)} will again trigger
27540 the same @code{EvalRules} rule and Calc will get into a loop inside
27541 the rewrite mechanism itself.)
27543 The @code{phase}, @code{schedule}, and @code{iterations} markers do
27544 not work in @code{EvalRules}. If the rule set is divided into phases,
27545 only the phase 1 rules are applied, and the schedule is ignored.
27546 The rules are always repeated as many times as possible.
27548 The @code{EvalRules} are applied to all function calls in a formula,
27549 but not to numbers (and other number-like objects like error forms),
27550 nor to vectors or individual variable names. (Though they will apply
27551 to @emph{components} of vectors and error forms when appropriate.) You
27552 might try to make a variable @code{phihat} which automatically expands
27553 to its definition without the need to press @kbd{=} by writing the
27554 rule @samp{quote(phihat) := (1-sqrt(5))/2}, but unfortunately this rule
27555 will not work as part of @code{EvalRules}.
27557 Finally, another limitation is that Calc sometimes calls its built-in
27558 functions directly rather than going through the default simplifications.
27559 When it does this, @code{EvalRules} will not be able to override those
27560 functions. For example, when you take the absolute value of the complex
27561 number @expr{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling
27562 the multiplication, addition, and square root functions directly rather
27563 than applying the default simplifications to this formula. So an
27564 @code{EvalRules} rule that (perversely) rewrites @samp{sqrt(13) := 6}
27565 would not apply. (However, if you put Calc into Symbolic mode so that
27566 @samp{sqrt(13)} will be left in symbolic form by the built-in square
27567 root function, your rule will be able to apply. But if the complex
27568 number were @expr{(3,4)}, so that @samp{sqrt(25)} must be calculated,
27569 then Symbolic mode will not help because @samp{sqrt(25)} can be
27570 evaluated exactly to 5.)
27572 One subtle restriction that normally only manifests itself with
27573 @code{EvalRules} is that while a given rewrite rule is in the process
27574 of being checked, that same rule cannot be recursively applied. Calc
27575 effectively removes the rule from its rule set while checking the rule,
27576 then puts it back once the match succeeds or fails. (The technical
27577 reason for this is that compiled pattern programs are not reentrant.)
27578 For example, consider the rule @samp{foo(x) := x :: foo(x/2) > 0}
27579 attempting to match @samp{foo(8)}. This rule will be inactive while
27580 the condition @samp{foo(4) > 0} is checked, even though it might be
27581 an integral part of evaluating that condition. Note that this is not
27582 a problem for the more usual recursive type of rule, such as
27583 @samp{foo(x) := foo(x/2)}, because there the rule has succeeded and
27584 been reactivated by the time the righthand side is evaluated.
27586 If @code{EvalRules} has no stored value (its default state), or if
27587 anything but a vector is stored in it, then it is ignored.
27589 Even though Calc's rewrite mechanism is designed to compare rewrite
27590 rules to formulas as quickly as possible, storing rules in
27591 @code{EvalRules} may make Calc run substantially slower. This is
27592 particularly true of rules where the top-level call is a commonly used
27593 function, or is not fixed. The rule @samp{f(n) := n f(n-1) :: n>0} will
27594 only activate the rewrite mechanism for calls to the function @code{f},
27595 but @samp{lg(n) + lg(m) := lg(n m)} will check every @samp{+} operator.
27598 apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10])
27602 may seem more ``efficient'' than two separate rules for @code{ln} and
27603 @code{log10}, but actually it is vastly less efficient because rules
27604 with @code{apply} as the top-level pattern must be tested against
27605 @emph{every} function call that is simplified.
27607 @cindex @code{AlgSimpRules} variable
27608 @vindex AlgSimpRules
27609 Suppose you want @samp{sin(a + b)} to be expanded out not all the time,
27610 but only when algebraic simplifications are used to simplify the
27611 formula. The variable @code{AlgSimpRules} holds rules for this purpose.
27612 The @kbd{a s} command will apply @code{EvalRules} and
27613 @code{AlgSimpRules} to the formula, as well as all of its built-in
27616 Most of the special limitations for @code{EvalRules} don't apply to
27617 @code{AlgSimpRules}. Calc simply does an @kbd{a r AlgSimpRules}
27618 command with an infinite repeat count as the first step of algebraic
27619 simplifications. It then applies its own built-in simplifications
27620 throughout the formula, and then repeats these two steps (along with
27621 applying the default simplifications) until no further changes are
27624 @cindex @code{ExtSimpRules} variable
27625 @cindex @code{UnitSimpRules} variable
27626 @vindex ExtSimpRules
27627 @vindex UnitSimpRules
27628 There are also @code{ExtSimpRules} and @code{UnitSimpRules} variables
27629 that are used by @kbd{a e} and @kbd{u s}, respectively; these commands
27630 also apply @code{EvalRules} and @code{AlgSimpRules}. The variable
27631 @code{IntegSimpRules} contains simplification rules that are used
27632 only during integration by @kbd{a i}.
27634 @node Debugging Rewrites, Examples of Rewrite Rules, Automatic Rewrites, Rewrite Rules
27635 @subsection Debugging Rewrites
27638 If a buffer named @samp{*Trace*} exists, the rewrite mechanism will
27639 record some useful information there as it operates. The original
27640 formula is written there, as is the result of each successful rewrite,
27641 and the final result of the rewriting. All phase changes are also
27644 Calc always appends to @samp{*Trace*}. You must empty this buffer
27645 yourself periodically if it is in danger of growing unwieldy.
27647 Note that the rewriting mechanism is substantially slower when the
27648 @samp{*Trace*} buffer exists, even if the buffer is not visible on
27649 the screen. Once you are done, you will probably want to kill this
27650 buffer (with @kbd{C-x k *Trace* @key{RET}}). If you leave it in
27651 existence and forget about it, all your future rewrite commands will
27652 be needlessly slow.
27654 @node Examples of Rewrite Rules, , Debugging Rewrites, Rewrite Rules
27655 @subsection Examples of Rewrite Rules
27658 Returning to the example of substituting the pattern
27659 @samp{sin(x)^2 + cos(x)^2} with 1, we saw that the rule
27660 @samp{opt(a) sin(x)^2 + opt(a) cos(x)^2 := a} does a good job of
27661 finding suitable cases. Another solution would be to use the rule
27662 @samp{cos(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification
27663 if necessary. This rule will be the most effective way to do the job,
27664 but at the expense of making some changes that you might not desire.
27666 Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}.
27667 To make this work with the @w{@kbd{j r}} command so that it can be
27668 easily targeted to a particular exponential in a large formula,
27669 you might wish to write the rule as @samp{select(exp(x+y)) :=
27670 select(exp(x) exp(y))}. The @samp{select} markers will be
27671 ignored by the regular @kbd{a r} command
27672 (@pxref{Selections with Rewrite Rules}).
27674 A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}.
27675 This will simplify the formula whenever @expr{b} and/or @expr{c} can
27676 be made simpler by squaring. For example, applying this rule to
27677 @samp{2 / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming
27678 Symbolic mode has been enabled to keep the square root from being
27679 evaluated to a floating-point approximation). This rule is also
27680 useful when working with symbolic complex numbers, e.g.,
27681 @samp{(a + b i) / (c + d i)}.
27683 As another example, we could define our own ``triangular numbers'' function
27684 with the rules @samp{[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]}. Enter
27685 this vector and store it in a variable: @kbd{@w{s t} trirules}. Now, given
27686 a suitable formula like @samp{tri(5)} on the stack, type @samp{a r trirules}
27687 to apply these rules repeatedly. After six applications, @kbd{a r} will
27688 stop with 15 on the stack. Once these rules are debugged, it would probably
27689 be most useful to add them to @code{EvalRules} so that Calc will evaluate
27690 the new @code{tri} function automatically. We could then use @kbd{Z K} on
27691 the keyboard macro @kbd{' tri($) @key{RET}} to make a command that applies
27692 @code{tri} to the value on the top of the stack. @xref{Programming}.
27694 @cindex Quaternions
27695 The following rule set, contributed by
27696 @texline Fran\c cois
27698 Pinard, implements @dfn{quaternions}, a generalization of the concept of
27699 complex numbers. Quaternions have four components, and are here
27700 represented by function calls @samp{quat(@var{w}, [@var{x}, @var{y},
27701 @var{z}])} with ``real part'' @var{w} and the three ``imaginary'' parts
27702 collected into a vector. Various arithmetical operations on quaternions
27703 are supported. To use these rules, either add them to @code{EvalRules},
27704 or create a command based on @kbd{a r} for simplifying quaternion
27705 formulas. A convenient way to enter quaternions would be a command
27706 defined by a keyboard macro containing: @kbd{' quat($$$$, [$$$, $$, $])
27710 [ quat(w, x, y, z) := quat(w, [x, y, z]),
27711 quat(w, [0, 0, 0]) := w,
27712 abs(quat(w, v)) := hypot(w, v),
27713 -quat(w, v) := quat(-w, -v),
27714 r + quat(w, v) := quat(r + w, v) :: real(r),
27715 r - quat(w, v) := quat(r - w, -v) :: real(r),
27716 quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
27717 r * quat(w, v) := quat(r * w, r * v) :: real(r),
27718 plain(quat(w1, v1) * quat(w2, v2))
27719 := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
27720 quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
27721 z / quat(w, v) := z * quatinv(quat(w, v)),
27722 quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
27723 quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
27724 quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
27725 :: integer(k) :: k > 0 :: k % 2 = 0,
27726 quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
27727 :: integer(k) :: k > 2,
27728 quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]
27731 Quaternions, like matrices, have non-commutative multiplication.
27732 In other words, @expr{q1 * q2 = q2 * q1} is not necessarily true if
27733 @expr{q1} and @expr{q2} are @code{quat} forms. The @samp{quat*quat}
27734 rule above uses @code{plain} to prevent Calc from rearranging the
27735 product. It may also be wise to add the line @samp{[quat(), matrix]}
27736 to the @code{Decls} matrix, to ensure that Calc's other algebraic
27737 operations will not rearrange a quaternion product. @xref{Declarations}.
27739 These rules also accept a four-argument @code{quat} form, converting
27740 it to the preferred form in the first rule. If you would rather see
27741 results in the four-argument form, just append the two items
27742 @samp{phase(2), quat(w, [x, y, z]) := quat(w, x, y, z)} to the end
27743 of the rule set. (But remember that multi-phase rule sets don't work
27744 in @code{EvalRules}.)
27746 @node Units, Store and Recall, Algebra, Top
27747 @chapter Operating on Units
27750 One special interpretation of algebraic formulas is as numbers with units.
27751 For example, the formula @samp{5 m / s^2} can be read ``five meters
27752 per second squared.'' The commands in this chapter help you
27753 manipulate units expressions in this form. Units-related commands
27754 begin with the @kbd{u} prefix key.
27757 * Basic Operations on Units::
27758 * The Units Table::
27759 * Predefined Units::
27760 * User-Defined Units::
27761 * Logarithmic Units::
27765 @node Basic Operations on Units, The Units Table, Units, Units
27766 @section Basic Operations on Units
27769 A @dfn{units expression} is a formula which is basically a number
27770 multiplied and/or divided by one or more @dfn{unit names}, which may
27771 optionally be raised to integer powers. Actually, the value part need not
27772 be a number; any product or quotient involving unit names is a units
27773 expression. Many of the units commands will also accept any formula,
27774 where the command applies to all units expressions which appear in the
27777 A unit name is a variable whose name appears in the @dfn{unit table},
27778 or a variable whose name is a prefix character like @samp{k} (for ``kilo'')
27779 or @samp{u} (for ``micro'') followed by a name in the unit table.
27780 A substantial table of built-in units is provided with Calc;
27781 @pxref{Predefined Units}. You can also define your own unit names;
27782 @pxref{User-Defined Units}.
27784 Note that if the value part of a units expression is exactly @samp{1},
27785 it will be removed by the Calculator's automatic algebra routines: The
27786 formula @samp{1 mm} is ``simplified'' to @samp{mm}. This is only a
27787 display anomaly, however; @samp{mm} will work just fine as a
27788 representation of one millimeter.
27790 You may find that Algebraic mode (@pxref{Algebraic Entry}) makes working
27791 with units expressions easier. Otherwise, you will have to remember
27792 to hit the apostrophe key every time you wish to enter units.
27795 @pindex calc-simplify-units
27797 @mindex usimpl@idots
27800 The @kbd{u s} (@code{calc-simplify-units}) [@code{usimplify}] command
27802 expression. It uses Calc's algebraic simplifications to simplify the
27803 expression first as a regular algebraic formula; it then looks for
27804 features that can be further simplified by converting one object's units
27805 to be compatible with another's. For example, @samp{5 m + 23 mm} will
27806 simplify to @samp{5.023 m}. When different but compatible units are
27807 added, the righthand term's units are converted to match those of the
27808 lefthand term. @xref{Simplification Modes}, for a way to have this done
27809 automatically at all times.
27811 Units simplification also handles quotients of two units with the same
27812 dimensionality, as in @w{@samp{2 in s/L cm}} to @samp{5.08 s/L}; fractional
27813 powers of unit expressions, as in @samp{sqrt(9 mm^2)} to @samp{3 mm} and
27814 @samp{sqrt(9 acre)} to a quantity in meters; and @code{floor},
27815 @code{ceil}, @code{round}, @code{rounde}, @code{roundu}, @code{trunc},
27816 @code{float}, @code{frac}, @code{abs}, and @code{clean}
27817 applied to units expressions, in which case
27818 the operation in question is applied only to the numeric part of the
27819 expression. Finally, trigonometric functions of quantities with units
27820 of angle are evaluated, regardless of the current angular mode.
27823 @pindex calc-convert-units
27824 The @kbd{u c} (@code{calc-convert-units}) command converts a units
27825 expression to new, compatible units. For example, given the units
27826 expression @samp{55 mph}, typing @kbd{u c m/s @key{RET}} produces
27827 @samp{24.5872 m/s}. If you have previously converted a units expression
27828 with the same type of units (in this case, distance over time), you will
27829 be offered the previous choice of new units as a default. Continuing
27830 the above example, entering the units expression @samp{100 km/hr} and
27831 typing @kbd{u c @key{RET}} (without specifying new units) produces
27832 @samp{27.7777777778 m/s}.
27835 @pindex calc-convert-temperature
27836 @cindex Temperature conversion
27837 The @kbd{u c} command treats temperature units (like @samp{degC} and
27838 @samp{K}) as relative temperatures. For example, @kbd{u c} converts
27839 @samp{10 degC} to @samp{18 degF}: A change of 10 degrees Celsius
27840 corresponds to a change of 18 degrees Fahrenheit. To convert absolute
27841 temperatures, you can use the @kbd{u t}
27842 (@code{calc-convert-temperature}) command. The value on the stack
27843 must be a simple units expression with units of temperature only.
27844 This command would convert @samp{10 degC} to @samp{50 degF}, the
27845 equivalent temperature on the Fahrenheit scale.
27847 While many of Calc's conversion factors are exact, some are necessarily
27848 approximate. If Calc is in fraction mode (@pxref{Fraction Mode}), then
27849 unit conversions will try to give exact, rational conversions, but it
27850 isn't always possible. Given @samp{55 mph} in fraction mode, typing
27851 @kbd{u c m/s @key{RET}} produces @samp{15367:625 m/s}, for example,
27852 while typing @kbd{u c au/yr @key{RET}} produces
27853 @samp{5.18665819999e-3 au/yr}.
27855 If the units you request are inconsistent with the original units, the
27856 number will be converted into your units times whatever ``remainder''
27857 units are left over. For example, converting @samp{55 mph} into acres
27858 produces @samp{6.08e-3 acre / m s}. (Recall that multiplication binds
27859 more strongly than division in Calc formulas, so the units here are
27860 acres per meter-second.) Remainder units are expressed in terms of
27861 ``fundamental'' units like @samp{m} and @samp{s}, regardless of the
27864 If you want to disallow using inconsistent units, you can set the customizable variable
27865 @code{calc-ensure-consistent-units} to @code{t} (@pxref{Customizing Calc}). In this case,
27866 if you request units which are inconsistent with the original units, you will be warned about
27867 it and no conversion will occur.
27869 One special exception is that if you specify a single unit name, and
27870 a compatible unit appears somewhere in the units expression, then
27871 that compatible unit will be converted to the new unit and the
27872 remaining units in the expression will be left alone. For example,
27873 given the input @samp{980 cm/s^2}, the command @kbd{u c ms} will
27874 change the @samp{s} to @samp{ms} to get @samp{9.8e-4 cm/ms^2}.
27875 The ``remainder unit'' @samp{cm} is left alone rather than being
27876 changed to the base unit @samp{m}.
27878 You can use explicit unit conversion instead of the @kbd{u s} command
27879 to gain more control over the units of the result of an expression.
27880 For example, given @samp{5 m + 23 mm}, you can type @kbd{u c m} or
27881 @kbd{u c mm} to express the result in either meters or millimeters.
27882 (For that matter, you could type @kbd{u c fath} to express the result
27883 in fathoms, if you preferred!)
27885 In place of a specific set of units, you can also enter one of the
27886 units system names @code{si}, @code{mks} (equivalent), or @code{cgs}.
27887 For example, @kbd{u c si @key{RET}} converts the expression into
27888 International System of Units (SI) base units. Also, @kbd{u c base}
27889 converts to Calc's base units, which are the same as @code{si} units
27890 except that @code{base} uses @samp{g} as the fundamental unit of mass
27891 whereas @code{si} uses @samp{kg}.
27893 @cindex Composite units
27894 The @kbd{u c} command also accepts @dfn{composite units}, which
27895 are expressed as the sum of several compatible unit names. For
27896 example, converting @samp{30.5 in} to units @samp{mi+ft+in} (miles,
27897 feet, and inches) produces @samp{2 ft + 6.5 in}. Calc first
27898 sorts the unit names into order of decreasing relative size.
27899 It then accounts for as much of the input quantity as it can
27900 using an integer number times the largest unit, then moves on
27901 to the next smaller unit, and so on. Only the smallest unit
27902 may have a non-integer amount attached in the result. A few
27903 standard unit names exist for common combinations, such as
27904 @code{mfi} for @samp{mi+ft+in}, and @code{tpo} for @samp{ton+lb+oz}.
27905 Composite units are expanded as if by @kbd{a x}, so that
27906 @samp{(ft+in)/hr} is first converted to @samp{ft/hr+in/hr}.
27908 If the value on the stack does not contain any units, @kbd{u c} will
27909 prompt first for the old units which this value should be considered
27910 to have, then for the new units. Assuming the old and new units you
27911 give are consistent with each other, the result also will not contain
27912 any units. For example, @kbd{@w{u c} cm @key{RET} in @key{RET}}
27913 converts the number 2 on the stack to 5.08.
27916 @pindex calc-base-units
27917 The @kbd{u b} (@code{calc-base-units}) command is shorthand for
27918 @kbd{u c base}; it converts the units expression on the top of the
27919 stack into @code{base} units. If @kbd{u s} does not simplify a
27920 units expression as far as you would like, try @kbd{u b}.
27922 Like the @kbd{u c} command, the @kbd{u b} command treats temperature
27923 units as relative temperatures.
27926 @pindex calc-remove-units
27928 @pindex calc-extract-units
27929 The @kbd{u r} (@code{calc-remove-units}) command removes units from the
27930 formula at the top of the stack. The @kbd{u x}
27931 (@code{calc-extract-units}) command extracts only the units portion of a
27932 formula. These commands essentially replace every term of the formula
27933 that does or doesn't (respectively) look like a unit name by the
27934 constant 1, then resimplify the formula.
27937 @pindex calc-autorange-units
27938 The @kbd{u a} (@code{calc-autorange-units}) command turns on and off a
27939 mode in which unit prefixes like @code{k} (``kilo'') are automatically
27940 applied to keep the numeric part of a units expression in a reasonable
27941 range. This mode affects @kbd{u s} and all units conversion commands
27942 except @kbd{u b}. For example, with autoranging on, @samp{12345 Hz}
27943 will be simplified to @samp{12.345 kHz}. Autoranging is useful for
27944 some kinds of units (like @code{Hz} and @code{m}), but is probably
27945 undesirable for non-metric units like @code{ft} and @code{tbsp}.
27946 (Composite units are more appropriate for those; see above.)
27948 Autoranging always applies the prefix to the leftmost unit name.
27949 Calc chooses the largest prefix that causes the number to be greater
27950 than or equal to 1.0. Thus an increasing sequence of adjusted times
27951 would be @samp{1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks}.
27952 Generally the rule of thumb is that the number will be adjusted
27953 to be in the interval @samp{[1 .. 1000)}, although there are several
27954 exceptions to this rule. First, if the unit has a power then this
27955 is not possible; @samp{0.1 s^2} simplifies to @samp{100000 ms^2}.
27956 Second, the ``centi-'' prefix is allowed to form @code{cm} (centimeters),
27957 but will not apply to other units. The ``deci-,'' ``deka-,'' and
27958 ``hecto-'' prefixes are never used. Thus the allowable interval is
27959 @samp{[1 .. 10)} for millimeters and @samp{[1 .. 100)} for centimeters.
27960 Finally, a prefix will not be added to a unit if the resulting name
27961 is also the actual name of another unit; @samp{1e-15 t} would normally
27962 be considered a ``femto-ton,'' but it is written as @samp{1000 at}
27963 (1000 atto-tons) instead because @code{ft} would be confused with feet.
27965 @node The Units Table, Predefined Units, Basic Operations on Units, Units
27966 @section The Units Table
27970 @pindex calc-enter-units-table
27971 The @kbd{u v} (@code{calc-enter-units-table}) command displays the units table
27972 in another buffer called @code{*Units Table*}. Each entry in this table
27973 gives the unit name as it would appear in an expression, the definition
27974 of the unit in terms of simpler units, and a full name or description of
27975 the unit. Fundamental units are defined as themselves; these are the
27976 units produced by the @kbd{u b} command. The fundamental units are
27977 meters, seconds, grams, kelvins, amperes, candelas, moles, radians,
27980 The Units Table buffer also displays the Unit Prefix Table. Note that
27981 two prefixes, ``kilo'' and ``hecto,'' accept either upper- or lower-case
27982 prefix letters. @samp{Meg} is also accepted as a synonym for the @samp{M}
27983 prefix. Whenever a unit name can be interpreted as either a built-in name
27984 or a prefix followed by another built-in name, the former interpretation
27985 wins. For example, @samp{2 pt} means two pints, not two pico-tons.
27987 The Units Table buffer, once created, is not rebuilt unless you define
27988 new units. To force the buffer to be rebuilt, give any numeric prefix
27989 argument to @kbd{u v}.
27992 @pindex calc-view-units-table
27993 The @kbd{u V} (@code{calc-view-units-table}) command is like @kbd{u v} except
27994 that the cursor is not moved into the Units Table buffer. You can
27995 type @kbd{u V} again to remove the Units Table from the display. To
27996 return from the Units Table buffer after a @kbd{u v}, type @kbd{C-x * c}
27997 again or use the regular Emacs @w{@kbd{C-x o}} (@code{other-window})
27998 command. You can also kill the buffer with @kbd{C-x k} if you wish;
27999 the actual units table is safely stored inside the Calculator.
28002 @pindex calc-get-unit-definition
28003 The @kbd{u g} (@code{calc-get-unit-definition}) command retrieves a unit's
28004 defining expression and pushes it onto the Calculator stack. For example,
28005 @kbd{u g in} will produce the expression @samp{2.54 cm}. This is the
28006 same definition for the unit that would appear in the Units Table buffer.
28007 Note that this command works only for actual unit names; @kbd{u g km}
28008 will report that no such unit exists, for example, because @code{km} is
28009 really the unit @code{m} with a @code{k} (``kilo'') prefix. To see a
28010 definition of a unit in terms of base units, it is easier to push the
28011 unit name on the stack and then reduce it to base units with @kbd{u b}.
28014 @pindex calc-explain-units
28015 The @kbd{u e} (@code{calc-explain-units}) command displays an English
28016 description of the units of the expression on the stack. For example,
28017 for the expression @samp{62 km^2 g / s^2 mol K}, the description is
28018 ``Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin).'' This
28019 command uses the English descriptions that appear in the righthand
28020 column of the Units Table.
28022 @node Predefined Units, User-Defined Units, The Units Table, Units
28023 @section Predefined Units
28026 The definitions of many units have changed over the years. For example,
28027 the meter was originally defined in 1791 as one ten-millionth of the
28028 distance from the equator to the north pole. In order to be more
28029 precise, the definition was adjusted several times, and now a meter is
28030 defined as the distance that light will travel in a vacuum in
28031 1/299792458 of a second; consequently, the speed of light in a
28032 vacuum is exactly 299792458 m/s. Many other units have been
28033 redefined in terms of fundamental physical processes; a second, for
28034 example, is currently defined as 9192631770 periods of a certain
28035 radiation related to the cesium-133 atom. The only SI unit that is not
28036 based on a fundamental physical process (although there are efforts to
28037 change this) is the kilogram, which was originally defined as the mass
28038 of one liter of water, but is now defined as the mass of the
28039 International Prototype Kilogram (IPK), a cylinder of platinum-iridium
28040 kept at the Bureau International des Poids et Mesures in S@`evres,
28041 France. (There are several copies of the IPK throughout the world.)
28042 The British imperial units, once defined in terms of physical objects,
28043 were redefined in 1963 in terms of SI units. The US customary units,
28044 which were the same as British units until the British imperial system
28045 was created in 1824, were also defined in terms of the SI units in 1893.
28046 Because of these redefinitions, conversions between metric, British
28047 Imperial, and US customary units can often be done precisely.
28049 Since the exact definitions of many kinds of units have evolved over the
28050 years, and since certain countries sometimes have local differences in
28051 their definitions, it is a good idea to examine Calc's definition of a
28052 unit before depending on its exact value. For example, there are three
28053 different units for gallons, corresponding to the US (@code{gal}),
28054 Canadian (@code{galC}), and British (@code{galUK}) definitions. Also,
28055 note that @code{oz} is a standard ounce of mass, @code{ozt} is a Troy
28056 ounce, and @code{ozfl} is a fluid ounce.
28058 The temperature units corresponding to degrees Kelvin and Centigrade
28059 (Celsius) are the same in this table, since most units commands treat
28060 temperatures as being relative. The @code{calc-convert-temperature}
28061 command has special rules for handling the different absolute magnitudes
28062 of the various temperature scales.
28064 The unit of volume ``liters'' can be referred to by either the lower-case
28065 @code{l} or the upper-case @code{L}.
28067 The unit @code{A} stands for Amperes; the name @code{Ang} is used
28075 The unit @code{pt} stands for pints; the name @code{point} stands for
28076 a typographical point, defined by @samp{72 point = 1 in}. This is
28077 slightly different than the point defined by the American Typefounder's
28078 Association in 1886, but the point used by Calc has become standard
28079 largely due to its use by the PostScript page description language.
28080 There is also @code{texpt}, which stands for a printer's point as
28081 defined by the @TeX{} typesetting system: @samp{72.27 texpt = 1 in}.
28082 Other units used by @TeX{} are available; they are @code{texpc} (a pica),
28083 @code{texbp} (a ``big point'', equal to a standard point which is larger
28084 than the point used by @TeX{}), @code{texdd} (a Didot point),
28085 @code{texcc} (a Cicero) and @code{texsp} (a scaled @TeX{} point,
28086 all dimensions representable in @TeX{} are multiples of this value).
28088 When Calc is using the @TeX{} or @LaTeX{} language mode (@pxref{TeX
28089 and LaTeX Language Modes}), the @TeX{} specific unit names will not
28090 use the @samp{tex} prefix; the unit name for a @TeX{} point will be
28091 @samp{pt} instead of @samp{texpt}, for example. To avoid conflicts,
28092 the unit names for pint and parsec will simply be @samp{pint} and
28093 @samp{parsec} instead of @samp{pt} and @samp{pc}.
28096 The unit @code{e} stands for the elementary (electron) unit of charge;
28097 because algebra command could mistake this for the special constant
28098 @expr{e}, Calc provides the alternate unit name @code{ech} which is
28099 preferable to @code{e}.
28101 The name @code{g} stands for one gram of mass; there is also @code{gf},
28102 one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.)
28103 Meanwhile, one ``@expr{g}'' of acceleration is denoted @code{ga}.
28105 The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is
28106 a metric ton of @samp{1000 kg}.
28108 The names @code{s} (or @code{sec}) and @code{min} refer to units of
28109 time; @code{arcsec} and @code{arcmin} are units of angle.
28111 Some ``units'' are really physical constants; for example, @code{c}
28112 represents the speed of light, and @code{h} represents Planck's
28113 constant. You can use these just like other units: converting
28114 @samp{.5 c} to @samp{m/s} expresses one-half the speed of light in
28115 meters per second. You can also use this merely as a handy reference;
28116 the @kbd{u g} command gets the definition of one of these constants
28117 in its normal terms, and @kbd{u b} expresses the definition in base
28120 Two units, @code{pi} and @code{alpha} (the fine structure constant,
28121 approximately @mathit{1/137}) are dimensionless. The units simplification
28122 commands simply treat these names as equivalent to their corresponding
28123 values. However you can, for example, use @kbd{u c} to convert a pure
28124 number into multiples of the fine structure constant, or @kbd{u b} to
28125 convert this back into a pure number. (When @kbd{u c} prompts for the
28126 ``old units,'' just enter a blank line to signify that the value
28127 really is unitless.)
28129 @c Describe angular units, luminosity vs. steradians problem.
28131 @node User-Defined Units, Logarithmic Units, Predefined Units, Units
28132 @section User-Defined Units
28135 Calc provides ways to get quick access to your selected ``favorite''
28136 units, as well as ways to define your own new units.
28139 @pindex calc-quick-units
28141 @cindex @code{Units} variable
28142 @cindex Quick units
28143 To select your favorite units, store a vector of unit names or
28144 expressions in the Calc variable @code{Units}. The @kbd{u 1}
28145 through @kbd{u 9} commands (@code{calc-quick-units}) provide access
28146 to these units. If the value on the top of the stack is a plain
28147 number (with no units attached), then @kbd{u 1} gives it the
28148 specified units. (Basically, it multiplies the number by the
28149 first item in the @code{Units} vector.) If the number on the
28150 stack @emph{does} have units, then @kbd{u 1} converts that number
28151 to the new units. For example, suppose the vector @samp{[in, ft]}
28152 is stored in @code{Units}. Then @kbd{30 u 1} will create the
28153 expression @samp{30 in}, and @kbd{u 2} will convert that expression
28156 The @kbd{u 0} command accesses the tenth element of @code{Units}.
28157 Only ten quick units may be defined at a time. If the @code{Units}
28158 variable has no stored value (the default), or if its value is not
28159 a vector, then the quick-units commands will not function. The
28160 @kbd{s U} command is a convenient way to edit the @code{Units}
28161 variable; @pxref{Operations on Variables}.
28164 @pindex calc-define-unit
28165 @cindex User-defined units
28166 The @kbd{u d} (@code{calc-define-unit}) command records the units
28167 expression on the top of the stack as the definition for a new,
28168 user-defined unit. For example, putting @samp{16.5 ft} on the stack and
28169 typing @kbd{u d rod} defines the new unit @samp{rod} to be equivalent to
28170 16.5 feet. The unit conversion and simplification commands will now
28171 treat @code{rod} just like any other unit of length. You will also be
28172 prompted for an optional English description of the unit, which will
28173 appear in the Units Table. If you wish the definition of this unit to
28174 be displayed in a special way in the Units Table buffer (such as with an
28175 asterisk to indicate an approximate value), then you can call this
28176 command with an argument, @kbd{C-u u d}; you will then also be prompted
28177 for a string that will be used to display the definition.
28180 @pindex calc-undefine-unit
28181 The @kbd{u u} (@code{calc-undefine-unit}) command removes a user-defined
28182 unit. It is not possible to remove one of the predefined units,
28185 If you define a unit with an existing unit name, your new definition
28186 will replace the original definition of that unit. If the unit was a
28187 predefined unit, the old definition will not be replaced, only
28188 ``shadowed.'' The built-in definition will reappear if you later use
28189 @kbd{u u} to remove the shadowing definition.
28191 To create a new fundamental unit, use either 1 or the unit name itself
28192 as the defining expression. Otherwise the expression can involve any
28193 other units that you like (except for composite units like @samp{mfi}).
28194 You can create a new composite unit with a sum of other units as the
28195 defining expression. The next unit operation like @kbd{u c} or @kbd{u v}
28196 will rebuild the internal unit table incorporating your modifications.
28197 Note that erroneous definitions (such as two units defined in terms of
28198 each other) will not be detected until the unit table is next rebuilt;
28199 @kbd{u v} is a convenient way to force this to happen.
28201 Temperature units are treated specially inside the Calculator; it is not
28202 possible to create user-defined temperature units.
28205 @pindex calc-permanent-units
28206 @cindex Calc init file, user-defined units
28207 The @kbd{u p} (@code{calc-permanent-units}) command stores the user-defined
28208 units in your Calc init file (the file given by the variable
28209 @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el}), so that the
28210 units will still be available in subsequent Emacs sessions. If there
28211 was already a set of user-defined units in your Calc init file, it
28212 is replaced by the new set. (@xref{General Mode Commands}, for a way to
28213 tell Calc to use a different file for the Calc init file.)
28215 @node Logarithmic Units, Musical Notes, User-Defined Units, Units
28216 @section Logarithmic Units
28218 The units @code{dB} (decibels) and @code{Np} (nepers) are logarithmic
28219 units which are manipulated differently than standard units. Calc
28220 provides commands to work with these logarithmic units.
28222 Decibels and nepers are used to measure power quantities as well as
28223 field quantities (quantities whose squares are proportional to power);
28224 these two types of quantities are handled slightly different from each
28225 other. By default the Calc commands work as if power quantities are
28226 being used; with the @kbd{H} prefix the Calc commands work as if field
28227 quantities are being used.
28229 The decibel level of a power
28230 @infoline @math{P1},
28231 @texline @math{P_1},
28232 relative to a reference power
28233 @infoline @math{P0},
28234 @texline @math{P_0},
28236 @infoline @math{10 log10(P1/P0) dB}.
28237 @texline @math{10 \log_{10}(P_{1}/P_{0}) {\rm dB}}.
28238 (The factor of 10 is because a decibel, as its name implies, is
28239 one-tenth of a bel. The bel, named after Alexander Graham Bell, was
28240 considered to be too large of a unit and was effectively replaced by
28241 the decibel.) If @math{F} is a field quantity with power
28242 @math{P=k F^2}, then a reference quantity of
28243 @infoline @math{F0}
28244 @texline @math{F_0}
28245 would correspond to a power of
28246 @infoline @math{P0=k F0^2}.
28247 @texline @math{P_{0}=kF_{0}^2}.
28249 @infoline @math{P1=k F1^2},
28250 @texline @math{P_{1}=kF_{1}^2},
28255 10 log10(P1/P0) = 10 log10(F1^2/F0^2) = 20 log10(F1/F0).
28259 $$ 10 \log_{10}(P_1/P_0) = 10 \log_{10}(F_1^2/F_0^2) = 20
28260 \log_{10}(F_1/F_0)$$
28264 In order to get the same decibel level regardless of whether a field
28265 quantity or the corresponding power quantity is used, the decibel
28266 level of a field quantity
28267 @infoline @math{F1},
28268 @texline @math{F_1},
28269 relative to a reference
28270 @infoline @math{F0},
28271 @texline @math{F_0},
28273 @infoline @math{20 log10(F1/F0) dB}.
28274 @texline @math{20 \log_{10}(F_{1}/F_{0}) {\rm dB}}.
28275 For example, the decibel value of a sound pressure level of
28276 @infoline @math{60 uPa}
28277 @texline @math{60 \mu{\rm Pa}}
28279 @infoline @math{20 uPa}
28280 @texline @math{20 \mu{\rm Pa}}
28281 (the threshold of human hearing) is
28282 @infoline @math{20 log10(60 uPa/ 20 uPa) dB = 20 log10(3) dB},
28283 @texline @math{20 \log_{10}(60 \mu{\rm Pa}/20 \mu{\rm Pa}) {\rm dB} = 20 \log_{10}(3) {\rm dB}},
28285 @infoline @math{9.54 dB}.
28286 @texline @math{9.54 {\rm dB}}.
28287 Note that in taking the ratio, the original units cancel and so these
28288 logarithmic units are dimensionless.
28290 Nepers (named after John Napier, who is credited with inventing the
28291 logarithm) are similar to bels except they use natural logarithms instead
28292 of common logarithms. The neper level of a power
28293 @infoline @math{P1},
28294 @texline @math{P_1},
28295 relative to a reference power
28296 @infoline @math{P0},
28297 @texline @math{P_0},
28299 @infoline @math{(1/2) ln(P1/P0) Np}.
28300 @texline @math{(1/2) \ln(P_1/P_0) {\rm Np}}.
28301 The neper level of a field
28302 @infoline @math{F1},
28303 @texline @math{F_1},
28304 relative to a reference field
28305 @infoline @math{F0},
28306 @texline @math{F_0},
28308 @infoline @math{ln(F1/F0) Np}.
28309 @texline @math{\ln(F_1/F_0) {\rm Np}}.
28311 @vindex calc-lu-power-reference
28312 @vindex calc-lu-field-reference
28313 For power quantities, Calc uses
28314 @infoline @math{1 mW}
28315 @texline @math{1 {\rm mW}}
28316 as the default reference quantity; this default can be changed by changing
28317 the value of the customizable variable
28318 @code{calc-lu-power-reference} (@pxref{Customizing Calc}).
28319 For field quantities, Calc uses
28320 @infoline @math{20 uPa}
28321 @texline @math{20 \mu{\rm Pa}}
28322 as the default reference quantity; this is the value used in acoustics
28323 which is where decibels are commonly encountered. This default can be
28324 changed by changing the value of the customizable variable
28325 @code{calc-lu-field-reference} (@pxref{Customizing Calc}). A
28326 non-default reference quantity will be read from the stack if the
28327 capital @kbd{O} prefix is used.
28330 @pindex calc-lu-quant
28333 The @kbd{l q} (@code{calc-lu-quant}) [@code{lupquant}]
28334 command computes the power quantity corresponding to a given number of
28335 logarithmic units. With the capital @kbd{O} prefix, @kbd{O l q}, the
28336 reference level will be read from the top of the stack. (In an
28337 algebraic formula, @code{lupquant} can be given an optional second
28338 argument which will be used for the reference level.) For example,
28339 @code{20 dB @key{RET} l q} will return @code{100 mW};
28340 @code{20 dB @key{RET} 4 W @key{RET} O l q} will return @code{400 W}.
28341 The @kbd{H l q} [@code{lufquant}] command behaves like @kbd{l q} but
28342 computes field quantities instead of power quantities.
28352 The @kbd{l d} (@code{calc-db}) [@code{dbpower}] command will compute
28353 the decibel level of a power quantity using the default reference
28354 level; @kbd{H l d} [@code{dbfield}] will compute the decibel level of
28355 a field quantity. The commands @kbd{l n} (@code{calc-np})
28356 [@code{nppower}] and @kbd{H l n} [@code{npfield}] will similarly
28357 compute neper levels. With the capital @kbd{O} prefix these commands
28358 will read a reference level from the stack; in an algebraic formula
28359 the reference level can be given as an optional second argument.
28362 @pindex calc-lu-plus
28366 @pindex calc-lu-minus
28370 @pindex calc-lu-times
28374 @pindex calc-lu-divide
28377 The sum of two power or field quantities doesn't correspond to the sum
28378 of the corresponding decibel or neper levels. If the powers
28379 corresponding to decibel levels
28380 @infoline @math{D1}
28381 @texline @math{D_1}
28383 @infoline @math{D2}
28384 @texline @math{D_2}
28385 are added, the corresponding decibel level ``sum'' will be
28389 10 log10(10^(D1/10) + 10^(D2/10)) dB.
28393 $$ 10 \log_{10}(10^{D_1/10} + 10^{D_2/10}) {\rm dB}.$$
28397 When field quantities are combined, it often means the corresponding
28398 powers are added and so the above formula might be used. In
28399 acoustics, for example, the sound pressure level is a field quantity
28400 and so the decibels are often defined using the field formula, but the
28401 sound pressure levels are combined as the sound power levels, and so
28402 the above formula should be used. If two field quantities themselves
28403 are added, the new decibel level will be
28407 20 log10(10^(D1/20) + 10^(D2/20)) dB.
28411 $$ 20 \log_{10}(10^{D_1/20} + 10^{D_2/20}) {\rm dB}.$$
28415 If the power corresponding to @math{D} dB is multiplied by a number @math{N},
28416 then the corresponding decibel level will be
28420 D + 10 log10(N) dB,
28424 $$ D + 10 \log_{10}(N) {\rm dB},$$
28428 if a field quantity is multiplied by @math{N} the corresponding decibel level
28433 D + 20 log10(N) dB.
28437 $$ D + 20 \log_{10}(N) {\rm dB}.$$
28441 There are similar formulas for combining nepers. The @kbd{l +}
28442 (@code{calc-lu-plus}) [@code{lupadd}] command will ``add'' two
28443 logarithmic unit power levels this way; with the @kbd{H} prefix,
28444 @kbd{H l +} [@code{lufadd}] will add logarithmic unit field levels.
28445 Similarly, logarithmic units can be ``subtracted'' with @kbd{l -}
28446 (@code{calc-lu-minus}) [@code{lupsub}] or @kbd{H l -} [@code{lufsub}].
28447 The @kbd{l *} (@code{calc-lu-times}) [@code{lupmul}] and @kbd{H l *}
28448 [@code{lufmul}] commands will ``multiply'' a logarithmic unit by a
28449 number; the @kbd{l /} (@code{calc-lu-divide}) [@code{lupdiv}] and
28450 @kbd{H l /} [@code{lufdiv}] commands will ``divide'' a logarithmic
28451 unit by a number. Note that the reference quantities don't play a role
28452 in this arithmetic.
28454 @node Musical Notes, , Logarithmic Units, Units
28455 @section Musical Notes
28457 Calc can convert between musical notes and their associated
28458 frequencies. Notes can be given using either scientific pitch
28459 notation or midi numbers. Since these note systems are basically
28460 logarithmic scales, Calc uses the @kbd{l} prefix for functions
28461 operating on notes.
28463 Scientific pitch notation refers to a note by giving a letter
28464 A through G, possibly followed by a flat or sharp) with a subscript
28465 indicating an octave number. Each octave starts with C and ends with
28467 @c increasing each note by a semitone will result
28468 @c in the sequence @expr{C}, @expr{C} sharp, @expr{D}, @expr{E} flat, @expr{E},
28469 @c @expr{F}, @expr{F} sharp, @expr{G}, @expr{A} flat, @expr{A}, @expr{B}
28470 @c flat and @expr{B}.
28471 the octave numbered 0 was chosen to correspond to the lowest
28472 audible frequency. Using this system, middle C (about 261.625 Hz)
28473 corresponds to the note @expr{C} in octave 4 and is denoted
28474 @expr{C_4}. Any frequency can be described by giving a note plus an
28475 offset in cents (where a cent is a ratio of frequencies so that a
28476 semitone consists of 100 cents).
28478 The midi note number system assigns numbers to notes so that
28479 @expr{C_(-1)} corresponds to the midi note number 0 and @expr{G_9}
28480 corresponds to the midi note number 127. A midi controller can have
28481 up to 128 keys and each midi note number from 0 to 127 corresponds to
28487 The @kbd{l s} (@code{calc-spn}) [@code{spn}] command converts either
28488 a frequency or a midi number to scientific pitch notation. For
28489 example, @code{500 Hz} gets converted to
28490 @code{B_4 + 21.3094853649 cents} and @code{84} to @code{C_6}.
28496 The @kbd{l m} (@code{calc-midi}) [@code{midi}] command converts either
28497 a frequency or a note given in scientific pitch notation to the
28498 corresponding midi number. For example, @code{C_6} gets converted to 84
28499 and @code{440 Hz} to 69.
28504 The @kbd{l f} (@code{calc-freq}) [@code{freq}] command converts either
28505 either a midi number or a note given in scientific pitch notation to
28506 the corresponding frequency. For example, @code{Asharp_2 + 30 cents}
28507 gets converted to @code{118.578040134 Hz} and @code{55} to
28508 @code{195.99771799 Hz}.
28510 Since the frequencies of notes are not usually given exactly (and are
28511 typically irrational), the customizable variable
28512 @code{calc-note-threshold} determines how close (in cents) a frequency
28513 needs to be to a note to be recognized as that note
28514 (@pxref{Customizing Calc}). This variable has a default value of
28515 @code{1}. For example, middle @var{C} is approximately
28516 @expr{261.625565302 Hz}; this frequency is often shortened to
28517 @expr{261.625 Hz}. Without @code{calc-note-threshold} (or a value of
28518 @expr{0}), Calc would convert @code{261.625 Hz} to scientific pitch
28519 notation @code{B_3 + 99.9962592773 cents}; with the default value of
28520 @code{1}, Calc converts @code{261.625 Hz} to @code{C_4}.
28524 @node Store and Recall, Graphics, Units, Top
28525 @chapter Storing and Recalling
28528 Calculator variables are really just Lisp variables that contain numbers
28529 or formulas in a form that Calc can understand. The commands in this
28530 section allow you to manipulate variables conveniently. Commands related
28531 to variables use the @kbd{s} prefix key.
28534 * Storing Variables::
28535 * Recalling Variables::
28536 * Operations on Variables::
28538 * Evaluates-To Operator::
28541 @node Storing Variables, Recalling Variables, Store and Recall, Store and Recall
28542 @section Storing Variables
28547 @cindex Storing variables
28548 @cindex Quick variables
28551 The @kbd{s s} (@code{calc-store}) command stores the value at the top of
28552 the stack into a specified variable. It prompts you to enter the
28553 name of the variable. If you press a single digit, the value is stored
28554 immediately in one of the ``quick'' variables @code{q0} through
28555 @code{q9}. Or you can enter any variable name.
28558 @pindex calc-store-into
28559 The @kbd{s s} command leaves the stored value on the stack. There is
28560 also an @kbd{s t} (@code{calc-store-into}) command, which removes a
28561 value from the stack and stores it in a variable.
28563 If the top of stack value is an equation @samp{a = 7} or assignment
28564 @samp{a := 7} with a variable on the lefthand side, then Calc will
28565 assign that variable with that value by default, i.e., if you type
28566 @kbd{s s @key{RET}} or @kbd{s t @key{RET}}. In this example, the
28567 value 7 would be stored in the variable @samp{a}. (If you do type
28568 a variable name at the prompt, the top-of-stack value is stored in
28569 its entirety, even if it is an equation: @samp{s s b @key{RET}}
28570 with @samp{a := 7} on the stack stores @samp{a := 7} in @code{b}.)
28572 In fact, the top of stack value can be a vector of equations or
28573 assignments with different variables on their lefthand sides; the
28574 default will be to store all the variables with their corresponding
28575 righthand sides simultaneously.
28577 It is also possible to type an equation or assignment directly at
28578 the prompt for the @kbd{s s} or @kbd{s t} command: @kbd{s s foo = 7}.
28579 In this case the expression to the right of the @kbd{=} or @kbd{:=}
28580 symbol is evaluated as if by the @kbd{=} command, and that value is
28581 stored in the variable. No value is taken from the stack; @kbd{s s}
28582 and @kbd{s t} are equivalent when used in this way.
28586 The prefix keys @kbd{s} and @kbd{t} may be followed immediately by a
28587 digit; @kbd{s 9} is equivalent to @kbd{s s 9}, and @kbd{t 9} is
28588 equivalent to @kbd{s t 9}. (The @kbd{t} prefix is otherwise used
28589 for trail and time/date commands.)
28625 @pindex calc-store-plus
28626 @pindex calc-store-minus
28627 @pindex calc-store-times
28628 @pindex calc-store-div
28629 @pindex calc-store-power
28630 @pindex calc-store-concat
28631 @pindex calc-store-neg
28632 @pindex calc-store-inv
28633 @pindex calc-store-decr
28634 @pindex calc-store-incr
28635 There are also several ``arithmetic store'' commands. For example,
28636 @kbd{s +} removes a value from the stack and adds it to the specified
28637 variable. The other arithmetic stores are @kbd{s -}, @kbd{s *}, @kbd{s /},
28638 @kbd{s ^}, and @w{@kbd{s |}} (vector concatenation), plus @kbd{s n} and
28639 @kbd{s &} which negate or invert the value in a variable, and @w{@kbd{s [}}
28640 and @kbd{s ]} which decrease or increase a variable by one.
28642 All the arithmetic stores accept the Inverse prefix to reverse the
28643 order of the operands. If @expr{v} represents the contents of the
28644 variable, and @expr{a} is the value drawn from the stack, then regular
28645 @w{@kbd{s -}} assigns
28646 @texline @math{v \coloneq v - a},
28647 @infoline @expr{v := v - a},
28648 but @kbd{I s -} assigns
28649 @texline @math{v \coloneq a - v}.
28650 @infoline @expr{v := a - v}.
