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 2.1 Manual
8 @comment %**end of header (This is for running Texinfo on a region.)
10 @c The following macros are used for conditional output for single lines.
12 @c `foo' will appear only in TeX output
14 @c `foo' will appear only in non-TeX output
16 @c @expr{expr} will typeset an expression;
17 @c $x$ in TeX, @samp{x} otherwise.
22 @alias infoline=comment
35 @alias texline=comment
36 @macro infoline{stuff}
52 % Suggested by Karl Berry <karl@@freefriends.org>
53 \gdef\!{\mskip-\thinmuskip}
56 @c Fix some other things specifically for this manual.
59 @mathcode`@:=`@: @c Make Calc fractions come out right in math mode
61 \gdef\coloneq{\mathrel{\mathord:\mathord=}}
63 \gdef\beforedisplay{\vskip-10pt}
64 \gdef\afterdisplay{\vskip-5pt}
65 \gdef\beforedisplayh{\vskip-25pt}
66 \gdef\afterdisplayh{\vskip-10pt}
68 @newdimen@kyvpos @kyvpos=0pt
69 @newdimen@kyhpos @kyhpos=0pt
70 @newcount@calcclubpenalty @calcclubpenalty=1000
73 @newtoks@calcoldeverypar @calcoldeverypar=@everypar
74 @everypar={@calceverypar@the@calcoldeverypar}
75 @ifx@turnoffactive@undefinedzzz@def@turnoffactive{}@fi
76 @ifx@ninett@undefinedzzz@font@ninett=cmtt9@fi
77 @catcode`@\=0 \catcode`\@=11
79 \catcode`\@=0 @catcode`@\=@active
84 This file documents Calc, the GNU Emacs calculator.
86 Copyright (C) 1990, 1991, 2001, 2002, 2005 Free Software Foundation, Inc.
89 Permission is granted to copy, distribute and/or modify this document
90 under the terms of the GNU Free Documentation License, Version 1.1 or
91 any later version published by the Free Software Foundation; with the
92 Invariant Sections being just ``GNU GENERAL PUBLIC LICENSE'', with the
93 Front-Cover texts being ``A GNU Manual,'' and with the Back-Cover
94 Texts as in (a) below.
96 (a) The FSF's Back-Cover Text is: ``You have freedom to copy and modify
97 this GNU Manual, like GNU software. Copies published by the Free
98 Software Foundation raise funds for GNU development.''
104 * Calc: (calc). Advanced desk calculator and mathematical tool.
109 @center @titlefont{Calc Manual}
111 @center GNU Emacs Calc Version 2.1
116 @center Dave Gillespie
117 @center daveg@@synaptics.com
120 @vskip 0pt plus 1filll
121 Copyright @copyright{} 1990, 1991, 2001, 2002, 2005
122 Free Software Foundation, Inc.
128 @node Top, , (dir), (dir)
129 @chapter The GNU Emacs Calculator
132 @dfn{Calc} is an advanced desk calculator and mathematical tool
133 that runs as part of the GNU Emacs environment.
135 This manual is divided into three major parts: ``Getting Started,''
136 the ``Calc Tutorial,'' and the ``Calc Reference.'' The Tutorial
137 introduces all the major aspects of Calculator use in an easy,
138 hands-on way. The remainder of the manual is a complete reference to
139 the features of the Calculator.
141 For help in the Emacs Info system (which you are using to read this
142 file), type @kbd{?}. (You can also type @kbd{h} to run through a
143 longer Info tutorial.)
147 * Copying:: How you can copy and share Calc.
149 * Getting Started:: General description and overview.
150 * Interactive Tutorial::
151 * Tutorial:: A step-by-step introduction for beginners.
153 * Introduction:: Introduction to the Calc reference manual.
154 * Data Types:: Types of objects manipulated by Calc.
155 * Stack and Trail:: Manipulating the stack and trail buffers.
156 * Mode Settings:: Adjusting display format and other modes.
157 * Arithmetic:: Basic arithmetic functions.
158 * Scientific Functions:: Transcendentals and other scientific functions.
159 * Matrix Functions:: Operations on vectors and matrices.
160 * Algebra:: Manipulating expressions algebraically.
161 * Units:: Operations on numbers with units.
162 * Store and Recall:: Storing and recalling variables.
163 * Graphics:: Commands for making graphs of data.
164 * Kill and Yank:: Moving data into and out of Calc.
165 * Keypad Mode:: Operating Calc from a keypad.
166 * Embedded Mode:: Working with formulas embedded in a file.
167 * Programming:: Calc as a programmable calculator.
169 * Customizable Variables:: Customizable Variables.
170 * Reporting Bugs:: How to report bugs and make suggestions.