28651 While @kbd{I s *} might seem pointless, it is
28652 useful if matrix multiplication is involved. Actually, all the
28653 arithmetic stores use formulas designed to behave usefully both
28654 forwards and backwards:
28658 s + v := v + a v := a + v
28659 s - v := v - a v := a - v
28660 s * v := v * a v := a * v
28661 s / v := v / a v := a / v
28662 s ^ v := v ^ a v := a ^ v
28663 s | v := v | a v := a | v
28664 s n v := v / (-1) v := (-1) / v
28665 s & v := v ^ (-1) v := (-1) ^ v
28666 s [ v := v - 1 v := 1 - v
28667 s ] v := v - (-1) v := (-1) - v
28671 In the last four cases, a numeric prefix argument will be used in
28672 place of the number one. (For example, @kbd{M-2 s ]} increases
28673 a variable by 2, and @kbd{M-2 I s ]} replaces a variable by
28674 minus-two minus the variable.
28676 The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -},
28677 etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous
28678 arithmetic stores that don't remove the value @expr{a} from the stack.
28680 All arithmetic stores report the new value of the variable in the
28681 Trail for your information. They signal an error if the variable
28682 previously had no stored value. If default simplifications have been
28683 turned off, the arithmetic stores temporarily turn them on for numeric
28684 arguments only (i.e., they temporarily do an @kbd{m N} command).
28685 @xref{Simplification Modes}. Large vectors put in the trail by
28686 these commands always use abbreviated (@kbd{t .}) mode.
28689 @pindex calc-store-map
28690 The @kbd{s m} command is a general way to adjust a variable's value
28691 using any Calc function. It is a ``mapping'' command analogous to
28692 @kbd{V M}, @kbd{V R}, etc. @xref{Reducing and Mapping}, to see
28693 how to specify a function for a mapping command. Basically,
28694 all you do is type the Calc command key that would invoke that
28695 function normally. For example, @kbd{s m n} applies the @kbd{n}
28696 key to negate the contents of the variable, so @kbd{s m n} is
28697 equivalent to @kbd{s n}. Also, @kbd{s m Q} takes the square root
28698 of the value stored in a variable, @kbd{s m v v} uses @kbd{v v} to
28699 reverse the vector stored in the variable, and @kbd{s m H I S}
28700 takes the hyperbolic arcsine of the variable contents.
28702 If the mapping function takes two or more arguments, the additional
28703 arguments are taken from the stack; the old value of the variable
28704 is provided as the first argument. Thus @kbd{s m -} with @expr{a}
28705 on the stack computes @expr{v - a}, just like @kbd{s -}. With the
28706 Inverse prefix, the variable's original value becomes the @emph{last}
28707 argument instead of the first. Thus @kbd{I s m -} is also
28708 equivalent to @kbd{I s -}.
28711 @pindex calc-store-exchange
28712 The @kbd{s x} (@code{calc-store-exchange}) command exchanges the value
28713 of a variable with the value on the top of the stack. Naturally, the
28714 variable must already have a stored value for this to work.
28716 You can type an equation or assignment at the @kbd{s x} prompt. The
28717 command @kbd{s x a=6} takes no values from the stack; instead, it
28718 pushes the old value of @samp{a} on the stack and stores @samp{a = 6}.
28721 @pindex calc-unstore
28722 @cindex Void variables
28723 @cindex Un-storing variables
28724 Until you store something in them, most variables are ``void,'' that is,
28725 they contain no value at all. If they appear in an algebraic formula
28726 they will be left alone even if you press @kbd{=} (@code{calc-evaluate}).
28727 The @kbd{s u} (@code{calc-unstore}) command returns a variable to the
28731 @pindex calc-copy-variable
28732 The @kbd{s c} (@code{calc-copy-variable}) command copies the stored
28733 value of one variable to another. One way it differs from a simple
28734 @kbd{s r} followed by an @kbd{s t} (aside from saving keystrokes) is
28735 that the value never goes on the stack and thus is never rounded,
28736 evaluated, or simplified in any way; it is not even rounded down to the
28739 The only variables with predefined values are the ``special constants''
28740 @code{pi}, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free
28741 to unstore these variables or to store new values into them if you like,
28742 although some of the algebraic-manipulation functions may assume these
28743 variables represent their standard values. Calc displays a warning if
28744 you change the value of one of these variables, or of one of the other
28745 special variables @code{inf}, @code{uinf}, and @code{nan} (which are
28748 Note that @code{pi} doesn't actually have 3.14159265359 stored in it,
28749 but rather a special magic value that evaluates to @cpi{} at the current
28750 precision. Likewise @code{e}, @code{i}, and @code{phi} evaluate
28751 according to the current precision or polar mode. If you recall a value
28752 from @code{pi} and store it back, this magic property will be lost. The
28753 magic property is preserved, however, when a variable is copied with
28757 @pindex calc-copy-special-constant
28758 If one of the ``special constants'' is redefined (or undefined) so that
28759 it no longer has its magic property, the property can be restored with
28760 @kbd{s k} (@code{calc-copy-special-constant}). This command will prompt
28761 for a special constant and a variable to store it in, and so a special
28762 constant can be stored in any variable. Here, the special constant that
28763 you enter doesn't depend on the value of the corresponding variable;
28764 @code{pi} will represent 3.14159@dots{} regardless of what is currently
28765 stored in the Calc variable @code{pi}. If one of the other special
28766 variables, @code{inf}, @code{uinf} or @code{nan}, is given a value, its
28767 original behavior can be restored by voiding it with @kbd{s u}.
28769 @node Recalling Variables, Operations on Variables, Storing Variables, Store and Recall
28770 @section Recalling Variables
28774 @pindex calc-recall
28775 @cindex Recalling variables
28776 The most straightforward way to extract the stored value from a variable
28777 is to use the @kbd{s r} (@code{calc-recall}) command. This command prompts
28778 for a variable name (similarly to @code{calc-store}), looks up the value
28779 of the specified variable, and pushes that value onto the stack. It is
28780 an error to try to recall a void variable.
28782 It is also possible to recall the value from a variable by evaluating a
28783 formula containing that variable. For example, @kbd{' a @key{RET} =} is
28784 the same as @kbd{s r a @key{RET}} except that if the variable is void, the
28785 former will simply leave the formula @samp{a} on the stack whereas the
28786 latter will produce an error message.
28789 The @kbd{r} prefix may be followed by a digit, so that @kbd{r 9} is
28790 equivalent to @kbd{s r 9}.
28792 @node Operations on Variables, Let Command, Recalling Variables, Store and Recall
28793 @section Other Operations on Variables
28797 @pindex calc-edit-variable
28798 The @kbd{s e} (@code{calc-edit-variable}) command edits the stored
28799 value of a variable without ever putting that value on the stack
28800 or simplifying or evaluating the value. It prompts for the name of
28801 the variable to edit. If the variable has no stored value, the
28802 editing buffer will start out empty. If the editing buffer is
28803 empty when you press @kbd{C-c C-c} to finish, the variable will
28804 be made void. @xref{Editing Stack Entries}, for a general
28805 description of editing.
28807 The @kbd{s e} command is especially useful for creating and editing
28808 rewrite rules which are stored in variables. Sometimes these rules
28809 contain formulas which must not be evaluated until the rules are
28810 actually used. (For example, they may refer to @samp{deriv(x,y)},
28811 where @code{x} will someday become some expression involving @code{y};
28812 if you let Calc evaluate the rule while you are defining it, Calc will
28813 replace @samp{deriv(x,y)} with 0 because the formula @code{x} does
28814 not itself refer to @code{y}.) By contrast, recalling the variable,
28815 editing with @kbd{`}, and storing will evaluate the variable's value
28816 as a side effect of putting the value on the stack.
28864 @pindex calc-store-AlgSimpRules
28865 @pindex calc-store-Decls
28866 @pindex calc-store-EvalRules
28867 @pindex calc-store-FitRules
28868 @pindex calc-store-GenCount
28869 @pindex calc-store-Holidays
28870 @pindex calc-store-IntegLimit
28871 @pindex calc-store-LineStyles
28872 @pindex calc-store-PointStyles
28873 @pindex calc-store-PlotRejects
28874 @pindex calc-store-TimeZone
28875 @pindex calc-store-Units
28876 @pindex calc-store-ExtSimpRules
28877 There are several special-purpose variable-editing commands that
28878 use the @kbd{s} prefix followed by a shifted letter:
28882 Edit @code{AlgSimpRules}. @xref{Algebraic Simplifications}.
28884 Edit @code{Decls}. @xref{Declarations}.
28886 Edit @code{EvalRules}. @xref{Basic Simplifications}.
28888 Edit @code{FitRules}. @xref{Curve Fitting}.
28890 Edit @code{GenCount}. @xref{Solving Equations}.
28892 Edit @code{Holidays}. @xref{Business Days}.
28894 Edit @code{IntegLimit}. @xref{Calculus}.
28896 Edit @code{LineStyles}. @xref{Graphics}.
28898 Edit @code{PointStyles}. @xref{Graphics}.
28900 Edit @code{PlotRejects}. @xref{Graphics}.
28902 Edit @code{TimeZone}. @xref{Time Zones}.
28904 Edit @code{Units}. @xref{User-Defined Units}.
28906 Edit @code{ExtSimpRules}. @xref{Unsafe Simplifications}.
28909 These commands are just versions of @kbd{s e} that use fixed variable
28910 names rather than prompting for the variable name.
28913 @pindex calc-permanent-variable
28914 @cindex Storing variables
28915 @cindex Permanent variables
28916 @cindex Calc init file, variables
28917 The @kbd{s p} (@code{calc-permanent-variable}) command saves a
28918 variable's value permanently in your Calc init file (the file given by
28919 the variable @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el}), so
28920 that its value will still be available in future Emacs sessions. You
28921 can re-execute @w{@kbd{s p}} later on to update the saved value, but the
28922 only way to remove a saved variable is to edit your calc init file
28923 by hand. (@xref{General Mode Commands}, for a way to tell Calc to
28924 use a different file for the Calc init file.)
28926 If you do not specify the name of a variable to save (i.e.,
28927 @kbd{s p @key{RET}}), all Calc variables with defined values
28928 are saved except for the special constants @code{pi}, @code{e},
28929 @code{i}, @code{phi}, and @code{gamma}; the variables @code{TimeZone}
28930 and @code{PlotRejects};
28931 @code{FitRules}, @code{DistribRules}, and other built-in rewrite
28932 rules; and @code{PlotData@var{n}} variables generated
28933 by the graphics commands. (You can still save these variables by
28934 explicitly naming them in an @kbd{s p} command.)
28937 @pindex calc-insert-variables
28938 The @kbd{s i} (@code{calc-insert-variables}) command writes
28939 the values of all Calc variables into a specified buffer.
28940 The variables are written with the prefix @code{var-} in the form of
28941 Lisp @code{setq} commands
28942 which store the values in string form. You can place these commands
28943 in your Calc init file (or @file{.emacs}) if you wish, though in this case it
28944 would be easier to use @kbd{s p @key{RET}}. (Note that @kbd{s i}
28945 omits the same set of variables as @w{@kbd{s p @key{RET}}}; the difference
28946 is that @kbd{s i} will store the variables in any buffer, and it also
28947 stores in a more human-readable format.)
28949 @node Let Command, Evaluates-To Operator, Operations on Variables, Store and Recall
28950 @section The Let Command
28955 @cindex Variables, temporary assignment
28956 @cindex Temporary assignment to variables
28957 If you have an expression like @samp{a+b^2} on the stack and you wish to
28958 compute its value where @expr{b=3}, you can simply store 3 in @expr{b} and
28959 then press @kbd{=} to reevaluate the formula. This has the side-effect
28960 of leaving the stored value of 3 in @expr{b} for future operations.
28962 The @kbd{s l} (@code{calc-let}) command evaluates a formula under a
28963 @emph{temporary} assignment of a variable. It stores the value on the
28964 top of the stack into the specified variable, then evaluates the
28965 second-to-top stack entry, then restores the original value (or lack of one)
28966 in the variable. Thus after @kbd{'@w{ }a+b^2 @key{RET} 3 s l b @key{RET}},
28967 the stack will contain the formula @samp{a + 9}. The subsequent command
28968 @kbd{@w{5 s l a} @key{RET}} will replace this formula with the number 14.
28969 The variables @samp{a} and @samp{b} are not permanently affected in any way
28972 The value on the top of the stack may be an equation or assignment, or
28973 a vector of equations or assignments, in which case the default will be
28974 analogous to the case of @kbd{s t @key{RET}}. @xref{Storing Variables}.
28976 Also, you can answer the variable-name prompt with an equation or
28977 assignment: @kbd{s l b=3 @key{RET}} is the same as storing 3 on the stack
28978 and typing @kbd{s l b @key{RET}}.
28980 The @kbd{a b} (@code{calc-substitute}) command is another way to substitute
28981 a variable with a value in a formula. It does an actual substitution
28982 rather than temporarily assigning the variable and evaluating. For
28983 example, letting @expr{n=2} in @samp{f(n pi)} with @kbd{a b} will
28984 produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)}
28985 since the evaluation step will also evaluate @code{pi}.
28987 @node Evaluates-To Operator, , Let Command, Store and Recall
28988 @section The Evaluates-To Operator
28993 @cindex Evaluates-to operator
28994 @cindex @samp{=>} operator
28995 The special algebraic symbol @samp{=>} is known as the @dfn{evaluates-to
28996 operator}. (It will show up as an @code{evalto} function call in
28997 other language modes like Pascal and @LaTeX{}.) This is a binary
28998 operator, that is, it has a lefthand and a righthand argument,
28999 although it can be entered with the righthand argument omitted.
29001 A formula like @samp{@var{a} => @var{b}} is evaluated by Calc as
29002 follows: First, @var{a} is not simplified or modified in any
29003 way. The previous value of argument @var{b} is thrown away; the
29004 formula @var{a} is then copied and evaluated as if by the @kbd{=}
29005 command according to all current modes and stored variable values,
29006 and the result is installed as the new value of @var{b}.
29008 For example, suppose you enter the algebraic formula @samp{2 + 3 => 17}.
29009 The number 17 is ignored, and the lefthand argument is left in its
29010 unevaluated form; the result is the formula @samp{2 + 3 => 5}.
29013 @pindex calc-evalto
29014 You can enter an @samp{=>} formula either directly using algebraic
29015 entry (in which case the righthand side may be omitted since it is
29016 going to be replaced right away anyhow), or by using the @kbd{s =}
29017 (@code{calc-evalto}) command, which takes @var{a} from the stack
29018 and replaces it with @samp{@var{a} => @var{b}}.
29020 Calc keeps track of all @samp{=>} operators on the stack, and
29021 recomputes them whenever anything changes that might affect their
29022 values, i.e., a mode setting or variable value. This occurs only
29023 if the @samp{=>} operator is at the top level of the formula, or
29024 if it is part of a top-level vector. In other words, pushing
29025 @samp{2 + (a => 17)} will change the 17 to the actual value of
29026 @samp{a} when you enter the formula, but the result will not be
29027 dynamically updated when @samp{a} is changed later because the
29028 @samp{=>} operator is buried inside a sum. However, a vector
29029 of @samp{=>} operators will be recomputed, since it is convenient
29030 to push a vector like @samp{[a =>, b =>, c =>]} on the stack to
29031 make a concise display of all the variables in your problem.
29032 (Another way to do this would be to use @samp{[a, b, c] =>},
29033 which provides a slightly different format of display. You
29034 can use whichever you find easiest to read.)
29037 @pindex calc-auto-recompute
29038 The @kbd{m C} (@code{calc-auto-recompute}) command allows you to
29039 turn this automatic recomputation on or off. If you turn
29040 recomputation off, you must explicitly recompute an @samp{=>}
29041 operator on the stack in one of the usual ways, such as by
29042 pressing @kbd{=}. Turning recomputation off temporarily can save
29043 a lot of time if you will be changing several modes or variables
29044 before you look at the @samp{=>} entries again.
29046 Most commands are not especially useful with @samp{=>} operators
29047 as arguments. For example, given @samp{x + 2 => 17}, it won't
29048 work to type @kbd{1 +} to get @samp{x + 3 => 18}. If you want
29049 to operate on the lefthand side of the @samp{=>} operator on
29050 the top of the stack, type @kbd{j 1} (that's the digit ``one'')
29051 to select the lefthand side, execute your commands, then type
29052 @kbd{j u} to unselect.
29054 All current modes apply when an @samp{=>} operator is computed,
29055 including the current simplification mode. Recall that the
29056 formula @samp{arcsin(sin(x))} will not be handled by Calc's algebraic
29057 simplifications, but Calc's unsafe simplifications will reduce it to
29058 @samp{x}. If you enter @samp{arcsin(sin(x)) =>} normally, the result
29059 will be @samp{arcsin(sin(x)) => arcsin(sin(x))}. If you change to
29060 Extended Simplification mode, the result will be
29061 @samp{arcsin(sin(x)) => x}. However, just pressing @kbd{a e}
29062 once will have no effect on @samp{arcsin(sin(x)) => arcsin(sin(x))},
29063 because the righthand side depends only on the lefthand side
29064 and the current mode settings, and the lefthand side is not
29065 affected by commands like @kbd{a e}.
29067 The ``let'' command (@kbd{s l}) has an interesting interaction
29068 with the @samp{=>} operator. The @kbd{s l} command evaluates the
29069 second-to-top stack entry with the top stack entry supplying
29070 a temporary value for a given variable. As you might expect,
29071 if that stack entry is an @samp{=>} operator its righthand
29072 side will temporarily show this value for the variable. In
29073 fact, all @samp{=>}s on the stack will be updated if they refer
29074 to that variable. But this change is temporary in the sense
29075 that the next command that causes Calc to look at those stack
29076 entries will make them revert to the old variable value.
29080 2: a => a 2: a => 17 2: a => a
29081 1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1
29084 17 s l a @key{RET} p 8 @key{RET}
29088 Here the @kbd{p 8} command changes the current precision,
29089 thus causing the @samp{=>} forms to be recomputed after the
29090 influence of the ``let'' is gone. The @kbd{d @key{SPC}} command
29091 (@code{calc-refresh}) is a handy way to force the @samp{=>}
29092 operators on the stack to be recomputed without any other
29096 @pindex calc-assign
29099 Embedded mode also uses @samp{=>} operators. In Embedded mode,
29100 the lefthand side of an @samp{=>} operator can refer to variables
29101 assigned elsewhere in the file by @samp{:=} operators. The
29102 assignment operator @samp{a := 17} does not actually do anything
29103 by itself. But Embedded mode recognizes it and marks it as a sort
29104 of file-local definition of the variable. You can enter @samp{:=}
29105 operators in Algebraic mode, or by using the @kbd{s :}
29106 (@code{calc-assign}) [@code{assign}] command which takes a variable
29107 and value from the stack and replaces them with an assignment.
29109 @xref{TeX and LaTeX Language Modes}, for the way @samp{=>} appears in
29110 @TeX{} language output. The @dfn{eqn} mode gives similar
29111 treatment to @samp{=>}.
29113 @node Graphics, Kill and Yank, Store and Recall, Top
29117 The commands for graphing data begin with the @kbd{g} prefix key. Calc
29118 uses GNUPLOT 2.0 or later to do graphics. These commands will only work
29119 if GNUPLOT is available on your system. (While GNUPLOT sounds like
29120 a relative of GNU Emacs, it is actually completely unrelated.
29121 However, it is free software. It can be obtained from
29122 @samp{http://www.gnuplot.info}.)
29124 @vindex calc-gnuplot-name
29125 If you have GNUPLOT installed on your system but Calc is unable to
29126 find it, you may need to set the @code{calc-gnuplot-name} variable in
29127 your Calc init file or @file{.emacs}. You may also need to set some
29128 Lisp variables to show Calc how to run GNUPLOT on your system; these
29129 are described under @kbd{g D} and @kbd{g O} below. If you are using
29130 the X window system or MS-Windows, Calc will configure GNUPLOT for you
29131 automatically. If you have GNUPLOT 3.0 or later and you are using a
29132 Unix or GNU system without X, Calc will configure GNUPLOT to display
29133 graphs using simple character graphics that will work on any
29134 Posix-compatible terminal.
29138 * Three Dimensional Graphics::
29139 * Managing Curves::
29140 * Graphics Options::
29144 @node Basic Graphics, Three Dimensional Graphics, Graphics, Graphics
29145 @section Basic Graphics
29149 @pindex calc-graph-fast
29150 The easiest graphics command is @kbd{g f} (@code{calc-graph-fast}).
29151 This command takes two vectors of equal length from the stack.
29152 The vector at the top of the stack represents the ``y'' values of
29153 the various data points. The vector in the second-to-top position
29154 represents the corresponding ``x'' values. This command runs
29155 GNUPLOT (if it has not already been started by previous graphing
29156 commands) and displays the set of data points. The points will
29157 be connected by lines, and there will also be some kind of symbol
29158 to indicate the points themselves.
29160 The ``x'' entry may instead be an interval form, in which case suitable
29161 ``x'' values are interpolated between the minimum and maximum values of
29162 the interval (whether the interval is open or closed is ignored).
29164 The ``x'' entry may also be a number, in which case Calc uses the
29165 sequence of ``x'' values @expr{x}, @expr{x+1}, @expr{x+2}, etc.
29166 (Generally the number 0 or 1 would be used for @expr{x} in this case.)
29168 The ``y'' entry may be any formula instead of a vector. Calc effectively
29169 uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula;
29170 the result of this must be a formula in a single (unassigned) variable.
29171 The formula is plotted with this variable taking on the various ``x''
29172 values. Graphs of formulas by default use lines without symbols at the
29173 computed data points. Note that if neither ``x'' nor ``y'' is a vector,
29174 Calc guesses at a reasonable number of data points to use. See the
29175 @kbd{g N} command below. (The ``x'' values must be either a vector
29176 or an interval if ``y'' is a formula.)
29182 If ``y'' is (or evaluates to) a formula of the form
29183 @samp{xy(@var{x}, @var{y})} then the result is a
29184 parametric plot. The two arguments of the fictitious @code{xy} function
29185 are used as the ``x'' and ``y'' coordinates of the curve, respectively.
29186 In this case the ``x'' vector or interval you specified is not directly
29187 visible in the graph. For example, if ``x'' is the interval @samp{[0..360]}
29188 and ``y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph
29191 Also, ``x'' and ``y'' may each be variable names, in which case Calc
29192 looks for suitable vectors, intervals, or formulas stored in those
29195 The ``x'' and ``y'' values for the data points (as pulled from the vectors,
29196 calculated from the formulas, or interpolated from the intervals) should
29197 be real numbers (integers, fractions, or floats). One exception to this
29198 is that the ``y'' entry can consist of a vector of numbers combined with
29199 error forms, in which case the points will be plotted with the
29200 appropriate error bars. Other than this, if either the ``x''
29201 value or the ``y'' value of a given data point is not a real number, that
29202 data point will be omitted from the graph. The points on either side
29203 of the invalid point will @emph{not} be connected by a line.
29205 See the documentation for @kbd{g a} below for a description of the way
29206 numeric prefix arguments affect @kbd{g f}.
29208 @cindex @code{PlotRejects} variable
29209 @vindex PlotRejects
29210 If you store an empty vector in the variable @code{PlotRejects}
29211 (i.e., @kbd{[ ] s t PlotRejects}), Calc will append information to
29212 this vector for every data point which was rejected because its
29213 ``x'' or ``y'' values were not real numbers. The result will be
29214 a matrix where each row holds the curve number, data point number,
29215 ``x'' value, and ``y'' value for a rejected data point.
29216 @xref{Evaluates-To Operator}, for a handy way to keep tabs on the
29217 current value of @code{PlotRejects}. @xref{Operations on Variables},
29218 for the @kbd{s R} command which is another easy way to examine
29219 @code{PlotRejects}.
29222 @pindex calc-graph-clear
29223 To clear the graphics display, type @kbd{g c} (@code{calc-graph-clear}).
29224 If the GNUPLOT output device is an X window, the window will go away.
29225 Effects on other kinds of output devices will vary. You don't need
29226 to use @kbd{g c} if you don't want to---if you give another @kbd{g f}
29227 or @kbd{g p} command later on, it will reuse the existing graphics
29228 window if there is one.
29230 @node Three Dimensional Graphics, Managing Curves, Basic Graphics, Graphics
29231 @section Three-Dimensional Graphics
29234 @pindex calc-graph-fast-3d
29235 The @kbd{g F} (@code{calc-graph-fast-3d}) command makes a three-dimensional
29236 graph. It works only if you have GNUPLOT 3.0 or later; with GNUPLOT 2.0,
29237 you will see a GNUPLOT error message if you try this command.
29239 The @kbd{g F} command takes three values from the stack, called ``x'',
29240 ``y'', and ``z'', respectively. As was the case for 2D graphs, there
29241 are several options for these values.
29243 In the first case, ``x'' and ``y'' are each vectors (not necessarily of
29244 the same length); either or both may instead be interval forms. The
29245 ``z'' value must be a matrix with the same number of rows as elements
29246 in ``x'', and the same number of columns as elements in ``y''. The
29247 result is a surface plot where
29248 @texline @math{z_{ij}}
29249 @infoline @expr{z_ij}
29250 is the height of the point
29251 at coordinate @expr{(x_i, y_j)} on the surface. The 3D graph will
29252 be displayed from a certain default viewpoint; you can change this
29253 viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*}
29254 buffer as described later. See the GNUPLOT documentation for a
29255 description of the @samp{set view} command.
29257 Each point in the matrix will be displayed as a dot in the graph,
29258 and these points will be connected by a grid of lines (@dfn{isolines}).
29260 In the second case, ``x'', ``y'', and ``z'' are all vectors of equal
29261 length. The resulting graph displays a 3D line instead of a surface,
29262 where the coordinates of points along the line are successive triplets
29263 of values from the input vectors.
29265 In the third case, ``x'' and ``y'' are vectors or interval forms, and
29266 ``z'' is any formula involving two variables (not counting variables
29267 with assigned values). These variables are sorted into alphabetical
29268 order; the first takes on values from ``x'' and the second takes on
29269 values from ``y'' to form a matrix of results that are graphed as a
29276 If the ``z'' formula evaluates to a call to the fictitious function
29277 @samp{xyz(@var{x}, @var{y}, @var{z})}, then the result is a
29278 ``parametric surface.'' In this case, the axes of the graph are
29279 taken from the @var{x} and @var{y} values in these calls, and the
29280 ``x'' and ``y'' values from the input vectors or intervals are used only
29281 to specify the range of inputs to the formula. For example, plotting
29282 @samp{[0..360], [0..180], xyz(sin(x)*sin(y), cos(x)*sin(y), cos(y))}
29283 will draw a sphere. (Since the default resolution for 3D plots is
29284 5 steps in each of ``x'' and ``y'', this will draw a very crude
29285 sphere. You could use the @kbd{g N} command, described below, to
29286 increase this resolution, or specify the ``x'' and ``y'' values as
29287 vectors with more than 5 elements.
29289 It is also possible to have a function in a regular @kbd{g f} plot
29290 evaluate to an @code{xyz} call. Since @kbd{g f} plots a line, not
29291 a surface, the result will be a 3D parametric line. For example,
29292 @samp{[[0..720], xyz(sin(x), cos(x), x)]} will plot two turns of a
29293 helix (a three-dimensional spiral).
29295 As for @kbd{g f}, each of ``x'', ``y'', and ``z'' may instead be
29296 variables containing the relevant data.
29298 @node Managing Curves, Graphics Options, Three Dimensional Graphics, Graphics
29299 @section Managing Curves
29302 The @kbd{g f} command is really shorthand for the following commands:
29303 @kbd{C-u g d g a g p}. Likewise, @w{@kbd{g F}} is shorthand for
29304 @kbd{C-u g d g A g p}. You can gain more control over your graph
29305 by using these commands directly.
29308 @pindex calc-graph-add
29309 The @kbd{g a} (@code{calc-graph-add}) command adds the ``curve''
29310 represented by the two values on the top of the stack to the current
29311 graph. You can have any number of curves in the same graph. When
29312 you give the @kbd{g p} command, all the curves will be drawn superimposed
29315 The @kbd{g a} command (and many others that affect the current graph)
29316 will cause a special buffer, @samp{*Gnuplot Commands*}, to be displayed
29317 in another window. This buffer is a template of the commands that will
29318 be sent to GNUPLOT when it is time to draw the graph. The first
29319 @kbd{g a} command adds a @code{plot} command to this buffer. Succeeding
29320 @kbd{g a} commands add extra curves onto that @code{plot} command.
29321 Other graph-related commands put other GNUPLOT commands into this
29322 buffer. In normal usage you never need to work with this buffer
29323 directly, but you can if you wish. The only constraint is that there
29324 must be only one @code{plot} command, and it must be the last command
29325 in the buffer. If you want to save and later restore a complete graph
29326 configuration, you can use regular Emacs commands to save and restore
29327 the contents of the @samp{*Gnuplot Commands*} buffer.
29331 If the values on the stack are not variable names, @kbd{g a} will invent
29332 variable names for them (of the form @samp{PlotData@var{n}}) and store
29333 the values in those variables. The ``x'' and ``y'' variables are what
29334 go into the @code{plot} command in the template. If you add a curve
29335 that uses a certain variable and then later change that variable, you
29336 can replot the graph without having to delete and re-add the curve.
29337 That's because the variable name, not the vector, interval or formula
29338 itself, is what was added by @kbd{g a}.
29340 A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way
29341 stack entries are interpreted as curves. With a positive prefix
29342 argument @expr{n}, the top @expr{n} stack entries are ``y'' values
29343 for @expr{n} different curves which share a common ``x'' value in
29344 the @expr{n+1}st stack entry. (Thus @kbd{g a} with no prefix
29345 argument is equivalent to @kbd{C-u 1 g a}.)
29347 A prefix of zero or plain @kbd{C-u} means to take two stack entries,
29348 ``x'' and ``y'' as usual, but to interpret ``y'' as a vector of
29349 ``y'' values for several curves that share a common ``x''.
29351 A negative prefix argument tells Calc to read @expr{n} vectors from
29352 the stack; each vector @expr{[x, y]} describes an independent curve.
29353 This is the only form of @kbd{g a} that creates several curves at once
29354 that don't have common ``x'' values. (Of course, the range of ``x''
29355 values covered by all the curves ought to be roughly the same if
29356 they are to look nice on the same graph.)
29358 For example, to plot
29359 @texline @math{\sin n x}
29360 @infoline @expr{sin(n x)}
29361 for integers @expr{n}
29362 from 1 to 5, you could use @kbd{v x} to create a vector of integers
29363 (@expr{n}), then @kbd{V M '} or @kbd{V M $} to map @samp{sin(n x)}
29364 across this vector. The resulting vector of formulas is suitable
29365 for use as the ``y'' argument to a @kbd{C-u g a} or @kbd{C-u g f}
29369 @pindex calc-graph-add-3d
29370 The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve
29371 to the graph. It is not valid to intermix 2D and 3D curves in a
29372 single graph. This command takes three arguments, ``x'', ``y'',
29373 and ``z'', from the stack. With a positive prefix @expr{n}, it
29374 takes @expr{n+2} arguments (common ``x'' and ``y'', plus @expr{n}
29375 separate ``z''s). With a zero prefix, it takes three stack entries
29376 but the ``z'' entry is a vector of curve values. With a negative
29377 prefix @expr{-n}, it takes @expr{n} vectors of the form @expr{[x, y, z]}.
29378 The @kbd{g A} command works by adding a @code{splot} (surface-plot)
29379 command to the @samp{*Gnuplot Commands*} buffer.
29381 (Although @kbd{g a} adds a 2D @code{plot} command to the
29382 @samp{*Gnuplot Commands*} buffer, Calc changes this to @code{splot}
29383 before sending it to GNUPLOT if it notices that the data points are
29384 evaluating to @code{xyz} calls. It will not work to mix 2D and 3D
29385 @kbd{g a} curves in a single graph, although Calc does not currently
29389 @pindex calc-graph-delete
29390 The @kbd{g d} (@code{calc-graph-delete}) command deletes the most
29391 recently added curve from the graph. It has no effect if there are
29392 no curves in the graph. With a numeric prefix argument of any kind,
29393 it deletes all of the curves from the graph.
29396 @pindex calc-graph-hide
29397 The @kbd{g H} (@code{calc-graph-hide}) command ``hides'' or ``unhides''
29398 the most recently added curve. A hidden curve will not appear in
29399 the actual plot, but information about it such as its name and line and
29400 point styles will be retained.
29403 @pindex calc-graph-juggle
29404 The @kbd{g j} (@code{calc-graph-juggle}) command moves the curve
29405 at the end of the list (the ``most recently added curve'') to the
29406 front of the list. The next-most-recent curve is thus exposed for
29407 @w{@kbd{g d}} or similar commands to use. With @kbd{g j} you can work
29408 with any curve in the graph even though curve-related commands only
29409 affect the last curve in the list.
29412 @pindex calc-graph-plot
29413 The @kbd{g p} (@code{calc-graph-plot}) command uses GNUPLOT to draw
29414 the graph described in the @samp{*Gnuplot Commands*} buffer. Any
29415 GNUPLOT parameters which are not defined by commands in this buffer
29416 are reset to their default values. The variables named in the @code{plot}
29417 command are written to a temporary data file and the variable names
29418 are then replaced by the file name in the template. The resulting
29419 plotting commands are fed to the GNUPLOT program. See the documentation
29420 for the GNUPLOT program for more specific information. All temporary
29421 files are removed when Emacs or GNUPLOT exits.
29423 If you give a formula for ``y'', Calc will remember all the values that
29424 it calculates for the formula so that later plots can reuse these values.
29425 Calc throws out these saved values when you change any circumstances
29426 that may affect the data, such as switching from Degrees to Radians
29427 mode, or changing the value of a parameter in the formula. You can
29428 force Calc to recompute the data from scratch by giving a negative
29429 numeric prefix argument to @kbd{g p}.
29431 Calc uses a fairly rough step size when graphing formulas over intervals.
29432 This is to ensure quick response. You can ``refine'' a plot by giving
29433 a positive numeric prefix argument to @kbd{g p}. Calc goes through
29434 the data points it has computed and saved from previous plots of the
29435 function, and computes and inserts a new data point midway between
29436 each of the existing points. You can refine a plot any number of times,
29437 but beware that the amount of calculation involved doubles each time.
29439 Calc does not remember computed values for 3D graphs. This means the
29440 numerix prefix argument, if any, to @kbd{g p} is effectively ignored if
29441 the current graph is three-dimensional.
29444 @pindex calc-graph-print
29445 The @kbd{g P} (@code{calc-graph-print}) command is like @kbd{g p},
29446 except that it sends the output to a printer instead of to the
29447 screen. More precisely, @kbd{g p} looks for @samp{set terminal}
29448 or @samp{set output} commands in the @samp{*Gnuplot Commands*} buffer;
29449 lacking these it uses the default settings. However, @kbd{g P}
29450 ignores @samp{set terminal} and @samp{set output} commands and
29451 uses a different set of default values. All of these values are
29452 controlled by the @kbd{g D} and @kbd{g O} commands discussed below.
29453 Provided everything is set up properly, @kbd{g p} will plot to
29454 the screen unless you have specified otherwise and @kbd{g P} will
29455 always plot to the printer.
29457 @node Graphics Options, Devices, Managing Curves, Graphics
29458 @section Graphics Options
29462 @pindex calc-graph-grid
29463 The @kbd{g g} (@code{calc-graph-grid}) command turns the ``grid''
29464 on and off. It is off by default; tick marks appear only at the
29465 edges of the graph. With the grid turned on, dotted lines appear
29466 across the graph at each tick mark. Note that this command only
29467 changes the setting in @samp{*Gnuplot Commands*}; to see the effects
29468 of the change you must give another @kbd{g p} command.
29471 @pindex calc-graph-border
29472 The @kbd{g b} (@code{calc-graph-border}) command turns the border
29473 (the box that surrounds the graph) on and off. It is on by default.
29474 This command will only work with GNUPLOT 3.0 and later versions.
29477 @pindex calc-graph-key
29478 The @kbd{g k} (@code{calc-graph-key}) command turns the ``key''
29479 on and off. The key is a chart in the corner of the graph that
29480 shows the correspondence between curves and line styles. It is
29481 off by default, and is only really useful if you have several
29482 curves on the same graph.
29485 @pindex calc-graph-num-points
29486 The @kbd{g N} (@code{calc-graph-num-points}) command allows you
29487 to select the number of data points in the graph. This only affects
29488 curves where neither ``x'' nor ``y'' is specified as a vector.
29489 Enter a blank line to revert to the default value (initially 15).
29490 With no prefix argument, this command affects only the current graph.
29491 With a positive prefix argument this command changes or, if you enter
29492 a blank line, displays the default number of points used for all
29493 graphs created by @kbd{g a} that don't specify the resolution explicitly.
29494 With a negative prefix argument, this command changes or displays
29495 the default value (initially 5) used for 3D graphs created by @kbd{g A}.
29496 Note that a 3D setting of 5 means that a total of @expr{5^2 = 25} points
29497 will be computed for the surface.
29499 Data values in the graph of a function are normally computed to a
29500 precision of five digits, regardless of the current precision at the
29501 time. This is usually more than adequate, but there are cases where
29502 it will not be. For example, plotting @expr{1 + x} with @expr{x} in the
29503 interval @samp{[0 ..@: 1e-6]} will round all the data points down
29504 to 1.0! Putting the command @samp{set precision @var{n}} in the
29505 @samp{*Gnuplot Commands*} buffer will cause the data to be computed
29506 at precision @var{n} instead of 5. Since this is such a rare case,
29507 there is no keystroke-based command to set the precision.
29510 @pindex calc-graph-header
29511 The @kbd{g h} (@code{calc-graph-header}) command sets the title
29512 for the graph. This will show up centered above the graph.
29513 The default title is blank (no title).
29516 @pindex calc-graph-name
29517 The @kbd{g n} (@code{calc-graph-name}) command sets the title of an
29518 individual curve. Like the other curve-manipulating commands, it
29519 affects the most recently added curve, i.e., the last curve on the
29520 list in the @samp{*Gnuplot Commands*} buffer. To set the title of
29521 the other curves you must first juggle them to the end of the list
29522 with @kbd{g j}, or edit the @samp{*Gnuplot Commands*} buffer by hand.
29523 Curve titles appear in the key; if the key is turned off they are
29528 @pindex calc-graph-title-x
29529 @pindex calc-graph-title-y
29530 The @kbd{g t} (@code{calc-graph-title-x}) and @kbd{g T}
29531 (@code{calc-graph-title-y}) commands set the titles on the ``x''
29532 and ``y'' axes, respectively. These titles appear next to the
29533 tick marks on the left and bottom edges of the graph, respectively.
29534 Calc does not have commands to control the tick marks themselves,
29535 but you can edit them into the @samp{*Gnuplot Commands*} buffer if
29536 you wish. See the GNUPLOT documentation for details.
29540 @pindex calc-graph-range-x
29541 @pindex calc-graph-range-y
29542 The @kbd{g r} (@code{calc-graph-range-x}) and @kbd{g R}
29543 (@code{calc-graph-range-y}) commands set the range of values on the
29544 ``x'' and ``y'' axes, respectively. You are prompted to enter a
29545 suitable range. This should be either a pair of numbers of the
29546 form, @samp{@var{min}:@var{max}}, or a blank line to revert to the
29547 default behavior of setting the range based on the range of values
29548 in the data, or @samp{$} to take the range from the top of the stack.
29549 Ranges on the stack can be represented as either interval forms or
29550 vectors: @samp{[@var{min} ..@: @var{max}]} or @samp{[@var{min}, @var{max}]}.
29554 @pindex calc-graph-log-x
29555 @pindex calc-graph-log-y
29556 The @kbd{g l} (@code{calc-graph-log-x}) and @kbd{g L} (@code{calc-graph-log-y})
29557 commands allow you to set either or both of the axes of the graph to
29558 be logarithmic instead of linear.
29563 @pindex calc-graph-log-z
29564 @pindex calc-graph-range-z
29565 @pindex calc-graph-title-z
29566 For 3D plots, @kbd{g C-t}, @kbd{g C-r}, and @kbd{g C-l} (those are
29567 letters with the Control key held down) are the corresponding commands
29568 for the ``z'' axis.
29572 @pindex calc-graph-zero-x
29573 @pindex calc-graph-zero-y
29574 The @kbd{g z} (@code{calc-graph-zero-x}) and @kbd{g Z}
29575 (@code{calc-graph-zero-y}) commands control whether a dotted line is
29576 drawn to indicate the ``x'' and/or ``y'' zero axes. (These are the same
29577 dotted lines that would be drawn there anyway if you used @kbd{g g} to
29578 turn the ``grid'' feature on.) Zero-axis lines are on by default, and
29579 may be turned off only in GNUPLOT 3.0 and later versions. They are
29580 not available for 3D plots.
29583 @pindex calc-graph-line-style
29584 The @kbd{g s} (@code{calc-graph-line-style}) command turns the connecting
29585 lines on or off for the most recently added curve, and optionally selects
29586 the style of lines to be used for that curve. Plain @kbd{g s} simply
29587 toggles the lines on and off. With a numeric prefix argument, @kbd{g s}
29588 turns lines on and sets a particular line style. Line style numbers
29589 start at one and their meanings vary depending on the output device.
29590 GNUPLOT guarantees that there will be at least six different line styles
29591 available for any device.
29594 @pindex calc-graph-point-style
29595 The @kbd{g S} (@code{calc-graph-point-style}) command similarly turns
29596 the symbols at the data points on or off, or sets the point style.
29597 If you turn both lines and points off, the data points will show as
29598 tiny dots. If the ``y'' values being plotted contain error forms and
29599 the connecting lines are turned off, then this command will also turn
29600 the error bars on or off.
29602 @cindex @code{LineStyles} variable
29603 @cindex @code{PointStyles} variable
29605 @vindex PointStyles
29606 Another way to specify curve styles is with the @code{LineStyles} and
29607 @code{PointStyles} variables. These variables initially have no stored
29608 values, but if you store a vector of integers in one of these variables,
29609 the @kbd{g a} and @kbd{g f} commands will use those style numbers
29610 instead of the defaults for new curves that are added to the graph.
29611 An entry should be a positive integer for a specific style, or 0 to let
29612 the style be chosen automatically, or @mathit{-1} to turn off lines or points
29613 altogether. If there are more curves than elements in the vector, the
29614 last few curves will continue to have the default styles. Of course,
29615 you can later use @kbd{g s} and @kbd{g S} to change any of these styles.
29617 For example, @kbd{'[2 -1 3] @key{RET} s t LineStyles} causes the first curve
29618 to have lines in style number 2, the second curve to have no connecting
29619 lines, and the third curve to have lines in style 3. Point styles will
29620 still be assigned automatically, but you could store another vector in
29621 @code{PointStyles} to define them, too.
29623 @node Devices, , Graphics Options, Graphics
29624 @section Graphical Devices
29628 @pindex calc-graph-device
29629 The @kbd{g D} (@code{calc-graph-device}) command sets the device name
29630 (or ``terminal name'' in GNUPLOT lingo) to be used by @kbd{g p} commands
29631 on this graph. It does not affect the permanent default device name.
29632 If you enter a blank name, the device name reverts to the default.
29633 Enter @samp{?} to see a list of supported devices.
29635 With a positive numeric prefix argument, @kbd{g D} instead sets
29636 the default device name, used by all plots in the future which do
29637 not override it with a plain @kbd{g D} command. If you enter a
29638 blank line this command shows you the current default. The special
29639 name @code{default} signifies that Calc should choose @code{x11} if
29640 the X window system is in use (as indicated by the presence of a
29641 @code{DISPLAY} environment variable), @code{windows} on MS-Windows, or
29642 otherwise @code{dumb} under GNUPLOT 3.0 and later, or
29643 @code{postscript} under GNUPLOT 2.0. This is the initial default
29646 The @code{dumb} device is an interface to ``dumb terminals,'' i.e.,
29647 terminals with no special graphics facilities. It writes a crude
29648 picture of the graph composed of characters like @code{-} and @code{|}
29649 to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays.
29650 The graph is made the same size as the Emacs screen, which on most
29651 dumb terminals will be
29652 @texline @math{80\times24}
29654 characters. The graph is displayed in
29655 an Emacs ``recursive edit''; type @kbd{q} or @kbd{C-c C-c} to exit
29656 the recursive edit and return to Calc. Note that the @code{dumb}
29657 device is present only in GNUPLOT 3.0 and later versions.
29659 The word @code{dumb} may be followed by two numbers separated by
29660 spaces. These are the desired width and height of the graph in
29661 characters. Also, the device name @code{big} is like @code{dumb}
29662 but creates a graph four times the width and height of the Emacs
29663 screen. You will then have to scroll around to view the entire
29664 graph. In the @samp{*Gnuplot Trail*} buffer, @key{SPC}, @key{DEL},
29665 @kbd{<}, and @kbd{>} are defined to scroll by one screenful in each
29666 of the four directions.
29668 With a negative numeric prefix argument, @kbd{g D} sets or displays
29669 the device name used by @kbd{g P} (@code{calc-graph-print}). This
29670 is initially @code{postscript}. If you don't have a PostScript
29671 printer, you may decide once again to use @code{dumb} to create a
29672 plot on any text-only printer.
29675 @pindex calc-graph-output
29676 The @kbd{g O} (@code{calc-graph-output}) command sets the name of the
29677 output file used by GNUPLOT@. For some devices, notably @code{x11} and
29678 @code{windows}, there is no output file and this information is not
29679 used. Many other ``devices'' are really file formats like
29680 @code{postscript}; in these cases the output in the desired format
29681 goes into the file you name with @kbd{g O}. Type @kbd{g O stdout
29682 @key{RET}} to set GNUPLOT to write to its standard output stream,
29683 i.e., to @samp{*Gnuplot Trail*}. This is the default setting.
29685 Another special output name is @code{tty}, which means that GNUPLOT
29686 is going to write graphics commands directly to its standard output,
29687 which you wish Emacs to pass through to your terminal. Tektronix
29688 graphics terminals, among other devices, operate this way. Calc does
29689 this by telling GNUPLOT to write to a temporary file, then running a
29690 sub-shell executing the command @samp{cat tempfile >/dev/tty}. On
29691 typical Unix systems, this will copy the temporary file directly to
29692 the terminal, bypassing Emacs entirely. You will have to type @kbd{C-l}
29693 to Emacs afterwards to refresh the screen.
29695 Once again, @kbd{g O} with a positive or negative prefix argument
29696 sets the default or printer output file names, respectively. In each
29697 case you can specify @code{auto}, which causes Calc to invent a temporary
29698 file name for each @kbd{g p} (or @kbd{g P}) command. This temporary file
29699 will be deleted once it has been displayed or printed. If the output file
29700 name is not @code{auto}, the file is not automatically deleted.
29702 The default and printer devices and output files can be saved
29703 permanently by the @kbd{m m} (@code{calc-save-modes}) command. The
29704 default number of data points (see @kbd{g N}) and the X geometry
29705 (see @kbd{g X}) are also saved. Other graph information is @emph{not}
29706 saved; you can save a graph's configuration simply by saving the contents
29707 of the @samp{*Gnuplot Commands*} buffer.
29709 @vindex calc-gnuplot-plot-command
29710 @vindex calc-gnuplot-default-device
29711 @vindex calc-gnuplot-default-output
29712 @vindex calc-gnuplot-print-command
29713 @vindex calc-gnuplot-print-device
29714 @vindex calc-gnuplot-print-output
29715 You may wish to configure the default and
29716 printer devices and output files for the whole system. The relevant
29717 Lisp variables are @code{calc-gnuplot-default-device} and @code{-output},
29718 and @code{calc-gnuplot-print-device} and @code{-output}. The output
29719 file names must be either strings as described above, or Lisp
29720 expressions which are evaluated on the fly to get the output file names.
29722 Other important Lisp variables are @code{calc-gnuplot-plot-command} and
29723 @code{calc-gnuplot-print-command}, which give the system commands to
29724 display or print the output of GNUPLOT, respectively. These may be
29725 @code{nil} if no command is necessary, or strings which can include
29726 @samp{%s} to signify the name of the file to be displayed or printed.
29727 Or, these variables may contain Lisp expressions which are evaluated
29728 to display or print the output. These variables are customizable
29729 (@pxref{Customizing Calc}).
29732 @pindex calc-graph-display
29733 The @kbd{g x} (@code{calc-graph-display}) command lets you specify
29734 on which X window system display your graphs should be drawn. Enter
29735 a blank line to see the current display name. This command has no
29736 effect unless the current device is @code{x11}.
29739 @pindex calc-graph-geometry
29740 The @kbd{g X} (@code{calc-graph-geometry}) command is a similar
29741 command for specifying the position and size of the X window.
29742 The normal value is @code{default}, which generally means your
29743 window manager will let you place the window interactively.
29744 Entering @samp{800x500+0+0} would create an 800-by-500 pixel
29745 window in the upper-left corner of the screen. This command has no
29746 effect if the current device is @code{windows}.
29748 The buffer called @samp{*Gnuplot Trail*} holds a transcript of the
29749 session with GNUPLOT@. This shows the commands Calc has ``typed'' to
29750 GNUPLOT and the responses it has received. Calc tries to notice when an
29751 error message has appeared here and display the buffer for you when
29752 this happens. You can check this buffer yourself if you suspect
29753 something has gone wrong@footnote{
29754 On MS-Windows, due to the peculiarities of how the Windows version of
29755 GNUPLOT (called @command{wgnuplot}) works, the GNUPLOT responses are
29756 not communicated back to Calc. Instead, you need to look them up in
29757 the GNUPLOT command window that is displayed as in normal interactive
29762 @pindex calc-graph-command
29763 The @kbd{g C} (@code{calc-graph-command}) command prompts you to
29764 enter any line of text, then simply sends that line to the current
29765 GNUPLOT process. The @samp{*Gnuplot Trail*} buffer looks deceptively
29766 like a Shell buffer but you can't type commands in it yourself.
29767 Instead, you must use @kbd{g C} for this purpose.
29771 @pindex calc-graph-view-commands
29772 @pindex calc-graph-view-trail
29773 The @kbd{g v} (@code{calc-graph-view-commands}) and @kbd{g V}
29774 (@code{calc-graph-view-trail}) commands display the @samp{*Gnuplot Commands*}
29775 and @samp{*Gnuplot Trail*} buffers, respectively, in another window.
29776 This happens automatically when Calc thinks there is something you
29777 will want to see in either of these buffers. If you type @kbd{g v}
29778 or @kbd{g V} when the relevant buffer is already displayed, the
29779 buffer is hidden again. (Note that on MS-Windows, the @samp{*Gnuplot
29780 Trail*} buffer will usually show nothing of interest, because
29781 GNUPLOT's responses are not communicated back to Calc.)
29783 One reason to use @kbd{g v} is to add your own commands to the
29784 @samp{*Gnuplot Commands*} buffer. Press @kbd{g v}, then use
29785 @kbd{C-x o} to switch into that window. For example, GNUPLOT has
29786 @samp{set label} and @samp{set arrow} commands that allow you to
29787 annotate your plots. Since Calc doesn't understand these commands,
29788 you have to add them to the @samp{*Gnuplot Commands*} buffer
29789 yourself, then use @w{@kbd{g p}} to replot using these new commands. Note
29790 that your commands must appear @emph{before} the @code{plot} command.
29791 To get help on any GNUPLOT feature, type, e.g., @kbd{g C help set label}.
29792 You may have to type @kbd{g C @key{RET}} a few times to clear the
29793 ``press return for more'' or ``subtopic of @dots{}'' requests.
29794 Note that Calc always sends commands (like @samp{set nolabel}) to
29795 reset all plotting parameters to the defaults before each plot, so
29796 to delete a label all you need to do is delete the @samp{set label}
29797 line you added (or comment it out with @samp{#}) and then replot
29801 @pindex calc-graph-quit
29802 You can use @kbd{g q} (@code{calc-graph-quit}) to kill the GNUPLOT
29803 process that is running. The next graphing command you give will
29804 start a fresh GNUPLOT process. The word @samp{Graph} appears in
29805 the Calc window's mode line whenever a GNUPLOT process is currently
29806 running. The GNUPLOT process is automatically killed when you
29807 exit Emacs if you haven't killed it manually by then.
29810 @pindex calc-graph-kill
29811 The @kbd{g K} (@code{calc-graph-kill}) command is like @kbd{g q}
29812 except that it also views the @samp{*Gnuplot Trail*} buffer so that
29813 you can see the process being killed. This is better if you are
29814 killing GNUPLOT because you think it has gotten stuck.
29816 @node Kill and Yank, Keypad Mode, Graphics, Top
29817 @chapter Kill and Yank Functions
29820 The commands in this chapter move information between the Calculator and
29821 other Emacs editing buffers.
29823 In many cases Embedded mode is an easier and more natural way to
29824 work with Calc from a regular editing buffer. @xref{Embedded Mode}.
29827 * Killing From Stack::
29828 * Yanking Into Stack::
29829 * Saving Into Registers::
29830 * Inserting From Registers::
29831 * Grabbing From Buffers::
29832 * Yanking Into Buffers::
29833 * X Cut and Paste::
29836 @node Killing From Stack, Yanking Into Stack, Kill and Yank, Kill and Yank
29837 @section Killing from the Stack
29843 @pindex calc-copy-as-kill
29845 @pindex calc-kill-region
29847 @pindex calc-copy-region-as-kill
29850 @dfn{Kill} commands are Emacs commands that insert text into the ``kill
29851 ring,'' from which it can later be ``yanked'' by a @kbd{C-y} command.
29852 Three common kill commands in normal Emacs are @kbd{C-k}, which kills
29853 one line, @kbd{C-w}, which kills the region between mark and point, and
29854 @kbd{M-w}, which puts the region into the kill ring without actually
29855 deleting it. All of these commands work in the Calculator, too,
29856 although in the Calculator they operate on whole stack entries, so they
29857 ``round up'' the specified region to encompass full lines. (To copy
29858 only parts of lines, the @kbd{M-C-w} command in the Calculator will copy
29859 the region to the kill ring without any ``rounding up'', just like the
29860 @kbd{M-w} command in normal Emacs.) Also, @kbd{M-k} has been provided
29861 to complete the set; it puts the current line into the kill ring without
29864 The kill commands are unusual in that they pay attention to the location
29865 of the cursor in the Calculator buffer. If the cursor is on or below
29866 the bottom line, the kill commands operate on the top of the stack.
29867 Otherwise, they operate on whatever stack element the cursor is on. The
29868 text is copied into the kill ring exactly as it appears on the screen,
29869 including line numbers if they are enabled.
29871 A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number
29872 of lines killed. A positive argument kills the current line and @expr{n-1}
29873 lines below it. A negative argument kills the @expr{-n} lines above the
29874 current line. Again this mirrors the behavior of the standard Emacs
29875 @kbd{C-k} command. Although a whole line is always deleted, @kbd{C-k}
29876 with no argument copies only the number itself into the kill ring, whereas
29877 @kbd{C-k} with a prefix argument of 1 copies the number with its trailing
29880 @node Yanking Into Stack, Saving Into Registers, Killing From Stack, Kill and Yank
29881 @section Yanking into the Stack
29886 The @kbd{C-y} command yanks the most recently killed text back into the
29887 Calculator. It pushes this value onto the top of the stack regardless of
29888 the cursor position. In general it re-parses the killed text as a number
29889 or formula (or a list of these separated by commas or newlines). However if
29890 the thing being yanked is something that was just killed from the Calculator
29891 itself, its full internal structure is yanked. For example, if you have
29892 set the floating-point display mode to show only four significant digits,
29893 then killing and re-yanking 3.14159 (which displays as 3.142) will yank the
29894 full 3.14159, even though yanking it into any other buffer would yank the
29895 number in its displayed form, 3.142. (Since the default display modes
29896 show all objects to their full precision, this feature normally makes no
29899 @node Saving Into Registers, Inserting From Registers, Yanking Into Stack, Kill and Yank
29900 @section Saving into Registers
29904 @pindex calc-copy-to-register
29905 @pindex calc-prepend-to-register
29906 @pindex calc-append-to-register
29908 An alternative to killing and yanking stack entries is using
29909 registers in Calc. Saving stack entries in registers is like
29910 saving text in normal Emacs registers; although, like Calc's kill
29911 commands, register commands always operate on whole stack
29914 Registers in Calc are places to store stack entries for later use;
29915 each register is indexed by a single character. To store the current
29916 region (rounded up, of course, to include full stack entries) into a
29917 register, use the command @kbd{r s} (@code{calc-copy-to-register}).
29918 You will then be prompted for a register to use, the next character
29919 you type will be the index for the register. To store the region in
29920 register @var{r}, the full command will be @kbd{r s @var{r}}. With an
29921 argument, @kbd{C-u r s @var{r}}, the region being copied to the
29922 register will be deleted from the Calc buffer.
29924 It is possible to add additional stack entries to a register. The
29925 command @kbd{M-x calc-append-to-register} will prompt for a register,
29926 then add the stack entries in the region to the end of the register
29927 contents. The command @kbd{M-x calc-prepend-to-register} will
29928 similarly prompt for a register and add the stack entries in the
29929 region to the beginning of the register contents. Both commands take
29930 @kbd{C-u} arguments, which will cause the region to be deleted after being
29931 added to the register.
29933 @node Inserting From Registers, Grabbing From Buffers, Saving Into Registers, Kill and Yank
29934 @section Inserting from Registers
29937 @pindex calc-insert-register
29938 The command @kbd{r i} (@code{calc-insert-register}) will prompt for a
29939 register, then insert the contents of that register into the
29940 Calculator. If the contents of the register were placed there from
29941 within Calc, then the full internal structure of the contents will be
29942 inserted into the Calculator, otherwise whatever text is in the
29943 register is reparsed and then inserted into the Calculator.
29945 @node Grabbing From Buffers, Yanking Into Buffers, Inserting From Registers, Kill and Yank
29946 @section Grabbing from Other Buffers
29950 @pindex calc-grab-region
29951 The @kbd{C-x * g} (@code{calc-grab-region}) command takes the text between
29952 point and mark in the current buffer and attempts to parse it as a
29953 vector of values. Basically, it wraps the text in vector brackets
29954 @samp{[ ]} unless the text already is enclosed in vector brackets,
29955 then reads the text as if it were an algebraic entry. The contents
29956 of the vector may be numbers, formulas, or any other Calc objects.
29957 If the @kbd{C-x * g} command works successfully, it does an automatic
29958 @kbd{C-x * c} to enter the Calculator buffer.
29960 A numeric prefix argument grabs the specified number of lines around
29961 point, ignoring the mark. A positive prefix grabs from point to the
29962 @expr{n}th following newline (so that @kbd{M-1 C-x * g} grabs from point
29963 to the end of the current line); a negative prefix grabs from point
29964 back to the @expr{n+1}st preceding newline. In these cases the text
29965 that is grabbed is exactly the same as the text that @kbd{C-k} would
29966 delete given that prefix argument.
29968 A prefix of zero grabs the current line; point may be anywhere on the
29971 A plain @kbd{C-u} prefix interprets the region between point and mark
29972 as a single number or formula rather than a vector. For example,
29973 @kbd{C-x * g} on the text @samp{2 a b} produces the vector of three
29974 values @samp{[2, a, b]}, but @kbd{C-u C-x * g} on the same region
29975 reads a formula which is a product of three things: @samp{2 a b}.
29976 (The text @samp{a + b}, on the other hand, will be grabbed as a
29977 vector of one element by plain @kbd{C-x * g} because the interpretation
29978 @samp{[a, +, b]} would be a syntax error.)
29980 If a different language has been specified (@pxref{Language Modes}),
29981 the grabbed text will be interpreted according to that language.
29984 @pindex calc-grab-rectangle
29985 The @kbd{C-x * r} (@code{calc-grab-rectangle}) command takes the text between
29986 point and mark and attempts to parse it as a matrix. If point and mark
29987 are both in the leftmost column, the lines in between are parsed in their
29988 entirety. Otherwise, point and mark define the corners of a rectangle
29989 whose contents are parsed.
29991 Each line of the grabbed area becomes a row of the matrix. The result
29992 will actually be a vector of vectors, which Calc will treat as a matrix
29993 only if every row contains the same number of values.
29995 If a line contains a portion surrounded by square brackets (or curly
29996 braces), that portion is interpreted as a vector which becomes a row
29997 of the matrix. Any text surrounding the bracketed portion on the line
30000 Otherwise, the entire line is interpreted as a row vector as if it
30001 were surrounded by square brackets. Leading line numbers (in the
30002 format used in the Calc stack buffer) are ignored. If you wish to
30003 force this interpretation (even if the line contains bracketed
30004 portions), give a negative numeric prefix argument to the
30005 @kbd{C-x * r} command.
30007 If you give a numeric prefix argument of zero or plain @kbd{C-u}, each
30008 line is instead interpreted as a single formula which is converted into
30009 a one-element vector. Thus the result of @kbd{C-u C-x * r} will be a
30010 one-column matrix. For example, suppose one line of the data is the
30011 expression @samp{2 a}. A plain @w{@kbd{C-x * r}} will interpret this as
30012 @samp{[2 a]}, which in turn is read as a two-element vector that forms
30013 one row of the matrix. But a @kbd{C-u C-x * r} will interpret this row
30016 If you give a positive numeric prefix argument @var{n}, then each line
30017 will be split up into columns of width @var{n}; each column is parsed
30018 separately as a matrix element. If a line contained
30019 @w{@samp{2 +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8
30020 would correctly split the line into two error forms.
30022 @xref{Matrix Functions}, to see how to pull the matrix apart into its
30023 constituent rows and columns. (If it is a
30024 @texline @math{1\times1}
30026 matrix, just hit @kbd{v u} (@code{calc-unpack}) twice.)
30030 @pindex calc-grab-sum-across
30031 @pindex calc-grab-sum-down
30032 @cindex Summing rows and columns of data
30033 The @kbd{C-x * :} (@code{calc-grab-sum-down}) command is a handy way to
30034 grab a rectangle of data and sum its columns. It is equivalent to
30035 typing @kbd{C-x * r}, followed by @kbd{V R : +} (the vector reduction
30036 command that sums the columns of a matrix; @pxref{Reducing}). The
30037 result of the command will be a vector of numbers, one for each column
30038 in the input data. The @kbd{C-x * _} (@code{calc-grab-sum-across}) command
30039 similarly grabs a rectangle and sums its rows by executing @w{@kbd{V R _ +}}.
30041 As well as being more convenient, @kbd{C-x * :} and @kbd{C-x * _} are also
30042 much faster because they don't actually place the grabbed vector on
30043 the stack. In a @kbd{C-x * r V R : +} sequence, formatting the vector
30044 for display on the stack takes a large fraction of the total time
30045 (unless you have planned ahead and used @kbd{v .} and @kbd{t .} modes).
30047 For example, suppose we have a column of numbers in a file which we
30048 wish to sum. Go to one corner of the column and press @kbd{C-@@} to
30049 set the mark; go to the other corner and type @kbd{C-x * :}. Since there
30050 is only one column, the result will be a vector of one number, the sum.
30051 (You can type @kbd{v u} to unpack this vector into a plain number if
30052 you want to do further arithmetic with it.)
30054 To compute the product of the column of numbers, we would have to do
30055 it ``by hand'' since there's no special grab-and-multiply command.
30056 Use @kbd{C-x * r} to grab the column of numbers into the calculator in
30057 the form of a column matrix. The statistics command @kbd{u *} is a
30058 handy way to find the product of a vector or matrix of numbers.
30059 @xref{Statistical Operations}. Another approach would be to use
30060 an explicit column reduction command, @kbd{V R : *}.
30062 @node Yanking Into Buffers, X Cut and Paste, Grabbing From Buffers, Kill and Yank
30063 @section Yanking into Other Buffers
30067 @pindex calc-copy-to-buffer
30068 The plain @kbd{y} (@code{calc-copy-to-buffer}) command inserts the number
30069 at the top of the stack into the most recently used normal editing buffer.
30070 (More specifically, this is the most recently used buffer which is displayed
30071 in a window and whose name does not begin with @samp{*}. If there is no
30072 such buffer, this is the most recently used buffer except for Calculator
30073 and Calc Trail buffers.) The number is inserted exactly as it appears and
30074 without a newline. (If line-numbering is enabled, the line number is
30075 normally not included.) The number is @emph{not} removed from the stack.
30077 With a prefix argument, @kbd{y} inserts several numbers, one per line.
30078 A positive argument inserts the specified number of values from the top
30079 of the stack. A negative argument inserts the @expr{n}th value from the
30080 top of the stack. An argument of zero inserts the entire stack. Note
30081 that @kbd{y} with an argument of 1 is slightly different from @kbd{y}
30082 with no argument; the former always copies full lines, whereas the
30083 latter strips off the trailing newline.
30085 With a lone @kbd{C-u} as a prefix argument, @kbd{y} @emph{replaces} the
30086 region in the other buffer with the yanked text, then quits the
30087 Calculator, leaving you in that buffer. A typical use would be to use
30088 @kbd{C-x * g} to read a region of data into the Calculator, operate on the
30089 data to produce a new matrix, then type @kbd{C-u y} to replace the
30090 original data with the new data. One might wish to alter the matrix
30091 display style (@pxref{Vector and Matrix Formats}) or change the current
30092 display language (@pxref{Language Modes}) before doing this. Also, note
30093 that this command replaces a linear region of text (as grabbed by
30094 @kbd{C-x * g}), not a rectangle (as grabbed by @kbd{C-x * r}).
30096 If the editing buffer is in overwrite (as opposed to insert) mode,
30097 and the @kbd{C-u} prefix was not used, then the yanked number will
30098 overwrite the characters following point rather than being inserted
30099 before those characters. The usual conventions of overwrite mode
30100 are observed; for example, characters will be inserted at the end of
30101 a line rather than overflowing onto the next line. Yanking a multi-line
30102 object such as a matrix in overwrite mode overwrites the next @var{n}
30103 lines in the buffer, lengthening or shortening each line as necessary.
30104 Finally, if the thing being yanked is a simple integer or floating-point
30105 number (like @samp{-1.2345e-3}) and the characters following point also
30106 make up such a number, then Calc will replace that number with the new
30107 number, lengthening or shortening as necessary. The concept of
30108 ``overwrite mode'' has thus been generalized from overwriting characters
30109 to overwriting one complete number with another.
30112 The @kbd{C-x * y} key sequence is equivalent to @kbd{y} except that
30113 it can be typed anywhere, not just in Calc. This provides an easy
30114 way to guarantee that Calc knows which editing buffer you want to use!
30116 @node X Cut and Paste, , Yanking Into Buffers, Kill and Yank
30117 @section X Cut and Paste
30120 If you are using Emacs with the X window system, there is an easier
30121 way to move small amounts of data into and out of the calculator:
30122 Use the mouse-oriented cut and paste facilities of X.
30124 The default bindings for a three-button mouse cause the left button
30125 to move the Emacs cursor to the given place, the right button to
30126 select the text between the cursor and the clicked location, and
30127 the middle button to yank the selection into the buffer at the
30128 clicked location. So, if you have a Calc window and an editing
30129 window on your Emacs screen, you can use left-click/right-click
30130 to select a number, vector, or formula from one window, then
30131 middle-click to paste that value into the other window. When you
30132 paste text into the Calc window, Calc interprets it as an algebraic
30133 entry. It doesn't matter where you click in the Calc window; the
30134 new value is always pushed onto the top of the stack.
30136 The @code{xterm} program that is typically used for general-purpose
30137 shell windows in X interprets the mouse buttons in the same way.
30138 So you can use the mouse to move data between Calc and any other
30139 Unix program. One nice feature of @code{xterm} is that a double
30140 left-click selects one word, and a triple left-click selects a
30141 whole line. So you can usually transfer a single number into Calc
30142 just by double-clicking on it in the shell, then middle-clicking
30143 in the Calc window.
30145 @node Keypad Mode, Embedded Mode, Kill and Yank, Top
30146 @chapter Keypad Mode
30150 @pindex calc-keypad
30151 The @kbd{C-x * k} (@code{calc-keypad}) command starts the Calculator
30152 and displays a picture of a calculator-style keypad. If you are using
30153 the X window system, you can click on any of the ``keys'' in the
30154 keypad using the left mouse button to operate the calculator.
30155 The original window remains the selected window; in Keypad mode
30156 you can type in your file while simultaneously performing
30157 calculations with the mouse.
30159 @pindex full-calc-keypad
30160 If you have used @kbd{C-x * b} first, @kbd{C-x * k} instead invokes
30161 the @code{full-calc-keypad} command, which takes over the whole
30162 Emacs screen and displays the keypad, the Calc stack, and the Calc
30163 trail all at once. This mode would normally be used when running
30164 Calc standalone (@pxref{Standalone Operation}).
30166 If you aren't using the X window system, you must switch into
30167 the @samp{*Calc Keypad*} window, place the cursor on the desired
30168 ``key,'' and type @key{SPC} or @key{RET}. If you think this
30169 is easier than using Calc normally, go right ahead.
30171 Calc commands are more or less the same in Keypad mode. Certain
30172 keypad keys differ slightly from the corresponding normal Calc
30173 keystrokes; all such deviations are described below.
30175 Keypad mode includes many more commands than will fit on the keypad
30176 at once. Click the right mouse button [@code{calc-keypad-menu}]
30177 to switch to the next menu. The bottom five rows of the keypad
30178 stay the same; the top three rows change to a new set of commands.
30179 To return to earlier menus, click the middle mouse button
30180 [@code{calc-keypad-menu-back}] or simply advance through the menus
30181 until you wrap around. Typing @key{TAB} inside the keypad window
30182 is equivalent to clicking the right mouse button there.
30184 You can always click the @key{EXEC} button and type any normal
30185 Calc key sequence. This is equivalent to switching into the
30186 Calc buffer, typing the keys, then switching back to your
30190 * Keypad Main Menu::
30191 * Keypad Functions Menu::
30192 * Keypad Binary Menu::
30193 * Keypad Vectors Menu::
30194 * Keypad Modes Menu::
30197 @node Keypad Main Menu, Keypad Functions Menu, Keypad Mode, Keypad Mode
30202 |----+----+--Calc---+----+----1
30203 |FLR |CEIL|RND |TRNC|CLN2|FLT |
30204 |----+----+----+----+----+----|
30205 | LN |EXP | |ABS |IDIV|MOD |
30206 |----+----+----+----+----+----|
30207 |SIN |COS |TAN |SQRT|y^x |1/x |
30208 |----+----+----+----+----+----|
30209 | ENTER |+/- |EEX |UNDO| <- |
30210 |-----+---+-+--+--+-+---++----|
30211 | INV | 7 | 8 | 9 | / |
30212 |-----+-----+-----+-----+-----|
30213 | HYP | 4 | 5 | 6 | * |
30214 |-----+-----+-----+-----+-----|
30215 |EXEC | 1 | 2 | 3 | - |
30216 |-----+-----+-----+-----+-----|
30217 | OFF | 0 | . | PI | + |
30218 |-----+-----+-----+-----+-----+
30223 This is the menu that appears the first time you start Keypad mode.
30224 It will show up in a vertical window on the right side of your screen.
30225 Above this menu is the traditional Calc stack display. On a 24-line
30226 screen you will be able to see the top three stack entries.
30228 The ten digit keys, decimal point, and @key{EEX} key are used for
30229 entering numbers in the obvious way. @key{EEX} begins entry of an
30230 exponent in scientific notation. Just as with regular Calc, the
30231 number is pushed onto the stack as soon as you press @key{ENTER}
30232 or any other function key.
30234 The @key{+/-} key corresponds to normal Calc's @kbd{n} key. During
30235 numeric entry it changes the sign of the number or of the exponent.
30236 At other times it changes the sign of the number on the top of the
30239 The @key{INV} and @key{HYP} keys modify other keys. As well as
30240 having the effects described elsewhere in this manual, Keypad mode
30241 defines several other ``inverse'' operations. These are described
30242 below and in the following sections.
30244 The @key{ENTER} key finishes the current numeric entry, or otherwise
30245 duplicates the top entry on the stack.
30247 The @key{UNDO} key undoes the most recent Calc operation.
30248 @kbd{INV UNDO} is the ``redo'' command, and @kbd{HYP UNDO} is
30249 ``last arguments'' (@kbd{M-@key{RET}}).
30251 The @key{<-} key acts as a ``backspace'' during numeric entry.
30252 At other times it removes the top stack entry. @kbd{INV <-}
30253 clears the entire stack. @kbd{HYP <-} takes an integer from
30254 the stack, then removes that many additional stack elements.
30256 The @key{EXEC} key prompts you to enter any keystroke sequence
30257 that would normally work in Calc mode. This can include a
30258 numeric prefix if you wish. It is also possible simply to
30259 switch into the Calc window and type commands in it; there is
30260 nothing ``magic'' about this window when Keypad mode is active.
30262 The other keys in this display perform their obvious calculator
30263 functions. @key{CLN2} rounds the top-of-stack by temporarily
30264 reducing the precision by 2 digits. @key{FLT} converts an
30265 integer or fraction on the top of the stack to floating-point.
30267 The @key{INV} and @key{HYP} keys combined with several of these keys
30268 give you access to some common functions even if the appropriate menu
30269 is not displayed. Obviously you don't need to learn these keys
30270 unless you find yourself wasting time switching among the menus.
30274 is the same as @key{1/x}.
30276 is the same as @key{SQRT}.
30278 is the same as @key{CONJ}.
30280 is the same as @key{y^x}.
30282 is the same as @key{INV y^x} (the @expr{x}th root of @expr{y}).
30284 are the same as @key{SIN} / @kbd{INV SIN}.
30286 are the same as @key{COS} / @kbd{INV COS}.
30288 are the same as @key{TAN} / @kbd{INV TAN}.
30290 are the same as @key{LN} / @kbd{HYP LN}.
30292 are the same as @key{EXP} / @kbd{HYP EXP}.
30294 is the same as @key{ABS}.
30296 is the same as @key{RND} (@code{calc-round}).
30298 is the same as @key{CLN2}.
30300 is the same as @key{FLT} (@code{calc-float}).
30302 is the same as @key{IMAG}.
30304 is the same as @key{PREC}.
30306 is the same as @key{SWAP}.
30308 is the same as @key{RLL3}.
30309 @item INV HYP ENTER
30310 is the same as @key{OVER}.
30312 packs the top two stack entries as an error form.
30314 packs the top two stack entries as a modulo form.
30316 creates an interval form; this removes an integer which is one
30317 of 0 @samp{[]}, 1 @samp{[)}, 2 @samp{(]} or 3 @samp{()}, followed
30318 by the two limits of the interval.
30321 The @kbd{OFF} key turns Calc off; typing @kbd{C-x * k} or @kbd{C-x * *}
30322 again has the same effect. This is analogous to typing @kbd{q} or
30323 hitting @kbd{C-x * c} again in the normal calculator. If Calc is
30324 running standalone (the @code{full-calc-keypad} command appeared in the
30325 command line that started Emacs), then @kbd{OFF} is replaced with
30326 @kbd{EXIT}; clicking on this actually exits Emacs itself.
30328 @node Keypad Functions Menu, Keypad Binary Menu, Keypad Main Menu, Keypad Mode
30329 @section Functions Menu
30333 |----+----+----+----+----+----2
30334 |IGAM|BETA|IBET|ERF |BESJ|BESY|
30335 |----+----+----+----+----+----|
30336 |IMAG|CONJ| RE |ATN2|RAND|RAGN|
30337 |----+----+----+----+----+----|
30338 |GCD |FACT|DFCT|BNOM|PERM|NXTP|
30339 |----+----+----+----+----+----|
30344 This menu provides various operations from the @kbd{f} and @kbd{k}
30347 @key{IMAG} multiplies the number on the stack by the imaginary
30348 number @expr{i = (0, 1)}.
30350 @key{RE} extracts the real part a complex number. @kbd{INV RE}
30351 extracts the imaginary part.
30353 @key{RAND} takes a number from the top of the stack and computes
30354 a random number greater than or equal to zero but less than that
30355 number. (@xref{Random Numbers}.) @key{RAGN} is the ``random
30356 again'' command; it computes another random number using the
30357 same limit as last time.
30359 @key{INV GCD} computes the LCM (least common multiple) function.
30361 @key{INV FACT} is the gamma function.
30362 @texline @math{\Gamma(x) = (x-1)!}.
30363 @infoline @expr{gamma(x) = (x-1)!}.
30365 @key{PERM} is the number-of-permutations function, which is on the
30366 @kbd{H k c} key in normal Calc.
30368 @key{NXTP} finds the next prime after a number. @kbd{INV NXTP}
30369 finds the previous prime.
30371 @node Keypad Binary Menu, Keypad Vectors Menu, Keypad Functions Menu, Keypad Mode
30372 @section Binary Menu
30376 |----+----+----+----+----+----3
30377 |AND | OR |XOR |NOT |LSH |RSH |
30378 |----+----+----+----+----+----|
30379 |DEC |HEX |OCT |BIN |WSIZ|ARSH|
30380 |----+----+----+----+----+----|
30381 | A | B | C | D | E | F |
30382 |----+----+----+----+----+----|
30387 The keys in this menu perform operations on binary integers.
30388 Note that both logical and arithmetic right-shifts are provided.
30389 @key{INV LSH} rotates one bit to the left.
30391 The ``difference'' function (normally on @kbd{b d}) is on @key{INV AND}.
30392 The ``clip'' function (normally on @w{@kbd{b c}}) is on @key{INV NOT}.
30394 The @key{DEC}, @key{HEX}, @key{OCT}, and @key{BIN} keys select the
30395 current radix for display and entry of numbers: Decimal, hexadecimal,
30396 octal, or binary. The six letter keys @key{A} through @key{F} are used
30397 for entering hexadecimal numbers.
30399 The @key{WSIZ} key displays the current word size for binary operations
30400 and allows you to enter a new word size. You can respond to the prompt
30401 using either the keyboard or the digits and @key{ENTER} from the keypad.
30402 The initial word size is 32 bits.
30404 @node Keypad Vectors Menu, Keypad Modes Menu, Keypad Binary Menu, Keypad Mode
30405 @section Vectors Menu
30409 |----+----+----+----+----+----4
30410 |SUM |PROD|MAX |MAP*|MAP^|MAP$|
30411 |----+----+----+----+----+----|
30412 |MINV|MDET|MTRN|IDNT|CRSS|"x" |
30413 |----+----+----+----+----+----|
30414 |PACK|UNPK|INDX|BLD |LEN |... |
30415 |----+----+----+----+----+----|
30420 The keys in this menu operate on vectors and matrices.
30422 @key{PACK} removes an integer @var{n} from the top of the stack;
30423 the next @var{n} stack elements are removed and packed into a vector,
30424 which is replaced onto the stack. Thus the sequence
30425 @kbd{1 ENTER 3 ENTER 5 ENTER 3 PACK} enters the vector
30426 @samp{[1, 3, 5]} onto the stack. To enter a matrix, build each row
30427 on the stack as a vector, then use a final @key{PACK} to collect the
30428 rows into a matrix.
30430 @key{UNPK} unpacks the vector on the stack, pushing each of its
30431 components separately.
30433 @key{INDX} removes an integer @var{n}, then builds a vector of
30434 integers from 1 to @var{n}. @kbd{INV INDX} takes three numbers
30435 from the stack: The vector size @var{n}, the starting number,
30436 and the increment. @kbd{BLD} takes an integer @var{n} and any
30437 value @var{x} and builds a vector of @var{n} copies of @var{x}.
30439 @key{IDNT} removes an integer @var{n}, then builds an @var{n}-by-@var{n}
30442 @key{LEN} replaces a vector by its length, an integer.
30444 @key{...} turns on or off ``abbreviated'' display mode for large vectors.
30446 @key{MINV}, @key{MDET}, @key{MTRN}, and @key{CROSS} are the matrix
30447 inverse, determinant, and transpose, and vector cross product.
30449 @key{SUM} replaces a vector by the sum of its elements. It is
30450 equivalent to @kbd{u +} in normal Calc (@pxref{Statistical Operations}).
30451 @key{PROD} computes the product of the elements of a vector, and
30452 @key{MAX} computes the maximum of all the elements of a vector.
30454 @key{INV SUM} computes the alternating sum of the first element
30455 minus the second, plus the third, minus the fourth, and so on.
30456 @key{INV MAX} computes the minimum of the vector elements.
30458 @key{HYP SUM} computes the mean of the vector elements.
30459 @key{HYP PROD} computes the sample standard deviation.
30460 @key{HYP MAX} computes the median.
30462 @key{MAP*} multiplies two vectors elementwise. It is equivalent
30463 to the @kbd{V M *} command. @key{MAP^} computes powers elementwise.
30464 The arguments must be vectors of equal length, or one must be a vector
30465 and the other must be a plain number. For example, @kbd{2 MAP^} squares
30466 all the elements of a vector.
30468 @key{MAP$} maps the formula on the top of the stack across the
30469 vector in the second-to-top position. If the formula contains
30470 several variables, Calc takes that many vectors starting at the
30471 second-to-top position and matches them to the variables in
30472 alphabetical order. The result is a vector of the same size as
30473 the input vectors, whose elements are the formula evaluated with
30474 the variables set to the various sets of numbers in those vectors.
30475 For example, you could simulate @key{MAP^} using @key{MAP$} with
30476 the formula @samp{x^y}.
30478 The @kbd{"x"} key pushes the variable name @expr{x} onto the
30479 stack. To build the formula @expr{x^2 + 6}, you would use the
30480 key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be
30481 suitable for use with the @key{MAP$} key described above.
30482 With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the
30483 @kbd{"x"} key pushes the variable names @expr{y}, @expr{z}, and
30484 @expr{t}, respectively.
30486 @node Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode
30487 @section Modes Menu
30491 |----+----+----+----+----+----5
30492 |FLT |FIX |SCI |ENG |GRP | |
30493 |----+----+----+----+----+----|
30494 |RAD |DEG |FRAC|POLR|SYMB|PREC|
30495 |----+----+----+----+----+----|
30496 |SWAP|RLL3|RLL4|OVER|STO |RCL |
30497 |----+----+----+----+----+----|
30502 The keys in this menu manipulate modes, variables, and the stack.
30504 The @key{FLT}, @key{FIX}, @key{SCI}, and @key{ENG} keys select
30505 floating-point, fixed-point, scientific, or engineering notation.
30506 @key{FIX} displays two digits after the decimal by default; the
30507 others display full precision. With the @key{INV} prefix, these
30508 keys pop a number-of-digits argument from the stack.
30510 The @key{GRP} key turns grouping of digits with commas on or off.
30511 @kbd{INV GRP} enables grouping to the right of the decimal point as
30512 well as to the left.
30514 The @key{RAD} and @key{DEG} keys switch between radians and degrees
30515 for trigonometric functions.
30517 The @key{FRAC} key turns Fraction mode on or off. This affects
30518 whether commands like @kbd{/} with integer arguments produce
30519 fractional or floating-point results.
30521 The @key{POLR} key turns Polar mode on or off, determining whether
30522 polar or rectangular complex numbers are used by default.
30524 The @key{SYMB} key turns Symbolic mode on or off, in which
30525 operations that would produce inexact floating-point results
30526 are left unevaluated as algebraic formulas.
30528 The @key{PREC} key selects the current precision. Answer with
30529 the keyboard or with the keypad digit and @key{ENTER} keys.
30531 The @key{SWAP} key exchanges the top two stack elements.
30532 The @key{RLL3} key rotates the top three stack elements upwards.
30533 The @key{RLL4} key rotates the top four stack elements upwards.
30534 The @key{OVER} key duplicates the second-to-top stack element.
30536 The @key{STO} and @key{RCL} keys are analogous to @kbd{s t} and
30537 @kbd{s r} in regular Calc. @xref{Store and Recall}. Click the
30538 @key{STO} or @key{RCL} key, then one of the ten digits. (Named
30539 variables are not available in Keypad mode.) You can also use,
30540 for example, @kbd{STO + 3} to add to register 3.
30542 @node Embedded Mode, Programming, Keypad Mode, Top
30543 @chapter Embedded Mode
30546 Embedded mode in Calc provides an alternative to copying numbers
30547 and formulas back and forth between editing buffers and the Calc
30548 stack. In Embedded mode, your editing buffer becomes temporarily
30549 linked to the stack and this copying is taken care of automatically.
30552 * Basic Embedded Mode::
30553 * More About Embedded Mode::
30554 * Assignments in Embedded Mode::
30555 * Mode Settings in Embedded Mode::
30556 * Customizing Embedded Mode::
30559 @node Basic Embedded Mode, More About Embedded Mode, Embedded Mode, Embedded Mode
30560 @section Basic Embedded Mode
30564 @pindex calc-embedded
30565 To enter Embedded mode, position the Emacs point (cursor) on a
30566 formula in any buffer and press @kbd{C-x * e} (@code{calc-embedded}).
30567 Note that @kbd{C-x * e} is not to be used in the Calc stack buffer
30568 like most Calc commands, but rather in regular editing buffers that
30569 are visiting your own files.
30571 Calc will try to guess an appropriate language based on the major mode
30572 of the editing buffer. (@xref{Language Modes}.) If the current buffer is
30573 in @code{latex-mode}, for example, Calc will set its language to @LaTeX{}.
30574 Similarly, Calc will use @TeX{} language for @code{tex-mode},
30575 @code{plain-tex-mode} and @code{context-mode}, C language for
30576 @code{c-mode} and @code{c++-mode}, FORTRAN language for
30577 @code{fortran-mode} and @code{f90-mode}, Pascal for @code{pascal-mode},
30578 and eqn for @code{nroff-mode} (@pxref{Customizing Calc}).
30579 These can be overridden with Calc's mode
30580 changing commands (@pxref{Mode Settings in Embedded Mode}). If no
30581 suitable language is available, Calc will continue with its current language.
30583 Calc normally scans backward and forward in the buffer for the
30584 nearest opening and closing @dfn{formula delimiters}. The simplest
30585 delimiters are blank lines. Other delimiters that Embedded mode
30590 The @TeX{} and @LaTeX{} math delimiters @samp{$ $}, @samp{$$ $$},
30591 @samp{\[ \]}, and @samp{\( \)};
30593 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
30595 Lines beginning with @samp{@@} (Texinfo delimiters).
30597 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
30599 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
30602 @xref{Customizing Embedded Mode}, to see how to make Calc recognize
30603 your own favorite delimiters. Delimiters like @samp{$ $} can appear
30604 on their own separate lines or in-line with the formula.
30606 If you give a positive or negative numeric prefix argument, Calc
30607 instead uses the current point as one end of the formula, and includes
30608 that many lines forward or backward (respectively, including the current
30609 line). Explicit delimiters are not necessary in this case.
30611 With a prefix argument of zero, Calc uses the current region (delimited
30612 by point and mark) instead of formula delimiters. With a prefix
30613 argument of @kbd{C-u} only, Calc uses the current line as the formula.
30616 @pindex calc-embedded-word
30617 The @kbd{C-x * w} (@code{calc-embedded-word}) command will start Embedded
30618 mode on the current ``word''; in this case Calc will scan for the first
30619 non-numeric character (i.e., the first character that is not a digit,
30620 sign, decimal point, or upper- or lower-case @samp{e}) forward and
30621 backward to delimit the formula.
30623 When you enable Embedded mode for a formula, Calc reads the text
30624 between the delimiters and tries to interpret it as a Calc formula.
30625 Calc can generally identify @TeX{} formulas and
30626 Big-style formulas even if the language mode is wrong. If Calc
30627 can't make sense of the formula, it beeps and refuses to enter
30628 Embedded mode. But if the current language is wrong, Calc can
30629 sometimes parse the formula successfully (but incorrectly);
30630 for example, the C expression @samp{atan(a[1])} can be parsed
30631 in Normal language mode, but the @code{atan} won't correspond to
30632 the built-in @code{arctan} function, and the @samp{a[1]} will be
30633 interpreted as @samp{a} times the vector @samp{[1]}!
30635 If you press @kbd{C-x * e} or @kbd{C-x * w} to activate an embedded
30636 formula which is blank, say with the cursor on the space between
30637 the two delimiters @samp{$ $}, Calc will immediately prompt for
30638 an algebraic entry.
30640 Only one formula in one buffer can be enabled at a time. If you
30641 move to another area of the current buffer and give Calc commands,
30642 Calc turns Embedded mode off for the old formula and then tries
30643 to restart Embedded mode at the new position. Other buffers are
30644 not affected by Embedded mode.
30646 When Embedded mode begins, Calc pushes the current formula onto
30647 the stack. No Calc stack window is created; however, Calc copies
30648 the top-of-stack position into the original buffer at all times.
30649 You can create a Calc window by hand with @kbd{C-x * o} if you
30650 find you need to see the entire stack.
30652 For example, typing @kbd{C-x * e} while somewhere in the formula
30653 @samp{n>2} in the following line enables Embedded mode on that
30657 We define $F_n = F_(n-1)+F_(n-2)$ for all $n>2$.
30661 The formula @expr{n>2} will be pushed onto the Calc stack, and
30662 the top of stack will be copied back into the editing buffer.
30663 This means that spaces will appear around the @samp{>} symbol
30664 to match Calc's usual display style:
30667 We define $F_n = F_(n-1)+F_(n-2)$ for all $n > 2$.
30671 No spaces have appeared around the @samp{+} sign because it's
30672 in a different formula, one which we have not yet touched with
30675 Now that Embedded mode is enabled, keys you type in this buffer
30676 are interpreted as Calc commands. At this point we might use
30677 the ``commute'' command @kbd{j C} to reverse the inequality.
30678 This is a selection-based command for which we first need to
30679 move the cursor onto the operator (@samp{>} in this case) that
30680 needs to be commuted.
30683 We define $F_n = F_(n-1)+F_(n-2)$ for all $2 < n$.
30686 The @kbd{C-x * o} command is a useful way to open a Calc window
30687 without actually selecting that window. Giving this command
30688 verifies that @samp{2 < n} is also on the Calc stack. Typing
30689 @kbd{17 @key{RET}} would produce:
30692 We define $F_n = F_(n-1)+F_(n-2)$ for all $17$.
30696 with @samp{2 < n} and @samp{17} on the stack; typing @key{TAB}
30697 at this point will exchange the two stack values and restore
30698 @samp{2 < n} to the embedded formula. Even though you can't
30699 normally see the stack in Embedded mode, it is still there and
30700 it still operates in the same way. But, as with old-fashioned
30701 RPN calculators, you can only see the value at the top of the
30702 stack at any given time (unless you use @kbd{C-x * o}).
30704 Typing @kbd{C-x * e} again turns Embedded mode off. The Calc
30705 window reveals that the formula @w{@samp{2 < n}} is automatically
30706 removed from the stack, but the @samp{17} is not. Entering
30707 Embedded mode always pushes one thing onto the stack, and
30708 leaving Embedded mode always removes one thing. Anything else
30709 that happens on the stack is entirely your business as far as
30710 Embedded mode is concerned.
30712 If you press @kbd{C-x * e} in the wrong place by accident, it is
30713 possible that Calc will be able to parse the nearby text as a
30714 formula and will mangle that text in an attempt to redisplay it
30715 ``properly'' in the current language mode. If this happens,
30716 press @kbd{C-x * e} again to exit Embedded mode, then give the
30717 regular Emacs ``undo'' command (@kbd{C-_} or @kbd{C-x u}) to put
30718 the text back the way it was before Calc edited it. Note that Calc's
30719 own Undo command (typed before you turn Embedded mode back off)
30720 will not do you any good, because as far as Calc is concerned
30721 you haven't done anything with this formula yet.
30723 @node More About Embedded Mode, Assignments in Embedded Mode, Basic Embedded Mode, Embedded Mode
30724 @section More About Embedded Mode
30727 When Embedded mode ``activates'' a formula, i.e., when it examines
30728 the formula for the first time since the buffer was created or
30729 loaded, Calc tries to sense the language in which the formula was
30730 written. If the formula contains any @LaTeX{}-like @samp{\} sequences,
30731 it is parsed (i.e., read) in @LaTeX{} mode. If the formula appears to
30732 be written in multi-line Big mode, it is parsed in Big mode. Otherwise,
30733 it is parsed according to the current language mode.
30735 Note that Calc does not change the current language mode according
30736 the formula it reads in. Even though it can read a @LaTeX{} formula when
30737 not in @LaTeX{} mode, it will immediately rewrite this formula using
30738 whatever language mode is in effect.
30745 @pindex calc-show-plain
30746 Calc's parser is unable to read certain kinds of formulas. For
30747 example, with @kbd{v ]} (@code{calc-matrix-brackets}) you can
30748 specify matrix display styles which the parser is unable to
30749 recognize as matrices. The @kbd{d p} (@code{calc-show-plain})
30750 command turns on a mode in which a ``plain'' version of a
30751 formula is placed in front of the fully-formatted version.
30752 When Calc reads a formula that has such a plain version in
30753 front, it reads the plain version and ignores the formatted
30756 Plain formulas are preceded and followed by @samp{%%%} signs
30757 by default. This notation has the advantage that the @samp{%}
30758 character begins a comment in @TeX{} and @LaTeX{}, so if your formula is
30759 embedded in a @TeX{} or @LaTeX{} document its plain version will be
30760 invisible in the final printed copy. Certain major modes have different
30761 delimiters to ensure that the ``plain'' version will be
30762 in a comment for those modes, also.
30763 See @ref{Customizing Embedded Mode} to see how to change the ``plain''
30764 formula delimiters.
30766 There are several notations which Calc's parser for ``big''
30767 formatted formulas can't yet recognize. In particular, it can't
30768 read the large symbols for @code{sum}, @code{prod}, and @code{integ},
30769 and it can't handle @samp{=>} with the righthand argument omitted.
30770 Also, Calc won't recognize special formats you have defined with
30771 the @kbd{Z C} command (@pxref{User-Defined Compositions}). In
30772 these cases it is important to use ``plain'' mode to make sure
30773 Calc will be able to read your formula later.
30775 Another example where ``plain'' mode is important is if you have
30776 specified a float mode with few digits of precision. Normally
30777 any digits that are computed but not displayed will simply be
30778 lost when you save and re-load your embedded buffer, but ``plain''
30779 mode allows you to make sure that the complete number is present
30780 in the file as well as the rounded-down number.
30786 Embedded buffers remember active formulas for as long as they
30787 exist in Emacs memory. Suppose you have an embedded formula
30788 which is @cpi{} to the normal 12 decimal places, and then
30789 type @w{@kbd{C-u 5 d n}} to display only five decimal places.
30790 If you then type @kbd{d n}, all 12 places reappear because the
30791 full number is still there on the Calc stack. More surprisingly,
30792 even if you exit Embedded mode and later re-enter it for that
30793 formula, typing @kbd{d n} will restore all 12 places because
30794 each buffer remembers all its active formulas. However, if you
30795 save the buffer in a file and reload it in a new Emacs session,
30796 all non-displayed digits will have been lost unless you used
30803 In some applications of Embedded mode, you will want to have a
30804 sequence of copies of a formula that show its evolution as you
30805 work on it. For example, you might want to have a sequence
30806 like this in your file (elaborating here on the example from
30807 the ``Getting Started'' chapter):
30816 @r{(the derivative of }ln(ln(x))@r{)}
30818 whose value at x = 2 is
30828 @pindex calc-embedded-duplicate
30829 The @kbd{C-x * d} (@code{calc-embedded-duplicate}) command is a
30830 handy way to make sequences like this. If you type @kbd{C-x * d},
30831 the formula under the cursor (which may or may not have Embedded
30832 mode enabled for it at the time) is copied immediately below and
30833 Embedded mode is then enabled for that copy.
30835 For this example, you would start with just
30844 and press @kbd{C-x * d} with the cursor on this formula. The result
30857 with the second copy of the formula enabled in Embedded mode.
30858 You can now press @kbd{a d x @key{RET}} to take the derivative, and
30859 @kbd{C-x * d C-x * d} to make two more copies of the derivative.
30860 To complete the computations, type @kbd{3 s l x @key{RET}} to evaluate
30861 the last formula, then move up to the second-to-last formula
30862 and type @kbd{2 s l x @key{RET}}.
30864 Finally, you would want to press @kbd{C-x * e} to exit Embedded
30865 mode, then go up and insert the necessary text in between the
30866 various formulas and numbers.
30874 @pindex calc-embedded-new-formula
30875 The @kbd{C-x * f} (@code{calc-embedded-new-formula}) command
30876 creates a new embedded formula at the current point. It inserts
30877 some default delimiters, which are usually just blank lines,
30878 and then does an algebraic entry to get the formula (which is
30879 then enabled for Embedded mode). This is just shorthand for
30880 typing the delimiters yourself, positioning the cursor between
30881 the new delimiters, and pressing @kbd{C-x * e}. The key sequence
30882 @kbd{C-x * '} is equivalent to @kbd{C-x * f}.
30886 @pindex calc-embedded-next
30887 @pindex calc-embedded-previous
30888 The @kbd{C-x * n} (@code{calc-embedded-next}) and @kbd{C-x * p}
30889 (@code{calc-embedded-previous}) commands move the cursor to the
30890 next or previous active embedded formula in the buffer. They
30891 can take positive or negative prefix arguments to move by several
30892 formulas. Note that these commands do not actually examine the
30893 text of the buffer looking for formulas; they only see formulas
30894 which have previously been activated in Embedded mode. In fact,
30895 @kbd{C-x * n} and @kbd{C-x * p} are a useful way to tell which
30896 embedded formulas are currently active. Also, note that these
30897 commands do not enable Embedded mode on the next or previous
30898 formula, they just move the cursor.
30901 @pindex calc-embedded-edit
30902 The @kbd{C-x * `} (@code{calc-embedded-edit}) command edits the
30903 embedded formula at the current point as if by @kbd{`} (@code{calc-edit}).
30904 Embedded mode does not have to be enabled for this to work. Press
30905 @kbd{C-c C-c} to finish the edit, or @kbd{C-x k} to cancel.
30907 @node Assignments in Embedded Mode, Mode Settings in Embedded Mode, More About Embedded Mode, Embedded Mode
30908 @section Assignments in Embedded Mode
30911 The @samp{:=} (assignment) and @samp{=>} (``evaluates-to'') operators
30912 are especially useful in Embedded mode. They allow you to make
30913 a definition in one formula, then refer to that definition in
30914 other formulas embedded in the same buffer.
30916 An embedded formula which is an assignment to a variable, as in
30923 records @expr{5} as the stored value of @code{foo} for the
30924 purposes of Embedded mode operations in the current buffer. It
30925 does @emph{not} actually store @expr{5} as the ``global'' value
30926 of @code{foo}, however. Regular Calc operations, and Embedded
30927 formulas in other buffers, will not see this assignment.
30929 One way to use this assigned value is simply to create an
30930 Embedded formula elsewhere that refers to @code{foo}, and to press
30931 @kbd{=} in that formula. However, this permanently replaces the
30932 @code{foo} in the formula with its current value. More interesting
30933 is to use @samp{=>} elsewhere:
30939 @xref{Evaluates-To Operator}, for a general discussion of @samp{=>}.