172 * Summary:: Summary of Calc commands and functions.
174 * Key Index:: The standard Calc key sequences.
175 * Command Index:: The interactive Calc commands.
176 * Function Index:: Functions (in algebraic formulas).
177 * Concept Index:: General concepts.
178 * Variable Index:: Variables used by Calc (both user and internal).
179 * Lisp Function Index:: Internal Lisp math functions.
182 @node Copying, Getting Started, Top, Top
183 @unnumbered GNU GENERAL PUBLIC LICENSE
184 @center Version 2, June 1991
186 @c This file is intended to be included in another file.
189 Copyright @copyright{} 1989, 1991 Free Software Foundation, Inc.
190 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA
192 Everyone is permitted to copy and distribute verbatim copies
193 of this license document, but changing it is not allowed.
196 @unnumberedsec Preamble
198 The licenses for most software are designed to take away your
199 freedom to share and change it. By contrast, the GNU General Public
200 License is intended to guarantee your freedom to share and change free
201 software---to make sure the software is free for all its users. This
202 General Public License applies to most of the Free Software
203 Foundation's software and to any other program whose authors commit to
204 using it. (Some other Free Software Foundation software is covered by
205 the GNU Library General Public License instead.) You can apply it to
208 When we speak of free software, we are referring to freedom, not
209 price. Our General Public Licenses are designed to make sure that you
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215 To protect your rights, we need to make restrictions that forbid
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217 These restrictions translate to certain responsibilities for you if you
218 distribute copies of the software, or if you modify it.
220 For example, if you distribute copies of such a program, whether
221 gratis or for a fee, you must give the recipients all the rights that
222 you have. You must make sure that they, too, receive or can get the
223 source code. And you must show them these terms so they know their
226 We protect your rights with two steps: (1) copyright the software, and
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243 The precise terms and conditions for copying, distribution and
247 @unnumberedsec TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
250 @center TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
255 This License applies to any program or other work which contains
256 a notice placed by the copyright holder saying it may be distributed
257 under the terms of this General Public License. The ``Program'', below,
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270 Whether that is true depends on what the Program does.
273 You may copy and distribute verbatim copies of the Program's
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498 POSSIBILITY OF SUCH DAMAGES.
502 @heading END OF TERMS AND CONDITIONS
505 @center END OF TERMS AND CONDITIONS
509 @unnumberedsec Appendix: How to Apply These Terms to Your New Programs
511 If you develop a new program, and you want it to be of the greatest
512 possible use to the public, the best way to achieve this is to make it
513 free software which everyone can redistribute and change under these terms.
515 To do so, attach the following notices to the program. It is safest
516 to attach them to the start of each source file to most effectively
517 convey the exclusion of warranty; and each file should have at least
518 the ``copyright'' line and a pointer to where the full notice is found.
521 @var{one line to give the program's name and a brief idea of what it does.}
522 Copyright (C) @var{yyyy} @var{name of author}
524 This program is free software; you can redistribute it and/or modify
525 it under the terms of the GNU General Public License as published by
526 the Free Software Foundation; either version 2 of the License, or
527 (at your option) any later version.
529 This program is distributed in the hope that it will be useful,
530 but WITHOUT ANY WARRANTY; without even the implied warranty of
531 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
532 GNU General Public License for more details.
534 You should have received a copy of the GNU General Public License
535 along with this program; if not, write to the Free Software
536 Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
539 Also add information on how to contact you by electronic and paper mail.
541 If the program is interactive, make it output a short notice like this
542 when it starts in an interactive mode:
545 Gnomovision version 69, Copyright (C) 19@var{yy} @var{name of author}
546 Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
547 This is free software, and you are welcome to redistribute it
548 under certain conditions; type `show c' for details.
551 The hypothetical commands @samp{show w} and @samp{show c} should show
552 the appropriate parts of the General Public License. Of course, the
553 commands you use may be called something other than @samp{show w} and
554 @samp{show c}; they could even be mouse-clicks or menu items---whatever
557 You should also get your employer (if you work as a programmer) or your
558 school, if any, to sign a ``copyright disclaimer'' for the program, if
559 necessary. Here is a sample; alter the names:
562 Yoyodyne, Inc., hereby disclaims all copyright interest in the program
563 `Gnomovision' (which makes passes at compilers) written by James Hacker.
565 @var{signature of Ty Coon}, 1 April 1989
566 Ty Coon, President of Vice
569 This General Public License does not permit incorporating your program into
570 proprietary programs. If your program is a subroutine library, you may
571 consider it more useful to permit linking proprietary applications with the
572 library. If this is what you want to do, use the GNU Library General
573 Public License instead of this License.
575 @node Getting Started, Tutorial, Copying, Top
576 @chapter Getting Started
578 This chapter provides a general overview of Calc, the GNU Emacs
579 Calculator: What it is, how to start it and how to exit from it,
580 and what are the various ways that it can be used.