30941 If you move back and change the assignment to @code{foo}, any
30942 @samp{=>} formulas which refer to it are automatically updated.
30950 The obvious question then is, @emph{how} can one easily change the
30951 assignment to @code{foo}? If you simply select the formula in
30952 Embedded mode and type 17, the assignment itself will be replaced
30953 by the 17. The effect on the other formula will be that the
30954 variable @code{foo} becomes unassigned:
30962 The right thing to do is first to use a selection command (@kbd{j 2}
30963 will do the trick) to select the righthand side of the assignment.
30964 Then, @kbd{17 @key{TAB} @key{DEL}} will swap the 17 into place (@pxref{Selecting
30965 Subformulas}, to see how this works).
30968 @pindex calc-embedded-select
30969 The @kbd{C-x * j} (@code{calc-embedded-select}) command provides an
30970 easy way to operate on assignments. It is just like @kbd{C-x * e},
30971 except that if the enabled formula is an assignment, it uses
30972 @kbd{j 2} to select the righthand side. If the enabled formula
30973 is an evaluates-to, it uses @kbd{j 1} to select the lefthand side.
30974 A formula can also be a combination of both:
30977 bar := foo + 3 => 20
30981 in which case @kbd{C-x * j} will select the middle part (@samp{foo + 3}).
30983 The formula is automatically deselected when you leave Embedded
30987 @pindex calc-embedded-update-formula
30988 Another way to change the assignment to @code{foo} would simply be
30989 to edit the number using regular Emacs editing rather than Embedded
30990 mode. Then, we have to find a way to get Embedded mode to notice
30991 the change. The @kbd{C-x * u} (@code{calc-embedded-update-formula})
30992 command is a convenient way to do this.
31000 Pressing @kbd{C-x * u} is much like pressing @kbd{C-x * e = C-x * e}, that
31001 is, temporarily enabling Embedded mode for the formula under the
31002 cursor and then evaluating it with @kbd{=}. But @kbd{C-x * u} does
31003 not actually use @kbd{C-x * e}, and in fact another formula somewhere
31004 else can be enabled in Embedded mode while you use @kbd{C-x * u} and
31005 that formula will not be disturbed.
31007 With a numeric prefix argument, @kbd{C-x * u} updates all active
31008 @samp{=>} formulas in the buffer. Formulas which have not yet
31009 been activated in Embedded mode, and formulas which do not have
31010 @samp{=>} as their top-level operator, are not affected by this.
31011 (This is useful only if you have used @kbd{m C}; see below.)
31013 With a plain @kbd{C-u} prefix, @kbd{C-u C-x * u} updates only in the
31014 region between mark and point rather than in the whole buffer.
31016 @kbd{C-x * u} is also a handy way to activate a formula, such as an
31017 @samp{=>} formula that has freshly been typed in or loaded from a
31021 @pindex calc-embedded-activate
31022 The @kbd{C-x * a} (@code{calc-embedded-activate}) command scans
31023 through the current buffer and activates all embedded formulas
31024 that contain @samp{:=} or @samp{=>} symbols. This does not mean
31025 that Embedded mode is actually turned on, but only that the
31026 formulas' positions are registered with Embedded mode so that
31027 the @samp{=>} values can be properly updated as assignments are
31030 It is a good idea to type @kbd{C-x * a} right after loading a file
31031 that uses embedded @samp{=>} operators. Emacs includes a nifty
31032 ``buffer-local variables'' feature that you can use to do this
31033 automatically. The idea is to place near the end of your file
31034 a few lines that look like this:
31037 --- Local Variables: ---
31038 --- eval:(calc-embedded-activate) ---
31043 where the leading and trailing @samp{---} can be replaced by
31044 any suitable strings (which must be the same on all three lines)
31045 or omitted altogether; in a @TeX{} or @LaTeX{} file, @samp{%} would be a good
31046 leading string and no trailing string would be necessary. In a
31047 C program, @samp{/*} and @samp{*/} would be good leading and
31050 When Emacs loads a file into memory, it checks for a Local Variables
31051 section like this one at the end of the file. If it finds this
31052 section, it does the specified things (in this case, running
31053 @kbd{C-x * a} automatically) before editing of the file begins.
31054 The Local Variables section must be within 3000 characters of the
31055 end of the file for Emacs to find it, and it must be in the last
31056 page of the file if the file has any page separators.
31057 @xref{File Variables, , Local Variables in Files, emacs, the
31060 Note that @kbd{C-x * a} does not update the formulas it finds.
31061 To do this, type, say, @kbd{M-1 C-x * u} after @w{@kbd{C-x * a}}.
31062 Generally this should not be a problem, though, because the
31063 formulas will have been up-to-date already when the file was
31066 Normally, @kbd{C-x * a} activates all the formulas it finds, but
31067 any previous active formulas remain active as well. With a
31068 positive numeric prefix argument, @kbd{C-x * a} first deactivates
31069 all current active formulas, then actives the ones it finds in
31070 its scan of the buffer. With a negative prefix argument,
31071 @kbd{C-x * a} simply deactivates all formulas.
31073 Embedded mode has two symbols, @samp{Active} and @samp{~Active},
31074 which it puts next to the major mode name in a buffer's mode line.
31075 It puts @samp{Active} if it has reason to believe that all
31076 formulas in the buffer are active, because you have typed @kbd{C-x * a}
31077 and Calc has not since had to deactivate any formulas (which can
31078 happen if Calc goes to update an @samp{=>} formula somewhere because
31079 a variable changed, and finds that the formula is no longer there
31080 due to some kind of editing outside of Embedded mode). Calc puts
31081 @samp{~Active} in the mode line if some, but probably not all,
31082 formulas in the buffer are active. This happens if you activate
31083 a few formulas one at a time but never use @kbd{C-x * a}, or if you
31084 used @kbd{C-x * a} but then Calc had to deactivate a formula
31085 because it lost track of it. If neither of these symbols appears
31086 in the mode line, no embedded formulas are active in the buffer
31087 (e.g., before Embedded mode has been used, or after a @kbd{M-- C-x * a}).
31089 Embedded formulas can refer to assignments both before and after them
31090 in the buffer. If there are several assignments to a variable, the
31091 nearest preceding assignment is used if there is one, otherwise the
31092 following assignment is used.
31106 As well as simple variables, you can also assign to subscript
31107 expressions of the form @samp{@var{var}_@var{number}} (as in
31108 @code{x_0}), or @samp{@var{var}_@var{var}} (as in @code{x_max}).
31109 Assignments to other kinds of objects can be represented by Calc,
31110 but the automatic linkage between assignments and references works
31111 only for plain variables and these two kinds of subscript expressions.
31113 If there are no assignments to a given variable, the global
31114 stored value for the variable is used (@pxref{Storing Variables}),
31115 or, if no value is stored, the variable is left in symbolic form.
31116 Note that global stored values will be lost when the file is saved
31117 and loaded in a later Emacs session, unless you have used the
31118 @kbd{s p} (@code{calc-permanent-variable}) command to save them;
31119 @pxref{Operations on Variables}.
31121 The @kbd{m C} (@code{calc-auto-recompute}) command turns automatic
31122 recomputation of @samp{=>} forms on and off. If you turn automatic
31123 recomputation off, you will have to use @kbd{C-x * u} to update these
31124 formulas manually after an assignment has been changed. If you
31125 plan to change several assignments at once, it may be more efficient
31126 to type @kbd{m C}, change all the assignments, then use @kbd{M-1 C-x * u}
31127 to update the entire buffer afterwards. The @kbd{m C} command also
31128 controls @samp{=>} formulas on the stack; @pxref{Evaluates-To
31129 Operator}. When you turn automatic recomputation back on, the
31130 stack will be updated but the Embedded buffer will not; you must
31131 use @kbd{C-x * u} to update the buffer by hand.
31133 @node Mode Settings in Embedded Mode, Customizing Embedded Mode, Assignments in Embedded Mode, Embedded Mode
31134 @section Mode Settings in Embedded Mode
31137 @pindex calc-embedded-preserve-modes
31139 The mode settings can be changed while Calc is in embedded mode, but
31140 by default they will revert to their original values when embedded mode
31141 is ended. However, the modes saved when the mode-recording mode is
31142 @code{Save} (see below) and the modes in effect when the @kbd{m e}
31143 (@code{calc-embedded-preserve-modes}) command is given
31144 will be preserved when embedded mode is ended.
31146 Embedded mode has a rather complicated mechanism for handling mode
31147 settings in Embedded formulas. It is possible to put annotations
31148 in the file that specify mode settings either global to the entire
31149 file or local to a particular formula or formulas. In the latter
31150 case, different modes can be specified for use when a formula
31151 is the enabled Embedded mode formula.
31153 When you give any mode-setting command, like @kbd{m f} (for Fraction
31154 mode) or @kbd{d s} (for scientific notation), Embedded mode adds
31155 a line like the following one to the file just before the opening
31156 delimiter of the formula.
31159 % [calc-mode: fractions: t]
31160 % [calc-mode: float-format: (sci 0)]
31163 When Calc interprets an embedded formula, it scans the text before
31164 the formula for mode-setting annotations like these and sets the
31165 Calc buffer to match these modes. Modes not explicitly described
31166 in the file are not changed. Calc scans all the way to the top of
31167 the file, or up to a line of the form
31174 which you can insert at strategic places in the file if this backward
31175 scan is getting too slow, or just to provide a barrier between one
31176 ``zone'' of mode settings and another.
31178 If the file contains several annotations for the same mode, the
31179 closest one before the formula is used. Annotations after the
31180 formula are never used (except for global annotations, described
31183 The scan does not look for the leading @samp{% }, only for the
31184 square brackets and the text they enclose. In fact, the leading
31185 characters are different for different major modes. You can edit the
31186 mode annotations to a style that works better in context if you wish.
31187 @xref{Customizing Embedded Mode}, to see how to change the style
31188 that Calc uses when it generates the annotations. You can write
31189 mode annotations into the file yourself if you know the syntax;
31190 the easiest way to find the syntax for a given mode is to let
31191 Calc write the annotation for it once and see what it does.
31193 If you give a mode-changing command for a mode that already has
31194 a suitable annotation just above the current formula, Calc will
31195 modify that annotation rather than generating a new, conflicting
31198 Mode annotations have three parts, separated by colons. (Spaces
31199 after the colons are optional.) The first identifies the kind
31200 of mode setting, the second is a name for the mode itself, and
31201 the third is the value in the form of a Lisp symbol, number,
31202 or list. Annotations with unrecognizable text in the first or
31203 second parts are ignored. The third part is not checked to make
31204 sure the value is of a valid type or range; if you write an
31205 annotation by hand, be sure to give a proper value or results
31206 will be unpredictable. Mode-setting annotations are case-sensitive.
31208 While Embedded mode is enabled, the word @code{Local} appears in
31209 the mode line. This is to show that mode setting commands generate
31210 annotations that are ``local'' to the current formula or set of
31211 formulas. The @kbd{m R} (@code{calc-mode-record-mode}) command
31212 causes Calc to generate different kinds of annotations. Pressing
31213 @kbd{m R} repeatedly cycles through the possible modes.
31215 @code{LocEdit} and @code{LocPerm} modes generate annotations
31216 that look like this, respectively:
31219 % [calc-edit-mode: float-format: (sci 0)]
31220 % [calc-perm-mode: float-format: (sci 5)]
31223 The first kind of annotation will be used only while a formula
31224 is enabled in Embedded mode. The second kind will be used only
31225 when the formula is @emph{not} enabled. (Whether the formula
31226 is ``active'' or not, i.e., whether Calc has seen this formula
31227 yet, is not relevant here.)
31229 @code{Global} mode generates an annotation like this at the end
31233 % [calc-global-mode: fractions t]
31236 Global mode annotations affect all formulas throughout the file,
31237 and may appear anywhere in the file. This allows you to tuck your
31238 mode annotations somewhere out of the way, say, on a new page of
31239 the file, as long as those mode settings are suitable for all
31240 formulas in the file.
31242 Enabling a formula with @kbd{C-x * e} causes a fresh scan for local
31243 mode annotations; you will have to use this after adding annotations
31244 above a formula by hand to get the formula to notice them. Updating
31245 a formula with @kbd{C-x * u} will also re-scan the local modes, but
31246 global modes are only re-scanned by @kbd{C-x * a}.
31248 Another way that modes can get out of date is if you add a local
31249 mode annotation to a formula that has another formula after it.
31250 In this example, we have used the @kbd{d s} command while the
31251 first of the two embedded formulas is active. But the second
31252 formula has not changed its style to match, even though by the
31253 rules of reading annotations the @samp{(sci 0)} applies to it, too.
31256 % [calc-mode: float-format: (sci 0)]
31262 We would have to go down to the other formula and press @kbd{C-x * u}
31263 on it in order to get it to notice the new annotation.
31265 Two more mode-recording modes selectable by @kbd{m R} are available
31266 which are also available outside of Embedded mode.
31267 (@pxref{General Mode Commands}.) They are @code{Save}, in which mode
31268 settings are recorded permanently in your Calc init file (the file given
31269 by the variable @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el})
31270 rather than by annotating the current document, and no-recording
31271 mode (where there is no symbol like @code{Save} or @code{Local} in
31272 the mode line), in which mode-changing commands do not leave any
31273 annotations at all.
31275 When Embedded mode is not enabled, mode-recording modes except
31276 for @code{Save} have no effect.
31278 @node Customizing Embedded Mode, , Mode Settings in Embedded Mode, Embedded Mode
31279 @section Customizing Embedded Mode
31282 You can modify Embedded mode's behavior by setting various Lisp
31283 variables described here. These variables are customizable
31284 (@pxref{Customizing Calc}), or you can use @kbd{M-x set-variable}
31285 or @kbd{M-x edit-options} to adjust a variable on the fly.
31286 (Another possibility would be to use a file-local variable annotation at
31287 the end of the file;
31288 @pxref{File Variables, , Local Variables in Files, emacs, the Emacs manual}.)
31289 Many of the variables given mentioned here can be set to depend on the
31290 major mode of the editing buffer (@pxref{Customizing Calc}).
31292 @vindex calc-embedded-open-formula
31293 The @code{calc-embedded-open-formula} variable holds a regular
31294 expression for the opening delimiter of a formula. @xref{Regexp Search,
31295 , Regular Expression Search, emacs, the Emacs manual}, to see
31296 how regular expressions work. Basically, a regular expression is a
31297 pattern that Calc can search for. A regular expression that considers
31298 blank lines, @samp{$}, and @samp{$$} to be opening delimiters is
31299 @code{"\\`\\|^\n\\|\\$\\$?"}. Just in case the meaning of this
31300 regular expression is not completely plain, let's go through it
31303 The surrounding @samp{" "} marks quote the text between them as a
31304 Lisp string. If you left them off, @code{set-variable} or
31305 @code{edit-options} would try to read the regular expression as a
31308 The most obvious property of this regular expression is that it
31309 contains indecently many backslashes. There are actually two levels
31310 of backslash usage going on here. First, when Lisp reads a quoted
31311 string, all pairs of characters beginning with a backslash are
31312 interpreted as special characters. Here, @code{\n} changes to a
31313 new-line character, and @code{\\} changes to a single backslash.
31314 So the actual regular expression seen by Calc is
31315 @samp{\`\|^ @r{(newline)} \|\$\$?}.
31317 Regular expressions also consider pairs beginning with backslash
31318 to have special meanings. Sometimes the backslash is used to quote
31319 a character that otherwise would have a special meaning in a regular
31320 expression, like @samp{$}, which normally means ``end-of-line,''
31321 or @samp{?}, which means that the preceding item is optional. So
31322 @samp{\$\$?} matches either one or two dollar signs.
31324 The other codes in this regular expression are @samp{^}, which matches
31325 ``beginning-of-line,'' @samp{\|}, which means ``or,'' and @samp{\`},
31326 which matches ``beginning-of-buffer.'' So the whole pattern means
31327 that a formula begins at the beginning of the buffer, or on a newline
31328 that occurs at the beginning of a line (i.e., a blank line), or at
31329 one or two dollar signs.
31331 The default value of @code{calc-embedded-open-formula} looks just
31332 like this example, with several more alternatives added on to
31333 recognize various other common kinds of delimiters.
31335 By the way, the reason to use @samp{^\n} rather than @samp{^$}
31336 or @samp{\n\n}, which also would appear to match blank lines,
31337 is that the former expression actually ``consumes'' only one
31338 newline character as @emph{part of} the delimiter, whereas the
31339 latter expressions consume zero or two newlines, respectively.
31340 The former choice gives the most natural behavior when Calc
31341 must operate on a whole formula including its delimiters.
31343 See the Emacs manual for complete details on regular expressions.
31344 But just for your convenience, here is a list of all characters
31345 which must be quoted with backslash (like @samp{\$}) to avoid
31346 some special interpretation: @samp{. * + ? [ ] ^ $ \}. (Note
31347 the backslash in this list; for example, to match @samp{\[} you
31348 must use @code{"\\\\\\["}. An exercise for the reader is to
31349 account for each of these six backslashes!)
31351 @vindex calc-embedded-close-formula
31352 The @code{calc-embedded-close-formula} variable holds a regular
31353 expression for the closing delimiter of a formula. A closing
31354 regular expression to match the above example would be
31355 @code{"\\'\\|\n$\\|\\$\\$?"}. This is almost the same as the
31356 other one, except it now uses @samp{\'} (``end-of-buffer'') and
31357 @samp{\n$} (newline occurring at end of line, yet another way
31358 of describing a blank line that is more appropriate for this
31361 @vindex calc-embedded-word-regexp
31362 The @code{calc-embedded-word-regexp} variable holds a regular expression
31363 used to define an expression to look for (a ``word'') when you type
31364 @kbd{C-x * w} to enable Embedded mode.
31366 @vindex calc-embedded-open-plain
31367 The @code{calc-embedded-open-plain} variable is a string which
31368 begins a ``plain'' formula written in front of the formatted
31369 formula when @kbd{d p} mode is turned on. Note that this is an
31370 actual string, not a regular expression, because Calc must be able
31371 to write this string into a buffer as well as to recognize it.
31372 The default string is @code{"%%% "} (note the trailing space), but may
31373 be different for certain major modes.
31375 @vindex calc-embedded-close-plain
31376 The @code{calc-embedded-close-plain} variable is a string which
31377 ends a ``plain'' formula. The default is @code{" %%%\n"}, but may be
31378 different for different major modes. Without
31379 the trailing newline here, the first line of a Big mode formula
31380 that followed might be shifted over with respect to the other lines.
31382 @vindex calc-embedded-open-new-formula
31383 The @code{calc-embedded-open-new-formula} variable is a string
31384 which is inserted at the front of a new formula when you type
31385 @kbd{C-x * f}. Its default value is @code{"\n\n"}. If this
31386 string begins with a newline character and the @kbd{C-x * f} is
31387 typed at the beginning of a line, @kbd{C-x * f} will skip this
31388 first newline to avoid introducing unnecessary blank lines in
31391 @vindex calc-embedded-close-new-formula
31392 The @code{calc-embedded-close-new-formula} variable is the corresponding
31393 string which is inserted at the end of a new formula. Its default
31394 value is also @code{"\n\n"}. The final newline is omitted by
31395 @w{@kbd{C-x * f}} if typed at the end of a line. (It follows that if
31396 @kbd{C-x * f} is typed on a blank line, both a leading opening
31397 newline and a trailing closing newline are omitted.)
31399 @vindex calc-embedded-announce-formula
31400 The @code{calc-embedded-announce-formula} variable is a regular
31401 expression which is sure to be followed by an embedded formula.
31402 The @kbd{C-x * a} command searches for this pattern as well as for
31403 @samp{=>} and @samp{:=} operators. Note that @kbd{C-x * a} will
31404 not activate just anything surrounded by formula delimiters; after
31405 all, blank lines are considered formula delimiters by default!
31406 But if your language includes a delimiter which can only occur
31407 actually in front of a formula, you can take advantage of it here.
31408 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, but may be
31409 different for different major modes.
31410 This pattern will check for @samp{%Embed} followed by any number of
31411 lines beginning with @samp{%} and a space. This last is important to
31412 make Calc consider mode annotations part of the pattern, so that the
31413 formula's opening delimiter really is sure to follow the pattern.
31415 @vindex calc-embedded-open-mode
31416 The @code{calc-embedded-open-mode} variable is a string (not a
31417 regular expression) which should precede a mode annotation.
31418 Calc never scans for this string; Calc always looks for the
31419 annotation itself. But this is the string that is inserted before
31420 the opening bracket when Calc adds an annotation on its own.
31421 The default is @code{"% "}, but may be different for different major
31424 @vindex calc-embedded-close-mode
31425 The @code{calc-embedded-close-mode} variable is a string which
31426 follows a mode annotation written by Calc. Its default value
31427 is simply a newline, @code{"\n"}, but may be different for different
31428 major modes. If you change this, it is a good idea still to end with a
31429 newline so that mode annotations will appear on lines by themselves.
31431 @node Programming, Copying, Embedded Mode, Top
31432 @chapter Programming
31435 There are several ways to ``program'' the Emacs Calculator, depending
31436 on the nature of the problem you need to solve.
31440 @dfn{Keyboard macros} allow you to record a sequence of keystrokes
31441 and play them back at a later time. This is just the standard Emacs
31442 keyboard macro mechanism, dressed up with a few more features such
31443 as loops and conditionals.
31446 @dfn{Algebraic definitions} allow you to use any formula to define a
31447 new function. This function can then be used in algebraic formulas or
31448 as an interactive command.
31451 @dfn{Rewrite rules} are discussed in the section on algebra commands.
31452 @xref{Rewrite Rules}. If you put your rewrite rules in the variable
31453 @code{EvalRules}, they will be applied automatically to all Calc
31454 results in just the same way as an internal ``rule'' is applied to
31455 evaluate @samp{sqrt(9)} to 3 and so on. @xref{Automatic Rewrites}.
31458 @dfn{Lisp} is the programming language that Calc (and most of Emacs)
31459 is written in. If the above techniques aren't powerful enough, you
31460 can write Lisp functions to do anything that built-in Calc commands
31461 can do. Lisp code is also somewhat faster than keyboard macros or
31466 Programming features are available through the @kbd{z} and @kbd{Z}
31467 prefix keys. New commands that you define are two-key sequences
31468 beginning with @kbd{z}. Commands for managing these definitions
31469 use the shift-@kbd{Z} prefix. (The @kbd{Z T} (@code{calc-timing})
31470 command is described elsewhere; @pxref{Troubleshooting Commands}.
31471 The @kbd{Z C} (@code{calc-user-define-composition}) command is also
31472 described elsewhere; @pxref{User-Defined Compositions}.)
31475 * Creating User Keys::
31476 * Keyboard Macros::
31477 * Invocation Macros::
31478 * Algebraic Definitions::
31479 * Lisp Definitions::
31482 @node Creating User Keys, Keyboard Macros, Programming, Programming
31483 @section Creating User Keys
31487 @pindex calc-user-define
31488 Any Calculator command may be bound to a key using the @kbd{Z D}
31489 (@code{calc-user-define}) command. Actually, it is bound to a two-key
31490 sequence beginning with the lower-case @kbd{z} prefix.
31492 The @kbd{Z D} command first prompts for the key to define. For example,
31493 press @kbd{Z D a} to define the new key sequence @kbd{z a}. You are then
31494 prompted for the name of the Calculator command that this key should
31495 run. For example, the @code{calc-sincos} command is not normally
31496 available on a key. Typing @kbd{Z D s sincos @key{RET}} programs the
31497 @kbd{z s} key sequence to run @code{calc-sincos}. This definition will remain
31498 in effect for the rest of this Emacs session, or until you redefine
31499 @kbd{z s} to be something else.
31501 You can actually bind any Emacs command to a @kbd{z} key sequence by
31502 backspacing over the @samp{calc-} when you are prompted for the command name.
31504 As with any other prefix key, you can type @kbd{z ?} to see a list of
31505 all the two-key sequences you have defined that start with @kbd{z}.
31506 Initially, no @kbd{z} sequences (except @kbd{z ?} itself) are defined.
31508 User keys are typically letters, but may in fact be any key.
31509 (@key{META}-keys are not permitted, nor are a terminal's special
31510 function keys which generate multi-character sequences when pressed.)
31511 You can define different commands on the shifted and unshifted versions
31512 of a letter if you wish.
31515 @pindex calc-user-undefine
31516 The @kbd{Z U} (@code{calc-user-undefine}) command unbinds a user key.
31517 For example, the key sequence @kbd{Z U s} will undefine the @code{sincos}
31518 key we defined above.
31521 @pindex calc-user-define-permanent
31522 @cindex Storing user definitions
31523 @cindex Permanent user definitions
31524 @cindex Calc init file, user-defined commands
31525 The @kbd{Z P} (@code{calc-user-define-permanent}) command makes a key
31526 binding permanent so that it will remain in effect even in future Emacs
31527 sessions. (It does this by adding a suitable bit of Lisp code into
31528 your Calc init file; that is, the file given by the variable
31529 @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el}.) For example,
31530 @kbd{Z P s} would register our @code{sincos} command permanently. If
31531 you later wish to unregister this command you must edit your Calc init
31532 file by hand. (@xref{General Mode Commands}, for a way to tell Calc to
31533 use a different file for the Calc init file.)
31535 The @kbd{Z P} command also saves the user definition, if any, for the
31536 command bound to the key. After @kbd{Z F} and @kbd{Z C}, a given user
31537 key could invoke a command, which in turn calls an algebraic function,
31538 which might have one or more special display formats. A single @kbd{Z P}
31539 command will save all of these definitions.
31540 To save an algebraic function, type @kbd{'} (the apostrophe)
31541 when prompted for a key, and type the function name. To save a command
31542 without its key binding, type @kbd{M-x} and enter a function name. (The
31543 @samp{calc-} prefix will automatically be inserted for you.)
31544 (If the command you give implies a function, the function will be saved,
31545 and if the function has any display formats, those will be saved, but
31546 not the other way around: Saving a function will not save any commands
31547 or key bindings associated with the function.)
31550 @pindex calc-user-define-edit
31551 @cindex Editing user definitions
31552 The @kbd{Z E} (@code{calc-user-define-edit}) command edits the definition
31553 of a user key. This works for keys that have been defined by either
31554 keyboard macros or formulas; further details are contained in the relevant
31555 following sections.
31557 @node Keyboard Macros, Invocation Macros, Creating User Keys, Programming
31558 @section Programming with Keyboard Macros
31562 @cindex Programming with keyboard macros
31563 @cindex Keyboard macros
31564 The easiest way to ``program'' the Emacs Calculator is to use standard
31565 keyboard macros. Press @w{@kbd{C-x (}} to begin recording a macro. From
31566 this point on, keystrokes you type will be saved away as well as
31567 performing their usual functions. Press @kbd{C-x )} to end recording.
31568 Press shift-@kbd{X} (or the standard Emacs key sequence @kbd{C-x e}) to
31569 execute your keyboard macro by replaying the recorded keystrokes.
31570 @xref{Keyboard Macros, , , emacs, the Emacs Manual}, for further
31573 When you use @kbd{X} to invoke a keyboard macro, the entire macro is
31574 treated as a single command by the undo and trail features. The stack
31575 display buffer is not updated during macro execution, but is instead
31576 fixed up once the macro completes. Thus, commands defined with keyboard
31577 macros are convenient and efficient. The @kbd{C-x e} command, on the
31578 other hand, invokes the keyboard macro with no special treatment: Each
31579 command in the macro will record its own undo information and trail entry,
31580 and update the stack buffer accordingly. If your macro uses features
31581 outside of Calc's control to operate on the contents of the Calc stack
31582 buffer, or if it includes Undo, Redo, or last-arguments commands, you
31583 must use @kbd{C-x e} to make sure the buffer and undo list are up-to-date
31584 at all times. You could also consider using @kbd{K} (@code{calc-keep-args})
31585 instead of @kbd{M-@key{RET}} (@code{calc-last-args}).
31587 Calc extends the standard Emacs keyboard macros in several ways.
31588 Keyboard macros can be used to create user-defined commands. Keyboard
31589 macros can include conditional and iteration structures, somewhat
31590 analogous to those provided by a traditional programmable calculator.
31593 * Naming Keyboard Macros::
31594 * Conditionals in Macros::
31595 * Loops in Macros::
31596 * Local Values in Macros::
31597 * Queries in Macros::
31600 @node Naming Keyboard Macros, Conditionals in Macros, Keyboard Macros, Keyboard Macros
31601 @subsection Naming Keyboard Macros
31605 @pindex calc-user-define-kbd-macro
31606 Once you have defined a keyboard macro, you can bind it to a @kbd{z}
31607 key sequence with the @kbd{Z K} (@code{calc-user-define-kbd-macro}) command.
31608 This command prompts first for a key, then for a command name. For
31609 example, if you type @kbd{C-x ( n @key{TAB} n @key{TAB} C-x )} you will
31610 define a keyboard macro which negates the top two numbers on the stack
31611 (@key{TAB} swaps the top two stack elements). Now you can type
31612 @kbd{Z K n @key{RET}} to define this keyboard macro onto the @kbd{z n} key
31613 sequence. The default command name (if you answer the second prompt with
31614 just the @key{RET} key as in this example) will be something like
31615 @samp{calc-User-n}. The keyboard macro will now be available as both
31616 @kbd{z n} and @kbd{M-x calc-User-n}. You can backspace and enter a more
31617 descriptive command name if you wish.
31619 Macros defined by @kbd{Z K} act like single commands; they are executed
31620 in the same way as by the @kbd{X} key. If you wish to define the macro
31621 as a standard no-frills Emacs macro (to be executed as if by @kbd{C-x e}),
31622 give a negative prefix argument to @kbd{Z K}.
31624 Once you have bound your keyboard macro to a key, you can use
31625 @kbd{Z P} to register it permanently with Emacs. @xref{Creating User Keys}.
31627 @cindex Keyboard macros, editing
31628 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
31629 been defined by a keyboard macro tries to use the @code{edmacro} package
31630 edit the macro. Type @kbd{C-c C-c} to finish editing and update
31631 the definition stored on the key, or, to cancel the edit, kill the
31632 buffer with @kbd{C-x k}.
31633 The special characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC},
31634 @code{DEL}, and @code{NUL} must be entered as these three character
31635 sequences, written in all uppercase, as must the prefixes @code{C-} and
31636 @code{M-}. Spaces and line breaks are ignored. Other characters are
31637 copied verbatim into the keyboard macro. Basically, the notation is the
31638 same as is used in all of this manual's examples, except that the manual
31639 takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}},
31640 we take it for granted that it is clear we really mean
31641 @kbd{' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}.
31644 @pindex read-kbd-macro
31645 The @kbd{C-x * m} (@code{read-kbd-macro}) command reads an Emacs ``region''
31646 of spelled-out keystrokes and defines it as the current keyboard macro.
31647 It is a convenient way to define a keyboard macro that has been stored
31648 in a file, or to define a macro without executing it at the same time.
31650 @node Conditionals in Macros, Loops in Macros, Naming Keyboard Macros, Keyboard Macros
31651 @subsection Conditionals in Keyboard Macros
31656 @pindex calc-kbd-if
31657 @pindex calc-kbd-else
31658 @pindex calc-kbd-else-if
31659 @pindex calc-kbd-end-if
31660 @cindex Conditional structures
31661 The @kbd{Z [} (@code{calc-kbd-if}) and @kbd{Z ]} (@code{calc-kbd-end-if})
31662 commands allow you to put simple tests in a keyboard macro. When Calc
31663 sees the @kbd{Z [}, it pops an object from the stack and, if the object is
31664 a non-zero value, continues executing keystrokes. But if the object is
31665 zero, or if it is not provably nonzero, Calc skips ahead to the matching
31666 @kbd{Z ]} keystroke. @xref{Logical Operations}, for a set of commands for
31667 performing tests which conveniently produce 1 for true and 0 for false.
31669 For example, @kbd{@key{RET} 0 a < Z [ n Z ]} implements an absolute-value
31670 function in the form of a keyboard macro. This macro duplicates the
31671 number on the top of the stack, pushes zero and compares using @kbd{a <}
31672 (@code{calc-less-than}), then, if the number was less than zero,
31673 executes @kbd{n} (@code{calc-change-sign}). Otherwise, the change-sign
31674 command is skipped.
31676 To program this macro, type @kbd{C-x (}, type the above sequence of
31677 keystrokes, then type @kbd{C-x )}. Note that the keystrokes will be
31678 executed while you are making the definition as well as when you later
31679 re-execute the macro by typing @kbd{X}. Thus you should make sure a
31680 suitable number is on the stack before defining the macro so that you
31681 don't get a stack-underflow error during the definition process.
31683 Conditionals can be nested arbitrarily. However, there should be exactly
31684 one @kbd{Z ]} for each @kbd{Z [} in a keyboard macro.
31687 The @kbd{Z :} (@code{calc-kbd-else}) command allows you to choose between
31688 two keystroke sequences. The general format is @kbd{@var{cond} Z [
31689 @var{then-part} Z : @var{else-part} Z ]}. If @var{cond} is true
31690 (i.e., if the top of stack contains a non-zero number after @var{cond}
31691 has been executed), the @var{then-part} will be executed and the
31692 @var{else-part} will be skipped. Otherwise, the @var{then-part} will
31693 be skipped and the @var{else-part} will be executed.
31696 The @kbd{Z |} (@code{calc-kbd-else-if}) command allows you to choose
31697 between any number of alternatives. For example,
31698 @kbd{@var{cond1} Z [ @var{part1} Z : @var{cond2} Z | @var{part2} Z :
31699 @var{part3} Z ]} will execute @var{part1} if @var{cond1} is true,
31700 otherwise it will execute @var{part2} if @var{cond2} is true, otherwise
31701 it will execute @var{part3}.
31703 More precisely, @kbd{Z [} pops a number and conditionally skips to the
31704 next matching @kbd{Z :} or @kbd{Z ]} key. @w{@kbd{Z ]}} has no effect when
31705 actually executed. @kbd{Z :} skips to the next matching @kbd{Z ]}.
31706 @kbd{Z |} pops a number and conditionally skips to the next matching
31707 @kbd{Z :} or @kbd{Z ]}; thus, @kbd{Z [} and @kbd{Z |} are functionally
31708 equivalent except that @kbd{Z [} participates in nesting but @kbd{Z |}
31711 Calc's conditional and looping constructs work by scanning the
31712 keyboard macro for occurrences of character sequences like @samp{Z:}
31713 and @samp{Z]}. One side-effect of this is that if you use these
31714 constructs you must be careful that these character pairs do not
31715 occur by accident in other parts of the macros. Since Calc rarely
31716 uses shift-@kbd{Z} for any purpose except as a prefix character, this
31717 is not likely to be a problem. Another side-effect is that it will
31718 not work to define your own custom key bindings for these commands.
31719 Only the standard shift-@kbd{Z} bindings will work correctly.
31722 If Calc gets stuck while skipping characters during the definition of a
31723 macro, type @kbd{Z C-g} to cancel the definition. (Typing plain @kbd{C-g}
31724 actually adds a @kbd{C-g} keystroke to the macro.)
31726 @node Loops in Macros, Local Values in Macros, Conditionals in Macros, Keyboard Macros
31727 @subsection Loops in Keyboard Macros
31732 @pindex calc-kbd-repeat
31733 @pindex calc-kbd-end-repeat
31734 @cindex Looping structures
31735 @cindex Iterative structures
31736 The @kbd{Z <} (@code{calc-kbd-repeat}) and @kbd{Z >}
31737 (@code{calc-kbd-end-repeat}) commands pop a number from the stack,
31738 which must be an integer, then repeat the keystrokes between the brackets
31739 the specified number of times. If the integer is zero or negative, the
31740 body is skipped altogether. For example, @kbd{1 @key{TAB} Z < 2 * Z >}
31741 computes two to a nonnegative integer power. First, we push 1 on the
31742 stack and then swap the integer argument back to the top. The @kbd{Z <}
31743 pops that argument leaving the 1 back on top of the stack. Then, we
31744 repeat a multiply-by-two step however many times.
31746 Once again, the keyboard macro is executed as it is being entered.
31747 In this case it is especially important to set up reasonable initial
31748 conditions before making the definition: Suppose the integer 1000 just
31749 happened to be sitting on the stack before we typed the above definition!
31750 Another approach is to enter a harmless dummy definition for the macro,
31751 then go back and edit in the real one with a @kbd{Z E} command. Yet
31752 another approach is to type the macro as written-out keystroke names
31753 in a buffer, then use @kbd{C-x * m} (@code{read-kbd-macro}) to read the
31758 The @kbd{Z /} (@code{calc-kbd-break}) command allows you to break out
31759 of a keyboard macro loop prematurely. It pops an object from the stack;
31760 if that object is true (a non-zero number), control jumps out of the
31761 innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues
31762 after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no
31763 effect. Thus @kbd{@var{cond} Z /} is similar to @samp{if (@var{cond}) break;}
31768 @pindex calc-kbd-for
31769 @pindex calc-kbd-end-for
31770 The @kbd{Z (} (@code{calc-kbd-for}) and @kbd{Z )} (@code{calc-kbd-end-for})
31771 commands are similar to @kbd{Z <} and @kbd{Z >}, except that they make the
31772 value of the counter available inside the loop. The general layout is
31773 @kbd{@var{init} @var{final} Z ( @var{body} @var{step} Z )}. The @kbd{Z (}
31774 command pops initial and final values from the stack. It then creates
31775 a temporary internal counter and initializes it with the value @var{init}.
31776 The @kbd{Z (} command then repeatedly pushes the counter value onto the
31777 stack and executes @var{body} and @var{step}, adding @var{step} to the
31778 counter each time until the loop finishes.
31780 @cindex Summations (by keyboard macros)
31781 By default, the loop finishes when the counter becomes greater than (or
31782 less than) @var{final}, assuming @var{initial} is less than (greater
31783 than) @var{final}. If @var{initial} is equal to @var{final}, the body
31784 executes exactly once. The body of the loop always executes at least
31785 once. For example, @kbd{0 1 10 Z ( 2 ^ + 1 Z )} computes the sum of the
31786 squares of the integers from 1 to 10, in steps of 1.
31788 If you give a numeric prefix argument of 1 to @kbd{Z (}, the loop is
31789 forced to use upward-counting conventions. In this case, if @var{initial}
31790 is greater than @var{final} the body will not be executed at all.
31791 Note that @var{step} may still be negative in this loop; the prefix
31792 argument merely constrains the loop-finished test. Likewise, a prefix
31793 argument of @mathit{-1} forces downward-counting conventions.
31797 @pindex calc-kbd-loop
31798 @pindex calc-kbd-end-loop
31799 The @kbd{Z @{} (@code{calc-kbd-loop}) and @kbd{Z @}}
31800 (@code{calc-kbd-end-loop}) commands are similar to @kbd{Z <} and
31801 @kbd{Z >}, except that they do not pop a count from the stack---they
31802 effectively create an infinite loop. Every @kbd{Z @{} @dots{} @kbd{Z @}}
31803 loop ought to include at least one @kbd{Z /} to make sure the loop
31804 doesn't run forever. (If any error message occurs which causes Emacs
31805 to beep, the keyboard macro will also be halted; this is a standard
31806 feature of Emacs. You can also generally press @kbd{C-g} to halt a
31807 running keyboard macro, although not all versions of Unix support
31810 The conditional and looping constructs are not actually tied to
31811 keyboard macros, but they are most often used in that context.
31812 For example, the keystrokes @kbd{10 Z < 23 @key{RET} Z >} push
31813 ten copies of 23 onto the stack. This can be typed ``live'' just
31814 as easily as in a macro definition.
31816 @xref{Conditionals in Macros}, for some additional notes about
31817 conditional and looping commands.
31819 @node Local Values in Macros, Queries in Macros, Loops in Macros, Keyboard Macros
31820 @subsection Local Values in Macros
31823 @cindex Local variables
31824 @cindex Restoring saved modes
31825 Keyboard macros sometimes want to operate under known conditions
31826 without affecting surrounding conditions. For example, a keyboard
31827 macro may wish to turn on Fraction mode, or set a particular
31828 precision, independent of the user's normal setting for those
31833 @pindex calc-kbd-push
31834 @pindex calc-kbd-pop
31835 Macros also sometimes need to use local variables. Assignments to
31836 local variables inside the macro should not affect any variables
31837 outside the macro. The @kbd{Z `} (@code{calc-kbd-push}) and @kbd{Z '}
31838 (@code{calc-kbd-pop}) commands give you both of these capabilities.
31840 When you type @kbd{Z `} (with a backquote or accent grave character),
31841 the values of various mode settings are saved away. The ten ``quick''
31842 variables @code{q0} through @code{q9} are also saved. When
31843 you type @w{@kbd{Z '}} (with an apostrophe), these values are restored.
31844 Pairs of @kbd{Z `} and @kbd{Z '} commands may be nested.
31846 If a keyboard macro halts due to an error in between a @kbd{Z `} and
31847 a @kbd{Z '}, the saved values will be restored correctly even though
31848 the macro never reaches the @kbd{Z '} command. Thus you can use
31849 @kbd{Z `} and @kbd{Z '} without having to worry about what happens
31850 in exceptional conditions.
31852 If you type @kbd{Z `} ``live'' (not in a keyboard macro), Calc puts
31853 you into a ``recursive edit.'' You can tell you are in a recursive
31854 edit because there will be extra square brackets in the mode line,
31855 as in @samp{[(Calculator)]}. These brackets will go away when you
31856 type the matching @kbd{Z '} command. The modes and quick variables
31857 will be saved and restored in just the same way as if actual keyboard
31858 macros were involved.
31860 The modes saved by @kbd{Z `} and @kbd{Z '} are the current precision
31861 and binary word size, the angular mode (Deg, Rad, or HMS), the
31862 simplification mode, Algebraic mode, Symbolic mode, Infinite mode,
31863 Matrix or Scalar mode, Fraction mode, and the current complex mode
31864 (Polar or Rectangular). The ten ``quick'' variables' values (or lack
31865 thereof) are also saved.
31867 Most mode-setting commands act as toggles, but with a numeric prefix
31868 they force the mode either on (positive prefix) or off (negative
31869 or zero prefix). Since you don't know what the environment might
31870 be when you invoke your macro, it's best to use prefix arguments
31871 for all mode-setting commands inside the macro.
31873 In fact, @kbd{C-u Z `} is like @kbd{Z `} except that it sets the modes
31874 listed above to their default values. As usual, the matching @kbd{Z '}
31875 will restore the modes to their settings from before the @kbd{C-u Z `}.
31876 Also, @w{@kbd{Z `}} with a negative prefix argument resets the algebraic mode
31877 to its default (off) but leaves the other modes the same as they were
31878 outside the construct.
31880 The contents of the stack and trail, values of non-quick variables, and
31881 other settings such as the language mode and the various display modes,
31882 are @emph{not} affected by @kbd{Z `} and @kbd{Z '}.
31884 @node Queries in Macros, , Local Values in Macros, Keyboard Macros
31885 @subsection Queries in Keyboard Macros
31889 @c @pindex calc-kbd-report
31890 @c The @kbd{Z =} (@code{calc-kbd-report}) command displays an informative
31891 @c message including the value on the top of the stack. You are prompted
31892 @c to enter a string. That string, along with the top-of-stack value,
31893 @c is displayed unless @kbd{m w} (@code{calc-working}) has been used
31894 @c to turn such messages off.
31898 @pindex calc-kbd-query
31899 The @kbd{Z #} (@code{calc-kbd-query}) command prompts for an algebraic
31900 entry which takes its input from the keyboard, even during macro
31901 execution. All the normal conventions of algebraic input, including the
31902 use of @kbd{$} characters, are supported. The prompt message itself is
31903 taken from the top of the stack, and so must be entered (as a string)
31904 before the @kbd{Z #} command. (Recall, as a string it can be entered by
31905 pressing the @kbd{"} key and will appear as a vector when it is put on
31906 the stack. The prompt message is only put on the stack to provide a
31907 prompt for the @kbd{Z #} command; it will not play any role in any
31908 subsequent calculations.) This command allows your keyboard macros to
31909 accept numbers or formulas as interactive input.
31912 @kbd{2 @key{RET} "Power: " @key{RET} Z # 3 @key{RET} ^} will prompt for
31913 input with ``Power: '' in the minibuffer, then return 2 to the provided
31914 power. (The response to the prompt that's given, 3 in this example,
31915 will not be part of the macro.)
31917 @xref{Keyboard Macro Query, , , emacs, the Emacs Manual}, for a description of
31918 @kbd{C-x q} (@code{kbd-macro-query}), the standard Emacs way to accept
31919 keyboard input during a keyboard macro. In particular, you can use
31920 @kbd{C-x q} to enter a recursive edit, which allows the user to perform
31921 any Calculator operations interactively before pressing @kbd{C-M-c} to
31922 return control to the keyboard macro.
31924 @node Invocation Macros, Algebraic Definitions, Keyboard Macros, Programming
31925 @section Invocation Macros
31929 @pindex calc-user-invocation
31930 @pindex calc-user-define-invocation
31931 Calc provides one special keyboard macro, called up by @kbd{C-x * z}
31932 (@code{calc-user-invocation}), that is intended to allow you to define
31933 your own special way of starting Calc. To define this ``invocation
31934 macro,'' create the macro in the usual way with @kbd{C-x (} and
31935 @kbd{C-x )}, then type @kbd{Z I} (@code{calc-user-define-invocation}).
31936 There is only one invocation macro, so you don't need to type any
31937 additional letters after @kbd{Z I}. From now on, you can type
31938 @kbd{C-x * z} at any time to execute your invocation macro.
31940 For example, suppose you find yourself often grabbing rectangles of
31941 numbers into Calc and multiplying their columns. You can do this
31942 by typing @kbd{C-x * r} to grab, and @kbd{V R : *} to multiply columns.
31943 To make this into an invocation macro, just type @kbd{C-x ( C-x * r
31944 V R : * C-x )}, then @kbd{Z I}. Then, to multiply a rectangle of data,
31945 just mark the data in its buffer in the usual way and type @kbd{C-x * z}.
31947 Invocation macros are treated like regular Emacs keyboard macros;
31948 all the special features described above for @kbd{Z K}-style macros
31949 do not apply. @kbd{C-x * z} is just like @kbd{C-x e}, except that it
31950 uses the macro that was last stored by @kbd{Z I}. (In fact, the
31951 macro does not even have to have anything to do with Calc!)
31953 The @kbd{m m} command saves the last invocation macro defined by
31954 @kbd{Z I} along with all the other Calc mode settings.
31955 @xref{General Mode Commands}.
31957 @node Algebraic Definitions, Lisp Definitions, Invocation Macros, Programming
31958 @section Programming with Formulas
31962 @pindex calc-user-define-formula
31963 @cindex Programming with algebraic formulas
31964 Another way to create a new Calculator command uses algebraic formulas.
31965 The @kbd{Z F} (@code{calc-user-define-formula}) command stores the
31966 formula at the top of the stack as the definition for a key. This
31967 command prompts for five things: The key, the command name, the function
31968 name, the argument list, and the behavior of the command when given
31969 non-numeric arguments.
31971 For example, suppose we type @kbd{' a+2b @key{RET}} to push the formula
31972 @samp{a + 2*b} onto the stack. We now type @kbd{Z F m} to define this
31973 formula on the @kbd{z m} key sequence. The next prompt is for a command
31974 name, beginning with @samp{calc-}, which should be the long (@kbd{M-x}) form
31975 for the new command. If you simply press @key{RET}, a default name like
31976 @code{calc-User-m} will be constructed. In our example, suppose we enter
31977 @kbd{spam @key{RET}} to define the new command as @code{calc-spam}.
31979 If you want to give the formula a long-style name only, you can press
31980 @key{SPC} or @key{RET} when asked which single key to use. For example
31981 @kbd{Z F @key{RET} spam @key{RET}} defines the new command as
31982 @kbd{M-x calc-spam}, with no keyboard equivalent.
31984 The third prompt is for an algebraic function name. The default is to
31985 use the same name as the command name but without the @samp{calc-}
31986 prefix. (If this is of the form @samp{User-m}, the hyphen is removed so
31987 it won't be taken for a minus sign in algebraic formulas.)
31988 This is the name you will use if you want to enter your
31989 new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}.
31990 Then the new function can be invoked by pushing two numbers on the
31991 stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic
31992 formula @samp{yow(x,y)}.
31994 The fourth prompt is for the function's argument list. This is used to
31995 associate values on the stack with the variables that appear in the formula.
31996 The default is a list of all variables which appear in the formula, sorted
31997 into alphabetical order. In our case, the default would be @samp{(a b)}.
31998 This means that, when the user types @kbd{z m}, the Calculator will remove
31999 two numbers from the stack, substitute these numbers for @samp{a} and
32000 @samp{b} (respectively) in the formula, then simplify the formula and
32001 push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m}
32002 would replace the 10 and 100 on the stack with the number 210, which is
32003 @expr{a + 2 b} with @expr{a=10} and @expr{b=100}. Likewise, the formula
32004 @samp{yow(10, 100)} will be evaluated by substituting @expr{a=10} and
32005 @expr{b=100} in the definition.
32007 You can rearrange the order of the names before pressing @key{RET} to
32008 control which stack positions go to which variables in the formula. If
32009 you remove a variable from the argument list, that variable will be left
32010 in symbolic form by the command. Thus using an argument list of @samp{(b)}
32011 for our function would cause @kbd{10 z m} to replace the 10 on the stack
32012 with the formula @samp{a + 20}. If we had used an argument list of
32013 @samp{(b a)}, the result with inputs 10 and 100 would have been 120.
32015 You can also put a nameless function on the stack instead of just a
32016 formula, as in @samp{<a, b : a + 2 b>}. @xref{Specifying Operators}.
32017 In this example, the command will be defined by the formula @samp{a + 2 b}
32018 using the argument list @samp{(a b)}.
32020 The final prompt is a y-or-n question concerning what to do if symbolic
32021 arguments are given to your function. If you answer @kbd{y}, then
32022 executing @kbd{z m} (using the original argument list @samp{(a b)}) with
32023 arguments @expr{10} and @expr{x} will leave the function in symbolic
32024 form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n},
32025 then the formula will always be expanded, even for non-constant
32026 arguments: @samp{10 + 2 x}. If you never plan to feed algebraic
32027 formulas to your new function, it doesn't matter how you answer this
32030 If you answered @kbd{y} to this question you can still cause a function
32031 call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}).
32032 Also, Calc will expand the function if necessary when you take a
32033 derivative or integral or solve an equation involving the function.
32036 @pindex calc-get-user-defn
32037 Once you have defined a formula on a key, you can retrieve this formula
32038 with the @kbd{Z G} (@code{calc-user-define-get-defn}) command. Press a
32039 key, and this command pushes the formula that was used to define that
32040 key onto the stack. Actually, it pushes a nameless function that
32041 specifies both the argument list and the defining formula. You will get
32042 an error message if the key is undefined, or if the key was not defined
32043 by a @kbd{Z F} command.
32045 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
32046 been defined by a formula uses a variant of the @code{calc-edit} command
32047 to edit the defining formula. Press @kbd{C-c C-c} to finish editing and
32048 store the new formula back in the definition, or kill the buffer with
32050 cancel the edit. (The argument list and other properties of the
32051 definition are unchanged; to adjust the argument list, you can use
32052 @kbd{Z G} to grab the function onto the stack, edit with @kbd{`}, and
32053 then re-execute the @kbd{Z F} command.)
32055 As usual, the @kbd{Z P} command records your definition permanently.
32056 In this case it will permanently record all three of the relevant
32057 definitions: the key, the command, and the function.
32059 You may find it useful to turn off the default simplifications with
32060 @kbd{m O} (@code{calc-no-simplify-mode}) when entering a formula to be
32061 used as a function definition. For example, the formula @samp{deriv(a^2,v)}
32062 which might be used to define a new function @samp{dsqr(a,v)} will be
32063 ``simplified'' to 0 immediately upon entry since @code{deriv} considers
32064 @expr{a} to be constant with respect to @expr{v}. Turning off
32065 default simplifications cures this problem: The definition will be stored
32066 in symbolic form without ever activating the @code{deriv} function. Press
32067 @kbd{m D} to turn the default simplifications back on afterwards.
32069 @node Lisp Definitions, , Algebraic Definitions, Programming
32070 @section Programming with Lisp
32073 The Calculator can be programmed quite extensively in Lisp. All you
32074 do is write a normal Lisp function definition, but with @code{defmath}
32075 in place of @code{defun}. This has the same form as @code{defun}, but it
32076 automagically replaces calls to standard Lisp functions like @code{+} and
32077 @code{zerop} with calls to the corresponding functions in Calc's own library.
32078 Thus you can write natural-looking Lisp code which operates on all of the
32079 standard Calculator data types. You can then use @kbd{Z D} if you wish to
32080 bind your new command to a @kbd{z}-prefix key sequence. The @kbd{Z E} command
32081 will not edit a Lisp-based definition.
32083 Emacs Lisp is described in the GNU Emacs Lisp Reference Manual. This section
32084 assumes a familiarity with Lisp programming concepts; if you do not know
32085 Lisp, you may find keyboard macros or rewrite rules to be an easier way
32086 to program the Calculator.
32088 This section first discusses ways to write commands, functions, or
32089 small programs to be executed inside of Calc. Then it discusses how
32090 your own separate programs are able to call Calc from the outside.
32091 Finally, there is a list of internal Calc functions and data structures
32092 for the true Lisp enthusiast.
32095 * Defining Functions::
32096 * Defining Simple Commands::
32097 * Defining Stack Commands::
32098 * Argument Qualifiers::
32099 * Example Definitions::
32101 * Calling Calc from Your Programs::
32105 @node Defining Functions, Defining Simple Commands, Lisp Definitions, Lisp Definitions
32106 @subsection Defining New Functions
32110 The @code{defmath} function (actually a Lisp macro) is like @code{defun}
32111 except that code in the body of the definition can make use of the full
32112 range of Calculator data types. The prefix @samp{calcFunc-} is added
32113 to the specified name to get the actual Lisp function name. As a simple
32117 (defmath myfact (n)
32119 (* n (myfact (1- n)))
32124 This actually expands to the code,
32127 (defun calcFunc-myfact (n)
32129 (math-mul n (calcFunc-myfact (math-add n -1)))
32134 This function can be used in algebraic expressions, e.g., @samp{myfact(5)}.
32136 The @samp{myfact} function as it is defined above has the bug that an
32137 expression @samp{myfact(a+b)} will be simplified to 1 because the
32138 formula @samp{a+b} is not considered to be @code{posp}. A robust
32139 factorial function would be written along the following lines:
32142 (defmath myfact (n)
32144 (* n (myfact (1- n)))
32147 nil))) ; this could be simplified as: (and (= n 0) 1)
32150 If a function returns @code{nil}, it is left unsimplified by the Calculator
32151 (except that its arguments will be simplified). Thus, @samp{myfact(a+1+2)}
32152 will be simplified to @samp{myfact(a+3)} but no further. Beware that every
32153 time the Calculator reexamines this formula it will attempt to resimplify
32154 it, so your function ought to detect the returning-@code{nil} case as
32155 efficiently as possible.
32157 The following standard Lisp functions are treated by @code{defmath}:
32158 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^} or
32159 @code{expt}, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=},
32160 @code{/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior}, @code{logxor},
32161 @code{logandc2}, @code{lognot}. Also, @code{~=} is an abbreviation for
32162 @code{math-nearly-equal}, which is useful in implementing Taylor series.
32164 For other functions @var{func}, if a function by the name
32165 @samp{calcFunc-@var{func}} exists it is used, otherwise if a function by the
32166 name @samp{math-@var{func}} exists it is used, otherwise if @var{func} itself
32167 is defined as a function it is used, otherwise @samp{calcFunc-@var{func}} is
32168 used on the assumption that this is a to-be-defined math function. Also, if
32169 the function name is quoted as in @samp{('integerp a)} the function name is
32170 always used exactly as written (but not quoted).
32172 Variable names have @samp{var-} prepended to them unless they appear in
32173 the function's argument list or in an enclosing @code{let}, @code{let*},
32174 @code{for}, or @code{foreach} form,
32175 or their names already contain a @samp{-} character. Thus a reference to
32176 @samp{foo} is the same as a reference to @samp{var-foo}.
32178 A few other Lisp extensions are available in @code{defmath} definitions:
32182 The @code{elt} function accepts any number of index variables.
32183 Note that Calc vectors are stored as Lisp lists whose first
32184 element is the symbol @code{vec}; thus, @samp{(elt v 2)} yields
32185 the second element of vector @code{v}, and @samp{(elt m i j)}
32186 yields one element of a Calc matrix.
32189 The @code{setq} function has been extended to act like the Common
32190 Lisp @code{setf} function. (The name @code{setf} is recognized as
32191 a synonym of @code{setq}.) Specifically, the first argument of
32192 @code{setq} can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form,
32193 in which case the effect is to store into the specified
32194 element of a list. Thus, @samp{(setq (elt m i j) x)} stores @expr{x}
32195 into one element of a matrix.
32198 A @code{for} looping construct is available. For example,
32199 @samp{(for ((i 0 10)) body)} executes @code{body} once for each
32200 binding of @expr{i} from zero to 10. This is like a @code{let}
32201 form in that @expr{i} is temporarily bound to the loop count
32202 without disturbing its value outside the @code{for} construct.
32203 Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)},
32204 are also available. For each value of @expr{i} from zero to 10,
32205 @expr{j} counts from 0 to @expr{i-1} in steps of two. Note that
32206 @code{for} has the same general outline as @code{let*}, except
32207 that each element of the header is a list of three or four
32208 things, not just two.
32211 The @code{foreach} construct loops over elements of a list.
32212 For example, @samp{(foreach ((x (cdr v))) body)} executes
32213 @code{body} with @expr{x} bound to each element of Calc vector
32214 @expr{v} in turn. The purpose of @code{cdr} here is to skip over
32215 the initial @code{vec} symbol in the vector.
32218 The @code{break} function breaks out of the innermost enclosing
32219 @code{while}, @code{for}, or @code{foreach} loop. If given a
32220 value, as in @samp{(break x)}, this value is returned by the
32221 loop. (Lisp loops otherwise always return @code{nil}.)
32224 The @code{return} function prematurely returns from the enclosing
32225 function. For example, @samp{(return (+ x y))} returns @expr{x+y}
32226 as the value of a function. You can use @code{return} anywhere
32227 inside the body of the function.
32230 Non-integer numbers (and extremely large integers) cannot be included
32231 directly into a @code{defmath} definition. This is because the Lisp
32232 reader will fail to parse them long before @code{defmath} ever gets control.
32233 Instead, use the notation, @samp{:"3.1415"}. In fact, any algebraic
32234 formula can go between the quotes. For example,
32237 (defmath sqexp (x) ; sqexp(x) == sqrt(exp(x)) == exp(x*0.5)
32245 (defun calcFunc-sqexp (x)
32246 (and (math-numberp x)
32247 (calcFunc-exp (math-mul x '(float 5 -1)))))
32250 Note the use of @code{numberp} as a guard to ensure that the argument is
32251 a number first, returning @code{nil} if not. The exponential function
32252 could itself have been included in the expression, if we had preferred:
32253 @samp{:"exp(x * 0.5)"}. As another example, the multiplication-and-recursion
32254 step of @code{myfact} could have been written
32260 A good place to put your @code{defmath} commands is your Calc init file
32261 (the file given by @code{calc-settings-file}, typically
32262 @file{~/.emacs.d/calc.el}), which will not be loaded until Calc starts.
32263 If a file named @file{.emacs} exists in your home directory, Emacs reads
32264 and executes the Lisp forms in this file as it starts up. While it may
32265 seem reasonable to put your favorite @code{defmath} commands there,
32266 this has the unfortunate side-effect that parts of the Calculator must be
32267 loaded in to process the @code{defmath} commands whether or not you will
32268 actually use the Calculator! If you want to put the @code{defmath}
32269 commands there (for example, if you redefine @code{calc-settings-file}
32270 to be @file{.emacs}), a better effect can be had by writing
32273 (put 'calc-define 'thing '(progn
32280 @vindex calc-define
32281 The @code{put} function adds a @dfn{property} to a symbol. Each Lisp
32282 symbol has a list of properties associated with it. Here we add a
32283 property with a name of @code{thing} and a @samp{(progn ...)} form as
32284 its value. When Calc starts up, and at the start of every Calc command,
32285 the property list for the symbol @code{calc-define} is checked and the
32286 values of any properties found are evaluated as Lisp forms. The
32287 properties are removed as they are evaluated. The property names
32288 (like @code{thing}) are not used; you should choose something like the
32289 name of your project so as not to conflict with other properties.
32291 The net effect is that you can put the above code in your @file{.emacs}
32292 file and it will not be executed until Calc is loaded. Or, you can put
32293 that same code in another file which you load by hand either before or
32294 after Calc itself is loaded.
32296 The properties of @code{calc-define} are evaluated in the same order
32297 that they were added. They can assume that the Calc modules @file{calc.el},
32298 @file{calc-ext.el}, and @file{calc-macs.el} have been fully loaded, and
32299 that the @samp{*Calculator*} buffer will be the current buffer.
32301 If your @code{calc-define} property only defines algebraic functions,
32302 you can be sure that it will have been evaluated before Calc tries to
32303 call your function, even if the file defining the property is loaded
32304 after Calc is loaded. But if the property defines commands or key
32305 sequences, it may not be evaluated soon enough. (Suppose it defines the
32306 new command @code{tweak-calc}; the user can load your file, then type
32307 @kbd{M-x tweak-calc} before Calc has had chance to do anything.) To
32308 protect against this situation, you can put
32311 (run-hooks 'calc-check-defines)
32314 @findex calc-check-defines
32316 at the end of your file. The @code{calc-check-defines} function is what
32317 looks for and evaluates properties on @code{calc-define}; @code{run-hooks}
32318 has the advantage that it is quietly ignored if @code{calc-check-defines}
32319 is not yet defined because Calc has not yet been loaded.
32321 Examples of things that ought to be enclosed in a @code{calc-define}
32322 property are @code{defmath} calls, @code{define-key} calls that modify
32323 the Calc key map, and any calls that redefine things defined inside Calc.
32324 Ordinary @code{defun}s need not be enclosed with @code{calc-define}.
32326 @node Defining Simple Commands, Defining Stack Commands, Defining Functions, Lisp Definitions
32327 @subsection Defining New Simple Commands
32330 @findex interactive
32331 If a @code{defmath} form contains an @code{interactive} clause, it defines
32332 a Calculator command. Actually such a @code{defmath} results in @emph{two}
32333 function definitions: One, a @samp{calcFunc-} function as was just described,
32334 with the @code{interactive} clause removed. Two, a @samp{calc-} function
32335 with a suitable @code{interactive} clause and some sort of wrapper to make
32336 the command work in the Calc environment.
32338 In the simple case, the @code{interactive} clause has the same form as
32339 for normal Emacs Lisp commands:
32342 (defmath increase-precision (delta)
32343 "Increase precision by DELTA." ; This is the "documentation string"
32344 (interactive "p") ; Register this as a M-x-able command
32345 (setq calc-internal-prec (+ calc-internal-prec delta)))
32348 This expands to the pair of definitions,
32351 (defun calc-increase-precision (delta)
32352 "Increase precision by DELTA."
32355 (setq calc-internal-prec (math-add calc-internal-prec delta))))
32357 (defun calcFunc-increase-precision (delta)
32358 "Increase precision by DELTA."
32359 (setq calc-internal-prec (math-add calc-internal-prec delta)))
32363 where in this case the latter function would never really be used! Note
32364 that since the Calculator stores small integers as plain Lisp integers,
32365 the @code{math-add} function will work just as well as the native
32366 @code{+} even when the intent is to operate on native Lisp integers.
32368 @findex calc-wrapper
32369 The @samp{calc-wrapper} call invokes a macro which surrounds the body of
32370 the function with code that looks roughly like this:
32373 (let ((calc-command-flags nil))
32375 (save-current-buffer
32376 (calc-select-buffer)
32377 @emph{body of function}
32378 @emph{renumber stack}
32379 @emph{clear} Working @emph{message})
32380 @emph{realign cursor and window}
32381 @emph{clear Inverse, Hyperbolic, and Keep Args flags}
32382 @emph{update Emacs mode line}))
32385 @findex calc-select-buffer
32386 The @code{calc-select-buffer} function selects the @samp{*Calculator*}
32387 buffer if necessary, say, because the command was invoked from inside
32388 the @samp{*Calc Trail*} window.
32390 @findex calc-set-command-flag
32391 You can call, for example, @code{(calc-set-command-flag 'no-align)} to
32392 set the above-mentioned command flags. Calc routines recognize the
32393 following command flags:
32397 Stack line numbers @samp{1:}, @samp{2:}, and so on must be renumbered
32398 after this command completes. This is set by routines like
32401 @item clear-message
32402 Calc should call @samp{(message "")} if this command completes normally
32403 (to clear a ``Working@dots{}'' message out of the echo area).
32406 Do not move the cursor back to the @samp{.} top-of-stack marker.
32408 @item position-point
32409 Use the variables @code{calc-position-point-line} and
32410 @code{calc-position-point-column} to position the cursor after
32411 this command finishes.
32414 Do not clear @code{calc-inverse-flag}, @code{calc-hyperbolic-flag},
32415 and @code{calc-keep-args-flag} at the end of this command.
32418 Switch to buffer @samp{*Calc Edit*} after this command.
32421 Do not move trail pointer to end of trail when something is recorded
32427 @vindex calc-Y-help-msgs
32428 Calc reserves a special prefix key, shift-@kbd{Y}, for user-written
32429 extensions to Calc. There are no built-in commands that work with
32430 this prefix key; you must call @code{define-key} from Lisp (probably
32431 from inside a @code{calc-define} property) to add to it. Initially only
32432 @kbd{Y ?} is defined; it takes help messages from a list of strings
32433 (initially @code{nil}) in the variable @code{calc-Y-help-msgs}. All
32434 other undefined keys except for @kbd{Y} are reserved for use by
32435 future versions of Calc.
32437 If you are writing a Calc enhancement which you expect to give to
32438 others, it is best to minimize the number of @kbd{Y}-key sequences
32439 you use. In fact, if you have more than one key sequence you should
32440 consider defining three-key sequences with a @kbd{Y}, then a key that
32441 stands for your package, then a third key for the particular command
32442 within your package.
32444 Users may wish to install several Calc enhancements, and it is possible
32445 that several enhancements will choose to use the same key. In the
32446 example below, a variable @code{inc-prec-base-key} has been defined
32447 to contain the key that identifies the @code{inc-prec} package. Its
32448 value is initially @code{"P"}, but a user can change this variable
32449 if necessary without having to modify the file.
32451 Here is a complete file, @file{inc-prec.el}, which makes a @kbd{Y P I}
32452 command that increases the precision, and a @kbd{Y P D} command that
32453 decreases the precision.
32456 ;;; Increase and decrease Calc precision. Dave Gillespie, 5/31/91.
32457 ;; (Include copyright or copyleft stuff here.)
32459 (defvar inc-prec-base-key "P"
32460 "Base key for inc-prec.el commands.")
32462 (put 'calc-define 'inc-prec '(progn
32464 (define-key calc-mode-map (format "Y%sI" inc-prec-base-key)
32465 'increase-precision)
32466 (define-key calc-mode-map (format "Y%sD" inc-prec-base-key)
32467 'decrease-precision)
32469 (setq calc-Y-help-msgs
32470 (cons (format "%s + Inc-prec, Dec-prec" inc-prec-base-key)
32473 (defmath increase-precision (delta)
32474 "Increase precision by DELTA."
32476 (setq calc-internal-prec (+ calc-internal-prec delta)))
32478 (defmath decrease-precision (delta)
32479 "Decrease precision by DELTA."
32481 (setq calc-internal-prec (- calc-internal-prec delta)))
32483 )) ; end of calc-define property
32485 (run-hooks 'calc-check-defines)
32488 @node Defining Stack Commands, Argument Qualifiers, Defining Simple Commands, Lisp Definitions
32489 @subsection Defining New Stack-Based Commands
32492 To define a new computational command which takes and/or leaves arguments
32493 on the stack, a special form of @code{interactive} clause is used.
32496 (interactive @var{num} @var{tag})
32500 where @var{num} is an integer, and @var{tag} is a string. The effect is
32501 to pop @var{num} values off the stack, resimplify them by calling
32502 @code{calc-normalize}, and hand them to your function according to the
32503 function's argument list. Your function may include @code{&optional} and
32504 @code{&rest} parameters, so long as calling the function with @var{num}
32505 parameters is valid.
32507 Your function must return either a number or a formula in a form
32508 acceptable to Calc, or a list of such numbers or formulas. These value(s)
32509 are pushed onto the stack when the function completes. They are also
32510 recorded in the Calc Trail buffer on a line beginning with @var{tag},
32511 a string of (normally) four characters or less. If you omit @var{tag}
32512 or use @code{nil} as a tag, the result is not recorded in the trail.
32514 As an example, the definition
32517 (defmath myfact (n)
32518 "Compute the factorial of the integer at the top of the stack."
32519 (interactive 1 "fact")
32521 (* n (myfact (1- n)))
32526 is a version of the factorial function shown previously which can be used
32527 as a command as well as an algebraic function. It expands to
32530 (defun calc-myfact ()
32531 "Compute the factorial of the integer at the top of the stack."
32534 (calc-enter-result 1 "fact"
32535 (cons 'calcFunc-myfact (calc-top-list-n 1)))))
32537 (defun calcFunc-myfact (n)
32538 "Compute the factorial of the integer at the top of the stack."
32540 (math-mul n (calcFunc-myfact (math-add n -1)))
32541 (and (math-zerop n) 1)))
32544 @findex calc-slow-wrapper
32545 The @code{calc-slow-wrapper} function is a version of @code{calc-wrapper}
32546 that automatically puts up a @samp{Working...} message before the
32547 computation begins. (This message can be turned off by the user
32548 with an @kbd{m w} (@code{calc-working}) command.)
32550 @findex calc-top-list-n
32551 The @code{calc-top-list-n} function returns a list of the specified number
32552 of values from the top of the stack. It resimplifies each value by
32553 calling @code{calc-normalize}. If its argument is zero it returns an
32554 empty list. It does not actually remove these values from the stack.
32556 @findex calc-enter-result
32557 The @code{calc-enter-result} function takes an integer @var{num} and string
32558 @var{tag} as described above, plus a third argument which is either a
32559 Calculator data object or a list of such objects. These objects are
32560 resimplified and pushed onto the stack after popping the specified number
32561 of values from the stack. If @var{tag} is non-@code{nil}, the values
32562 being pushed are also recorded in the trail.
32564 Note that if @code{calcFunc-myfact} returns @code{nil} this represents
32565 ``leave the function in symbolic form.'' To return an actual empty list,
32566 in the sense that @code{calc-enter-result} will push zero elements back
32567 onto the stack, you should return the special value @samp{'(nil)}, a list
32568 containing the single symbol @code{nil}.
32570 The @code{interactive} declaration can actually contain a limited
32571 Emacs-style code string as well which comes just before @var{num} and
32572 @var{tag}. Currently the only Emacs code supported is @samp{"p"}, as in
32575 (defmath foo (a b &optional c)
32576 (interactive "p" 2 "foo")
32580 In this example, the command @code{calc-foo} will evaluate the expression
32581 @samp{foo(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if
32582 executed with a numeric prefix argument of @expr{n}.
32584 The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"}
32585 code as used with @code{defun}). It uses the numeric prefix argument as the
32586 number of objects to remove from the stack and pass to the function.
32587 In this case, the integer @var{num} serves as a default number of
32588 arguments to be used when no prefix is supplied.
32590 @node Argument Qualifiers, Example Definitions, Defining Stack Commands, Lisp Definitions
32591 @subsection Argument Qualifiers
32594 Anywhere a parameter name can appear in the parameter list you can also use
32595 an @dfn{argument qualifier}. Thus the general form of a definition is:
32598 (defmath @var{name} (@var{param} @var{param...}
32599 &optional @var{param} @var{param...}
32605 where each @var{param} is either a symbol or a list of the form
32608 (@var{qual} @var{param})
32611 The following qualifiers are recognized:
32616 The argument must not be an incomplete vector, interval, or complex number.
32617 (This is rarely needed since the Calculator itself will never call your
32618 function with an incomplete argument. But there is nothing stopping your
32619 own Lisp code from calling your function with an incomplete argument.)
32623 The argument must be an integer. If it is an integer-valued float
32624 it will be accepted but converted to integer form. Non-integers and
32625 formulas are rejected.
32629 Like @samp{integer}, but the argument must be non-negative.
32633 Like @samp{integer}, but the argument must fit into a native Lisp integer,
32634 which on most systems means less than 2^23 in absolute value. The
32635 argument is converted into Lisp-integer form if necessary.
32639 The argument is converted to floating-point format if it is a number or
32640 vector. If it is a formula it is left alone. (The argument is never
32641 actually rejected by this qualifier.)
32644 The argument must satisfy predicate @var{pred}, which is one of the
32645 standard Calculator predicates. @xref{Predicates}.
32647 @item not-@var{pred}
32648 The argument must @emph{not} satisfy predicate @var{pred}.
32654 (defmath foo (a (constp (not-matrixp b)) &optional (float c)
32663 (defun calcFunc-foo (a b &optional c &rest d)
32664 (and (math-matrixp b)
32665 (math-reject-arg b 'not-matrixp))
32666 (or (math-constp b)
32667 (math-reject-arg b 'constp))
32668 (and c (setq c (math-check-float c)))
32669 (setq d (mapcar 'math-check-integer d))
32674 which performs the necessary checks and conversions before executing the
32675 body of the function.
32677 @node Example Definitions, Calling Calc from Your Programs, Argument Qualifiers, Lisp Definitions
32678 @subsection Example Definitions
32681 This section includes some Lisp programming examples on a larger scale.
32682 These programs make use of some of the Calculator's internal functions;
32686 * Bit Counting Example::
32690 @node Bit Counting Example, Sine Example, Example Definitions, Example Definitions
32691 @subsubsection Bit-Counting
32698 Calc does not include a built-in function for counting the number of
32699 ``one'' bits in a binary integer. It's easy to invent one using @kbd{b u}
32700 to convert the integer to a set, and @kbd{V #} to count the elements of
32701 that set; let's write a function that counts the bits without having to
32702 create an intermediate set.
32705 (defmath bcount ((natnum n))
32706 (interactive 1 "bcnt")
32710 (setq count (1+ count)))
32711 (setq n (lsh n -1)))
32716 When this is expanded by @code{defmath}, it will become the following
32717 Emacs Lisp function:
32720 (defun calcFunc-bcount (n)
32721 (setq n (math-check-natnum n))
32723 (while (math-posp n)
32725 (setq count (math-add count 1)))
32726 (setq n (calcFunc-lsh n -1)))
32730 If the input numbers are large, this function involves a fair amount
32731 of arithmetic. A binary right shift is essentially a division by two;
32732 recall that Calc stores integers in decimal form so bit shifts must
32733 involve actual division.
32735 To gain a bit more efficiency, we could divide the integer into
32736 @var{n}-bit chunks, each of which can be handled quickly because
32737 they fit into Lisp integers. It turns out that Calc's arithmetic
32738 routines are especially fast when dividing by an integer less than
32739 1000, so we can set @var{n = 9} bits and use repeated division by 512:
32742 (defmath bcount ((natnum n))
32743 (interactive 1 "bcnt")
32745 (while (not (fixnump n))
32746 (let ((qr (idivmod n 512)))
32747 (setq count (+ count (bcount-fixnum (cdr qr)))
32749 (+ count (bcount-fixnum n))))
32751 (defun bcount-fixnum (n)
32754 (setq count (+ count (logand n 1))
32760 Note that the second function uses @code{defun}, not @code{defmath}.
32761 Because this function deals only with native Lisp integers (``fixnums''),
32762 it can use the actual Emacs @code{+} and related functions rather
32763 than the slower but more general Calc equivalents which @code{defmath}
32766 The @code{idivmod} function does an integer division, returning both
32767 the quotient and the remainder at once. Again, note that while it
32768 might seem that @samp{(logand n 511)} and @samp{(lsh n -9)} are
32769 more efficient ways to split off the bottom nine bits of @code{n},
32770 actually they are less efficient because each operation is really
32771 a division by 512 in disguise; @code{idivmod} allows us to do the
32772 same thing with a single division by 512.
32774 @node Sine Example, , Bit Counting Example, Example Definitions
32775 @subsubsection The Sine Function
32782 A somewhat limited sine function could be defined as follows, using the
32783 well-known Taylor series expansion for
32784 @texline @math{\sin x}:
32785 @infoline @samp{sin(x)}:
32788 (defmath mysin ((float (anglep x)))
32789 (interactive 1 "mysn")
32790 (setq x (to-radians x)) ; Convert from current angular mode.
32791 (let ((sum x) ; Initial term of Taylor expansion of sin.
32793 (nfact 1) ; "nfact" equals "n" factorial at all times.
32794 (xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2.
32795 (for ((n 3 100 2)) ; Upper limit of 100 is a good precaution.
32796 (working "mysin" sum) ; Display "Working" message, if enabled.
32797 (setq nfact (* nfact (1- n) n)
32799 newsum (+ sum (/ x nfact)))
32800 (if (~= newsum sum) ; If newsum is "nearly equal to" sum,
32801 (break)) ; then we are done.
32806 The actual @code{sin} function in Calc works by first reducing the problem
32807 to a sine or cosine of a nonnegative number less than @cpiover{4}. This
32808 ensures that the Taylor series will converge quickly. Also, the calculation
32809 is carried out with two extra digits of precision to guard against cumulative
32810 round-off in @samp{sum}. Finally, complex arguments are allowed and handled
32811 by a separate algorithm.
32814 (defmath mysin ((float (scalarp x)))
32815 (interactive 1 "mysn")
32816 (setq x (to-radians x)) ; Convert from current angular mode.
32817 (with-extra-prec 2 ; Evaluate with extra precision.
32818 (cond ((complexp x)
32821 (- (mysin-raw (- x))) ; Always call mysin-raw with x >= 0.
32822 (t (mysin-raw x))))))
32824 (defmath mysin-raw (x)
32826 (mysin-raw (% x (two-pi)))) ; Now x < 7.
32828 (- (mysin-raw (- x (pi))))) ; Now -pi/2 <= x <= pi/2.
32830 (mycos-raw (- x (pi-over-2)))) ; Now -pi/2 <= x <= pi/4.
32831 ((< x (- (pi-over-4)))
32832 (- (mycos-raw (+ x (pi-over-2))))) ; Now -pi/4 <= x <= pi/4,
32833 (t (mysin-series x)))) ; so the series will be efficient.
32837 where @code{mysin-complex} is an appropriate function to handle complex
32838 numbers, @code{mysin-series} is the routine to compute the sine Taylor
32839 series as before, and @code{mycos-raw} is a function analogous to
32840 @code{mysin-raw} for cosines.
32842 The strategy is to ensure that @expr{x} is nonnegative before calling
32843 @code{mysin-raw}. This function then recursively reduces its argument
32844 to a suitable range, namely, plus-or-minus @cpiover{4}. Note that each
32845 test, and particularly the first comparison against 7, is designed so
32846 that small roundoff errors cannot produce an infinite loop. (Suppose
32847 we compared with @samp{(two-pi)} instead; if due to roundoff problems
32848 the modulo operator ever returned @samp{(two-pi)} exactly, an infinite
32849 recursion could result!) We use modulo only for arguments that will
32850 clearly get reduced, knowing that the next rule will catch any reductions
32851 that this rule misses.
32853 If a program is being written for general use, it is important to code
32854 it carefully as shown in this second example. For quick-and-dirty programs,
32855 when you know that your own use of the sine function will never encounter
32856 a large argument, a simpler program like the first one shown is fine.
32858 @node Calling Calc from Your Programs, Internals, Example Definitions, Lisp Definitions
32859 @subsection Calling Calc from Your Lisp Programs
32862 A later section (@pxref{Internals}) gives a full description of
32863 Calc's internal Lisp functions. It's not hard to call Calc from
32864 inside your programs, but the number of these functions can be daunting.
32865 So Calc provides one special ``programmer-friendly'' function called
32866 @code{calc-eval} that can be made to do just about everything you
32867 need. It's not as fast as the low-level Calc functions, but it's
32868 much simpler to use!
32870 It may seem that @code{calc-eval} itself has a daunting number of
32871 options, but they all stem from one simple operation.
32873 In its simplest manifestation, @samp{(calc-eval "1+2")} parses the
32874 string @code{"1+2"} as if it were a Calc algebraic entry and returns
32875 the result formatted as a string: @code{"3"}.
32877 Since @code{calc-eval} is on the list of recommended @code{autoload}
32878 functions, you don't need to make any special preparations to load
32879 Calc before calling @code{calc-eval} the first time. Calc will be
32880 loaded and initialized for you.
32882 All the Calc modes that are currently in effect will be used when
32883 evaluating the expression and formatting the result.
32890 @subsubsection Additional Arguments to @code{calc-eval}
32893 If the input string parses to a list of expressions, Calc returns
32894 the results separated by @code{", "}. You can specify a different
32895 separator by giving a second string argument to @code{calc-eval}:
32896 @samp{(calc-eval "1+2,3+4" ";")} returns @code{"3;7"}.
32898 The ``separator'' can also be any of several Lisp symbols which
32899 request other behaviors from @code{calc-eval}. These are discussed
32902 You can give additional arguments to be substituted for
32903 @samp{$}, @samp{$$}, and so on in the main expression. For
32904 example, @samp{(calc-eval "$/$$" nil "7" "1+1")} evaluates the
32905 expression @code{"7/(1+1)"} to yield the result @code{"3.5"}
32906 (assuming Fraction mode is not in effect). Note the @code{nil}
32907 used as a placeholder for the item-separator argument.
32914 @subsubsection Error Handling
32917 If @code{calc-eval} encounters an error, it returns a list containing
32918 the character position of the error, plus a suitable message as a
32919 string. Note that @samp{1 / 0} is @emph{not} an error by Calc's
32920 standards; it simply returns the string @code{"1 / 0"} which is the
32921 division left in symbolic form. But @samp{(calc-eval "1/")} will
32922 return the list @samp{(2 "Expected a number")}.
32924 If you bind the variable @code{calc-eval-error} to @code{t}
32925 using a @code{let} form surrounding the call to @code{calc-eval},
32926 errors instead call the Emacs @code{error} function which aborts
32927 to the Emacs command loop with a beep and an error message.
32929 If you bind this variable to the symbol @code{string}, error messages
32930 are returned as strings instead of lists. The character position is
32933 As a courtesy to other Lisp code which may be using Calc, be sure
32934 to bind @code{calc-eval-error} using @code{let} rather than changing
32935 it permanently with @code{setq}.
32942 @subsubsection Numbers Only
32945 Sometimes it is preferable to treat @samp{1 / 0} as an error
32946 rather than returning a symbolic result. If you pass the symbol
32947 @code{num} as the second argument to @code{calc-eval}, results
32948 that are not constants are treated as errors. The error message
32949 reported is the first @code{calc-why} message if there is one,
32950 or otherwise ``Number expected.''
32952 A result is ``constant'' if it is a number, vector, or other
32953 object that does not include variables or function calls. If it
32954 is a vector, the components must themselves be constants.
32961 @subsubsection Default Modes
32964 If the first argument to @code{calc-eval} is a list whose first
32965 element is a formula string, then @code{calc-eval} sets all the
32966 various Calc modes to their default values while the formula is
32967 evaluated and formatted. For example, the precision is set to 12
32968 digits, digit grouping is turned off, and the Normal language
32971 This same principle applies to the other options discussed below.
32972 If the first argument would normally be @var{x}, then it can also
32973 be the list @samp{(@var{x})} to use the default mode settings.
32975 If there are other elements in the list, they are taken as
32976 variable-name/value pairs which override the default mode
32977 settings. Look at the documentation at the front of the
32978 @file{calc.el} file to find the names of the Lisp variables for
32979 the various modes. The mode settings are restored to their
32980 original values when @code{calc-eval} is done.
32982 For example, @samp{(calc-eval '("$+$$" calc-internal-prec 8) 'num a b)}
32983 computes the sum of two numbers, requiring a numeric result, and
32984 using default mode settings except that the precision is 8 instead
32985 of the default of 12.
32987 It's usually best to use this form of @code{calc-eval} unless your
32988 program actually considers the interaction with Calc's mode settings
32989 to be a feature. This will avoid all sorts of potential ``gotchas'';
32990 consider what happens with @samp{(calc-eval "sqrt(2)" 'num)}
32991 when the user has left Calc in Symbolic mode or No-Simplify mode.
32993 As another example, @samp{(equal (calc-eval '("$<$$") nil a b) "1")}
32994 checks if the number in string @expr{a} is less than the one in
32995 string @expr{b}. Without using a list, the integer 1 might
32996 come out in a variety of formats which would be hard to test for
32997 conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But
32998 see ``Predicates'' mode, below.)
33005 @subsubsection Raw Numbers
33008 Normally all input and output for @code{calc-eval} is done with strings.
33009 You can do arithmetic with, say, @samp{(calc-eval "$+$$" nil a b)}
33010 in place of @samp{(+ a b)}, but this is very inefficient since the
33011 numbers must be converted to and from string format as they are passed
33012 from one @code{calc-eval} to the next.
33014 If the separator is the symbol @code{raw}, the result will be returned
33015 as a raw Calc data structure rather than a string. You can read about
33016 how these objects look in the following sections, but usually you can
33017 treat them as ``black box'' objects with no important internal
33020 There is also a @code{rawnum} symbol, which is a combination of
33021 @code{raw} (returning a raw Calc object) and @code{num} (signaling
33022 an error if that object is not a constant).
33024 You can pass a raw Calc object to @code{calc-eval} in place of a
33025 string, either as the formula itself or as one of the @samp{$}
33026 arguments. Thus @samp{(calc-eval "$+$$" 'raw a b)} is an
33027 addition function that operates on raw Calc objects. Of course
33028 in this case it would be easier to call the low-level @code{math-add}
33029 function in Calc, if you can remember its name.
33031 In particular, note that a plain Lisp integer is acceptable to Calc
33032 as a raw object. (All Lisp integers are accepted on input, but
33033 integers of more than six decimal digits are converted to ``big-integer''
33034 form for output. @xref{Data Type Formats}.)
33036 When it comes time to display the object, just use @samp{(calc-eval a)}
33037 to format it as a string.
33039 It is an error if the input expression evaluates to a list of
33040 values. The separator symbol @code{list} is like @code{raw}
33041 except that it returns a list of one or more raw Calc objects.
33043 Note that a Lisp string is not a valid Calc object, nor is a list
33044 containing a string. Thus you can still safely distinguish all the
33045 various kinds of error returns discussed above.
33052 @subsubsection Predicates
33055 If the separator symbol is @code{pred}, the result of the formula is
33056 treated as a true/false value; @code{calc-eval} returns @code{t} or
33057 @code{nil}, respectively. A value is considered ``true'' if it is a
33058 non-zero number, or false if it is zero or if it is not a number.
33060 For example, @samp{(calc-eval "$<$$" 'pred a b)} tests whether
33061 one value is less than another.
33063 As usual, it is also possible for @code{calc-eval} to return one of
33064 the error indicators described above. Lisp will interpret such an
33065 indicator as ``true'' if you don't check for it explicitly. If you
33066 wish to have an error register as ``false'', use something like
33067 @samp{(eq (calc-eval ...) t)}.
33074 @subsubsection Variable Values
33077 Variables in the formula passed to @code{calc-eval} are not normally
33078 replaced by their values. If you wish this, you can use the
33079 @code{evalv} function (@pxref{Algebraic Manipulation}). For example,
33080 if 4 is stored in Calc variable @code{a} (i.e., in Lisp variable
33081 @code{var-a}), then @samp{(calc-eval "a+pi")} will return the
33082 formula @code{"a + pi"}, but @samp{(calc-eval "evalv(a+pi)")}
33083 will return @code{"7.14159265359"}.
33085 To store in a Calc variable, just use @code{setq} to store in the
33086 corresponding Lisp variable. (This is obtained by prepending
33087 @samp{var-} to the Calc variable name.) Calc routines will
33088 understand either string or raw form values stored in variables,
33089 although raw data objects are much more efficient. For example,
33090 to increment the Calc variable @code{a}:
33093 (setq var-a (calc-eval "evalv(a+1)" 'raw))
33101 @subsubsection Stack Access
33104 If the separator symbol is @code{push}, the formula argument is
33105 evaluated (with possible @samp{$} expansions, as usual). The
33106 result is pushed onto the Calc stack. The return value is @code{nil}
33107 (unless there is an error from evaluating the formula, in which
33108 case the return value depends on @code{calc-eval-error} in the
33111 If the separator symbol is @code{pop}, the first argument to
33112 @code{calc-eval} must be an integer instead of a string. That
33113 many values are popped from the stack and thrown away. A negative
33114 argument deletes the entry at that stack level. The return value
33115 is the number of elements remaining in the stack after popping;
33116 @samp{(calc-eval 0 'pop)} is a good way to measure the size of
33119 If the separator symbol is @code{top}, the first argument to
33120 @code{calc-eval} must again be an integer. The value at that
33121 stack level is formatted as a string and returned. Thus
33122 @samp{(calc-eval 1 'top)} returns the top-of-stack value. If the
33123 integer is out of range, @code{nil} is returned.
33125 The separator symbol @code{rawtop} is just like @code{top} except
33126 that the stack entry is returned as a raw Calc object instead of
33129 In all of these cases the first argument can be made a list in
33130 order to force the default mode settings, as described above.
33131 Thus @samp{(calc-eval '(2 calc-number-radix 16) 'top)} returns the
33132 second-to-top stack entry, formatted as a string using the default
33133 instead of current display modes, except that the radix is
33134 hexadecimal instead of decimal.
33136 It is, of course, polite to put the Calc stack back the way you
33137 found it when you are done, unless the user of your program is
33138 actually expecting it to affect the stack.
33140 Note that you do not actually have to switch into the @samp{*Calculator*}
33141 buffer in order to use @code{calc-eval}; it temporarily switches into
33142 the stack buffer if necessary.
33149 @subsubsection Keyboard Macros
33152 If the separator symbol is @code{macro}, the first argument must be a
33153 string of characters which Calc can execute as a sequence of keystrokes.
33154 This switches into the Calc buffer for the duration of the macro.
33155 For example, @samp{(calc-eval "vx5\rVR+" 'macro)} pushes the
33156 vector @samp{[1,2,3,4,5]} on the stack and then replaces it
33157 with the sum of those numbers. Note that @samp{\r} is the Lisp
33158 notation for the carriage-return, @key{RET}, character.
33160 If your keyboard macro wishes to pop the stack, @samp{\C-d} is
33161 safer than @samp{\177} (the @key{DEL} character) because some
33162 installations may have switched the meanings of @key{DEL} and
33163 @kbd{C-h}. Calc always interprets @kbd{C-d} as a synonym for
33164 ``pop-stack'' regardless of key mapping.
33166 If you provide a third argument to @code{calc-eval}, evaluation
33167 of the keyboard macro will leave a record in the Trail using
33168 that argument as a tag string. Normally the Trail is unaffected.
33170 The return value in this case is always @code{nil}.
33177 @subsubsection Lisp Evaluation
33180 Finally, if the separator symbol is @code{eval}, then the Lisp
33181 @code{eval} function is called on the first argument, which must
33182 be a Lisp expression rather than a Calc formula. Remember to
33183 quote the expression so that it is not evaluated until inside
33186 The difference from plain @code{eval} is that @code{calc-eval}
33187 switches to the Calc buffer before evaluating the expression.
33188 For example, @samp{(calc-eval '(setq calc-internal-prec 17) 'eval)}
33189 will correctly affect the buffer-local Calc precision variable.
33191 An alternative would be @samp{(calc-eval '(calc-precision 17) 'eval)}.
33192 This is evaluating a call to the function that is normally invoked
33193 by the @kbd{p} key, giving it 17 as its ``numeric prefix argument.''
33194 Note that this function will leave a message in the echo area as
33195 a side effect. Also, all Calc functions switch to the Calc buffer
33196 automatically if not invoked from there, so the above call is
33197 also equivalent to @samp{(calc-precision 17)} by itself.
33198 In all cases, Calc uses @code{save-excursion} to switch back to
33199 your original buffer when it is done.
33201 As usual the first argument can be a list that begins with a Lisp
33202 expression to use default instead of current mode settings.
33204 The result of @code{calc-eval} in this usage is just the result
33205 returned by the evaluated Lisp expression.
33212 @subsubsection Example
33215 @findex convert-temp
33216 Here is a sample Emacs command that uses @code{calc-eval}. Suppose
33217 you have a document with lots of references to temperatures on the
33218 Fahrenheit scale, say ``98.6 F'', and you wish to convert these
33219 references to Centigrade. The following command does this conversion.
33220 Place the Emacs cursor right after the letter ``F'' and invoke the
33221 command to change ``98.6 F'' to ``37 C''. Or, if the temperature is
33222 already in Centigrade form, the command changes it back to Fahrenheit.
33225 (defun convert-temp ()
33228 (re-search-backward "[^-.0-9]\\([-.0-9]+\\) *\\([FC]\\)")
33229 (let* ((top1 (match-beginning 1))
33230 (bot1 (match-end 1))
33231 (number (buffer-substring top1 bot1))
33232 (top2 (match-beginning 2))
33233 (bot2 (match-end 2))
33234 (type (buffer-substring top2 bot2)))
33235 (if (equal type "F")
33237 number (calc-eval "($ - 32)*5/9" nil number))
33239 number (calc-eval "$*9/5 + 32" nil number)))
33241 (delete-region top2 bot2)
33242 (insert-before-markers type)
33244 (delete-region top1 bot1)
33245 (if (string-match "\\.$" number) ; change "37." to "37"
33246 (setq number (substring number 0 -1)))
33250 Note the use of @code{insert-before-markers} when changing between
33251 ``F'' and ``C'', so that the character winds up before the cursor
33252 instead of after it.
33254 @node Internals, , Calling Calc from Your Programs, Lisp Definitions
33255 @subsection Calculator Internals
33258 This section describes the Lisp functions defined by the Calculator that
33259 may be of use to user-written Calculator programs (as described in the
33260 rest of this chapter). These functions are shown by their names as they
33261 conventionally appear in @code{defmath}. Their full Lisp names are
33262 generally gotten by prepending @samp{calcFunc-} or @samp{math-} to their
33263 apparent names. (Names that begin with @samp{calc-} are already in
33264 their full Lisp form.) You can use the actual full names instead if you
33265 prefer them, or if you are calling these functions from regular Lisp.
33267 The functions described here are scattered throughout the various
33268 Calc component files. Note that @file{calc.el} includes @code{autoload}s
33269 for only a few component files; when Calc wants to call an advanced
33270 function it calls @samp{(calc-extensions)} first; this function
33271 autoloads @file{calc-ext.el}, which in turn autoloads all the functions
33272 in the remaining component files.
33274 Because @code{defmath} itself uses the extensions, user-written code
33275 generally always executes with the extensions already loaded, so
33276 normally you can use any Calc function and be confident that it will
33277 be autoloaded for you when necessary. If you are doing something
33278 special, check carefully to make sure each function you are using is
33279 from @file{calc.el} or its components, and call @samp{(calc-extensions)}
33280 before using any function based in @file{calc-ext.el} if you can't
33281 prove this file will already be loaded.
33284 * Data Type Formats::
33285 * Interactive Lisp Functions::
33286 * Stack Lisp Functions::
33288 * Computational Lisp Functions::
33289 * Vector Lisp Functions::
33290 * Symbolic Lisp Functions::
33291 * Formatting Lisp Functions::
33295 @node Data Type Formats, Interactive Lisp Functions, Internals, Internals
33296 @subsubsection Data Type Formats
33299 Integers are stored in either of two ways, depending on their magnitude.
33300 Integers less than one million in absolute value are stored as standard
33301 Lisp integers. This is the only storage format for Calc data objects
33302 which is not a Lisp list.
33304 Large integers are stored as lists of the form @samp{(bigpos @var{d0}
33305 @var{d1} @var{d2} @dots{})} for positive integers 1000000 or more, or
33306 @samp{(bigneg @var{d0} @var{d1} @var{d2} @dots{})} for negative integers
33307 @mathit{-1000000} or less. Each @var{d} is a base-1000 ``digit,'' a Lisp integer
33308 from 0 to 999. The least significant digit is @var{d0}; the last digit,
33309 @var{dn}, which is always nonzero, is the most significant digit. For
33310 example, the integer @mathit{-12345678} is stored as @samp{(bigneg 678 345 12)}.
33312 The distinction between small and large integers is entirely hidden from
33313 the user. In @code{defmath} definitions, the Lisp predicate @code{integerp}
33314 returns true for either kind of integer, and in general both big and small
33315 integers are accepted anywhere the word ``integer'' is used in this manual.
33316 If the distinction must be made, native Lisp integers are called @dfn{fixnums}
33317 and large integers are called @dfn{bignums}.
33319 Fractions are stored as a list of the form, @samp{(frac @var{n} @var{d})}
33320 where @var{n} is an integer (big or small) numerator, @var{d} is an
33321 integer denominator greater than one, and @var{n} and @var{d} are relatively
33322 prime. Note that fractions where @var{d} is one are automatically converted
33323 to plain integers by all math routines; fractions where @var{d} is negative
33324 are normalized by negating the numerator and denominator.
33326 Floating-point numbers are stored in the form, @samp{(float @var{mant}
33327 @var{exp})}, where @var{mant} (the ``mantissa'') is an integer less than
33328 @samp{10^@var{p}} in absolute value (@var{p} represents the current
33329 precision), and @var{exp} (the ``exponent'') is a fixnum. The value of
33330 the float is @samp{@var{mant} * 10^@var{exp}}. For example, the number
33331 @mathit{-3.14} is stored as @samp{(float -314 -2) = -314*10^-2}. Other constraints
33332 are that the number 0.0 is always stored as @samp{(float 0 0)}, and,
33333 except for the 0.0 case, the rightmost base-10 digit of @var{mant} is
33334 always nonzero. (If the rightmost digit is zero, the number is
33335 rearranged by dividing @var{mant} by ten and incrementing @var{exp}.)
33337 Rectangular complex numbers are stored in the form @samp{(cplx @var{re}
33338 @var{im})}, where @var{re} and @var{im} are each real numbers, either
33339 integers, fractions, or floats. The value is @samp{@var{re} + @var{im}i}.
33340 The @var{im} part is nonzero; complex numbers with zero imaginary
33341 components are converted to real numbers automatically.