584 * About This Manual::
585 * Notations Used in This Manual::
587 * Demonstration of Calc::
588 * History and Acknowledgements::
591 @node What is Calc, About This Manual, Getting Started, Getting Started
592 @section What is Calc?
595 @dfn{Calc} is an advanced calculator and mathematical tool that runs as
596 part of the GNU Emacs environment. Very roughly based on the HP-28/48
597 series of calculators, its many features include:
601 Choice of algebraic or RPN (stack-based) entry of calculations.
604 Arbitrary precision integers and floating-point numbers.
607 Arithmetic on rational numbers, complex numbers (rectangular and polar),
608 error forms with standard deviations, open and closed intervals, vectors
609 and matrices, dates and times, infinities, sets, quantities with units,
610 and algebraic formulas.
613 Mathematical operations such as logarithms and trigonometric functions.
616 Programmer's features (bitwise operations, non-decimal numbers).
619 Financial functions such as future value and internal rate of return.
622 Number theoretical features such as prime factorization and arithmetic
623 modulo @var{m} for any @var{m}.
626 Algebraic manipulation features, including symbolic calculus.
629 Moving data to and from regular editing buffers.
632 Embedded mode for manipulating Calc formulas and data directly
633 inside any editing buffer.
636 Graphics using GNUPLOT, a versatile (and free) plotting program.
639 Easy programming using keyboard macros, algebraic formulas,
640 algebraic rewrite rules, or extended Emacs Lisp.
643 Calc tries to include a little something for everyone; as a result it is
644 large and might be intimidating to the first-time user. If you plan to
645 use Calc only as a traditional desk calculator, all you really need to
646 read is the ``Getting Started'' chapter of this manual and possibly the
647 first few sections of the tutorial. As you become more comfortable with
648 the program you can learn its additional features. Calc does not
649 have the scope and depth of a fully-functional symbolic math package,
650 but Calc has the advantages of convenience, portability, and freedom.
652 @node About This Manual, Notations Used in This Manual, What is Calc, Getting Started
653 @section About This Manual
656 This document serves as a complete description of the GNU Emacs
657 Calculator. It works both as an introduction for novices, and as
658 a reference for experienced users. While it helps to have some
659 experience with GNU Emacs in order to get the most out of Calc,
660 this manual ought to be readable even if you don't know or use Emacs
664 The manual is divided into three major parts:@: the ``Getting
665 Started'' chapter you are reading now, the Calc tutorial (chapter 2),
666 and the Calc reference manual (the remaining chapters and appendices).
669 The manual is divided into three major parts:@: the ``Getting
670 Started'' chapter you are reading now, the Calc tutorial (chapter 2),
671 and the Calc reference manual (the remaining chapters and appendices).
673 @c This manual has been printed in two volumes, the @dfn{Tutorial} and the
674 @c @dfn{Reference}. Both volumes include a copy of the ``Getting Started''
678 If you are in a hurry to use Calc, there is a brief ``demonstration''
679 below which illustrates the major features of Calc in just a couple of
680 pages. If you don't have time to go through the full tutorial, this
681 will show you everything you need to know to begin.
682 @xref{Demonstration of Calc}.
684 The tutorial chapter walks you through the various parts of Calc
685 with lots of hands-on examples and explanations. If you are new
686 to Calc and you have some time, try going through at least the
687 beginning of the tutorial. The tutorial includes about 70 exercises
688 with answers. These exercises give you some guided practice with
689 Calc, as well as pointing out some interesting and unusual ways
692 The reference section discusses Calc in complete depth. You can read
693 the reference from start to finish if you want to learn every aspect
694 of Calc. Or, you can look in the table of contents or the Concept
695 Index to find the parts of the manual that discuss the things you
698 @cindex Marginal notes
699 Every Calc keyboard command is listed in the Calc Summary, and also
700 in the Key Index. Algebraic functions, @kbd{M-x} commands, and
701 variables also have their own indices.
703 @infoline In the printed manual, each
704 paragraph that is referenced in the Key or Function Index is marked
705 in the margin with its index entry.
707 @c [fix-ref Help Commands]
708 You can access this manual on-line at any time within Calc by
709 pressing the @kbd{h i} key sequence. Outside of the Calc window,
710 you can press @kbd{M-# i} to read the manual on-line. Also, you
711 can jump directly to the Tutorial by pressing @kbd{h t} or @kbd{M-# t},
712 or to the Summary by pressing @kbd{h s} or @kbd{M-# s}. Within Calc,
713 you can also go to the part of the manual describing any Calc key,
714 function, or variable using @w{@kbd{h k}}, @kbd{h f}, or @kbd{h v},
715 respectively. @xref{Help Commands}.
717 The Calc manual can be printed, but because the manual is so large, you
718 should only make a printed copy if you really need it. To print the
719 manual, you will need the @TeX{} typesetting program (this is a free
720 program by Donald Knuth at Stanford University) as well as the
721 @file{texindex} program and @file{texinfo.tex} file, both of which can
722 be obtained from the FSF as part of the @code{texinfo} package.
723 To print the Calc manual in one huge tome, you will need the
724 source code to this manual, @file{calc.texi}, available as part of the
725 Emacs source. Once you have this file, type @kbd{texi2dvi calc.texi}.
726 Alternatively, change to the @file{man} subdirectory of the Emacs
727 source distribution, and type @kbd{make calc.dvi}. (Don't worry if you
728 get some ``overfull box'' warnings while @TeX{} runs.)
729 The result will be a device-independent output file called
730 @file{calc.dvi}, which you must print in whatever way is right
731 for your system. On many systems, the command is
744 @c Printed copies of this manual are also available from the Free Software
747 @node Notations Used in This Manual, Demonstration of Calc, About This Manual, Getting Started
748 @section Notations Used in This Manual
751 This section describes the various notations that are used
752 throughout the Calc manual.
754 In keystroke sequences, uppercase letters mean you must hold down
755 the shift key while typing the letter. Keys pressed with Control
756 held down are shown as @kbd{C-x}. Keys pressed with Meta held down
757 are shown as @kbd{M-x}. Other notations are @key{RET} for the
758 Return key, @key{SPC} for the space bar, @key{TAB} for the Tab key,
759 @key{DEL} for the Delete key, and @key{LFD} for the Line-Feed key.
760 The @key{DEL} key is called Backspace on some keyboards, it is
761 whatever key you would use to correct a simple typing error when
762 regularly using Emacs.
764 (If you don't have the @key{LFD} or @key{TAB} keys on your keyboard,
765 the @kbd{C-j} and @kbd{C-i} keys are equivalent to them, respectively.
766 If you don't have a Meta key, look for Alt or Extend Char. You can
767 also press @key{ESC} or @key{C-[} first to get the same effect, so
768 that @kbd{M-x}, @kbd{@key{ESC} x}, and @kbd{C-[ x} are all equivalent.)
770 Sometimes the @key{RET} key is not shown when it is ``obvious''
771 that you must press @key{RET} to proceed. For example, the @key{RET}
772 is usually omitted in key sequences like @kbd{M-x calc-keypad @key{RET}}.
774 Commands are generally shown like this: @kbd{p} (@code{calc-precision})
775 or @kbd{M-# k} (@code{calc-keypad}). This means that the command is
776 normally used by pressing the @kbd{p} key or @kbd{M-# k} key sequence,
777 but it also has the full-name equivalent shown, e.g., @kbd{M-x calc-precision}.
779 Commands that correspond to functions in algebraic notation
780 are written: @kbd{C} (@code{calc-cos}) [@code{cos}]. This means
781 the @kbd{C} key is equivalent to @kbd{M-x calc-cos}, and that
782 the corresponding function in an algebraic-style formula would
783 be @samp{cos(@var{x})}.
785 A few commands don't have key equivalents: @code{calc-sincos}
788 @node Demonstration of Calc, Using Calc, Notations Used in This Manual, Getting Started
789 @section A Demonstration of Calc
792 @cindex Demonstration of Calc
793 This section will show some typical small problems being solved with
794 Calc. The focus is more on demonstration than explanation, but
795 everything you see here will be covered more thoroughly in the
798 To begin, start Emacs if necessary (usually the command @code{emacs}
799 does this), and type @kbd{M-# c} (or @kbd{@key{ESC} # c}) to start the
800 Calculator. (@xref{Starting Calc}, if this doesn't work for you.)
802 Be sure to type all the sample input exactly, especially noting the
803 difference between lower-case and upper-case letters. Remember,
804 @key{RET}, @key{TAB}, @key{DEL}, and @key{SPC} are the Return, Tab,
805 Delete, and Space keys.
807 @strong{RPN calculation.} In RPN, you type the input number(s) first,
808 then the command to operate on the numbers.
811 Type @kbd{2 @key{RET} 3 + Q} to compute
812 @texline @math{\sqrt{2+3} = 2.2360679775}.
813 @infoline the square root of 2+3, which is 2.2360679775.
816 Type @kbd{P 2 ^} to compute
817 @texline @math{\pi^2 = 9.86960440109}.
818 @infoline the value of `pi' squared, 9.86960440109.