33343 Polar complex numbers are stored in the form @samp{(polar @var{r}
33344 @var{theta})}, where @var{r} is a positive real value and @var{theta}
33345 is a real value or HMS form representing an angle. This angle is
33346 usually normalized to lie in the interval @samp{(-180 ..@: 180)} degrees,
33347 or @samp{(-pi ..@: pi)} radians, according to the current angular mode.
33348 If the angle is 0 the value is converted to a real number automatically.
33349 (If the angle is 180 degrees, the value is usually also converted to a
33350 negative real number.)
33352 Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m}
33353 @var{s})}, where @var{h} is an integer or an integer-valued float (i.e.,
33354 a float with @samp{@var{exp} >= 0}), @var{m} is an integer or integer-valued
33355 float in the range @w{@samp{[0 ..@: 60)}}, and @var{s} is any real number
33356 in the range @samp{[0 ..@: 60)}.
33358 Date forms are stored as @samp{(date @var{n})}, where @var{n} is
33359 a real number that counts days since midnight on the morning of
33360 January 1, 1 AD@. If @var{n} is an integer, this is a pure date
33361 form. If @var{n} is a fraction or float, this is a date/time form.
33363 Modulo forms are stored as @samp{(mod @var{n} @var{m})}, where @var{m} is a
33364 positive real number or HMS form, and @var{n} is a real number or HMS
33365 form in the range @samp{[0 ..@: @var{m})}.
33367 Error forms are stored as @samp{(sdev @var{x} @var{sigma})}, where @var{x}
33368 is the mean value and @var{sigma} is the standard deviation. Each
33369 component is either a number, an HMS form, or a symbolic object
33370 (a variable or function call). If @var{sigma} is zero, the value is
33371 converted to a plain real number. If @var{sigma} is negative or
33372 complex, it is automatically normalized to be a positive real.
33374 Interval forms are stored as @samp{(intv @var{mask} @var{lo} @var{hi})},
33375 where @var{mask} is one of the integers 0, 1, 2, or 3, and @var{lo} and
33376 @var{hi} are real numbers, HMS forms, or symbolic objects. The @var{mask}
33377 is a binary integer where 1 represents the fact that the interval is
33378 closed on the high end, and 2 represents the fact that it is closed on
33379 the low end. (Thus 3 represents a fully closed interval.) The interval
33380 @w{@samp{(intv 3 @var{x} @var{x})}} is converted to the plain number @var{x};
33381 intervals @samp{(intv @var{mask} @var{x} @var{x})} for any other @var{mask}
33382 represent empty intervals. If @var{hi} is less than @var{lo}, the interval
33383 is converted to a standard empty interval by replacing @var{hi} with @var{lo}.
33385 Vectors are stored as @samp{(vec @var{v1} @var{v2} @dots{})}, where @var{v1}
33386 is the first element of the vector, @var{v2} is the second, and so on.
33387 An empty vector is stored as @samp{(vec)}. A matrix is simply a vector
33388 where all @var{v}'s are themselves vectors of equal lengths. Note that
33389 Calc vectors are unrelated to the Emacs Lisp ``vector'' type, which is
33390 generally unused by Calc data structures.
33392 Variables are stored as @samp{(var @var{name} @var{sym})}, where
33393 @var{name} is a Lisp symbol whose print name is used as the visible name
33394 of the variable, and @var{sym} is a Lisp symbol in which the variable's
33395 value is actually stored. Thus, @samp{(var pi var-pi)} represents the
33396 special constant @samp{pi}. Almost always, the form is @samp{(var
33397 @var{v} var-@var{v})}. If the variable name was entered with @code{#}
33398 signs (which are converted to hyphens internally), the form is
33399 @samp{(var @var{u} @var{v})}, where @var{u} is a symbol whose name
33400 contains @code{#} characters, and @var{v} is a symbol that contains
33401 @code{-} characters instead. The value of a variable is the Calc
33402 object stored in its @var{sym} symbol's value cell. If the symbol's
33403 value cell is void or if it contains @code{nil}, the variable has no
33404 value. Special constants have the form @samp{(special-const
33405 @var{value})} stored in their value cell, where @var{value} is a formula
33406 which is evaluated when the constant's value is requested. Variables
33407 which represent units are not stored in any special way; they are units
33408 only because their names appear in the units table. If the value
33409 cell contains a string, it is parsed to get the variable's value when
33410 the variable is used.
33412 A Lisp list with any other symbol as the first element is a function call.
33413 The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^},
33414 and @code{|} represent special binary operators; these lists are always
33415 of the form @samp{(@var{op} @var{lhs} @var{rhs})} where @var{lhs} is the
33416 sub-formula on the lefthand side and @var{rhs} is the sub-formula on the
33417 right. The symbol @code{neg} represents unary negation; this list is always
33418 of the form @samp{(neg @var{arg})}. Any other symbol @var{func} represents a
33419 function that would be displayed in function-call notation; the symbol
33420 @var{func} is in general always of the form @samp{calcFunc-@var{name}}.
33421 The function cell of the symbol @var{func} should contain a Lisp function
33422 for evaluating a call to @var{func}. This function is passed the remaining
33423 elements of the list (themselves already evaluated) as arguments; such
33424 functions should return @code{nil} or call @code{reject-arg} to signify
33425 that they should be left in symbolic form, or they should return a Calc
33426 object which represents their value, or a list of such objects if they
33427 wish to return multiple values. (The latter case is allowed only for
33428 functions which are the outer-level call in an expression whose value is
33429 about to be pushed on the stack; this feature is considered obsolete
33430 and is not used by any built-in Calc functions.)
33432 @node Interactive Lisp Functions, Stack Lisp Functions, Data Type Formats, Internals
33433 @subsubsection Interactive Functions
33436 The functions described here are used in implementing interactive Calc
33437 commands. Note that this list is not exhaustive! If there is an
33438 existing command that behaves similarly to the one you want to define,
33439 you may find helpful tricks by checking the source code for that command.
33441 @defun calc-set-command-flag flag
33442 Set the command flag @var{flag}. This is generally a Lisp symbol, but
33443 may in fact be anything. The effect is to add @var{flag} to the list
33444 stored in the variable @code{calc-command-flags}, unless it is already
33445 there. @xref{Defining Simple Commands}.
33448 @defun calc-clear-command-flag flag
33449 If @var{flag} appears among the list of currently-set command flags,
33450 remove it from that list.
33453 @defun calc-record-undo rec
33454 Add the ``undo record'' @var{rec} to the list of steps to take if the
33455 current operation should need to be undone. Stack push and pop functions
33456 automatically call @code{calc-record-undo}, so the kinds of undo records
33457 you might need to create take the form @samp{(set @var{sym} @var{value})},
33458 which says that the Lisp variable @var{sym} was changed and had previously
33459 contained @var{value}; @samp{(store @var{var} @var{value})} which says that
33460 the Calc variable @var{var} (a string which is the name of the symbol that
33461 contains the variable's value) was stored and its previous value was
33462 @var{value} (either a Calc data object, or @code{nil} if the variable was
33463 previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})},
33464 which means that to undo requires calling the function @samp{(@var{undo}
33465 @var{args} @dots{})} and, if the undo is later redone, calling
33466 @samp{(@var{redo} @var{args} @dots{})}.
33469 @defun calc-record-why msg args
33470 Record the error or warning message @var{msg}, which is normally a string.
33471 This message will be replayed if the user types @kbd{w} (@code{calc-why});
33472 if the message string begins with a @samp{*}, it is considered important
33473 enough to display even if the user doesn't type @kbd{w}. If one or more
33474 @var{args} are present, the displayed message will be of the form,
33475 @samp{@var{msg}: @var{arg1}, @var{arg2}, @dots{}}, where the arguments are
33476 formatted on the assumption that they are either strings or Calc objects of
33477 some sort. If @var{msg} is a symbol, it is the name of a Calc predicate
33478 (such as @code{integerp} or @code{numvecp}) which the arguments did not
33479 satisfy; it is expanded to a suitable string such as ``Expected an
33480 integer.'' The @code{reject-arg} function calls @code{calc-record-why}
33481 automatically; @pxref{Predicates}.
33484 @defun calc-is-inverse
33485 This predicate returns true if the current command is inverse,
33486 i.e., if the Inverse (@kbd{I} key) flag was set.
33489 @defun calc-is-hyperbolic
33490 This predicate is the analogous function for the @kbd{H} key.
33493 @node Stack Lisp Functions, Predicates, Interactive Lisp Functions, Internals
33494 @subsubsection Stack-Oriented Functions
33497 The functions described here perform various operations on the Calc
33498 stack and trail. They are to be used in interactive Calc commands.
33500 @defun calc-push-list vals n
33501 Push the Calc objects in list @var{vals} onto the stack at stack level
33502 @var{n}. If @var{n} is omitted it defaults to 1, so that the elements
33503 are pushed at the top of the stack. If @var{n} is greater than 1, the
33504 elements will be inserted into the stack so that the last element will
33505 end up at level @var{n}, the next-to-last at level @var{n}+1, etc.
33506 The elements of @var{vals} are assumed to be valid Calc objects, and
33507 are not evaluated, rounded, or renormalized in any way. If @var{vals}
33508 is an empty list, nothing happens.
33510 The stack elements are pushed without any sub-formula selections.
33511 You can give an optional third argument to this function, which must
33512 be a list the same size as @var{vals} of selections. Each selection
33513 must be @code{eq} to some sub-formula of the corresponding formula
33514 in @var{vals}, or @code{nil} if that formula should have no selection.
33517 @defun calc-top-list n m
33518 Return a list of the @var{n} objects starting at level @var{m} of the
33519 stack. If @var{m} is omitted it defaults to 1, so that the elements are
33520 taken from the top of the stack. If @var{n} is omitted, it also
33521 defaults to 1, so that the top stack element (in the form of a
33522 one-element list) is returned. If @var{m} is greater than 1, the
33523 @var{m}th stack element will be at the end of the list, the @var{m}+1st
33524 element will be next-to-last, etc. If @var{n} or @var{m} are out of
33525 range, the command is aborted with a suitable error message. If @var{n}
33526 is zero, the function returns an empty list. The stack elements are not
33527 evaluated, rounded, or renormalized.
33529 If any stack elements contain selections, and selections have not
33530 been disabled by the @kbd{j e} (@code{calc-enable-selections}) command,
33531 this function returns the selected portions rather than the entire
33532 stack elements. It can be given a third ``selection-mode'' argument
33533 which selects other behaviors. If it is the symbol @code{t}, then
33534 a selection in any of the requested stack elements produces an
33535 ``invalid operation on selections'' error. If it is the symbol @code{full},
33536 the whole stack entry is always returned regardless of selections.
33537 If it is the symbol @code{sel}, the selected portion is always returned,
33538 or @code{nil} if there is no selection. (This mode ignores the @kbd{j e}
33539 command.) If the symbol is @code{entry}, the complete stack entry in
33540 list form is returned; the first element of this list will be the whole
33541 formula, and the third element will be the selection (or @code{nil}).
33544 @defun calc-pop-stack n m
33545 Remove the specified elements from the stack. The parameters @var{n}
33546 and @var{m} are defined the same as for @code{calc-top-list}. The return
33547 value of @code{calc-pop-stack} is uninteresting.
33549 If there are any selected sub-formulas among the popped elements, and
33550 @kbd{j e} has not been used to disable selections, this produces an
33551 error without changing the stack. If you supply an optional third
33552 argument of @code{t}, the stack elements are popped even if they
33553 contain selections.
33556 @defun calc-record-list vals tag
33557 This function records one or more results in the trail. The @var{vals}
33558 are a list of strings or Calc objects. The @var{tag} is the four-character
33559 tag string to identify the values. If @var{tag} is omitted, a blank tag
33563 @defun calc-normalize n
33564 This function takes a Calc object and ``normalizes'' it. At the very
33565 least this involves re-rounding floating-point values according to the
33566 current precision and other similar jobs. Also, unless the user has
33567 selected No-Simplify mode (@pxref{Simplification Modes}), this involves
33568 actually evaluating a formula object by executing the function calls
33569 it contains, and possibly also doing algebraic simplification, etc.
33572 @defun calc-top-list-n n m
33573 This function is identical to @code{calc-top-list}, except that it calls
33574 @code{calc-normalize} on the values that it takes from the stack. They
33575 are also passed through @code{check-complete}, so that incomplete
33576 objects will be rejected with an error message. All computational
33577 commands should use this in preference to @code{calc-top-list}; the only
33578 standard Calc commands that operate on the stack without normalizing
33579 are stack management commands like @code{calc-enter} and @code{calc-roll-up}.
33580 This function accepts the same optional selection-mode argument as
33581 @code{calc-top-list}.
33584 @defun calc-top-n m
33585 This function is a convenient form of @code{calc-top-list-n} in which only
33586 a single element of the stack is taken and returned, rather than a list
33587 of elements. This also accepts an optional selection-mode argument.
33590 @defun calc-enter-result n tag vals
33591 This function is a convenient interface to most of the above functions.
33592 The @var{vals} argument should be either a single Calc object, or a list
33593 of Calc objects; the object or objects are normalized, and the top @var{n}
33594 stack entries are replaced by the normalized objects. If @var{tag} is
33595 non-@code{nil}, the normalized objects are also recorded in the trail.
33596 A typical stack-based computational command would take the form,
33599 (calc-enter-result @var{n} @var{tag} (cons 'calcFunc-@var{func}
33600 (calc-top-list-n @var{n})))
33603 If any of the @var{n} stack elements replaced contain sub-formula
33604 selections, and selections have not been disabled by @kbd{j e},
33605 this function takes one of two courses of action. If @var{n} is
33606 equal to the number of elements in @var{vals}, then each element of
33607 @var{vals} is spliced into the corresponding selection; this is what
33608 happens when you use the @key{TAB} key, or when you use a unary
33609 arithmetic operation like @code{sqrt}. If @var{vals} has only one
33610 element but @var{n} is greater than one, there must be only one
33611 selection among the top @var{n} stack elements; the element from
33612 @var{vals} is spliced into that selection. This is what happens when
33613 you use a binary arithmetic operation like @kbd{+}. Any other
33614 combination of @var{n} and @var{vals} is an error when selections
33618 @defun calc-unary-op tag func arg
33619 This function implements a unary operator that allows a numeric prefix
33620 argument to apply the operator over many stack entries. If the prefix
33621 argument @var{arg} is @code{nil}, this uses @code{calc-enter-result}
33622 as outlined above. Otherwise, it maps the function over several stack
33623 elements; @pxref{Prefix Arguments}. For example,
33626 (defun calc-zeta (arg)
33628 (calc-unary-op "zeta" 'calcFunc-zeta arg))
33632 @defun calc-binary-op tag func arg ident unary
33633 This function implements a binary operator, analogously to
33634 @code{calc-unary-op}. The optional @var{ident} and @var{unary}
33635 arguments specify the behavior when the prefix argument is zero or
33636 one, respectively. If the prefix is zero, the value @var{ident}
33637 is pushed onto the stack, if specified, otherwise an error message
33638 is displayed. If the prefix is one, the unary function @var{unary}
33639 is applied to the top stack element, or, if @var{unary} is not
33640 specified, nothing happens. When the argument is two or more,
33641 the binary function @var{func} is reduced across the top @var{arg}
33642 stack elements; when the argument is negative, the function is
33643 mapped between the next-to-top @mathit{-@var{arg}} stack elements and the
33647 @defun calc-stack-size
33648 Return the number of elements on the stack as an integer. This count
33649 does not include elements that have been temporarily hidden by stack
33650 truncation; @pxref{Truncating the Stack}.
33653 @defun calc-cursor-stack-index n
33654 Move the point to the @var{n}th stack entry. If @var{n} is zero, this
33655 will be the @samp{.} line. If @var{n} is from 1 to the current stack size,
33656 this will be the beginning of the first line of that stack entry's display.
33657 If line numbers are enabled, this will move to the first character of the
33658 line number, not the stack entry itself.
33661 @defun calc-substack-height n
33662 Return the number of lines between the beginning of the @var{n}th stack
33663 entry and the bottom of the buffer. If @var{n} is zero, this
33664 will be one (assuming no stack truncation). If all stack entries are
33665 one line long (i.e., no matrices are displayed), the return value will
33666 be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big
33667 mode, the return value includes the blank lines that separate stack
33671 @defun calc-refresh
33672 Erase the @code{*Calculator*} buffer and reformat its contents from memory.
33673 This must be called after changing any parameter, such as the current
33674 display radix, which might change the appearance of existing stack
33675 entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing
33676 is suppressed, but a flag is set so that the entire stack will be refreshed
33677 rather than just the top few elements when the macro finishes.)
33680 @node Predicates, Computational Lisp Functions, Stack Lisp Functions, Internals
33681 @subsubsection Predicates
33684 The functions described here are predicates, that is, they return a
33685 true/false value where @code{nil} means false and anything else means
33686 true. These predicates are expanded by @code{defmath}, for example,
33687 from @code{zerop} to @code{math-zerop}. In many cases they correspond
33688 to native Lisp functions by the same name, but are extended to cover
33689 the full range of Calc data types.
33692 Returns true if @var{x} is numerically zero, in any of the Calc data
33693 types. (Note that for some types, such as error forms and intervals,
33694 it never makes sense to return true.) In @code{defmath}, the expression
33695 @samp{(= x 0)} will automatically be converted to @samp{(math-zerop x)},
33696 and @samp{(/= x 0)} will be converted to @samp{(not (math-zerop x))}.
33700 Returns true if @var{x} is negative. This accepts negative real numbers
33701 of various types, negative HMS and date forms, and intervals in which
33702 all included values are negative. In @code{defmath}, the expression
33703 @samp{(< x 0)} will automatically be converted to @samp{(math-negp x)},
33704 and @samp{(>= x 0)} will be converted to @samp{(not (math-negp x))}.
33708 Returns true if @var{x} is positive (and non-zero). For complex
33709 numbers, none of these three predicates will return true.
33712 @defun looks-negp x
33713 Returns true if @var{x} is ``negative-looking.'' This returns true if
33714 @var{x} is a negative number, or a formula with a leading minus sign
33715 such as @samp{-a/b}. In other words, this is an object which can be
33716 made simpler by calling @code{(- @var{x})}.
33720 Returns true if @var{x} is an integer of any size.
33724 Returns true if @var{x} is a native Lisp integer.
33728 Returns true if @var{x} is a nonnegative integer of any size.
33731 @defun fixnatnump x
33732 Returns true if @var{x} is a nonnegative Lisp integer.
33735 @defun num-integerp x
33736 Returns true if @var{x} is numerically an integer, i.e., either a
33737 true integer or a float with no significant digits to the right of
33741 @defun messy-integerp x
33742 Returns true if @var{x} is numerically, but not literally, an integer.
33743 A value is @code{num-integerp} if it is @code{integerp} or
33744 @code{messy-integerp} (but it is never both at once).
33747 @defun num-natnump x
33748 Returns true if @var{x} is numerically a nonnegative integer.
33752 Returns true if @var{x} is an even integer.
33755 @defun looks-evenp x
33756 Returns true if @var{x} is an even integer, or a formula with a leading
33757 multiplicative coefficient which is an even integer.
33761 Returns true if @var{x} is an odd integer.
33765 Returns true if @var{x} is a rational number, i.e., an integer or a
33770 Returns true if @var{x} is a real number, i.e., an integer, fraction,
33771 or floating-point number.
33775 Returns true if @var{x} is a real number or HMS form.
33779 Returns true if @var{x} is a float, or a complex number, error form,
33780 interval, date form, or modulo form in which at least one component
33785 Returns true if @var{x} is a rectangular or polar complex number
33786 (but not a real number).
33789 @defun rect-complexp x
33790 Returns true if @var{x} is a rectangular complex number.
33793 @defun polar-complexp x
33794 Returns true if @var{x} is a polar complex number.
33798 Returns true if @var{x} is a real number or a complex number.
33802 Returns true if @var{x} is a real or complex number or an HMS form.
33806 Returns true if @var{x} is a vector (this simply checks if its argument
33807 is a list whose first element is the symbol @code{vec}).
33811 Returns true if @var{x} is a number or vector.
33815 Returns true if @var{x} is a matrix, i.e., a vector of one or more vectors,
33816 all of the same size.
33819 @defun square-matrixp x
33820 Returns true if @var{x} is a square matrix.
33824 Returns true if @var{x} is any numeric Calc object, including real and
33825 complex numbers, HMS forms, date forms, error forms, intervals, and
33826 modulo forms. (Note that error forms and intervals may include formulas
33827 as their components; see @code{constp} below.)
33831 Returns true if @var{x} is an object or a vector. This also accepts
33832 incomplete objects, but it rejects variables and formulas (except as
33833 mentioned above for @code{objectp}).
33837 Returns true if @var{x} is a ``primitive'' or ``atomic'' Calc object,
33838 i.e., one whose components cannot be regarded as sub-formulas. This
33839 includes variables, and all @code{objectp} types except error forms
33844 Returns true if @var{x} is constant, i.e., a real or complex number,
33845 HMS form, date form, or error form, interval, or vector all of whose
33846 components are @code{constp}.
33850 Returns true if @var{x} is numerically less than @var{y}. Returns false
33851 if @var{x} is greater than or equal to @var{y}, or if the order is
33852 undefined or cannot be determined. Generally speaking, this works
33853 by checking whether @samp{@var{x} - @var{y}} is @code{negp}. In
33854 @code{defmath}, the expression @samp{(< x y)} will automatically be
33855 converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=},
33856 and @code{>=} are similarly converted in terms of @code{lessp}.
33860 Returns true if @var{x} comes before @var{y} in a canonical ordering
33861 of Calc objects. If @var{x} and @var{y} are both real numbers, this
33862 will be the same as @code{lessp}. But whereas @code{lessp} considers
33863 other types of objects to be unordered, @code{beforep} puts any two
33864 objects into a definite, consistent order. The @code{beforep}
33865 function is used by the @kbd{V S} vector-sorting command, and also
33866 by Calc's algebraic simplifications to put the terms of a product into
33867 canonical order: This allows @samp{x y + y x} to be simplified easily to
33872 This is the standard Lisp @code{equal} predicate; it returns true if
33873 @var{x} and @var{y} are structurally identical. This is the usual way
33874 to compare numbers for equality, but note that @code{equal} will treat
33875 0 and 0.0 as different.
33878 @defun math-equal x y
33879 Returns true if @var{x} and @var{y} are numerically equal, either because
33880 they are @code{equal}, or because their difference is @code{zerop}. In
33881 @code{defmath}, the expression @samp{(= x y)} will automatically be
33882 converted to @samp{(math-equal x y)}.
33885 @defun equal-int x n
33886 Returns true if @var{x} and @var{n} are numerically equal, where @var{n}
33887 is a fixnum which is not a multiple of 10. This will automatically be
33888 used by @code{defmath} in place of the more general @code{math-equal}
33892 @defun nearly-equal x y
33893 Returns true if @var{x} and @var{y}, as floating-point numbers, are
33894 equal except possibly in the last decimal place. For example,
33895 314.159 and 314.166 are considered nearly equal if the current
33896 precision is 6 (since they differ by 7 units), but not if the current
33897 precision is 7 (since they differ by 70 units). Most functions which
33898 use series expansions use @code{with-extra-prec} to evaluate the
33899 series with 2 extra digits of precision, then use @code{nearly-equal}
33900 to decide when the series has converged; this guards against cumulative
33901 error in the series evaluation without doing extra work which would be
33902 lost when the result is rounded back down to the current precision.
33903 In @code{defmath}, this can be written @samp{(~= @var{x} @var{y})}.
33904 The @var{x} and @var{y} can be numbers of any kind, including complex.
33907 @defun nearly-zerop x y
33908 Returns true if @var{x} is nearly zero, compared to @var{y}. This
33909 checks whether @var{x} plus @var{y} would by be @code{nearly-equal}
33910 to @var{y} itself, to within the current precision, in other words,
33911 if adding @var{x} to @var{y} would have a negligible effect on @var{y}
33912 due to roundoff error. @var{X} may be a real or complex number, but
33913 @var{y} must be real.
33917 Return true if the formula @var{x} represents a true value in
33918 Calc, not Lisp, terms. It tests if @var{x} is a non-zero number
33919 or a provably non-zero formula.
33922 @defun reject-arg val pred
33923 Abort the current function evaluation due to unacceptable argument values.
33924 This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a
33925 Lisp error which @code{normalize} will trap. The net effect is that the
33926 function call which led here will be left in symbolic form.
33929 @defun inexact-value
33930 If Symbolic mode is enabled, this will signal an error that causes
33931 @code{normalize} to leave the formula in symbolic form, with the message
33932 ``Inexact result.'' (This function has no effect when not in Symbolic mode.)
33933 Note that if your function calls @samp{(sin 5)} in Symbolic mode, the
33934 @code{sin} function will call @code{inexact-value}, which will cause your
33935 function to be left unsimplified. You may instead wish to call
33936 @samp{(normalize (list 'calcFunc-sin 5))}, which in Symbolic mode will
33937 return the formula @samp{sin(5)} to your function.
33941 This signals an error that will be reported as a floating-point overflow.
33945 This signals a floating-point underflow.
33948 @node Computational Lisp Functions, Vector Lisp Functions, Predicates, Internals
33949 @subsubsection Computational Functions
33952 The functions described here do the actual computational work of the
33953 Calculator. In addition to these, note that any function described in
33954 the main body of this manual may be called from Lisp; for example, if
33955 the documentation refers to the @code{calc-sqrt} [@code{sqrt}] command,
33956 this means @code{calc-sqrt} is an interactive stack-based square-root
33957 command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt})
33958 is the actual Lisp function for taking square roots.
33960 The functions @code{math-add}, @code{math-sub}, @code{math-mul},
33961 @code{math-div}, @code{math-mod}, and @code{math-neg} are not included
33962 in this list, since @code{defmath} allows you to write native Lisp
33963 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-},
33964 respectively, instead.
33966 @defun normalize val
33967 (Full form: @code{math-normalize}.)
33968 Reduce the value @var{val} to standard form. For example, if @var{val}
33969 is a fixnum, it will be converted to a bignum if it is too large, and
33970 if @var{val} is a bignum it will be normalized by clipping off trailing
33971 (i.e., most-significant) zero digits and converting to a fixnum if it is
33972 small. All the various data types are similarly converted to their standard
33973 forms. Variables are left alone, but function calls are actually evaluated
33974 in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will
33977 If a function call fails, because the function is void or has the wrong
33978 number of parameters, or because it returns @code{nil} or calls
33979 @code{reject-arg} or @code{inexact-result}, @code{normalize} returns
33980 the formula still in symbolic form.
33982 If the current simplification mode is ``none'' or ``numeric arguments
33983 only,'' @code{normalize} will act appropriately. However, the more
33984 powerful simplification modes (like Algebraic Simplification) are
33985 not handled by @code{normalize}. They are handled by @code{calc-normalize},
33986 which calls @code{normalize} and possibly some other routines, such
33987 as @code{simplify} or @code{simplify-units}. Programs generally will
33988 never call @code{calc-normalize} except when popping or pushing values
33992 @defun evaluate-expr expr
33993 Replace all variables in @var{expr} that have values with their values,
33994 then use @code{normalize} to simplify the result. This is what happens
33995 when you press the @kbd{=} key interactively.
33998 @defmac with-extra-prec n body
33999 Evaluate the Lisp forms in @var{body} with precision increased by @var{n}
34000 digits. This is a macro which expands to
34004 (let ((calc-internal-prec (+ calc-internal-prec @var{n})))
34008 The surrounding call to @code{math-normalize} causes a floating-point
34009 result to be rounded down to the original precision afterwards. This
34010 is important because some arithmetic operations assume a number's
34011 mantissa contains no more digits than the current precision allows.
34014 @defun make-frac n d
34015 Build a fraction @samp{@var{n}:@var{d}}. This is equivalent to calling
34016 @samp{(normalize (list 'frac @var{n} @var{d}))}, but more efficient.
34019 @defun make-float mant exp
34020 Build a floating-point value out of @var{mant} and @var{exp}, both
34021 of which are arbitrary integers. This function will return a
34022 properly normalized float value, or signal an overflow or underflow
34023 if @var{exp} is out of range.
34026 @defun make-sdev x sigma
34027 Build an error form out of @var{x} and the absolute value of @var{sigma}.
34028 If @var{sigma} is zero, the result is the number @var{x} directly.
34029 If @var{sigma} is negative or complex, its absolute value is used.
34030 If @var{x} or @var{sigma} is not a valid type of object for use in
34031 error forms, this calls @code{reject-arg}.
34034 @defun make-intv mask lo hi
34035 Build an interval form out of @var{mask} (which is assumed to be an
34036 integer from 0 to 3), and the limits @var{lo} and @var{hi}. If
34037 @var{lo} is greater than @var{hi}, an empty interval form is returned.
34038 This calls @code{reject-arg} if @var{lo} or @var{hi} is unsuitable.
34041 @defun sort-intv mask lo hi
34042 Build an interval form, similar to @code{make-intv}, except that if
34043 @var{lo} is less than @var{hi} they are simply exchanged, and the
34044 bits of @var{mask} are swapped accordingly.
34047 @defun make-mod n m
34048 Build a modulo form out of @var{n} and the modulus @var{m}. Since modulo
34049 forms do not allow formulas as their components, if @var{n} or @var{m}
34050 is not a real number or HMS form the result will be a formula which
34051 is a call to @code{makemod}, the algebraic version of this function.
34055 Convert @var{x} to floating-point form. Integers and fractions are
34056 converted to numerically equivalent floats; components of complex
34057 numbers, vectors, HMS forms, date forms, error forms, intervals, and
34058 modulo forms are recursively floated. If the argument is a variable
34059 or formula, this calls @code{reject-arg}.
34063 Compare the numbers @var{x} and @var{y}, and return @mathit{-1} if
34064 @samp{(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})},
34065 0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is
34066 undefined or cannot be determined.
34070 Return the number of digits of integer @var{n}, effectively
34071 @samp{ceil(log10(@var{n}))}, but much more efficient. Zero is
34072 considered to have zero digits.
34075 @defun scale-int x n
34076 Shift integer @var{x} left @var{n} decimal digits, or right @mathit{-@var{n}}
34077 digits with truncation toward zero.
34080 @defun scale-rounding x n
34081 Like @code{scale-int}, except that a right shift rounds to the nearest
34082 integer rather than truncating.
34086 Return the integer @var{n} as a fixnum, i.e., a native Lisp integer.
34087 If @var{n} is outside the permissible range for Lisp integers (usually
34088 24 binary bits) the result is undefined.
34092 Compute the square of @var{x}; short for @samp{(* @var{x} @var{x})}.
34095 @defun quotient x y
34096 Divide integer @var{x} by integer @var{y}; return an integer quotient
34097 and discard the remainder. If @var{x} or @var{y} is negative, the
34098 direction of rounding is undefined.
34102 Perform an integer division; if @var{x} and @var{y} are both nonnegative
34103 integers, this uses the @code{quotient} function, otherwise it computes
34104 @samp{floor(@var{x}/@var{y})}. Thus the result is well-defined but
34105 slower than for @code{quotient}.
34109 Divide integer @var{x} by integer @var{y}; return the integer remainder
34110 and discard the quotient. Like @code{quotient}, this works only for
34111 integer arguments and is not well-defined for negative arguments.
34112 For a more well-defined result, use @samp{(% @var{x} @var{y})}.
34116 Divide integer @var{x} by integer @var{y}; return a cons cell whose
34117 @code{car} is @samp{(quotient @var{x} @var{y})} and whose @code{cdr}
34118 is @samp{(imod @var{x} @var{y})}.
34122 Compute @var{x} to the power @var{y}. In @code{defmath} code, this can
34123 also be written @samp{(^ @var{x} @var{y})} or
34124 @w{@samp{(expt @var{x} @var{y})}}.
34127 @defun abs-approx x
34128 Compute a fast approximation to the absolute value of @var{x}. For
34129 example, for a rectangular complex number the result is the sum of
34130 the absolute values of the components.
34134 @findex gamma-const
34140 @findex pi-over-180
34141 @findex sqrt-two-pi
34145 The function @samp{(pi)} computes @samp{pi} to the current precision.
34146 Other related constant-generating functions are @code{two-pi},
34147 @code{pi-over-2}, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi},
34148 @code{e}, @code{sqrt-e}, @code{ln-2}, @code{ln-10}, @code{phi} and
34149 @code{gamma-const}. Each function returns a floating-point value in the
34150 current precision, and each uses caching so that all calls after the
34151 first are essentially free.
34154 @defmac math-defcache @var{func} @var{initial} @var{form}
34155 This macro, usually used as a top-level call like @code{defun} or
34156 @code{defvar}, defines a new cached constant analogous to @code{pi}, etc.
34157 It defines a function @code{func} which returns the requested value;
34158 if @var{initial} is non-@code{nil} it must be a @samp{(float @dots{})}
34159 form which serves as an initial value for the cache. If @var{func}
34160 is called when the cache is empty or does not have enough digits to
34161 satisfy the current precision, the Lisp expression @var{form} is evaluated
34162 with the current precision increased by four, and the result minus its
34163 two least significant digits is stored in the cache. For example,
34164 calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34
34165 digits, rounds it down to 32 digits for future use, then rounds it
34166 again to 30 digits for use in the present request.
34169 @findex half-circle
34170 @findex quarter-circle
34171 @defun full-circle symb
34172 If the current angular mode is Degrees or HMS, this function returns the
34173 integer 360. In Radians mode, this function returns either the
34174 corresponding value in radians to the current precision, or the formula
34175 @samp{2*pi}, depending on the Symbolic mode. There are also similar
34176 function @code{half-circle} and @code{quarter-circle}.
34179 @defun power-of-2 n
34180 Compute two to the integer power @var{n}, as a (potentially very large)
34181 integer. Powers of two are cached, so only the first call for a
34182 particular @var{n} is expensive.
34185 @defun integer-log2 n
34186 Compute the base-2 logarithm of @var{n}, which must be an integer which
34187 is a power of two. If @var{n} is not a power of two, this function will
34191 @defun div-mod a b m
34192 Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if
34193 there is no solution, or if any of the arguments are not integers.
34196 @defun pow-mod a b m
34197 Compute @var{a} to the power @var{b}, modulo @var{m}. If @var{a},
34198 @var{b}, and @var{m} are integers, this uses an especially efficient
34199 algorithm. Otherwise, it simply computes @samp{(% (^ a b) m)}.
34203 Compute the integer square root of @var{n}. This is the square root
34204 of @var{n} rounded down toward zero, i.e., @samp{floor(sqrt(@var{n}))}.
34205 If @var{n} is itself an integer, the computation is especially efficient.
34208 @defun to-hms a ang
34209 Convert the argument @var{a} into an HMS form. If @var{ang} is specified,
34210 it is the angular mode in which to interpret @var{a}, either @code{deg}
34211 or @code{rad}. Otherwise, the current angular mode is used. If @var{a}
34212 is already an HMS form it is returned as-is.
34215 @defun from-hms a ang
34216 Convert the HMS form @var{a} into a real number. If @var{ang} is specified,
34217 it is the angular mode in which to express the result, otherwise the
34218 current angular mode is used. If @var{a} is already a real number, it
34222 @defun to-radians a
34223 Convert the number or HMS form @var{a} to radians from the current
34227 @defun from-radians a
34228 Convert the number @var{a} from radians to the current angular mode.
34229 If @var{a} is a formula, this returns the formula @samp{deg(@var{a})}.
34232 @defun to-radians-2 a
34233 Like @code{to-radians}, except that in Symbolic mode a degrees to
34234 radians conversion yields a formula like @samp{@var{a}*pi/180}.
34237 @defun from-radians-2 a
34238 Like @code{from-radians}, except that in Symbolic mode a radians to
34239 degrees conversion yields a formula like @samp{@var{a}*180/pi}.
34242 @defun random-digit
34243 Produce a random base-1000 digit in the range 0 to 999.
34246 @defun random-digits n
34247 Produce a random @var{n}-digit integer; this will be an integer
34248 in the interval @samp{[0, 10^@var{n})}.
34251 @defun random-float
34252 Produce a random float in the interval @samp{[0, 1)}.
34255 @defun prime-test n iters
34256 Determine whether the integer @var{n} is prime. Return a list which has
34257 one of these forms: @samp{(nil @var{f})} means the number is non-prime
34258 because it was found to be divisible by @var{f}; @samp{(nil)} means it
34259 was found to be non-prime by table look-up (so no factors are known);
34260 @samp{(nil unknown)} means it is definitely non-prime but no factors
34261 are known because @var{n} was large enough that Fermat's probabilistic
34262 test had to be used; @samp{(t)} means the number is definitely prime;
34263 and @samp{(maybe @var{i} @var{p})} means that Fermat's test, after @var{i}
34264 iterations, is @var{p} percent sure that the number is prime. The
34265 @var{iters} parameter is the number of Fermat iterations to use, in the
34266 case that this is necessary. If @code{prime-test} returns ``maybe,''
34267 you can call it again with the same @var{n} to get a greater certainty;
34268 @code{prime-test} remembers where it left off.
34271 @defun to-simple-fraction f
34272 If @var{f} is a floating-point number which can be represented exactly
34273 as a small rational number. return that number, else return @var{f}.
34274 For example, 0.75 would be converted to 3:4. This function is very
34278 @defun to-fraction f tol
34279 Find a rational approximation to floating-point number @var{f} to within
34280 a specified tolerance @var{tol}; this corresponds to the algebraic
34281 function @code{frac}, and can be rather slow.
34284 @defun quarter-integer n
34285 If @var{n} is an integer or integer-valued float, this function
34286 returns zero. If @var{n} is a half-integer (i.e., an integer plus
34287 @mathit{1:2} or 0.5), it returns 2. If @var{n} is a quarter-integer,
34288 it returns 1 or 3. If @var{n} is anything else, this function
34289 returns @code{nil}.
34292 @node Vector Lisp Functions, Symbolic Lisp Functions, Computational Lisp Functions, Internals
34293 @subsubsection Vector Functions
34296 The functions described here perform various operations on vectors and
34299 @defun math-concat x y
34300 Do a vector concatenation; this operation is written @samp{@var{x} | @var{y}}
34301 in a symbolic formula. @xref{Building Vectors}.
34304 @defun vec-length v
34305 Return the length of vector @var{v}. If @var{v} is not a vector, the
34306 result is zero. If @var{v} is a matrix, this returns the number of
34307 rows in the matrix.
34310 @defun mat-dimens m
34311 Determine the dimensions of vector or matrix @var{m}. If @var{m} is not
34312 a vector, the result is an empty list. If @var{m} is a plain vector
34313 but not a matrix, the result is a one-element list containing the length
34314 of the vector. If @var{m} is a matrix with @var{r} rows and @var{c} columns,
34315 the result is the list @samp{(@var{r} @var{c})}. Higher-order tensors
34316 produce lists of more than two dimensions. Note that the object
34317 @samp{[[1, 2, 3], [4, 5]]} is a vector of vectors not all the same size,
34318 and is treated by this and other Calc routines as a plain vector of two
34322 @defun dimension-error
34323 Abort the current function with a message of ``Dimension error.''
34324 The Calculator will leave the function being evaluated in symbolic
34325 form; this is really just a special case of @code{reject-arg}.
34328 @defun build-vector args
34329 Return a Calc vector with @var{args} as elements.
34330 For example, @samp{(build-vector 1 2 3)} returns the Calc vector
34331 @samp{[1, 2, 3]}, stored internally as the list @samp{(vec 1 2 3)}.
34334 @defun make-vec obj dims
34335 Return a Calc vector or matrix all of whose elements are equal to
34336 @var{obj}. For example, @samp{(make-vec 27 3 4)} returns a 3x4 matrix
34340 @defun row-matrix v
34341 If @var{v} is a plain vector, convert it into a row matrix, i.e.,
34342 a matrix whose single row is @var{v}. If @var{v} is already a matrix,
34346 @defun col-matrix v
34347 If @var{v} is a plain vector, convert it into a column matrix, i.e., a
34348 matrix with each element of @var{v} as a separate row. If @var{v} is
34349 already a matrix, leave it alone.
34353 Map the Lisp function @var{f} over the Calc vector @var{v}. For example,
34354 @samp{(map-vec 'math-floor v)} returns a vector of the floored components
34358 @defun map-vec-2 f a b
34359 Map the Lisp function @var{f} over the two vectors @var{a} and @var{b}.
34360 If @var{a} and @var{b} are vectors of equal length, the result is a
34361 vector of the results of calling @samp{(@var{f} @var{ai} @var{bi})}
34362 for each pair of elements @var{ai} and @var{bi}. If either @var{a} or
34363 @var{b} is a scalar, it is matched with each value of the other vector.
34364 For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v}
34365 with each element increased by one. Note that using @samp{'+} would not
34366 work here, since @code{defmath} does not expand function names everywhere,
34367 just where they are in the function position of a Lisp expression.
34370 @defun reduce-vec f v
34371 Reduce the function @var{f} over the vector @var{v}. For example, if
34372 @var{v} is @samp{[10, 20, 30, 40]}, this calls @samp{(f (f (f 10 20) 30) 40)}.
34373 If @var{v} is a matrix, this reduces over the rows of @var{v}.
34376 @defun reduce-cols f m
34377 Reduce the function @var{f} over the columns of matrix @var{m}. For
34378 example, if @var{m} is @samp{[[1, 2], [3, 4], [5, 6]]}, the result
34379 is a vector of the two elements @samp{(f (f 1 3) 5)} and @samp{(f (f 2 4) 6)}.
34383 Return the @var{n}th row of matrix @var{m}. This is equivalent to
34384 @samp{(elt m n)}. For a slower but safer version, use @code{mrow}.
34385 (@xref{Extracting Elements}.)
34389 Return the @var{n}th column of matrix @var{m}, in the form of a vector.
34390 The arguments are not checked for correctness.
34393 @defun mat-less-row m n
34394 Return a copy of matrix @var{m} with its @var{n}th row deleted. The
34395 number @var{n} must be in range from 1 to the number of rows in @var{m}.
34398 @defun mat-less-col m n
34399 Return a copy of matrix @var{m} with its @var{n}th column deleted.
34403 Return the transpose of matrix @var{m}.
34406 @defun flatten-vector v
34407 Flatten nested vector @var{v} into a vector of scalars. For example,
34408 if @var{v} is @samp{[[1, 2, 3], [4, 5]]} the result is @samp{[1, 2, 3, 4, 5]}.
34411 @defun copy-matrix m
34412 If @var{m} is a matrix, return a copy of @var{m}. This maps
34413 @code{copy-sequence} over the rows of @var{m}; in Lisp terms, each
34414 element of the result matrix will be @code{eq} to the corresponding
34415 element of @var{m}, but none of the @code{cons} cells that make up
34416 the structure of the matrix will be @code{eq}. If @var{m} is a plain
34417 vector, this is the same as @code{copy-sequence}.
34420 @defun swap-rows m r1 r2
34421 Exchange rows @var{r1} and @var{r2} of matrix @var{m} in-place. In
34422 other words, unlike most of the other functions described here, this
34423 function changes @var{m} itself rather than building up a new result
34424 matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)}
34425 is true, with the side effect of exchanging the first two rows of
34429 @node Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp Functions, Internals
34430 @subsubsection Symbolic Functions
34433 The functions described here operate on symbolic formulas in the
34436 @defun calc-prepare-selection num
34437 Prepare a stack entry for selection operations. If @var{num} is
34438 omitted, the stack entry containing the cursor is used; otherwise,
34439 it is the number of the stack entry to use. This function stores
34440 useful information about the current stack entry into a set of
34441 variables. @code{calc-selection-cache-num} contains the number of
34442 the stack entry involved (equal to @var{num} if you specified it);
34443 @code{calc-selection-cache-entry} contains the stack entry as a
34444 list (such as @code{calc-top-list} would return with @code{entry}
34445 as the selection mode); and @code{calc-selection-cache-comp} contains
34446 a special ``tagged'' composition (@pxref{Formatting Lisp Functions})
34447 which allows Calc to relate cursor positions in the buffer with
34448 their corresponding sub-formulas.
34450 A slight complication arises in the selection mechanism because
34451 formulas may contain small integers. For example, in the vector
34452 @samp{[1, 2, 1]} the first and last elements are @code{eq} to each
34453 other; selections are recorded as the actual Lisp object that
34454 appears somewhere in the tree of the whole formula, but storing
34455 @code{1} would falsely select both @code{1}'s in the vector. So
34456 @code{calc-prepare-selection} also checks the stack entry and
34457 replaces any plain integers with ``complex number'' lists of the form
34458 @samp{(cplx @var{n} 0)}. This list will be displayed the same as a
34459 plain @var{n} and the change will be completely invisible to the
34460 user, but it will guarantee that no two sub-formulas of the stack
34461 entry will be @code{eq} to each other. Next time the stack entry
34462 is involved in a computation, @code{calc-normalize} will replace
34463 these lists with plain numbers again, again invisibly to the user.
34466 @defun calc-encase-atoms x
34467 This modifies the formula @var{x} to ensure that each part of the
34468 formula is a unique atom, using the @samp{(cplx @var{n} 0)} trick
34469 described above. This function may use @code{setcar} to modify
34470 the formula in-place.
34473 @defun calc-find-selected-part
34474 Find the smallest sub-formula of the current formula that contains
34475 the cursor. This assumes @code{calc-prepare-selection} has been
34476 called already. If the cursor is not actually on any part of the
34477 formula, this returns @code{nil}.
34480 @defun calc-change-current-selection selection
34481 Change the currently prepared stack element's selection to
34482 @var{selection}, which should be @code{eq} to some sub-formula
34483 of the stack element, or @code{nil} to unselect the formula.
34484 The stack element's appearance in the Calc buffer is adjusted
34485 to reflect the new selection.
34488 @defun calc-find-nth-part expr n
34489 Return the @var{n}th sub-formula of @var{expr}. This function is used
34490 by the selection commands, and (unless @kbd{j b} has been used) treats
34491 sums and products as flat many-element formulas. Thus if @var{expr}
34492 is @samp{((a + b) - c) + d}, calling @code{calc-find-nth-part} with
34493 @var{n} equal to four will return @samp{d}.
34496 @defun calc-find-parent-formula expr part
34497 Return the sub-formula of @var{expr} which immediately contains
34498 @var{part}. If @var{expr} is @samp{a*b + (c+1)*d} and @var{part}
34499 is @code{eq} to the @samp{c+1} term of @var{expr}, then this function
34500 will return @samp{(c+1)*d}. If @var{part} turns out not to be a
34501 sub-formula of @var{expr}, the function returns @code{nil}. If
34502 @var{part} is @code{eq} to @var{expr}, the function returns @code{t}.
34503 This function does not take associativity into account.
34506 @defun calc-find-assoc-parent-formula expr part
34507 This is the same as @code{calc-find-parent-formula}, except that
34508 (unless @kbd{j b} has been used) it continues widening the selection
34509 to contain a complete level of the formula. Given @samp{a} from
34510 @samp{((a + b) - c) + d}, @code{calc-find-parent-formula} will
34511 return @samp{a + b} but @code{calc-find-assoc-parent-formula} will
34512 return the whole expression.
34515 @defun calc-grow-assoc-formula expr part
34516 This expands sub-formula @var{part} of @var{expr} to encompass a
34517 complete level of the formula. If @var{part} and its immediate
34518 parent are not compatible associative operators, or if @kbd{j b}
34519 has been used, this simply returns @var{part}.
34522 @defun calc-find-sub-formula expr part
34523 This finds the immediate sub-formula of @var{expr} which contains
34524 @var{part}. It returns an index @var{n} such that
34525 @samp{(calc-find-nth-part @var{expr} @var{n})} would return @var{part}.
34526 If @var{part} is not a sub-formula of @var{expr}, it returns @code{nil}.
34527 If @var{part} is @code{eq} to @var{expr}, it returns @code{t}. This
34528 function does not take associativity into account.
34531 @defun calc-replace-sub-formula expr old new
34532 This function returns a copy of formula @var{expr}, with the
34533 sub-formula that is @code{eq} to @var{old} replaced by @var{new}.
34536 @defun simplify expr
34537 Simplify the expression @var{expr} by applying Calc's algebraic
34538 simplifications. This always returns a copy of the expression; the
34539 structure @var{expr} points to remains unchanged in memory.
34541 More precisely, here is what @code{simplify} does: The expression is
34542 first normalized and evaluated by calling @code{normalize}. If any
34543 @code{AlgSimpRules} have been defined, they are then applied. Then
34544 the expression is traversed in a depth-first, bottom-up fashion; at
34545 each level, any simplifications that can be made are made until no
34546 further changes are possible. Once the entire formula has been
34547 traversed in this way, it is compared with the original formula (from
34548 before the call to @code{normalize}) and, if it has changed,
34549 the entire procedure is repeated (starting with @code{normalize})
34550 until no further changes occur. Usually only two iterations are
34551 needed: one to simplify the formula, and another to verify that no
34552 further simplifications were possible.
34555 @defun simplify-extended expr
34556 Simplify the expression @var{expr}, with additional rules enabled that
34557 help do a more thorough job, while not being entirely ``safe'' in all
34558 circumstances. (For example, this mode will simplify @samp{sqrt(x^2)}
34559 to @samp{x}, which is only valid when @var{x} is positive.) This is
34560 implemented by temporarily binding the variable @code{math-living-dangerously}
34561 to @code{t} (using a @code{let} form) and calling @code{simplify}.