821 Type @key{TAB} to exchange the order of these two results.
824 Type @kbd{- I H S} to subtract these results and compute the Inverse
825 Hyperbolic sine of the difference, 2.72996136574.
828 Type @key{DEL} to erase this result.
830 @strong{Algebraic calculation.} You can also enter calculations using
831 conventional ``algebraic'' notation. To enter an algebraic formula,
832 use the apostrophe key.
835 Type @kbd{' sqrt(2+3) @key{RET}} to compute
836 @texline @math{\sqrt{2+3}}.
837 @infoline the square root of 2+3.
840 Type @kbd{' pi^2 @key{RET}} to enter
841 @texline @math{\pi^2}.
842 @infoline `pi' squared.
843 To evaluate this symbolic formula as a number, type @kbd{=}.
846 Type @kbd{' arcsinh($ - $$) @key{RET}} to subtract the second-most-recent
847 result from the most-recent and compute the Inverse Hyperbolic sine.
849 @strong{Keypad mode.} If you are using the X window system, press
850 @w{@kbd{M-# k}} to get Keypad mode. (If you don't use X, skip to
854 Click on the @key{2}, @key{ENTER}, @key{3}, @key{+}, and @key{SQRT}
855 ``buttons'' using your left mouse button.
858 Click on @key{PI}, @key{2}, and @tfn{y^x}.
861 Click on @key{INV}, then @key{ENTER} to swap the two results.
864 Click on @key{-}, @key{INV}, @key{HYP}, and @key{SIN}.
867 Click on @key{<-} to erase the result, then click @key{OFF} to turn
868 the Keypad Calculator off.
870 @strong{Grabbing data.} Type @kbd{M-# x} if necessary to exit Calc.
871 Now select the following numbers as an Emacs region: ``Mark'' the
872 front of the list by typing @kbd{C-@key{SPC}} or @kbd{C-@@} there,
873 then move to the other end of the list. (Either get this list from
874 the on-line copy of this manual, accessed by @w{@kbd{M-# i}}, or just
875 type these numbers into a scratch file.) Now type @kbd{M-# g} to
876 ``grab'' these numbers into Calc.
887 The result @samp{[1.23, 1.97, 1.6, 2, 1.19, 1.08]} is a Calc ``vector.''
888 Type @w{@kbd{V R +}} to compute the sum of these numbers.
891 Type @kbd{U} to Undo this command, then type @kbd{V R *} to compute
892 the product of the numbers.
895 You can also grab data as a rectangular matrix. Place the cursor on
896 the upper-leftmost @samp{1} and set the mark, then move to just after
897 the lower-right @samp{8} and press @kbd{M-# r}.
900 Type @kbd{v t} to transpose this
901 @texline @math{3\times2}
904 @texline @math{2\times3}
906 matrix. Type @w{@kbd{v u}} to unpack the rows into two separate
907 vectors. Now type @w{@kbd{V R + @key{TAB} V R +}} to compute the sums
908 of the two original columns. (There is also a special
909 grab-and-sum-columns command, @kbd{M-# :}.)
911 @strong{Units conversion.} Units are entered algebraically.
912 Type @w{@kbd{' 43 mi/hr @key{RET}}} to enter the quantity 43 miles-per-hour.
913 Type @w{@kbd{u c km/hr @key{RET}}}. Type @w{@kbd{u c m/s @key{RET}}}.
915 @strong{Date arithmetic.} Type @kbd{t N} to get the current date and
916 time. Type @kbd{90 +} to find the date 90 days from now. Type
917 @kbd{' <25 dec 87> @key{RET}} to enter a date, then @kbd{- 7 /} to see how
918 many weeks have passed since then.
920 @strong{Algebra.} Algebraic entries can also include formulas
921 or equations involving variables. Type @kbd{@w{' [x + y} = a, x y = 1] @key{RET}}
922 to enter a pair of equations involving three variables.
923 (Note the leading apostrophe in this example; also, note that the space
924 between @samp{x y} is required.) Type @w{@kbd{a S x,y @key{RET}}} to solve
925 these equations for the variables @expr{x} and @expr{y}.
928 Type @kbd{d B} to view the solutions in more readable notation.
929 Type @w{@kbd{d C}} to view them in C language notation, @kbd{d T}
930 to view them in the notation for the @TeX{} typesetting system,
931 and @kbd{d L} to view them in the notation for the La@TeX{} typesetting
932 system. Type @kbd{d N} to return to normal notation.
935 Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @expr{a = 7.5} in these formulas.
936 (That's a letter @kbd{l}, not a numeral @kbd{1}.)
939 @strong{Help functions.} You can read about any command in the on-line
940 manual. Type @kbd{M-# c} to return to Calc after each of these
941 commands: @kbd{h k t N} to read about the @kbd{t N} command,
942 @kbd{h f sqrt @key{RET}} to read about the @code{sqrt} function, and
943 @kbd{h s} to read the Calc summary.
946 @strong{Help functions.} You can read about any command in the on-line
947 manual. Remember to type the letter @kbd{l}, then @kbd{M-# c}, to
948 return here after each of these commands: @w{@kbd{h k t N}} to read
949 about the @w{@kbd{t N}} command, @kbd{h f sqrt @key{RET}} to read about the
950 @code{sqrt} function, and @kbd{h s} to read the Calc summary.
953 Press @key{DEL} repeatedly to remove any leftover results from the stack.
954 To exit from Calc, press @kbd{q} or @kbd{M-# c} again.
956 @node Using Calc, History and Acknowledgements, Demonstration of Calc, Getting Started
960 Calc has several user interfaces that are specialized for
961 different kinds of tasks. As well as Calc's standard interface,
962 there are Quick mode, Keypad mode, and Embedded mode.
966 * The Standard Interface::
967 * Quick Mode Overview::
968 * Keypad Mode Overview::
969 * Standalone Operation::
970 * Embedded Mode Overview::
971 * Other M-# Commands::
974 @node Starting Calc, The Standard Interface, Using Calc, Using Calc
975 @subsection Starting Calc
978 On most systems, you can type @kbd{M-#} to start the Calculator.
979 The notation @kbd{M-#} is short for Meta-@kbd{#}. On most
980 keyboards this means holding down the Meta (or Alt) and
981 Shift keys while typing @kbd{3}.
984 Once again, if you don't have a Meta key on your keyboard you can type
985 @key{ESC} first, then @kbd{#}, to accomplish the same thing. If you
986 don't even have an @key{ESC} key, you can fake it by holding down
987 Control or @key{CTRL} while typing a left square bracket
988 (that's @kbd{C-[} in Emacs notation).
990 @kbd{M-#} is a @dfn{prefix key}; when you press it, Emacs waits for
991 you to press a second key to complete the command. In this case,
992 you will follow @kbd{M-#} with a letter (upper- or lower-case, it
993 doesn't matter for @kbd{M-#}) that says which Calc interface you
996 To get Calc's standard interface, type @kbd{M-# c}. To get
997 Keypad mode, type @kbd{M-# k}. Type @kbd{M-# ?} to get a brief
998 list of the available options, and type a second @kbd{?} to get
1001 To ease typing, @kbd{M-# M-#} (or @kbd{M-# #} if that's easier)
1002 also works to start Calc. It starts the same interface (either
1003 @kbd{M-# c} or @w{@kbd{M-# k}}) that you last used, selecting the
1004 @kbd{M-# c} interface by default. (If your installation has
1005 a special function key set up to act like @kbd{M-#}, hitting that
1006 function key twice is just like hitting @kbd{M-# M-#}.)
1008 If @kbd{M-#} doesn't work for you, you can always type explicit
1009 commands like @kbd{M-x calc} (for the standard user interface) or
1010 @w{@kbd{M-x calc-keypad}} (for Keypad mode). First type @kbd{M-x}
1011 (that's Meta with the letter @kbd{x}), then, at the prompt,
1012 type the full command (like @kbd{calc-keypad}) and press Return.
1014 The same commands (like @kbd{M-# c} or @kbd{M-# M-#}) that start
1015 the Calculator also turn it off if it is already on.
1017 @node The Standard Interface, Quick Mode Overview, Starting Calc, Using Calc
1018 @subsection The Standard Calc Interface
1021 @cindex Standard user interface
1022 Calc's standard interface acts like a traditional RPN calculator,
1023 operated by the normal Emacs keyboard. When you type @kbd{M-# c}
1024 to start the Calculator, the Emacs screen splits into two windows
1025 with the file you were editing on top and Calc on the bottom.
1031 --**-Emacs: myfile (Fundamental)----All----------------------
1032 --- Emacs Calculator Mode --- |Emacs Calc Mode v2.1 ...
1040 --%%-Calc: 12 Deg (Calculator)----All----- --%%-Emacs: *Calc Trail*
1044 In this figure, the mode-line for @file{myfile} has moved up and the
1045 ``Calculator'' window has appeared below it. As you can see, Calc
1046 actually makes two windows side-by-side. The lefthand one is
1047 called the @dfn{stack window} and the righthand one is called the
1048 @dfn{trail window.} The stack holds the numbers involved in the
1049 calculation you are currently performing. The trail holds a complete
1050 record of all calculations you have done. In a desk calculator with
1051 a printer, the trail corresponds to the paper tape that records what
1054 In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
1055 were first entered into the Calculator, then the 2 and 4 were
1056 multiplied to get 8, then the 3 and 8 were subtracted to get @mathit{-5}.