34562 Dangerous simplification rules are written to check this variable
34563 before taking any action.
34566 @defun simplify-units expr
34567 Simplify the expression @var{expr}, treating variable names as units
34568 whenever possible. This works by binding the variable
34569 @code{math-simplifying-units} to @code{t} while calling @code{simplify}.
34572 @defmac math-defsimplify funcs body
34573 Register a new simplification rule; this is normally called as a top-level
34574 form, like @code{defun} or @code{defmath}. If @var{funcs} is a symbol
34575 (like @code{+} or @code{calcFunc-sqrt}), this simplification rule is
34576 applied to the formulas which are calls to the specified function. Or,
34577 @var{funcs} can be a list of such symbols; the rule applies to all
34578 functions on the list. The @var{body} is written like the body of a
34579 function with a single argument called @code{expr}. The body will be
34580 executed with @code{expr} bound to a formula which is a call to one of
34581 the functions @var{funcs}. If the function body returns @code{nil}, or
34582 if it returns a result @code{equal} to the original @code{expr}, it is
34583 ignored and Calc goes on to try the next simplification rule that applies.
34584 If the function body returns something different, that new formula is
34585 substituted for @var{expr} in the original formula.
34587 At each point in the formula, rules are tried in the order of the
34588 original calls to @code{math-defsimplify}; the search stops after the
34589 first rule that makes a change. Thus later rules for that same
34590 function will not have a chance to trigger until the next iteration
34591 of the main @code{simplify} loop.
34593 Note that, since @code{defmath} is not being used here, @var{body} must
34594 be written in true Lisp code without the conveniences that @code{defmath}
34595 provides. If you prefer, you can have @var{body} simply call another
34596 function (defined with @code{defmath}) which does the real work.
34598 The arguments of a function call will already have been simplified
34599 before any rules for the call itself are invoked. Since a new argument
34600 list is consed up when this happens, this means that the rule's body is
34601 allowed to rearrange the function's arguments destructively if that is
34602 convenient. Here is a typical example of a simplification rule:
34605 (math-defsimplify calcFunc-arcsinh
34606 (or (and (math-looks-negp (nth 1 expr))
34607 (math-neg (list 'calcFunc-arcsinh
34608 (math-neg (nth 1 expr)))))
34609 (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh)
34610 (or math-living-dangerously
34611 (math-known-realp (nth 1 (nth 1 expr))))
34612 (nth 1 (nth 1 expr)))))
34615 This is really a pair of rules written with one @code{math-defsimplify}
34616 for convenience; the first replaces @samp{arcsinh(-x)} with
34617 @samp{-arcsinh(x)}, and the second, which is safe only for real @samp{x},
34618 replaces @samp{arcsinh(sinh(x))} with @samp{x}.
34621 @defun common-constant-factor expr
34622 Check @var{expr} to see if it is a sum of terms all multiplied by the
34623 same rational value. If so, return this value. If not, return @code{nil}.
34624 For example, if called on @samp{6x + 9y + 12z}, it would return 3, since
34625 3 is a common factor of all the terms.
34628 @defun cancel-common-factor expr factor
34629 Assuming @var{expr} is a sum with @var{factor} as a common factor,
34630 divide each term of the sum by @var{factor}. This is done by
34631 destructively modifying parts of @var{expr}, on the assumption that
34632 it is being used by a simplification rule (where such things are
34633 allowed; see above). For example, consider this built-in rule for
34637 (math-defsimplify calcFunc-sqrt
34638 (let ((fac (math-common-constant-factor (nth 1 expr))))
34639 (and fac (not (eq fac 1))
34640 (math-mul (math-normalize (list 'calcFunc-sqrt fac))
34642 (list 'calcFunc-sqrt
34643 (math-cancel-common-factor
34644 (nth 1 expr) fac)))))))
34648 @defun frac-gcd a b
34649 Compute a ``rational GCD'' of @var{a} and @var{b}, which must both be
34650 rational numbers. This is the fraction composed of the GCD of the
34651 numerators of @var{a} and @var{b}, over the GCD of the denominators.
34652 It is used by @code{common-constant-factor}. Note that the standard
34653 @code{gcd} function uses the LCM to combine the denominators.
34656 @defun map-tree func expr many
34657 Try applying Lisp function @var{func} to various sub-expressions of
34658 @var{expr}. Initially, call @var{func} with @var{expr} itself as an
34659 argument. If this returns an expression which is not @code{equal} to
34660 @var{expr}, apply @var{func} again until eventually it does return
34661 @var{expr} with no changes. Then, if @var{expr} is a function call,
34662 recursively apply @var{func} to each of the arguments. This keeps going
34663 until no changes occur anywhere in the expression; this final expression
34664 is returned by @code{map-tree}. Note that, unlike simplification rules,
34665 @var{func} functions may @emph{not} make destructive changes to
34666 @var{expr}. If a third argument @var{many} is provided, it is an
34667 integer which says how many times @var{func} may be applied; the
34668 default, as described above, is infinitely many times.
34671 @defun compile-rewrites rules
34672 Compile the rewrite rule set specified by @var{rules}, which should
34673 be a formula that is either a vector or a variable name. If the latter,
34674 the compiled rules are saved so that later @code{compile-rules} calls
34675 for that same variable can return immediately. If there are problems
34676 with the rules, this function calls @code{error} with a suitable
34680 @defun apply-rewrites expr crules heads
34681 Apply the compiled rewrite rule set @var{crules} to the expression
34682 @var{expr}. This will make only one rewrite and only checks at the
34683 top level of the expression. The result @code{nil} if no rules
34684 matched, or if the only rules that matched did not actually change
34685 the expression. The @var{heads} argument is optional; if is given,
34686 it should be a list of all function names that (may) appear in
34687 @var{expr}. The rewrite compiler tags each rule with the
34688 rarest-looking function name in the rule; if you specify @var{heads},
34689 @code{apply-rewrites} can use this information to narrow its search
34690 down to just a few rules in the rule set.
34693 @defun rewrite-heads expr
34694 Compute a @var{heads} list for @var{expr} suitable for use with
34695 @code{apply-rewrites}, as discussed above.
34698 @defun rewrite expr rules many
34699 This is an all-in-one rewrite function. It compiles the rule set
34700 specified by @var{rules}, then uses @code{map-tree} to apply the
34701 rules throughout @var{expr} up to @var{many} (default infinity)
34705 @defun match-patterns pat vec not-flag
34706 Given a Calc vector @var{vec} and an uncompiled pattern set or
34707 pattern set variable @var{pat}, this function returns a new vector
34708 of all elements of @var{vec} which do (or don't, if @var{not-flag} is
34709 non-@code{nil}) match any of the patterns in @var{pat}.
34712 @defun deriv expr var value symb
34713 Compute the derivative of @var{expr} with respect to variable @var{var}
34714 (which may actually be any sub-expression). If @var{value} is specified,
34715 the derivative is evaluated at the value of @var{var}; otherwise, the
34716 derivative is left in terms of @var{var}. If the expression contains
34717 functions for which no derivative formula is known, new derivative
34718 functions are invented by adding primes to the names; @pxref{Calculus}.
34719 However, if @var{symb} is non-@code{nil}, the presence of nondifferentiable
34720 functions in @var{expr} instead cancels the whole differentiation, and
34721 @code{deriv} returns @code{nil} instead.
34723 Derivatives of an @var{n}-argument function can be defined by
34724 adding a @code{math-derivative-@var{n}} property to the property list
34725 of the symbol for the function's derivative, which will be the
34726 function name followed by an apostrophe. The value of the property
34727 should be a Lisp function; it is called with the same arguments as the
34728 original function call that is being differentiated. It should return
34729 a formula for the derivative. For example, the derivative of @code{ln}
34733 (put 'calcFunc-ln\' 'math-derivative-1
34734 (function (lambda (u) (math-div 1 u))))
34737 The two-argument @code{log} function has two derivatives,
34739 (put 'calcFunc-log\' 'math-derivative-2 ; d(log(x,b)) / dx
34740 (function (lambda (x b) ... )))
34741 (put 'calcFunc-log\'2 'math-derivative-2 ; d(log(x,b)) / db
34742 (function (lambda (x b) ... )))
34746 @defun tderiv expr var value symb
34747 Compute the total derivative of @var{expr}. This is the same as
34748 @code{deriv}, except that variables other than @var{var} are not
34749 assumed to be constant with respect to @var{var}.
34752 @defun integ expr var low high
34753 Compute the integral of @var{expr} with respect to @var{var}.
34754 @xref{Calculus}, for further details.
34757 @defmac math-defintegral funcs body
34758 Define a rule for integrating a function or functions of one argument;
34759 this macro is very similar in format to @code{math-defsimplify}.
34760 The main difference is that here @var{body} is the body of a function
34761 with a single argument @code{u} which is bound to the argument to the
34762 function being integrated, not the function call itself. Also, the
34763 variable of integration is available as @code{math-integ-var}. If
34764 evaluation of the integral requires doing further integrals, the body
34765 should call @samp{(math-integral @var{x})} to find the integral of
34766 @var{x} with respect to @code{math-integ-var}; this function returns
34767 @code{nil} if the integral could not be done. Some examples:
34770 (math-defintegral calcFunc-conj
34771 (let ((int (math-integral u)))
34773 (list 'calcFunc-conj int))))
34775 (math-defintegral calcFunc-cos
34776 (and (equal u math-integ-var)
34777 (math-from-radians-2 (list 'calcFunc-sin u))))
34780 In the @code{cos} example, we define only the integral of @samp{cos(x) dx},
34781 relying on the general integration-by-substitution facility to handle
34782 cosines of more complicated arguments. An integration rule should return
34783 @code{nil} if it can't do the integral; if several rules are defined for
34784 the same function, they are tried in order until one returns a non-@code{nil}
34788 @defmac math-defintegral-2 funcs body
34789 Define a rule for integrating a function or functions of two arguments.
34790 This is exactly analogous to @code{math-defintegral}, except that @var{body}
34791 is written as the body of a function with two arguments, @var{u} and
34795 @defun solve-for lhs rhs var full
34796 Attempt to solve the equation @samp{@var{lhs} = @var{rhs}} by isolating
34797 the variable @var{var} on the lefthand side; return the resulting righthand
34798 side, or @code{nil} if the equation cannot be solved. The variable
34799 @var{var} must appear at least once in @var{lhs} or @var{rhs}. Note that
34800 the return value is a formula which does not contain @var{var}; this is
34801 different from the user-level @code{solve} and @code{finv} functions,
34802 which return a rearranged equation or a functional inverse, respectively.
34803 If @var{full} is non-@code{nil}, a full solution including dummy signs
34804 and dummy integers will be produced. User-defined inverses are provided
34805 as properties in a manner similar to derivatives:
34808 (put 'calcFunc-ln 'math-inverse
34809 (function (lambda (x) (list 'calcFunc-exp x))))
34812 This function can call @samp{(math-solve-get-sign @var{x})} to create
34813 a new arbitrary sign variable, returning @var{x} times that sign, and
34814 @samp{(math-solve-get-int @var{x})} to create a new arbitrary integer
34815 variable multiplied by @var{x}. These functions simply return @var{x}
34816 if the caller requested a non-``full'' solution.
34819 @defun solve-eqn expr var full
34820 This version of @code{solve-for} takes an expression which will
34821 typically be an equation or inequality. (If it is not, it will be
34822 interpreted as the equation @samp{@var{expr} = 0}.) It returns an
34823 equation or inequality, or @code{nil} if no solution could be found.
34826 @defun solve-system exprs vars full
34827 This function solves a system of equations. Generally, @var{exprs}
34828 and @var{vars} will be vectors of equal length.
34829 @xref{Solving Systems of Equations}, for other options.
34832 @defun expr-contains expr var
34833 Returns a non-@code{nil} value if @var{var} occurs as a subexpression
34836 This function might seem at first to be identical to
34837 @code{calc-find-sub-formula}. The key difference is that
34838 @code{expr-contains} uses @code{equal} to test for matches, whereas
34839 @code{calc-find-sub-formula} uses @code{eq}. In the formula
34840 @samp{f(a, a)}, the two @samp{a}s will be @code{equal} but not
34841 @code{eq} to each other.
34844 @defun expr-contains-count expr var
34845 Returns the number of occurrences of @var{var} as a subexpression
34846 of @var{expr}, or @code{nil} if there are no occurrences.
34849 @defun expr-depends expr var
34850 Returns true if @var{expr} refers to any variable the occurs in @var{var}.
34851 In other words, it checks if @var{expr} and @var{var} have any variables
34855 @defun expr-contains-vars expr
34856 Return true if @var{expr} contains any variables, or @code{nil} if @var{expr}
34857 contains only constants and functions with constant arguments.
34860 @defun expr-subst expr old new
34861 Returns a copy of @var{expr}, with all occurrences of @var{old} replaced
34862 by @var{new}. This treats @code{lambda} forms specially with respect
34863 to the dummy argument variables, so that the effect is always to return
34864 @var{expr} evaluated at @var{old} = @var{new}.
34867 @defun multi-subst expr old new
34868 This is like @code{expr-subst}, except that @var{old} and @var{new}
34869 are lists of expressions to be substituted simultaneously. If one
34870 list is shorter than the other, trailing elements of the longer list
34874 @defun expr-weight expr
34875 Returns the ``weight'' of @var{expr}, basically a count of the total
34876 number of objects and function calls that appear in @var{expr}. For
34877 ``primitive'' objects, this will be one.
34880 @defun expr-height expr
34881 Returns the ``height'' of @var{expr}, which is the deepest level to
34882 which function calls are nested. (Note that @samp{@var{a} + @var{b}}
34883 counts as a function call.) For primitive objects, this returns zero.
34886 @defun polynomial-p expr var
34887 Check if @var{expr} is a polynomial in variable (or sub-expression)
34888 @var{var}. If so, return the degree of the polynomial, that is, the
34889 highest power of @var{var} that appears in @var{expr}. For example,
34890 for @samp{(x^2 + 3)^3 + 4} this would return 6. This function returns
34891 @code{nil} unless @var{expr}, when expanded out by @kbd{a x}
34892 (@code{calc-expand}), would consist of a sum of terms in which @var{var}
34893 appears only raised to nonnegative integer powers. Note that if
34894 @var{var} does not occur in @var{expr}, then @var{expr} is considered
34895 a polynomial of degree 0.
34898 @defun is-polynomial expr var degree loose
34899 Check if @var{expr} is a polynomial in variable or sub-expression
34900 @var{var}, and, if so, return a list representation of the polynomial
34901 where the elements of the list are coefficients of successive powers of
34902 @var{var}: @samp{@var{a} + @var{b} x + @var{c} x^3} would produce the
34903 list @samp{(@var{a} @var{b} 0 @var{c})}, and @samp{(x + 1)^2} would
34904 produce the list @samp{(1 2 1)}. The highest element of the list will
34905 be non-zero, with the special exception that if @var{expr} is the
34906 constant zero, the returned value will be @samp{(0)}. Return @code{nil}
34907 if @var{expr} is not a polynomial in @var{var}. If @var{degree} is
34908 specified, this will not consider polynomials of degree higher than that
34909 value. This is a good precaution because otherwise an input of
34910 @samp{(x+1)^1000} will cause a huge coefficient list to be built. If
34911 @var{loose} is non-@code{nil}, then a looser definition of a polynomial
34912 is used in which coefficients are no longer required not to depend on
34913 @var{var}, but are only required not to take the form of polynomials
34914 themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose
34915 polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin
34916 x))}. The result will never be @code{nil} in loose mode, since any
34917 expression can be interpreted as a ``constant'' loose polynomial.
34920 @defun polynomial-base expr pred
34921 Check if @var{expr} is a polynomial in any variable that occurs in it;
34922 if so, return that variable. (If @var{expr} is a multivariate polynomial,
34923 this chooses one variable arbitrarily.) If @var{pred} is specified, it should
34924 be a Lisp function which is called as @samp{(@var{pred} @var{subexpr})},
34925 and which should return true if @code{mpb-top-expr} (a global name for
34926 the original @var{expr}) is a suitable polynomial in @var{subexpr}.
34927 The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})};
34928 you can use @var{pred} to specify additional conditions. Or, you could
34929 have @var{pred} build up a list of every suitable @var{subexpr} that
34933 @defun poly-simplify poly
34934 Simplify polynomial coefficient list @var{poly} by (destructively)
34935 clipping off trailing zeros.
34938 @defun poly-mix a ac b bc
34939 Mix two polynomial lists @var{a} and @var{b} (in the form returned by
34940 @code{is-polynomial}) in a linear combination with coefficient expressions
34941 @var{ac} and @var{bc}. The result is a (not necessarily simplified)
34942 polynomial list representing @samp{@var{ac} @var{a} + @var{bc} @var{b}}.
34945 @defun poly-mul a b
34946 Multiply two polynomial coefficient lists @var{a} and @var{b}. The
34947 result will be in simplified form if the inputs were simplified.
34950 @defun build-polynomial-expr poly var
34951 Construct a Calc formula which represents the polynomial coefficient
34952 list @var{poly} applied to variable @var{var}. The @kbd{a c}
34953 (@code{calc-collect}) command uses @code{is-polynomial} to turn an
34954 expression into a coefficient list, then @code{build-polynomial-expr}
34955 to turn the list back into an expression in regular form.
34958 @defun check-unit-name var
34959 Check if @var{var} is a variable which can be interpreted as a unit
34960 name. If so, return the units table entry for that unit. This
34961 will be a list whose first element is the unit name (not counting
34962 prefix characters) as a symbol and whose second element is the
34963 Calc expression which defines the unit. (Refer to the Calc sources
34964 for details on the remaining elements of this list.) If @var{var}
34965 is not a variable or is not a unit name, return @code{nil}.
34968 @defun units-in-expr-p expr sub-exprs
34969 Return true if @var{expr} contains any variables which can be
34970 interpreted as units. If @var{sub-exprs} is @code{t}, the entire
34971 expression is searched. If @var{sub-exprs} is @code{nil}, this
34972 checks whether @var{expr} is directly a units expression.
34975 @defun single-units-in-expr-p expr
34976 Check whether @var{expr} contains exactly one units variable. If so,
34977 return the units table entry for the variable. If @var{expr} does
34978 not contain any units, return @code{nil}. If @var{expr} contains
34979 two or more units, return the symbol @code{wrong}.
34982 @defun to-standard-units expr which
34983 Convert units expression @var{expr} to base units. If @var{which}
34984 is @code{nil}, use Calc's native base units. Otherwise, @var{which}
34985 can specify a units system, which is a list of two-element lists,
34986 where the first element is a Calc base symbol name and the second
34987 is an expression to substitute for it.
34990 @defun remove-units expr
34991 Return a copy of @var{expr} with all units variables replaced by ones.
34992 This expression is generally normalized before use.
34995 @defun extract-units expr
34996 Return a copy of @var{expr} with everything but units variables replaced
35000 @node Formatting Lisp Functions, Hooks, Symbolic Lisp Functions, Internals
35001 @subsubsection I/O and Formatting Functions
35004 The functions described here are responsible for parsing and formatting
35005 Calc numbers and formulas.
35007 @defun calc-eval str sep arg1 arg2 @dots{}
35008 This is the simplest interface to the Calculator from another Lisp program.
35009 @xref{Calling Calc from Your Programs}.
35012 @defun read-number str
35013 If string @var{str} contains a valid Calc number, either integer,
35014 fraction, float, or HMS form, this function parses and returns that
35015 number. Otherwise, it returns @code{nil}.
35018 @defun read-expr str
35019 Read an algebraic expression from string @var{str}. If @var{str} does
35020 not have the form of a valid expression, return a list of the form
35021 @samp{(error @var{pos} @var{msg})} where @var{pos} is an integer index
35022 into @var{str} of the general location of the error, and @var{msg} is
35023 a string describing the problem.
35026 @defun read-exprs str
35027 Read a list of expressions separated by commas, and return it as a
35028 Lisp list. If an error occurs in any expressions, an error list as
35029 shown above is returned instead.
35032 @defun calc-do-alg-entry initial prompt no-norm
35033 Read an algebraic formula or formulas using the minibuffer. All
35034 conventions of regular algebraic entry are observed. The return value
35035 is a list of Calc formulas; there will be more than one if the user
35036 entered a list of values separated by commas. The result is @code{nil}
35037 if the user presses Return with a blank line. If @var{initial} is
35038 given, it is a string which the minibuffer will initially contain.
35039 If @var{prompt} is given, it is the prompt string to use; the default
35040 is ``Algebraic:''. If @var{no-norm} is @code{t}, the formulas will
35041 be returned exactly as parsed; otherwise, they will be passed through
35042 @code{calc-normalize} first.
35044 To support the use of @kbd{$} characters in the algebraic entry, use
35045 @code{let} to bind @code{calc-dollar-values} to a list of the values
35046 to be substituted for @kbd{$}, @kbd{$$}, and so on, and bind
35047 @code{calc-dollar-used} to 0. Upon return, @code{calc-dollar-used}
35048 will have been changed to the highest number of consecutive @kbd{$}s
35049 that actually appeared in the input.
35052 @defun format-number a
35053 Convert the real or complex number or HMS form @var{a} to string form.
35056 @defun format-flat-expr a prec
35057 Convert the arbitrary Calc number or formula @var{a} to string form,
35058 in the style used by the trail buffer and the @code{calc-edit} command.
35059 This is a simple format designed
35060 mostly to guarantee the string is of a form that can be re-parsed by
35061 @code{read-expr}. Most formatting modes, such as digit grouping,
35062 complex number format, and point character, are ignored to ensure the
35063 result will be re-readable. The @var{prec} parameter is normally 0; if
35064 you pass a large integer like 1000 instead, the expression will be
35065 surrounded by parentheses unless it is a plain number or variable name.
35068 @defun format-nice-expr a width
35069 This is like @code{format-flat-expr} (with @var{prec} equal to 0),
35070 except that newlines will be inserted to keep lines down to the
35071 specified @var{width}, and vectors that look like matrices or rewrite
35072 rules are written in a pseudo-matrix format. The @code{calc-edit}
35073 command uses this when only one stack entry is being edited.
35076 @defun format-value a width
35077 Convert the Calc number or formula @var{a} to string form, using the
35078 format seen in the stack buffer. Beware the string returned may
35079 not be re-readable by @code{read-expr}, for example, because of digit
35080 grouping. Multi-line objects like matrices produce strings that
35081 contain newline characters to separate the lines. The @var{w}
35082 parameter, if given, is the target window size for which to format
35083 the expressions. If @var{w} is omitted, the width of the Calculator
35087 @defun compose-expr a prec
35088 Format the Calc number or formula @var{a} according to the current
35089 language mode, returning a ``composition.'' To learn about the
35090 structure of compositions, see the comments in the Calc source code.
35091 You can specify the format of a given type of function call by putting
35092 a @code{math-compose-@var{lang}} property on the function's symbol,
35093 whose value is a Lisp function that takes @var{a} and @var{prec} as
35094 arguments and returns a composition. Here @var{lang} is a language
35095 mode name, one of @code{normal}, @code{big}, @code{c}, @code{pascal},
35096 @code{fortran}, @code{tex}, @code{eqn}, @code{math}, or @code{maple}.
35097 In Big mode, Calc actually tries @code{math-compose-big} first, then
35098 tries @code{math-compose-normal}. If this property does not exist,
35099 or if the function returns @code{nil}, the function is written in the
35100 normal function-call notation for that language.
35103 @defun composition-to-string c w
35104 Convert a composition structure returned by @code{compose-expr} into
35105 a string. Multi-line compositions convert to strings containing
35106 newline characters. The target window size is given by @var{w}.
35107 The @code{format-value} function basically calls @code{compose-expr}
35108 followed by @code{composition-to-string}.
35111 @defun comp-width c
35112 Compute the width in characters of composition @var{c}.
35115 @defun comp-height c
35116 Compute the height in lines of composition @var{c}.
35119 @defun comp-ascent c
35120 Compute the portion of the height of composition @var{c} which is on or
35121 above the baseline. For a one-line composition, this will be one.
35124 @defun comp-descent c
35125 Compute the portion of the height of composition @var{c} which is below
35126 the baseline. For a one-line composition, this will be zero.
35129 @defun comp-first-char c
35130 If composition @var{c} is a ``flat'' composition, return the first
35131 (leftmost) character of the composition as an integer. Otherwise,
35135 @defun comp-last-char c
35136 If composition @var{c} is a ``flat'' composition, return the last
35137 (rightmost) character, otherwise return @code{nil}.
35140 @comment @node Lisp Variables, Hooks, Formatting Lisp Functions, Internals
35141 @comment @subsubsection Lisp Variables
35144 @comment (This section is currently unfinished.)
35146 @node Hooks, , Formatting Lisp Functions, Internals
35147 @subsubsection Hooks
35150 Hooks are variables which contain Lisp functions (or lists of functions)
35151 which are called at various times. Calc defines a number of hooks
35152 that help you to customize it in various ways. Calc uses the Lisp
35153 function @code{run-hooks} to invoke the hooks shown below. Several
35154 other customization-related variables are also described here.
35156 @defvar calc-load-hook
35157 This hook is called at the end of @file{calc.el}, after the file has
35158 been loaded, before any functions in it have been called, but after
35159 @code{calc-mode-map} and similar variables have been set up.
35162 @defvar calc-ext-load-hook
35163 This hook is called at the end of @file{calc-ext.el}.
35166 @defvar calc-start-hook
35167 This hook is called as the last step in a @kbd{M-x calc} command.
35168 At this point, the Calc buffer has been created and initialized if
35169 necessary, the Calc window and trail window have been created,
35170 and the ``Welcome to Calc'' message has been displayed.
35173 @defvar calc-mode-hook
35174 This hook is called when the Calc buffer is being created. Usually
35175 this will only happen once per Emacs session. The hook is called
35176 after Emacs has switched to the new buffer, the mode-settings file
35177 has been read if necessary, and all other buffer-local variables
35178 have been set up. After this hook returns, Calc will perform a
35179 @code{calc-refresh} operation, set up the mode line display, then
35180 evaluate any deferred @code{calc-define} properties that have not
35181 been evaluated yet.
35184 @defvar calc-trail-mode-hook
35185 This hook is called when the Calc Trail buffer is being created.
35186 It is called as the very last step of setting up the Trail buffer.
35187 Like @code{calc-mode-hook}, this will normally happen only once
35191 @defvar calc-end-hook
35192 This hook is called by @code{calc-quit}, generally because the user
35193 presses @kbd{q} or @kbd{C-x * c} while in Calc. The Calc buffer will
35194 be the current buffer. The hook is called as the very first
35195 step, before the Calc window is destroyed.
35198 @defvar calc-window-hook
35199 If this hook is non-@code{nil}, it is called to create the Calc window.
35200 Upon return, this new Calc window should be the current window.
35201 (The Calc buffer will already be the current buffer when the
35202 hook is called.) If the hook is not defined, Calc will
35203 generally use @code{split-window}, @code{set-window-buffer},
35204 and @code{select-window} to create the Calc window.
35207 @defvar calc-trail-window-hook
35208 If this hook is non-@code{nil}, it is called to create the Calc Trail
35209 window. The variable @code{calc-trail-buffer} will contain the buffer
35210 which the window should use. Unlike @code{calc-window-hook}, this hook
35211 must @emph{not} switch into the new window.
35214 @defvar calc-embedded-mode-hook
35215 This hook is called the first time that Embedded mode is entered.
35218 @defvar calc-embedded-new-buffer-hook
35219 This hook is called each time that Embedded mode is entered in a
35223 @defvar calc-embedded-new-formula-hook
35224 This hook is called each time that Embedded mode is enabled for a
35228 @defvar calc-edit-mode-hook
35229 This hook is called by @code{calc-edit} (and the other ``edit''
35230 commands) when the temporary editing buffer is being created.
35231 The buffer will have been selected and set up to be in
35232 @code{calc-edit-mode}, but will not yet have been filled with
35233 text. (In fact it may still have leftover text from a previous
35234 @code{calc-edit} command.)
35237 @defvar calc-mode-save-hook
35238 This hook is called by the @code{calc-save-modes} command,
35239 after Calc's own mode features have been inserted into the
35240 Calc init file and just before the ``End of mode settings''
35241 message is inserted.
35244 @defvar calc-reset-hook
35245 This hook is called after @kbd{C-x * 0} (@code{calc-reset}) has
35246 reset all modes. The Calc buffer will be the current buffer.
35249 @defvar calc-other-modes
35250 This variable contains a list of strings. The strings are
35251 concatenated at the end of the modes portion of the Calc
35252 mode line (after standard modes such as ``Deg'', ``Inv'' and
35253 ``Hyp''). Each string should be a short, single word followed
35254 by a space. The variable is @code{nil} by default.
35257 @defvar calc-mode-map
35258 This is the keymap that is used by Calc mode. The best time
35259 to adjust it is probably in a @code{calc-mode-hook}. If the
35260 Calc extensions package (@file{calc-ext.el}) has not yet been
35261 loaded, many of these keys will be bound to @code{calc-missing-key},
35262 which is a command that loads the extensions package and
35263 ``retypes'' the key. If your @code{calc-mode-hook} rebinds
35264 one of these keys, it will probably be overridden when the
35265 extensions are loaded.
35268 @defvar calc-digit-map
35269 This is the keymap that is used during numeric entry. Numeric
35270 entry uses the minibuffer, but this map binds every non-numeric
35271 key to @code{calcDigit-nondigit} which generally calls
35272 @code{exit-minibuffer} and ``retypes'' the key.
35275 @defvar calc-alg-ent-map
35276 This is the keymap that is used during algebraic entry. This is
35277 mostly a copy of @code{minibuffer-local-map}.
35280 @defvar calc-store-var-map
35281 This is the keymap that is used during entry of variable names for
35282 commands like @code{calc-store} and @code{calc-recall}. This is
35283 mostly a copy of @code{minibuffer-local-completion-map}.
35286 @defvar calc-edit-mode-map
35287 This is the (sparse) keymap used by @code{calc-edit} and other
35288 temporary editing commands. It binds @key{RET}, @key{LFD},
35289 and @kbd{C-c C-c} to @code{calc-edit-finish}.
35292 @defvar calc-mode-var-list
35293 This is a list of variables which are saved by @code{calc-save-modes}.
35294 Each entry is a list of two items, the variable (as a Lisp symbol)
35295 and its default value. When modes are being saved, each variable
35296 is compared with its default value (using @code{equal}) and any
35297 non-default variables are written out.
35300 @defvar calc-local-var-list
35301 This is a list of variables which should be buffer-local to the
35302 Calc buffer. Each entry is a variable name (as a Lisp symbol).
35303 These variables also have their default values manipulated by
35304 the @code{calc} and @code{calc-quit} commands; @pxref{Multiple Calculators}.
35305 Since @code{calc-mode-hook} is called after this list has been
35306 used the first time, your hook should add a variable to the
35307 list and also call @code{make-local-variable} itself.
35310 @node Copying, GNU Free Documentation License, Programming, Top
35311 @appendix GNU GENERAL PUBLIC LICENSE
35314 @node GNU Free Documentation License, Customizing Calc, Copying, Top
35315 @appendix GNU Free Documentation License
35316 @include doclicense.texi
35318 @node Customizing Calc, Reporting Bugs, GNU Free Documentation License, Top
35319 @appendix Customizing Calc
35321 The usual prefix for Calc is the key sequence @kbd{C-x *}. If you wish
35322 to use a different prefix, you can put
35325 (global-set-key "NEWPREFIX" 'calc-dispatch)
35329 in your .emacs file.
35330 (@xref{Key Bindings,,Customizing Key Bindings,emacs,
35331 The GNU Emacs Manual}, for more information on binding keys.)
35332 A convenient way to start Calc is with @kbd{C-x * *}; to make it equally
35333 convenient for users who use a different prefix, the prefix can be
35334 followed by @kbd{=}, @kbd{&}, @kbd{#}, @kbd{\}, @kbd{/}, @kbd{+} or
35335 @kbd{-} as well as @kbd{*} to start Calc, and so in many cases the last
35336 character of the prefix can simply be typed twice.
35338 Calc is controlled by many variables, most of which can be reset
35339 from within Calc. Some variables are less involved with actual
35340 calculation and can be set outside of Calc using Emacs's
35341 customization facilities. These variables are listed below.
35342 Typing @kbd{M-x customize-variable RET @var{variable-name} RET}
35343 will bring up a buffer in which the variable's value can be redefined.
35344 Typing @kbd{M-x customize-group RET calc RET} will bring up a buffer which
35345 contains all of Calc's customizable variables. (These variables can
35346 also be reset by putting the appropriate lines in your .emacs file;
35347 @xref{Init File, ,Init File, emacs, The GNU Emacs Manual}.)
35349 Some of the customizable variables are regular expressions. A regular
35350 expression is basically a pattern that Calc can search for.
35351 See @ref{Regexp Search,, Regular Expression Search, emacs, The GNU Emacs Manual}
35352 to see how regular expressions work.
35354 @defvar calc-settings-file
35355 The variable @code{calc-settings-file} holds the file name in
35356 which commands like @kbd{m m} and @kbd{Z P} store ``permanent''
35358 If @code{calc-settings-file} is not your user init file (typically
35359 @file{~/.emacs}) and if the variable @code{calc-loaded-settings-file} is
35360 @code{nil}, then Calc will automatically load your settings file (if it
35361 exists) the first time Calc is invoked.
35363 The default value for this variable is @code{"~/.emacs.d/calc.el"}
35364 unless the file @file{~/.calc.el} exists, in which case the default
35365 value will be @code{"~/.calc.el"}.
35368 @defvar calc-gnuplot-name
35369 See @ref{Graphics}.@*
35370 The variable @code{calc-gnuplot-name} should be the name of the
35371 GNUPLOT program (a string). If you have GNUPLOT installed on your
35372 system but Calc is unable to find it, you may need to set this
35373 variable. You may also need to set some Lisp variables to show Calc how
35374 to run GNUPLOT on your system, see @ref{Devices, ,Graphical Devices} .
35375 The default value of @code{calc-gnuplot-name} is @code{"gnuplot"}.
35378 @defvar calc-gnuplot-plot-command
35379 @defvarx calc-gnuplot-print-command
35380 See @ref{Devices, ,Graphical Devices}.@*
35381 The variables @code{calc-gnuplot-plot-command} and
35382 @code{calc-gnuplot-print-command} represent system commands to
35383 display and print the output of GNUPLOT, respectively. These may be
35384 @code{nil} if no command is necessary, or strings which can include
35385 @samp{%s} to signify the name of the file to be displayed or printed.
35386 Or, these variables may contain Lisp expressions which are evaluated
35387 to display or print the output.
35389 The default value of @code{calc-gnuplot-plot-command} is @code{nil},
35390 and the default value of @code{calc-gnuplot-print-command} is
35394 @defvar calc-language-alist
35395 See @ref{Basic Embedded Mode}.@*
35396 The variable @code{calc-language-alist} controls the languages that
35397 Calc will associate with major modes. When Calc embedded mode is
35398 enabled, it will try to use the current major mode to
35399 determine what language should be used. (This can be overridden using
35400 Calc's mode changing commands, @xref{Mode Settings in Embedded Mode}.)
35401 The variable @code{calc-language-alist} consists of a list of pairs of
35402 the form @code{(@var{MAJOR-MODE} . @var{LANGUAGE})}; for example,
35403 @code{(latex-mode . latex)} is one such pair. If Calc embedded is
35404 activated in a buffer whose major mode is @var{MAJOR-MODE}, it will set itself
35405 to use the language @var{LANGUAGE}.
35407 The default value of @code{calc-language-alist} is
35409 ((latex-mode . latex)
35411 (plain-tex-mode . tex)
35412 (context-mode . tex)
35414 (pascal-mode . pascal)
35417 (fortran-mode . fortran)
35418 (f90-mode . fortran))
35422 @defvar calc-embedded-announce-formula
35423 @defvarx calc-embedded-announce-formula-alist
35424 See @ref{Customizing Embedded Mode}.@*
35425 The variable @code{calc-embedded-announce-formula} helps determine
35426 what formulas @kbd{C-x * a} will activate in a buffer. It is a
35427 regular expression, and when activating embedded formulas with
35428 @kbd{C-x * a}, it will tell Calc that what follows is a formula to be
35429 activated. (Calc also uses other patterns to find formulas, such as
35430 @samp{=>} and @samp{:=}.)
35432 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which checks
35433 for @samp{%Embed} followed by any number of lines beginning with
35434 @samp{%} and a space.
35436 The variable @code{calc-embedded-announce-formula-alist} is used to
35437 set @code{calc-embedded-announce-formula} to different regular
35438 expressions depending on the major mode of the editing buffer.
35439 It consists of a list of pairs of the form @code{(@var{MAJOR-MODE} .
35440 @var{REGEXP})}, and its default value is
35442 ((c++-mode . "//Embed\n\\(// .*\n\\)*")
35443 (c-mode . "/\\*Embed\\*/\n\\(/\\* .*\\*/\n\\)*")
35444 (f90-mode . "!Embed\n\\(! .*\n\\)*")
35445 (fortran-mode . "C Embed\n\\(C .*\n\\)*")
35446 (html-helper-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
35447 (html-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
35448 (nroff-mode . "\\\\\"Embed\n\\(\\\\\" .*\n\\)*")
35449 (pascal-mode . "@{Embed@}\n\\(@{.*@}\n\\)*")
35450 (sgml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
35451 (xml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
35452 (texinfo-mode . "@@c Embed\n\\(@@c .*\n\\)*"))
35454 Any major modes added to @code{calc-embedded-announce-formula-alist}
35455 should also be added to @code{calc-embedded-open-close-plain-alist}
35456 and @code{calc-embedded-open-close-mode-alist}.
35459 @defvar calc-embedded-open-formula
35460 @defvarx calc-embedded-close-formula
35461 @defvarx calc-embedded-open-close-formula-alist
35462 See @ref{Customizing Embedded Mode}.@*
35463 The variables @code{calc-embedded-open-formula} and
35464 @code{calc-embedded-close-formula} control the region that Calc will
35465 activate as a formula when Embedded mode is entered with @kbd{C-x * e}.
35466 They are regular expressions;
35467 Calc normally scans backward and forward in the buffer for the
35468 nearest text matching these regular expressions to be the ``formula
35471 The simplest delimiters are blank lines. Other delimiters that
35472 Embedded mode understands by default are:
35475 The @TeX{} and @LaTeX{} math delimiters @samp{$ $}, @samp{$$ $$},
35476 @samp{\[ \]}, and @samp{\( \)};
35478 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
35480 Lines beginning with @samp{@@} (Texinfo delimiters).
35482 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
35484 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
35487 The variable @code{calc-embedded-open-close-formula-alist} is used to
35488 set @code{calc-embedded-open-formula} and
35489 @code{calc-embedded-close-formula} to different regular
35490 expressions depending on the major mode of the editing buffer.
35491 It consists of a list of lists of the form
35492 @code{(@var{MAJOR-MODE} @var{OPEN-FORMULA-REGEXP}
35493 @var{CLOSE-FORMULA-REGEXP})}, and its default value is
35497 @defvar calc-embedded-word-regexp
35498 @defvarx calc-embedded-word-regexp-alist
35499 See @ref{Customizing Embedded Mode}.@*
35500 The variable @code{calc-embedded-word-regexp} determines the expression
35501 that Calc will activate when Embedded mode is entered with @kbd{C-x *
35502 w}. It is a regular expressions.
35504 The default value of @code{calc-embedded-word-regexp} is
35505 @code{"[-+]?[0-9]+\\(\\.[0-9]+\\)?\\([eE][-+]?[0-9]+\\)?"}.
35507 The variable @code{calc-embedded-word-regexp-alist} is used to
35508 set @code{calc-embedded-word-regexp} to a different regular
35509 expression depending on the major mode of the editing buffer.
35510 It consists of a list of lists of the form
35511 @code{(@var{MAJOR-MODE} @var{WORD-REGEXP})}, and its default value is
35515 @defvar calc-embedded-open-plain
35516 @defvarx calc-embedded-close-plain
35517 @defvarx calc-embedded-open-close-plain-alist
35518 See @ref{Customizing Embedded Mode}.@*
35519 The variables @code{calc-embedded-open-plain} and
35520 @code{calc-embedded-open-plain} are used to delimit ``plain''
35521 formulas. Note that these are actual strings, not regular
35522 expressions, because Calc must be able to write these string into a
35523 buffer as well as to recognize them.
35525 The default string for @code{calc-embedded-open-plain} is
35526 @code{"%%% "}, note the trailing space. The default string for
35527 @code{calc-embedded-close-plain} is @code{" %%%\n"}, without
35528 the trailing newline here, the first line of a Big mode formula
35529 that followed might be shifted over with respect to the other lines.
35531 The variable @code{calc-embedded-open-close-plain-alist} is used to
35532 set @code{calc-embedded-open-plain} and
35533 @code{calc-embedded-close-plain} to different strings
35534 depending on the major mode of the editing buffer.
35535 It consists of a list of lists of the form
35536 @code{(@var{MAJOR-MODE} @var{OPEN-PLAIN-STRING}
35537 @var{CLOSE-PLAIN-STRING})}, and its default value is
35539 ((c++-mode "// %% " " %%\n")
35540 (c-mode "/* %% " " %% */\n")
35541 (f90-mode "! %% " " %%\n")
35542 (fortran-mode "C %% " " %%\n")
35543 (html-helper-mode "<!-- %% " " %% -->\n")
35544 (html-mode "<!-- %% " " %% -->\n")
35545 (nroff-mode "\\\" %% " " %%\n")
35546 (pascal-mode "@{%% " " %%@}\n")
35547 (sgml-mode "<!-- %% " " %% -->\n")
35548 (xml-mode "<!-- %% " " %% -->\n")
35549 (texinfo-mode "@@c %% " " %%\n"))
35551 Any major modes added to @code{calc-embedded-open-close-plain-alist}
35552 should also be added to @code{calc-embedded-announce-formula-alist}
35553 and @code{calc-embedded-open-close-mode-alist}.
35556 @defvar calc-embedded-open-new-formula
35557 @defvarx calc-embedded-close-new-formula
35558 @defvarx calc-embedded-open-close-new-formula-alist
35559 See @ref{Customizing Embedded Mode}.@*
35560 The variables @code{calc-embedded-open-new-formula} and
35561 @code{calc-embedded-close-new-formula} are strings which are
35562 inserted before and after a new formula when you type @kbd{C-x * f}.
35564 The default value of @code{calc-embedded-open-new-formula} is
35565 @code{"\n\n"}. If this string begins with a newline character and the
35566 @kbd{C-x * f} is typed at the beginning of a line, @kbd{C-x * f} will skip
35567 this first newline to avoid introducing unnecessary blank lines in the
35568 file. The default value of @code{calc-embedded-close-new-formula} is
35569 also @code{"\n\n"}. The final newline is omitted by @w{@kbd{C-x * f}}
35570 if typed at the end of a line. (It follows that if @kbd{C-x * f} is
35571 typed on a blank line, both a leading opening newline and a trailing
35572 closing newline are omitted.)
35574 The variable @code{calc-embedded-open-close-new-formula-alist} is used to
35575 set @code{calc-embedded-open-new-formula} and
35576 @code{calc-embedded-close-new-formula} to different strings
35577 depending on the major mode of the editing buffer.
35578 It consists of a list of lists of the form
35579 @code{(@var{MAJOR-MODE} @var{OPEN-NEW-FORMULA-STRING}
35580 @var{CLOSE-NEW-FORMULA-STRING})}, and its default value is
35584 @defvar calc-embedded-open-mode
35585 @defvarx calc-embedded-close-mode
35586 @defvarx calc-embedded-open-close-mode-alist
35587 See @ref{Customizing Embedded Mode}.@*
35588 The variables @code{calc-embedded-open-mode} and
35589 @code{calc-embedded-close-mode} are strings which Calc will place before
35590 and after any mode annotations that it inserts. Calc never scans for
35591 these strings; Calc always looks for the annotation itself, so it is not
35592 necessary to add them to user-written annotations.
35594 The default value of @code{calc-embedded-open-mode} is @code{"% "}
35595 and the default value of @code{calc-embedded-close-mode} is
35597 If you change the value of @code{calc-embedded-close-mode}, it is a good
35598 idea still to end with a newline so that mode annotations will appear on
35599 lines by themselves.
35601 The variable @code{calc-embedded-open-close-mode-alist} is used to
35602 set @code{calc-embedded-open-mode} and
35603 @code{calc-embedded-close-mode} to different strings
35604 expressions depending on the major mode of the editing buffer.
35605 It consists of a list of lists of the form
35606 @code{(@var{MAJOR-MODE} @var{OPEN-MODE-STRING}
35607 @var{CLOSE-MODE-STRING})}, and its default value is
35609 ((c++-mode "// " "\n")
35610 (c-mode "/* " " */\n")
35611 (f90-mode "! " "\n")
35612 (fortran-mode "C " "\n")
35613 (html-helper-mode "<!-- " " -->\n")
35614 (html-mode "<!-- " " -->\n")
35615 (nroff-mode "\\\" " "\n")
35616 (pascal-mode "@{ " " @}\n")
35617 (sgml-mode "<!-- " " -->\n")
35618 (xml-mode "<!-- " " -->\n")
35619 (texinfo-mode "@@c " "\n"))
35621 Any major modes added to @code{calc-embedded-open-close-mode-alist}
35622 should also be added to @code{calc-embedded-announce-formula-alist}
35623 and @code{calc-embedded-open-close-plain-alist}.
35626 @defvar calc-lu-power-reference
35627 @defvarx calc-lu-field-reference
35628 See @ref{Logarithmic Units}.@*
35629 The variables @code{calc-lu-power-reference} and
35630 @code{calc-lu-field-reference} are unit expressions (written as
35631 strings) which Calc will use as reference quantities for logarithmic
35634 The default value of @code{calc-lu-power-reference} is @code{"mW"}
35635 and the default value of @code{calc-lu-field-reference} is
35639 @defvar calc-note-threshold
35640 See @ref{Musical Notes}.@*
35641 The variable @code{calc-note-threshold} is a number (written as a
35642 string) which determines how close (in cents) a frequency needs to be
35643 to a note to be recognized as that note.
35645 The default value of @code{calc-note-threshold} is 1.
35648 @defvar calc-highlight-selections-with-faces
35649 @defvarx calc-selected-face
35650 @defvarx calc-nonselected-face
35651 See @ref{Displaying Selections}.@*
35652 The variable @code{calc-highlight-selections-with-faces}
35653 determines how selected sub-formulas are distinguished.
35654 If @code{calc-highlight-selections-with-faces} is nil, then
35655 a selected sub-formula is distinguished either by changing every
35656 character not part of the sub-formula with a dot or by changing every
35657 character in the sub-formula with a @samp{#} sign.
35658 If @code{calc-highlight-selections-with-faces} is t,
35659 then a selected sub-formula is distinguished either by displaying the
35660 non-selected portion of the formula with @code{calc-nonselected-face}
35661 or by displaying the selected sub-formula with
35662 @code{calc-nonselected-face}.
35665 @defvar calc-multiplication-has-precedence
35666 The variable @code{calc-multiplication-has-precedence} determines
35667 whether multiplication has precedence over division in algebraic
35668 formulas in normal language modes. If
35669 @code{calc-multiplication-has-precedence} is non-@code{nil}, then
35670 multiplication has precedence (and, for certain obscure reasons, is
35671 right associative), and so for example @samp{a/b*c} will be interpreted
35672 as @samp{a/(b*c)}. If @code{calc-multiplication-has-precedence} is
35673 @code{nil}, then multiplication has the same precedence as division
35674 (and, like division, is left associative), and so for example
35675 @samp{a/b*c} will be interpreted as @samp{(a/b)*c}. The default value
35676 of @code{calc-multiplication-has-precedence} is @code{t}.
35679 @defvar calc-ensure-consistent-units
35680 When converting units, the variable @code{calc-ensure-consistent-units}
35681 determines whether or not the target units need to be consistent with the
35682 original units. If @code{calc-ensure-consistent-units} is @code{nil}, then
35683 the target units don't need to have the same dimensions as the original units;
35684 for example, converting @samp{100 ft/s} to @samp{m} will produce @samp{30.48 m/s}.
35685 If @code{calc-ensure-consistent-units} is non-@code{nil}, then the target units
35686 need to have the same dimensions as the original units; for example, converting
35687 @samp{100 ft/s} to @samp{m} will result in an error, since @samp{ft/s} and @samp{m}
35688 have different dimensions. The default value of @code{calc-ensure-consistent-units}
35692 @defvar calc-undo-length
35693 The variable @code{calc-undo-length} determines the number of undo
35694 steps that Calc will keep track of when @code{calc-quit} is called.
35695 If @code{calc-undo-length} is a non-negative integer, then this is the
35696 number of undo steps that will be preserved; if
35697 @code{calc-undo-length} has any other value, then all undo steps will
35698 be preserved. The default value of @code{calc-undo-length} is @expr{100}.
35701 @defvar calc-gregorian-switch
35702 See @ref{Date Forms}.@*
35703 The variable @code{calc-gregorian-switch} is either a list of integers
35704 @code{(@var{YEAR} @var{MONTH} @var{DAY})} or @code{nil}.
35705 If it is @code{nil}, then Calc's date forms always represent Gregorian dates.
35706 Otherwise, @code{calc-gregorian-switch} represents the date that the
35707 calendar switches from Julian dates to Gregorian dates;
35708 @code{(@var{YEAR} @var{MONTH} @var{DAY})} will be the first Gregorian
35709 date. The customization buffer will offer several standard dates to
35710 choose from, or the user can enter their own date.