1057 (The @samp{>} symbol shows that this was the most recent calculation.)
1058 The net result is the two numbers 17.3 and @mathit{-5} sitting on the stack.
1060 Most Calculator commands deal explicitly with the stack only, but
1061 there is a set of commands that allow you to search back through
1062 the trail and retrieve any previous result.
1064 Calc commands use the digits, letters, and punctuation keys.
1065 Shifted (i.e., upper-case) letters are different from lowercase
1066 letters. Some letters are @dfn{prefix} keys that begin two-letter
1067 commands. For example, @kbd{e} means ``enter exponent'' and shifted
1068 @kbd{E} means @expr{e^x}. With the @kbd{d} (``display modes'') prefix
1069 the letter ``e'' takes on very different meanings: @kbd{d e} means
1070 ``engineering notation'' and @kbd{d E} means ``@dfn{eqn} language mode.''
1072 There is nothing stopping you from switching out of the Calc
1073 window and back into your editing window, say by using the Emacs
1074 @w{@kbd{C-x o}} (@code{other-window}) command. When the cursor is
1075 inside a regular window, Emacs acts just like normal. When the
1076 cursor is in the Calc stack or trail windows, keys are interpreted
1079 When you quit by pressing @kbd{M-# c} a second time, the Calculator
1080 windows go away but the actual Stack and Trail are not gone, just
1081 hidden. When you press @kbd{M-# c} once again you will get the
1082 same stack and trail contents you had when you last used the
1085 The Calculator does not remember its state between Emacs sessions.
1086 Thus if you quit Emacs and start it again, @kbd{M-# c} will give you
1087 a fresh stack and trail. There is a command (@kbd{m m}) that lets
1088 you save your favorite mode settings between sessions, though.
1089 One of the things it saves is which user interface (standard or
1090 Keypad) you last used; otherwise, a freshly started Emacs will
1091 always treat @kbd{M-# M-#} the same as @kbd{M-# c}.
1093 The @kbd{q} key is another equivalent way to turn the Calculator off.
1095 If you type @kbd{M-# b} first and then @kbd{M-# c}, you get a
1096 full-screen version of Calc (@code{full-calc}) in which the stack and
1097 trail windows are still side-by-side but are now as tall as the whole
1098 Emacs screen. When you press @kbd{q} or @kbd{M-# c} again to quit,
1099 the file you were editing before reappears. The @kbd{M-# b} key
1100 switches back and forth between ``big'' full-screen mode and the
1101 normal partial-screen mode.
1103 Finally, @kbd{M-# o} (@code{calc-other-window}) is like @kbd{M-# c}
1104 except that the Calc window is not selected. The buffer you were
1105 editing before remains selected instead. @kbd{M-# o} is a handy
1106 way to switch out of Calc momentarily to edit your file; type
1107 @kbd{M-# c} to switch back into Calc when you are done.
1109 @node Quick Mode Overview, Keypad Mode Overview, The Standard Interface, Using Calc
1110 @subsection Quick Mode (Overview)
1113 @dfn{Quick mode} is a quick way to use Calc when you don't need the
1114 full complexity of the stack and trail. To use it, type @kbd{M-# q}
1115 (@code{quick-calc}) in any regular editing buffer.
1117 Quick mode is very simple: It prompts you to type any formula in
1118 standard algebraic notation (like @samp{4 - 2/3}) and then displays
1119 the result at the bottom of the Emacs screen (@mathit{3.33333333333}
1120 in this case). You are then back in the same editing buffer you
1121 were in before, ready to continue editing or to type @kbd{M-# q}
1122 again to do another quick calculation. The result of the calculation
1123 will also be in the Emacs ``kill ring'' so that a @kbd{C-y} command
1124 at this point will yank the result into your editing buffer.
1126 Calc mode settings affect Quick mode, too, though you will have to
1127 go into regular Calc (with @kbd{M-# c}) to change the mode settings.
1129 @c [fix-ref Quick Calculator mode]
1130 @xref{Quick Calculator}, for further information.
1132 @node Keypad Mode Overview, Standalone Operation, Quick Mode Overview, Using Calc
1133 @subsection Keypad Mode (Overview)
1136 @dfn{Keypad mode} is a mouse-based interface to the Calculator.
1137 It is designed for use with terminals that support a mouse. If you
1138 don't have a mouse, you will have to operate Keypad mode with your
1139 arrow keys (which is probably more trouble than it's worth).