35712 The default value of @code{calc-gregorian-switch} is @code{nil}.
35715 @node Reporting Bugs, Summary, Customizing Calc, Top
35716 @appendix Reporting Bugs
35719 If you find a bug in Calc, send e-mail to Jay Belanger,
35722 jay.p.belanger@@gmail.com
35726 There is an automatic command @kbd{M-x report-calc-bug} which helps
35727 you to report bugs. This command prompts you for a brief subject
35728 line, then leaves you in a mail editing buffer. Type @kbd{C-c C-c} to
35729 send your mail. Make sure your subject line indicates that you are
35730 reporting a Calc bug; this command sends mail to the maintainer's
35733 If you have suggestions for additional features for Calc, please send
35734 them. Some have dared to suggest that Calc is already top-heavy with
35735 features; this obviously cannot be the case, so if you have ideas, send
35738 At the front of the source file, @file{calc.el}, is a list of ideas for
35739 future work. If any enthusiastic souls wish to take it upon themselves
35740 to work on these, please send a message (using @kbd{M-x report-calc-bug})
35741 so any efforts can be coordinated.
35743 The latest version of Calc is available from Savannah, in the Emacs
35744 repository. See @uref{http://savannah.gnu.org/projects/emacs}.
35747 @node Summary, Key Index, Reporting Bugs, Top
35748 @appendix Calc Summary
35751 This section includes a complete list of Calc keystroke commands.
35752 Each line lists the stack entries used by the command (top-of-stack
35753 last), the keystrokes themselves, the prompts asked by the command,
35754 and the result of the command (also with top-of-stack last).
35755 The result is expressed using the equivalent algebraic function.
35756 Commands which put no results on the stack show the full @kbd{M-x}
35757 command name in that position. Numbers preceding the result or
35758 command name refer to notes at the end.
35760 Algebraic functions and @kbd{M-x} commands that don't have corresponding
35761 keystrokes are not listed in this summary.
35762 @xref{Command Index}. @xref{Function Index}.
35767 \vskip-2\baselineskip \null
35768 \gdef\sumrow#1{\sumrowx#1\relax}%
35769 \gdef\sumrowx#1\:#2\:#3\:#4\:#5\:#6\relax{%
35772 \hbox to5em{\sl\hss#1}%
35773 \hbox to5em{\tt#2\hss}%
35774 \hbox to4em{\sl#3\hss}%
35775 \hbox to5em{\rm\hss#4}%
35780 \gdef\sumlpar{{\rm(}}%
35781 \gdef\sumrpar{{\rm)}}%
35782 \gdef\sumcomma{{\rm,\thinspace}}%
35783 \gdef\sumexcl{{\rm!}}%
35784 \gdef\sumbreak{\vskip-2.5\baselineskip\goodbreak}%
35785 \gdef\minus#1{{\tt-}}%
35789 @catcode`@(=@active @let(=@sumlpar
35790 @catcode`@)=@active @let)=@sumrpar
35791 @catcode`@,=@active @let,=@sumcomma
35792 @catcode`@!=@active @let!=@sumexcl
35796 @advance@baselineskip-2.5pt
35799 @r{ @: C-x * a @: @: 33 @:calc-embedded-activate@:}
35800 @r{ @: C-x * b @: @: @:calc-big-or-small@:}
35801 @r{ @: C-x * c @: @: @:calc@:}
35802 @r{ @: C-x * d @: @: @:calc-embedded-duplicate@:}
35803 @r{ @: C-x * e @: @: 34 @:calc-embedded@:}
35804 @r{ @: C-x * f @:formula @: @:calc-embedded-new-formula@:}
35805 @r{ @: C-x * g @: @: 35 @:calc-grab-region@:}
35806 @r{ @: C-x * i @: @: @:calc-info@:}
35807 @r{ @: C-x * j @: @: @:calc-embedded-select@:}
35808 @r{ @: C-x * k @: @: @:calc-keypad@:}
35809 @r{ @: C-x * l @: @: @:calc-load-everything@:}
35810 @r{ @: C-x * m @: @: @:read-kbd-macro@:}
35811 @r{ @: C-x * n @: @: 4 @:calc-embedded-next@:}
35812 @r{ @: C-x * o @: @: @:calc-other-window@:}
35813 @r{ @: C-x * p @: @: 4 @:calc-embedded-previous@:}
35814 @r{ @: C-x * q @:formula @: @:quick-calc@:}
35815 @r{ @: C-x * r @: @: 36 @:calc-grab-rectangle@:}
35816 @r{ @: C-x * s @: @: @:calc-info-summary@:}
35817 @r{ @: C-x * t @: @: @:calc-tutorial@:}
35818 @r{ @: C-x * u @: @: @:calc-embedded-update-formula@:}
35819 @r{ @: C-x * w @: @: @:calc-embedded-word@:}
35820 @r{ @: C-x * x @: @: @:calc-quit@:}
35821 @r{ @: C-x * y @: @:1,28,49 @:calc-copy-to-buffer@:}
35822 @r{ @: C-x * z @: @: @:calc-user-invocation@:}
35823 @r{ @: C-x * : @: @: 36 @:calc-grab-sum-down@:}
35824 @r{ @: C-x * _ @: @: 36 @:calc-grab-sum-across@:}
35825 @r{ @: C-x * ` @:editing @: 30 @:calc-embedded-edit@:}
35826 @r{ @: C-x * 0 @:(zero) @: @:calc-reset@:}
35829 @r{ @: 0-9 @:number @: @:@:number}
35830 @r{ @: . @:number @: @:@:0.number}
35831 @r{ @: _ @:number @: @:-@:number}
35832 @r{ @: e @:number @: @:@:1e number}
35833 @r{ @: # @:number @: @:@:current-radix@tfn{#}number}
35834 @r{ @: P @:(in number) @: @:+/-@:}
35835 @r{ @: M @:(in number) @: @:mod@:}
35836 @r{ @: @@ ' " @: (in number)@: @:@:HMS form}
35837 @r{ @: h m s @: (in number)@: @:@:HMS form}
35840 @r{ @: ' @:formula @: 37,46 @:@:formula}
35841 @r{ @: $ @:formula @: 37,46 @:$@:formula}
35842 @r{ @: " @:string @: 37,46 @:@:string}
35845 @r{ a b@: + @: @: 2 @:add@:(a,b) a+b}
35846 @r{ a b@: - @: @: 2 @:sub@:(a,b) a@minus{}b}
35847 @r{ a b@: * @: @: 2 @:mul@:(a,b) a b, a*b}
35848 @r{ a b@: / @: @: 2 @:div@:(a,b) a/b}
35849 @r{ a b@: ^ @: @: 2 @:pow@:(a,b) a^b}
35850 @r{ a b@: I ^ @: @: 2 @:nroot@:(a,b) a^(1/b)}
35851 @r{ a b@: % @: @: 2 @:mod@:(a,b) a%b}
35852 @r{ a b@: \ @: @: 2 @:idiv@:(a,b) a\b}
35853 @r{ a b@: : @: @: 2 @:fdiv@:(a,b)}
35854 @r{ a b@: | @: @: 2 @:vconcat@:(a,b) a|b}
35855 @r{ a b@: I | @: @: @:vconcat@:(b,a) b|a}
35856 @r{ a b@: H | @: @: 2 @:append@:(a,b)}
35857 @r{ a b@: I H | @: @: @:append@:(b,a)}
35858 @r{ a@: & @: @: 1 @:inv@:(a) 1/a}
35859 @r{ a@: ! @: @: 1 @:fact@:(a) a!}
35860 @r{ a@: = @: @: 1 @:evalv@:(a)}
35861 @r{ a@: M-% @: @: @:percent@:(a) a%}
35864 @r{ ... a@: @summarykey{RET} @: @: 1 @:@:... a a}
35865 @r{ ... a@: @summarykey{SPC} @: @: 1 @:@:... a a}
35866 @r{... a b@: @summarykey{TAB} @: @: 3 @:@:... b a}
35867 @r{. a b c@: M-@summarykey{TAB} @: @: 3 @:@:... b c a}
35868 @r{... a b@: @summarykey{LFD} @: @: 1 @:@:... a b a}
35869 @r{ ... a@: @summarykey{DEL} @: @: 1 @:@:...}
35870 @r{... a b@: M-@summarykey{DEL} @: @: 1 @:@:... b}
35871 @r{ @: M-@summarykey{RET} @: @: 4 @:calc-last-args@:}
35872 @r{ a@: ` @:editing @: 1,30 @:calc-edit@:}
35875 @r{ ... a@: C-d @: @: 1 @:@:...}
35876 @r{ @: C-k @: @: 27 @:calc-kill@:}
35877 @r{ @: C-w @: @: 27 @:calc-kill-region@:}
35878 @r{ @: C-y @: @: @:calc-yank@:}
35879 @r{ @: C-_ @: @: 4 @:calc-undo@:}
35880 @r{ @: M-k @: @: 27 @:calc-copy-as-kill@:}
35881 @r{ @: M-w @: @: 27 @:calc-copy-region-as-kill@:}
35884 @r{ @: [ @: @: @:@:[...}
35885 @r{[.. a b@: ] @: @: @:@:[a,b]}
35886 @r{ @: ( @: @: @:@:(...}
35887 @r{(.. a b@: ) @: @: @:@:(a,b)}
35888 @r{ @: , @: @: @:@:vector or rect complex}
35889 @r{ @: ; @: @: @:@:matrix or polar complex}
35890 @r{ @: .. @: @: @:@:interval}
35893 @r{ @: ~ @: @: @:calc-num-prefix@:}
35894 @r{ @: < @: @: 4 @:calc-scroll-left@:}
35895 @r{ @: > @: @: 4 @:calc-scroll-right@:}
35896 @r{ @: @{ @: @: 4 @:calc-scroll-down@:}
35897 @r{ @: @} @: @: 4 @:calc-scroll-up@:}
35898 @r{ @: ? @: @: @:calc-help@:}
35901 @r{ a@: n @: @: 1 @:neg@:(a) @minus{}a}
35902 @r{ @: o @: @: 4 @:calc-realign@:}
35903 @r{ @: p @:precision @: 31 @:calc-precision@:}
35904 @r{ @: q @: @: @:calc-quit@:}
35905 @r{ @: w @: @: @:calc-why@:}
35906 @r{ @: x @:command @: @:M-x calc-@:command}
35907 @r{ a@: y @: @:1,28,49 @:calc-copy-to-buffer@:}
35910 @r{ a@: A @: @: 1 @:abs@:(a)}
35911 @r{ a b@: B @: @: 2 @:log@:(a,b)}
35912 @r{ a b@: I B @: @: 2 @:alog@:(a,b) b^a}
35913 @r{ a@: C @: @: 1 @:cos@:(a)}
35914 @r{ a@: I C @: @: 1 @:arccos@:(a)}
35915 @r{ a@: H C @: @: 1 @:cosh@:(a)}
35916 @r{ a@: I H C @: @: 1 @:arccosh@:(a)}
35917 @r{ @: D @: @: 4 @:calc-redo@:}
35918 @r{ a@: E @: @: 1 @:exp@:(a)}
35919 @r{ a@: H E @: @: 1 @:exp10@:(a) 10.^a}
35920 @r{ a@: F @: @: 1,11 @:floor@:(a,d)}
35921 @r{ a@: I F @: @: 1,11 @:ceil@:(a,d)}
35922 @r{ a@: H F @: @: 1,11 @:ffloor@:(a,d)}
35923 @r{ a@: I H F @: @: 1,11 @:fceil@:(a,d)}
35924 @r{ a@: G @: @: 1 @:arg@:(a)}
35925 @r{ @: H @:command @: 32 @:@:Hyperbolic}
35926 @r{ @: I @:command @: 32 @:@:Inverse}
35927 @r{ a@: J @: @: 1 @:conj@:(a)}
35928 @r{ @: K @:command @: 32 @:@:Keep-args}
35929 @r{ a@: L @: @: 1 @:ln@:(a)}
35930 @r{ a@: H L @: @: 1 @:log10@:(a)}
35931 @r{ @: M @: @: @:calc-more-recursion-depth@:}
35932 @r{ @: I M @: @: @:calc-less-recursion-depth@:}
35933 @r{ a@: N @: @: 5 @:evalvn@:(a)}
35934 @r{ @: O @:command @: 32 @:@:Option}
35935 @r{ @: P @: @: @:@:pi}
35936 @r{ @: I P @: @: @:@:gamma}
35937 @r{ @: H P @: @: @:@:e}
35938 @r{ @: I H P @: @: @:@:phi}
35939 @r{ a@: Q @: @: 1 @:sqrt@:(a)}
35940 @r{ a@: I Q @: @: 1 @:sqr@:(a) a^2}
35941 @r{ a@: R @: @: 1,11 @:round@:(a,d)}
35942 @r{ a@: I R @: @: 1,11 @:trunc@:(a,d)}
35943 @r{ a@: H R @: @: 1,11 @:fround@:(a,d)}
35944 @r{ a@: I H R @: @: 1,11 @:ftrunc@:(a,d)}
35945 @r{ a@: S @: @: 1 @:sin@:(a)}
35946 @r{ a@: I S @: @: 1 @:arcsin@:(a)}
35947 @r{ a@: H S @: @: 1 @:sinh@:(a)}
35948 @r{ a@: I H S @: @: 1 @:arcsinh@:(a)}
35949 @r{ a@: T @: @: 1 @:tan@:(a)}
35950 @r{ a@: I T @: @: 1 @:arctan@:(a)}
35951 @r{ a@: H T @: @: 1 @:tanh@:(a)}
35952 @r{ a@: I H T @: @: 1 @:arctanh@:(a)}
35953 @r{ @: U @: @: 4 @:calc-undo@:}
35954 @r{ @: X @: @: 4 @:calc-call-last-kbd-macro@:}
35957 @r{ a b@: a = @: @: 2 @:eq@:(a,b) a=b}
35958 @r{ a b@: a # @: @: 2 @:neq@:(a,b) a!=b}
35959 @r{ a b@: a < @: @: 2 @:lt@:(a,b) a<b}
35960 @r{ a b@: a > @: @: 2 @:gt@:(a,b) a>b}
35961 @r{ a b@: a [ @: @: 2 @:leq@:(a,b) a<=b}
35962 @r{ a b@: a ] @: @: 2 @:geq@:(a,b) a>=b}
35963 @r{ a b@: a @{ @: @: 2 @:in@:(a,b)}
35964 @r{ a b@: a & @: @: 2,45 @:land@:(a,b) a&&b}
35965 @r{ a b@: a | @: @: 2,45 @:lor@:(a,b) a||b}
35966 @r{ a@: a ! @: @: 1,45 @:lnot@:(a) !a}
35967 @r{ a b c@: a : @: @: 45 @:if@:(a,b,c) a?b:c}
35968 @r{ a@: a . @: @: 1 @:rmeq@:(a)}
35969 @r{ a@: a " @: @: 7,8 @:calc-expand-formula@:}
35972 @r{ a@: a + @:i, l, h @: 6,38 @:sum@:(a,i,l,h)}
35973 @r{ a@: a - @:i, l, h @: 6,38 @:asum@:(a,i,l,h)}
35974 @r{ a@: a * @:i, l, h @: 6,38 @:prod@:(a,i,l,h)}
35975 @r{ a b@: a _ @: @: 2 @:subscr@:(a,b) a_b}
35978 @r{ a b@: a \ @: @: 2 @:pdiv@:(a,b)}
35979 @r{ a b@: a % @: @: 2 @:prem@:(a,b)}
35980 @r{ a b@: a / @: @: 2 @:pdivrem@:(a,b) [q,r]}
35981 @r{ a b@: H a / @: @: 2 @:pdivide@:(a,b) q+r/b}
35984 @r{ a@: a a @: @: 1 @:apart@:(a)}
35985 @r{ a@: a b @:old, new @: 38 @:subst@:(a,old,new)}
35986 @r{ a@: a c @:v @: 38 @:collect@:(a,v)}
35987 @r{ a@: a d @:v @: 4,38 @:deriv@:(a,v)}
35988 @r{ a@: H a d @:v @: 4,38 @:tderiv@:(a,v)}
35989 @r{ a@: a e @: @: @:esimplify@:(a)}
35990 @r{ a@: a f @: @: 1 @:factor@:(a)}
35991 @r{ a@: H a f @: @: 1 @:factors@:(a)}
35992 @r{ a b@: a g @: @: 2 @:pgcd@:(a,b)}
35993 @r{ a@: a i @:v @: 38 @:integ@:(a,v)}
35994 @r{ a@: a m @:pats @: 38 @:match@:(a,pats)}
35995 @r{ a@: I a m @:pats @: 38 @:matchnot@:(a,pats)}
35996 @r{ data x@: a p @: @: 28 @:polint@:(data,x)}
35997 @r{ data x@: H a p @: @: 28 @:ratint@:(data,x)}
35998 @r{ a@: a n @: @: 1 @:nrat@:(a)}
35999 @r{ a@: a r @:rules @:4,8,38 @:rewrite@:(a,rules,n)}
36000 @r{ a@: a s @: @: @:simplify@:(a)}
36001 @r{ a@: a t @:v, n @: 31,39 @:taylor@:(a,v,n)}
36002 @r{ a@: a v @: @: 7,8 @:calc-alg-evaluate@:}
36003 @r{ a@: a x @: @: 4,8 @:expand@:(a)}
36006 @r{ data@: a F @:model, vars @: 48 @:fit@:(m,iv,pv,data)}
36007 @r{ data@: I a F @:model, vars @: 48 @:xfit@:(m,iv,pv,data)}
36008 @r{ data@: H a F @:model, vars @: 48 @:efit@:(m,iv,pv,data)}
36009 @r{ a@: a I @:v, l, h @: 38 @:ninteg@:(a,v,l,h)}
36010 @r{ a b@: a M @:op @: 22 @:mapeq@:(op,a,b)}
36011 @r{ a b@: I a M @:op @: 22 @:mapeqr@:(op,a,b)}
36012 @r{ a b@: H a M @:op @: 22 @:mapeqp@:(op,a,b)}
36013 @r{ a g@: a N @:v @: 38 @:minimize@:(a,v,g)}
36014 @r{ a g@: H a N @:v @: 38 @:wminimize@:(a,v,g)}
36015 @r{ a@: a P @:v @: 38 @:roots@:(a,v)}
36016 @r{ a g@: a R @:v @: 38 @:root@:(a,v,g)}
36017 @r{ a g@: H a R @:v @: 38 @:wroot@:(a,v,g)}
36018 @r{ a@: a S @:v @: 38 @:solve@:(a,v)}
36019 @r{ a@: I a S @:v @: 38 @:finv@:(a,v)}
36020 @r{ a@: H a S @:v @: 38 @:fsolve@:(a,v)}
36021 @r{ a@: I H a S @:v @: 38 @:ffinv@:(a,v)}
36022 @r{ a@: a T @:i, l, h @: 6,38 @:table@:(a,i,l,h)}
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36362 @r{ @: m x @: @: @:calc-always-load-extensions@:}
36365 @r{ @: m A @: @: 12 @:calc-alg-simplify-mode@:}
36366 @r{ @: m B @: @: 12 @:calc-bin-simplify-mode@:}
36367 @r{ @: m C @: @: 12 @:calc-auto-recompute@:}
36368 @r{ @: m D @: @: @:calc-default-simplify-mode@:}
36369 @r{ @: m E @: @: 12 @:calc-ext-simplify-mode@:}
36370 @r{ @: m F @:filename @: 13 @:calc-settings-file-name@:}
36371 @r{ @: m N @: @: 12 @:calc-num-simplify-mode@:}
36372 @r{ @: m O @: @: 12 @:calc-no-simplify-mode@:}
36373 @r{ @: m R @: @: 12,13 @:calc-mode-record-mode@:}
36374 @r{ @: m S @: @: 12 @:calc-shift-prefix@:}
36375 @r{ @: m U @: @: 12 @:calc-units-simplify-mode@:}
36378 @r{ @: r s @:register @: 27 @:calc-copy-to-register@:}
36379 @r{ @: r i @:register @: @:calc-insert-register@:}
36382 @r{ @: s c @:var1, var2 @: 29 @:calc-copy-variable@:}
36383 @r{ @: s d @:var, decl @: @:calc-declare-variable@:}
36384 @r{ @: s e @:var, editing @: 29,30 @:calc-edit-variable@:}
36385 @r{ @: s i @:buffer @: @:calc-insert-variables@:}
36386 @r{ @: s k @:const, var @: 29 @:calc-copy-special-constant@:}
36387 @r{ a b@: s l @:var @: 29 @:@:a (letting var=b)}
36388 @r{ a ...@: s m @:op, var @: 22,29 @:calc-store-map@:}
36389 @r{ @: s n @:var @: 29,47 @:calc-store-neg@: (v/-1)}
36390 @r{ @: s p @:var @: 29 @:calc-permanent-variable@:}
36391 @r{ @: s r @:var @: 29 @:@:v (recalled value)}
36392 @r{ @: r 0-9 @: @: @:calc-recall-quick@:}
36393 @r{ a@: s s @:var @: 28,29 @:calc-store@:}
36394 @r{ a@: s 0-9 @: @: @:calc-store-quick@:}
36395 @r{ a@: s t @:var @: 29 @:calc-store-into@:}
36396 @r{ a@: t 0-9 @: @: @:calc-store-into-quick@:}
36397 @r{ @: s u @:var @: 29 @:calc-unstore@:}
36398 @r{ a@: s x @:var @: 29 @:calc-store-exchange@:}
36401 @r{ @: s A @:editing @: 30 @:calc-edit-AlgSimpRules@:}
36402 @r{ @: s D @:editing @: 30 @:calc-edit-Decls@:}
36403 @r{ @: s E @:editing @: 30 @:calc-edit-EvalRules@:}
36404 @r{ @: s F @:editing @: 30 @:calc-edit-FitRules@:}
36405 @r{ @: s G @:editing @: 30 @:calc-edit-GenCount@:}
36406 @r{ @: s H @:editing @: 30 @:calc-edit-Holidays@:}
36407 @r{ @: s I @:editing @: 30 @:calc-edit-IntegLimit@:}
36408 @r{ @: s L @:editing @: 30 @:calc-edit-LineStyles@:}
36409 @r{ @: s P @:editing @: 30 @:calc-edit-PointStyles@:}
36410 @r{ @: s R @:editing @: 30 @:calc-edit-PlotRejects@:}
36411 @r{ @: s T @:editing @: 30 @:calc-edit-TimeZone@:}
36412 @r{ @: s U @:editing @: 30 @:calc-edit-Units@:}
36413 @r{ @: s X @:editing @: 30 @:calc-edit-ExtSimpRules@:}
36416 @r{ a@: s + @:var @: 29,47 @:calc-store-plus@: (v+a)}
36417 @r{ a@: s - @:var @: 29,47 @:calc-store-minus@: (v-a)}
36418 @r{ a@: s * @:var @: 29,47 @:calc-store-times@: (v*a)}
36419 @r{ a@: s / @:var @: 29,47 @:calc-store-div@: (v/a)}
36420 @r{ a@: s ^ @:var @: 29,47 @:calc-store-power@: (v^a)}
36421 @r{ a@: s | @:var @: 29,47 @:calc-store-concat@: (v|a)}
36422 @r{ @: s & @:var @: 29,47 @:calc-store-inv@: (v^-1)}
36423 @r{ @: s [ @:var @: 29,47 @:calc-store-decr@: (v-1)}
36424 @r{ @: s ] @:var @: 29,47 @:calc-store-incr@: (v-(-1))}
36425 @r{ a b@: s : @: @: 2 @:assign@:(a,b) a @tfn{:=} b}
36426 @r{ a@: s = @: @: 1 @:evalto@:(a,b) a @tfn{=>}}
36429 @r{ @: t [ @: @: 4 @:calc-trail-first@:}
36430 @r{ @: t ] @: @: 4 @:calc-trail-last@:}
36431 @r{ @: t < @: @: 4 @:calc-trail-scroll-left@:}
36432 @r{ @: t > @: @: 4 @:calc-trail-scroll-right@:}
36433 @r{ @: t . @: @: 12 @:calc-full-trail-vectors@:}
36436 @r{ @: t b @: @: 4 @:calc-trail-backward@:}
36437 @r{ @: t d @: @: 12,50 @:calc-trail-display@:}
36438 @r{ @: t f @: @: 4 @:calc-trail-forward@:}
36439 @r{ @: t h @: @: @:calc-trail-here@:}
36440 @r{ @: t i @: @: @:calc-trail-in@:}
36441 @r{ @: t k @: @: 4 @:calc-trail-kill@:}
36442 @r{ @: t m @:string @: @:calc-trail-marker@:}
36443 @r{ @: t n @: @: 4 @:calc-trail-next@:}
36444 @r{ @: t o @: @: @:calc-trail-out@:}
36445 @r{ @: t p @: @: 4 @:calc-trail-previous@:}
36446 @r{ @: t r @:string @: @:calc-trail-isearch-backward@:}
36447 @r{ @: t s @:string @: @:calc-trail-isearch-forward@:}
36448 @r{ @: t y @: @: 4 @:calc-trail-yank@:}
36451 @r{ d@: t C @:oz, nz @: @:tzconv@:(d,oz,nz)}
36452 @r{d oz nz@: t C @:$ @: @:tzconv@:(d,oz,nz)}
36453 @r{ d@: t D @: @: 15 @:date@:(d)}
36454 @r{ d@: t I @: @: 4 @:incmonth@:(d,n)}
36455 @r{ d@: t J @: @: 16 @:julian@:(d,z)}
36456 @r{ d@: t M @: @: 17 @:newmonth@:(d,n)}
36457 @r{ @: t N @: @: 16 @:now@:(z)}
36458 @r{ d@: t P @:1 @: 31 @:year@:(d)}
36459 @r{ d@: t P @:2 @: 31 @:month@:(d)}
36460 @r{ d@: t P @:3 @: 31 @:day@:(d)}
36461 @r{ d@: t P @:4 @: 31 @:hour@:(d)}
36462 @r{ d@: t P @:5 @: 31 @:minute@:(d)}
36463 @r{ d@: t P @:6 @: 31 @:second@:(d)}
36464 @r{ d@: t P @:7 @: 31 @:weekday@:(d)}
36465 @r{ d@: t P @:8 @: 31 @:yearday@:(d)}
36466 @r{ d@: t P @:9 @: 31 @:time@:(d)}
36467 @r{ d@: t U @: @: 16 @:unixtime@:(d,z)}
36468 @r{ d@: t W @: @: 17 @:newweek@:(d,w)}
36469 @r{ d@: t Y @: @: 17 @:newyear@:(d,n)}
36472 @r{ a b@: t + @: @: 2 @:badd@:(a,b)}
36473 @r{ a b@: t - @: @: 2 @:bsub@:(a,b)}
36476 @r{ @: u a @: @: 12 @:calc-autorange-units@:}
36477 @r{ a@: u b @: @: @:calc-base-units@:}
36478 @r{ a@: u c @:units @: 18 @:calc-convert-units@:}
36479 @r{ defn@: u d @:unit, descr @: @:calc-define-unit@:}
36480 @r{ @: u e @: @: @:calc-explain-units@:}
36481 @r{ @: u g @:unit @: @:calc-get-unit-definition@:}
36482 @r{ @: u p @: @: @:calc-permanent-units@:}
36483 @r{ a@: u r @: @: @:calc-remove-units@:}
36484 @r{ a@: u s @: @: @:usimplify@:(a)}
36485 @r{ a@: u t @:units @: 18 @:calc-convert-temperature@:}
36486 @r{ @: u u @:unit @: @:calc-undefine-unit@:}
36487 @r{ @: u v @: @: @:calc-enter-units-table@:}
36488 @r{ a@: u x @: @: @:calc-extract-units@:}
36489 @r{ a@: u 0-9 @: @: @:calc-quick-units@:}
36492 @r{ v1 v2@: u C @: @: 20 @:vcov@:(v1,v2)}
36493 @r{ v1 v2@: I u C @: @: 20 @:vpcov@:(v1,v2)}
36494 @r{ v1 v2@: H u C @: @: 20 @:vcorr@:(v1,v2)}
36495 @r{ v@: u G @: @: 19 @:vgmean@:(v)}
36496 @r{ a b@: H u G @: @: 2 @:agmean@:(a,b)}
36497 @r{ v@: u M @: @: 19 @:vmean@:(v)}
36498 @r{ v@: I u M @: @: 19 @:vmeane@:(v)}
36499 @r{ v@: H u M @: @: 19 @:vmedian@:(v)}
36500 @r{ v@: I H u M @: @: 19 @:vhmean@:(v)}
36501 @r{ v@: u N @: @: 19 @:vmin@:(v)}
36502 @r{ v@: u S @: @: 19 @:vsdev@:(v)}
36503 @r{ v@: I u S @: @: 19 @:vpsdev@:(v)}
36504 @r{ v@: H u S @: @: 19 @:vvar@:(v)}
36505 @r{ v@: I H u S @: @: 19 @:vpvar@:(v)}
36506 @r{ @: u V @: @: @:calc-view-units-table@:}
36507 @r{ v@: u X @: @: 19 @:vmax@:(v)}
36510 @r{ v@: u + @: @: 19 @:vsum@:(v)}
36511 @r{ v@: u * @: @: 19 @:vprod@:(v)}
36512 @r{ v@: u # @: @: 19 @:vcount@:(v)}
36515 @r{ @: V ( @: @: 50 @:calc-vector-parens@:}
36516 @r{ @: V @{ @: @: 50 @:calc-vector-braces@:}
36517 @r{ @: V [ @: @: 50 @:calc-vector-brackets@:}
36518 @r{ @: V ] @:ROCP @: 50 @:calc-matrix-brackets@:}
36519 @r{ @: V , @: @: 50 @:calc-vector-commas@:}
36520 @r{ @: V < @: @: 50 @:calc-matrix-left-justify@:}
36521 @r{ @: V = @: @: 50 @:calc-matrix-center-justify@:}
36522 @r{ @: V > @: @: 50 @:calc-matrix-right-justify@:}
36523 @r{ @: V / @: @: 12,50 @:calc-break-vectors@:}
36524 @r{ @: V . @: @: 12,50 @:calc-full-vectors@:}
36527 @r{ s t@: V ^ @: @: 2 @:vint@:(s,t)}
36528 @r{ s t@: V - @: @: 2 @:vdiff@:(s,t)}
36529 @r{ s@: V ~ @: @: 1 @:vcompl@:(s)}
36530 @r{ s@: V # @: @: 1 @:vcard@:(s)}
36531 @r{ s@: V : @: @: 1 @:vspan@:(s)}
36532 @r{ s@: V + @: @: 1 @:rdup@:(s)}
36535 @r{ m@: V & @: @: 1 @:inv@:(m) 1/m}
36538 @r{ v@: v a @:n @: @:arrange@:(v,n)}
36539 @r{ a@: v b @:n @: @:cvec@:(a,n)}
36540 @r{ v@: v c @:n >0 @: 21,31 @:mcol@:(v,n)}
36541 @r{ v@: v c @:n <0 @: 31 @:mrcol@:(v,-n)}
36542 @r{ m@: v c @:0 @: 31 @:getdiag@:(m)}
36543 @r{ v@: v d @: @: 25 @:diag@:(v,n)}
36544 @r{ v m@: v e @: @: 2 @:vexp@:(v,m)}
36545 @r{ v m f@: H v e @: @: 2 @:vexp@:(v,m,f)}
36546 @r{ v a@: v f @: @: 26 @:find@:(v,a,n)}
36547 @r{ v@: v h @: @: 1 @:head@:(v)}
36548 @r{ v@: I v h @: @: 1 @:tail@:(v)}
36549 @r{ v@: H v h @: @: 1 @:rhead@:(v)}
36550 @r{ v@: I H v h @: @: 1 @:rtail@:(v)}
36551 @r{ @: v i @:n @: 31 @:idn@:(1,n)}
36552 @r{ @: v i @:0 @: 31 @:idn@:(1)}
36553 @r{ h t@: v k @: @: 2 @:cons@:(h,t)}
36554 @r{ h t@: H v k @: @: 2 @:rcons@:(h,t)}
36555 @r{ v@: v l @: @: 1 @:vlen@:(v)}
36556 @r{ v@: H v l @: @: 1 @:mdims@:(v)}
36557 @r{ v m@: v m @: @: 2 @:vmask@:(v,m)}
36558 @r{ v@: v n @: @: 1 @:rnorm@:(v)}
36559 @r{ a b c@: v p @: @: 24 @:calc-pack@:}
36560 @r{ v@: v r @:n >0 @: 21,31 @:mrow@:(v,n)}
36561 @r{ v@: v r @:n <0 @: 31 @:mrrow@:(v,-n)}
36562 @r{ m@: v r @:0 @: 31 @:getdiag@:(m)}
36563 @r{ v i j@: v s @: @: @:subvec@:(v,i,j)}
36564 @r{ v i j@: I v s @: @: @:rsubvec@:(v,i,j)}
36565 @r{ m@: v t @: @: 1 @:trn@:(m)}
36566 @r{ v@: v u @: @: 24 @:calc-unpack@:}
36567 @r{ v@: v v @: @: 1 @:rev@:(v)}
36568 @r{ @: v x @:n @: 31 @:index@:(n)}
36569 @r{ n s i@: C-u v x @: @: @:index@:(n,s,i)}
36572 @r{ v@: V A @:op @: 22 @:apply@:(op,v)}
36573 @r{ v1 v2@: V C @: @: 2 @:cross@:(v1,v2)}
36574 @r{ m@: V D @: @: 1 @:det@:(m)}
36575 @r{ s@: V E @: @: 1 @:venum@:(s)}
36576 @r{ s@: V F @: @: 1 @:vfloor@:(s)}
36577 @r{ v@: V G @: @: @:grade@:(v)}
36578 @r{ v@: I V G @: @: @:rgrade@:(v)}
36579 @r{ v@: V H @:n @: 31 @:histogram@:(v,n)}
36580 @r{ v w@: H V H @:n @: 31 @:histogram@:(v,w,n)}
36581 @r{ v1 v2@: V I @:mop aop @: 22 @:inner@:(mop,aop,v1,v2)}
36582 @r{ m@: V J @: @: 1 @:ctrn@:(m)}
36583 @r{ m1 m2@: V K @: @: @:kron@:(m1,m2)}
36584 @r{ m@: V L @: @: 1 @:lud@:(m)}
36585 @r{ v@: V M @:op @: 22,23 @:map@:(op,v)}
36586 @r{ v@: V N @: @: 1 @:cnorm@:(v)}
36587 @r{ v1 v2@: V O @:op @: 22 @:outer@:(op,v1,v2)}
36588 @r{ v@: V R @:op @: 22,23 @:reduce@:(op,v)}
36589 @r{ v@: I V R @:op @: 22,23 @:rreduce@:(op,v)}
36590 @r{ a n@: H V R @:op @: 22 @:nest@:(op,a,n)}
36591 @r{ a@: I H V R @:op @: 22 @:fixp@:(op,a)}
36592 @r{ v@: V S @: @: @:sort@:(v)}
36593 @r{ v@: I V S @: @: @:rsort@:(v)}
36594 @r{ m@: V T @: @: 1 @:tr@:(m)}
36595 @r{ v@: V U @:op @: 22 @:accum@:(op,v)}
36596 @r{ v@: I V U @:op @: 22 @:raccum@:(op,v)}
36597 @r{ a n@: H V U @:op @: 22 @:anest@:(op,a,n)}
36598 @r{ a@: I H V U @:op @: 22 @:afixp@:(op,a)}
36599 @r{ s t@: V V @: @: 2 @:vunion@:(s,t)}
36600 @r{ s t@: V X @: @: 2 @:vxor@:(s,t)}
36603 @r{ @: Y @: @: @:@:user commands}
36606 @r{ @: z @: @: @:@:user commands}
36609 @r{ c@: Z [ @: @: 45 @:calc-kbd-if@:}
36610 @r{ c@: Z | @: @: 45 @:calc-kbd-else-if@:}
36611 @r{ @: Z : @: @: @:calc-kbd-else@:}
36612 @r{ @: Z ] @: @: @:calc-kbd-end-if@:}
36615 @r{ @: Z @{ @: @: 4 @:calc-kbd-loop@:}
36616 @r{ c@: Z / @: @: 45 @:calc-kbd-break@:}
36617 @r{ @: Z @} @: @: @:calc-kbd-end-loop@:}
36618 @r{ n@: Z < @: @: @:calc-kbd-repeat@:}
36619 @r{ @: Z > @: @: @:calc-kbd-end-repeat@:}
36620 @r{ n m@: Z ( @: @: @:calc-kbd-for@:}
36621 @r{ s@: Z ) @: @: @:calc-kbd-end-for@:}
36624 @r{ @: Z C-g @: @: @:@:cancel if/loop command}
36627 @r{ @: Z ` @: @: @:calc-kbd-push@:}
36628 @r{ @: Z ' @: @: @:calc-kbd-pop@:}
36629 @r{ @: Z # @: @: @:calc-kbd-query@:}
36632 @r{ comp@: Z C @:func, args @: 50 @:calc-user-define-composition@:}
36633 @r{ @: Z D @:key, command @: @:calc-user-define@:}
36634 @r{ @: Z E @:key, editing @: 30 @:calc-user-define-edit@:}
36635 @r{ defn@: Z F @:k, c, f, a, n@: 28 @:calc-user-define-formula@:}
36636 @r{ @: Z G @:key @: @:calc-get-user-defn@:}
36637 @r{ @: Z I @: @: @:calc-user-define-invocation@:}
36638 @r{ @: Z K @:key, command @: @:calc-user-define-kbd-macro@:}
36639 @r{ @: Z P @:key @: @:calc-user-define-permanent@:}
36640 @r{ @: Z S @: @: 30 @:calc-edit-user-syntax@:}
36641 @r{ @: Z T @: @: 12 @:calc-timing@:}
36642 @r{ @: Z U @:key @: @:calc-user-undefine@:}
36652 Positive prefix arguments apply to @expr{n} stack entries.
36653 Negative prefix arguments apply to the @expr{-n}th stack entry.
36654 A prefix of zero applies to the entire stack. (For @key{LFD} and
36655 @kbd{M-@key{DEL}}, the meaning of the sign is reversed.)
36659 Positive prefix arguments apply to @expr{n} stack entries.
36660 Negative prefix arguments apply to the top stack entry
36661 and the next @expr{-n} stack entries.
36665 Positive prefix arguments rotate top @expr{n} stack entries by one.
36666 Negative prefix arguments rotate the entire stack by @expr{-n}.
36667 A prefix of zero reverses the entire stack.
36671 Prefix argument specifies a repeat count or distance.
36675 Positive prefix arguments specify a precision @expr{p}.
36676 Negative prefix arguments reduce the current precision by @expr{-p}.
36680 A prefix argument is interpreted as an additional step-size parameter.
36681 A plain @kbd{C-u} prefix means to prompt for the step size.
36685 A prefix argument specifies simplification level and depth.
36686 1=Basic simplifications, 2=Algebraic simplifications, 3=Extended simplifications
36690 A negative prefix operates only on the top level of the input formula.
36694 Positive prefix arguments specify a word size of @expr{w} bits, unsigned.
36695 Negative prefix arguments specify a word size of @expr{w} bits, signed.
36699 Prefix arguments specify the shift amount @expr{n}. The @expr{w} argument
36700 cannot be specified in the keyboard version of this command.
36704 From the keyboard, @expr{d} is omitted and defaults to zero.
36708 Mode is toggled; a positive prefix always sets the mode, and a negative
36709 prefix always clears the mode.
36713 Some prefix argument values provide special variations of the mode.
36717 A prefix argument, if any, is used for @expr{m} instead of taking
36718 @expr{m} from the stack. @expr{M} may take any of these values:
36720 {@advance@tableindent10pt
36724 Random integer in the interval @expr{[0 .. m)}.
36726 Random floating-point number in the interval @expr{[0 .. m)}.
36728 Gaussian with mean 1 and standard deviation 0.
36730 Gaussian with specified mean and standard deviation.
36732 Random integer or floating-point number in that interval.
36734 Random element from the vector.
36742 A prefix argument from 1 to 6 specifies number of date components
36743 to remove from the stack. @xref{Date Conversions}.
36747 A prefix argument specifies a time zone; @kbd{C-u} says to take the
36748 time zone number or name from the top of the stack. @xref{Time Zones}.
36752 A prefix argument specifies a day number (0-6, 0-31, or 0-366).
36756 If the input has no units, you will be prompted for both the old and
36761 With a prefix argument, collect that many stack entries to form the
36762 input data set. Each entry may be a single value or a vector of values.
36766 With a prefix argument of 1, take a single
36767 @texline @var{n}@math{\times2}
36768 @infoline @mathit{@var{N}x2}
36769 matrix from the stack instead of two separate data vectors.
36773 The row or column number @expr{n} may be given as a numeric prefix
36774 argument instead. A plain @kbd{C-u} prefix says to take @expr{n}
36775 from the top of the stack. If @expr{n} is a vector or interval,
36776 a subvector/submatrix of the input is created.
36780 The @expr{op} prompt can be answered with the key sequence for the
36781 desired function, or with @kbd{x} or @kbd{z} followed by a function name,
36782 or with @kbd{$} to take a formula from the top of the stack, or with
36783 @kbd{'} and a typed formula. In the last two cases, the formula may
36784 be a nameless function like @samp{<#1+#2>} or @samp{<x, y : x+y>}, or it
36785 may include @kbd{$}, @kbd{$$}, etc. (where @kbd{$} will correspond to the
36786 last argument of the created function), or otherwise you will be
36787 prompted for an argument list. The number of vectors popped from the
36788 stack by @kbd{V M} depends on the number of arguments of the function.
36792 One of the mapping direction keys @kbd{_} (horizontal, i.e., map
36793 by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or
36794 reduce down), or @kbd{=} (map or reduce by rows) may be used before
36795 entering @expr{op}; these modify the function name by adding the letter
36796 @code{r} for ``rows,'' @code{c} for ``columns,'' @code{a} for ``across,''
36797 or @code{d} for ``down.''
36801 The prefix argument specifies a packing mode. A nonnegative mode
36802 is the number of items (for @kbd{v p}) or the number of levels
36803 (for @kbd{v u}). A negative mode is as described below. With no
36804 prefix argument, the mode is taken from the top of the stack and
36805 may be an integer or a vector of integers.
36807 {@advance@tableindent-20pt
36811 (@var{2}) Rectangular complex number.
36813 (@var{2}) Polar complex number.
36815 (@var{3}) HMS form.
36817 (@var{2}) Error form.
36819 (@var{2}) Modulo form.
36821 (@var{2}) Closed interval.
36823 (@var{2}) Closed .. open interval.
36825 (@var{2}) Open .. closed interval.
36827 (@var{2}) Open interval.
36829 (@var{2}) Fraction.
36831 (@var{2}) Float with integer mantissa.
36833 (@var{2}) Float with mantissa in @expr{[1 .. 10)}.
36835 (@var{1}) Date form (using date numbers).
36837 (@var{3}) Date form (using year, month, day).
36839 (@var{6}) Date form (using year, month, day, hour, minute, second).
36847 A prefix argument specifies the size @expr{n} of the matrix. With no
36848 prefix argument, @expr{n} is omitted and the size is inferred from
36853 The prefix argument specifies the starting position @expr{n} (default 1).
36857 Cursor position within stack buffer affects this command.
36861 Arguments are not actually removed from the stack by this command.
36865 Variable name may be a single digit or a full name.
36869 Editing occurs in a separate buffer. Press @kbd{C-c C-c} (or
36870 @key{LFD}, or in some cases @key{RET}) to finish the edit, or kill the
36871 buffer with @kbd{C-x k} to cancel the edit. The @key{LFD} key prevents evaluation
36872 of the result of the edit.
36876 The number prompted for can also be provided as a prefix argument.
36880 Press this key a second time to cancel the prefix.
36884 With a negative prefix, deactivate all formulas. With a positive
36885 prefix, deactivate and then reactivate from scratch.
36889 Default is to scan for nearest formula delimiter symbols. With a
36890 prefix of zero, formula is delimited by mark and point. With a
36891 non-zero prefix, formula is delimited by scanning forward or
36892 backward by that many lines.
36896 Parse the region between point and mark as a vector. A nonzero prefix
36897 parses @var{n} lines before or after point as a vector. A zero prefix
36898 parses the current line as a vector. A @kbd{C-u} prefix parses the
36899 region between point and mark as a single formula.
36903 Parse the rectangle defined by point and mark as a matrix. A positive
36904 prefix @var{n} divides the rectangle into columns of width @var{n}.
36905 A zero or @kbd{C-u} prefix parses each line as one formula. A negative
36906 prefix suppresses special treatment of bracketed portions of a line.
36910 A numeric prefix causes the current language mode to be ignored.
36914 Responding to a prompt with a blank line answers that and all
36915 later prompts by popping additional stack entries.
36919 Answer for @expr{v} may also be of the form @expr{v = v_0} or
36924 With a positive prefix argument, stack contains many @expr{y}'s and one
36925 common @expr{x}. With a zero prefix, stack contains a vector of
36926 @expr{y}s and a common @expr{x}. With a negative prefix, stack
36927 contains many @expr{[x,y]} vectors. (For 3D plots, substitute
36928 @expr{z} for @expr{y} and @expr{x,y} for @expr{x}.)
36932 With any prefix argument, all curves in the graph are deleted.
36936 With a positive prefix, refines an existing plot with more data points.
36937 With a negative prefix, forces recomputation of the plot data.
36941 With any prefix argument, set the default value instead of the
36942 value for this graph.
36946 With a negative prefix argument, set the value for the printer.
36950 Condition is considered ``true'' if it is a nonzero real or complex
36951 number, or a formula whose value is known to be nonzero; it is ``false''
36956 Several formulas separated by commas are pushed as multiple stack
36957 entries. Trailing @kbd{)}, @kbd{]}, @kbd{@}}, @kbd{>}, and @kbd{"}
36958 delimiters may be omitted. The notation @kbd{$$$} refers to the value
36959 in stack level three, and causes the formula to replace the top three
36960 stack levels. The notation @kbd{$3} refers to stack level three without
36961 causing that value to be removed from the stack. Use @key{LFD} in place
36962 of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET}
36963 to evaluate variables.
36967 The variable is replaced by the formula shown on the right. The
36968 Inverse flag reverses the order of the operands, e.g., @kbd{I s - x}
36970 @texline @math{x \coloneq a-x}.
36971 @infoline @expr{x := a-x}.
36975 Press @kbd{?} repeatedly to see how to choose a model. Answer the
36976 variables prompt with @expr{iv} or @expr{iv;pv} to specify
36977 independent and parameter variables. A positive prefix argument
36978 takes @mathit{@var{n}+1} vectors from the stack; a zero prefix takes a matrix
36979 and a vector from the stack.
36983 With a plain @kbd{C-u} prefix, replace the current region of the
36984 destination buffer with the yanked text instead of inserting.
36988 All stack entries are reformatted; the @kbd{H} prefix inhibits this.
36989 The @kbd{I} prefix sets the mode temporarily, redraws the top stack
36990 entry, then restores the original setting of the mode.
36994 A negative prefix sets the default 3D resolution instead of the
36995 default 2D resolution.
36999 This grabs a vector of the form [@var{prec}, @var{wsize}, @var{ssize},
37000 @var{radix}, @var{flfmt}, @var{ang}, @var{frac}, @var{symb}, @var{polar},
37001 @var{matrix}, @var{simp}, @var{inf}]. A prefix argument from 1 to 12
37002 grabs the @var{n}th mode value only.
37006 (Space is provided below for you to keep your own written notes.)
37014 @node Key Index, Command Index, Summary, Top
37015 @unnumbered Index of Key Sequences
37019 @node Command Index, Function Index, Key Index, Top
37020 @unnumbered Index of Calculator Commands
37022 Since all Calculator commands begin with the prefix @samp{calc-}, the
37023 @kbd{x} key has been provided as a variant of @kbd{M-x} which automatically
37024 types @samp{calc-} for you. Thus, @kbd{x last-args} is short for
37025 @kbd{M-x calc-last-args}.
37029 @node Function Index, Concept Index, Command Index, Top
37030 @unnumbered Index of Algebraic Functions
37032 This is a list of built-in functions and operators usable in algebraic
37033 expressions. Their full Lisp names are derived by adding the prefix
37034 @samp{calcFunc-}, as in @code{calcFunc-sqrt}.
37036 All functions except those noted with ``*'' have corresponding
37037 Calc keystrokes and can also be found in the Calc Summary.
37042 @node Concept Index, Variable Index, Function Index, Top
37043 @unnumbered Concept Index
37047 @node Variable Index, Lisp Function Index, Concept Index, Top
37048 @unnumbered Index of Variables
37050 The variables in this list that do not contain dashes are accessible
37051 as Calc variables. Add a @samp{var-} prefix to get the name of the
37052 corresponding Lisp variable.
37054 The remaining variables are Lisp variables suitable for @code{setq}ing
37055 in your Calc init file or @file{.emacs} file.
37059 @node Lisp Function Index, , Variable Index, Top
37060 @unnumbered Index of Lisp Math Functions
37062 The following functions are meant to be used with @code{defmath}, not
37063 @code{defun} definitions. For names that do not start with @samp{calc-},
37064 the corresponding full Lisp name is derived by adding a prefix of