1141 Type @kbd{M-# k} to turn Keypad mode on or off. Once again you
1142 get two new windows, this time on the righthand side of the screen
1143 instead of at the bottom. The upper window is the familiar Calc
1144 Stack; the lower window is a picture of a typical calculator keypad.
1148 \advance \dimen0 by 24\baselineskip%
1149 \ifdim \dimen0>\pagegoal \vfill\eject \fi%
1154 |--- Emacs Calculator Mode ---
1158 |--%%-Calc: 12 Deg (Calcul
1159 |----+-----Calc 2.1------+----1
1160 |FLR |CEIL|RND |TRNC|CLN2|FLT |
1161 |----+----+----+----+----+----|
1162 | LN |EXP | |ABS |IDIV|MOD |
1163 |----+----+----+----+----+----|
1164 |SIN |COS |TAN |SQRT|y^x |1/x |
1165 |----+----+----+----+----+----|
1166 | ENTER |+/- |EEX |UNDO| <- |
1167 |-----+---+-+--+--+-+---++----|
1168 | INV | 7 | 8 | 9 | / |
1169 |-----+-----+-----+-----+-----|
1170 | HYP | 4 | 5 | 6 | * |
1171 |-----+-----+-----+-----+-----|
1172 |EXEC | 1 | 2 | 3 | - |
1173 |-----+-----+-----+-----+-----|
1174 | OFF | 0 | . | PI | + |
1175 |-----+-----+-----+-----+-----+
1179 Keypad mode is much easier for beginners to learn, because there
1180 is no need to memorize lots of obscure key sequences. But not all
1181 commands in regular Calc are available on the Keypad. You can
1182 always switch the cursor into the Calc stack window to use
1183 standard Calc commands if you need. Serious Calc users, though,
1184 often find they prefer the standard interface over Keypad mode.
1186 To operate the Calculator, just click on the ``buttons'' of the
1187 keypad using your left mouse button. To enter the two numbers
1188 shown here you would click @w{@kbd{1 7 .@: 3 ENTER 5 +/- ENTER}}; to
1189 add them together you would then click @kbd{+} (to get 12.3 on
1192 If you click the right mouse button, the top three rows of the
1193 keypad change to show other sets of commands, such as advanced
1194 math functions, vector operations, and operations on binary
1197 Because Keypad mode doesn't use the regular keyboard, Calc leaves
1198 the cursor in your original editing buffer. You can type in
1199 this buffer in the usual way while also clicking on the Calculator
1200 keypad. One advantage of Keypad mode is that you don't need an
1201 explicit command to switch between editing and calculating.
1203 If you press @kbd{M-# b} first, you get a full-screen Keypad mode
1204 (@code{full-calc-keypad}) with three windows: The keypad in the lower
1205 left, the stack in the lower right, and the trail on top.
1207 @c [fix-ref Keypad Mode]
1208 @xref{Keypad Mode}, for further information.
1210 @node Standalone Operation, Embedded Mode Overview, Keypad Mode Overview, Using Calc
1211 @subsection Standalone Operation
1214 @cindex Standalone Operation
1215 If you are not in Emacs at the moment but you wish to use Calc,
1216 you must start Emacs first. If all you want is to run Calc, you
1217 can give the commands:
1227 emacs -f full-calc-keypad
1231 which run a full-screen Calculator (as if by @kbd{M-# b M-# c}) or
1232 a full-screen X-based Calculator (as if by @kbd{M-# b M-# k}).
1233 In standalone operation, quitting the Calculator (by pressing
1234 @kbd{q} or clicking on the keypad @key{EXIT} button) quits Emacs
1237 @node Embedded Mode Overview, Other M-# Commands, Standalone Operation, Using Calc
1238 @subsection Embedded Mode (Overview)
1241 @dfn{Embedded mode} is a way to use Calc directly from inside an
1242 editing buffer. Suppose you have a formula written as part of a
1256 and you wish to have Calc compute and format the derivative for
1257 you and store this derivative in the buffer automatically. To
1258 do this with Embedded mode, first copy the formula down to where
1259 you want the result to be:
1273 Now, move the cursor onto this new formula and press @kbd{M-# e}.
1274 Calc will read the formula (using the surrounding blank lines to
1275 tell how much text to read), then push this formula (invisibly)
1276 onto the Calc stack. The cursor will stay on the formula in the
1277 editing buffer, but the buffer's mode line will change to look
1278 like the Calc mode line (with mode indicators like @samp{12 Deg}
1279 and so on). Even though you are still in your editing buffer,
1280 the keyboard now acts like the Calc keyboard, and any new result
1281 you get is copied from the stack back into the buffer. To take
1282 the derivative, you would type @kbd{a d x @key{RET}}.
1296 To make this look nicer, you might want to press @kbd{d =} to center
1297 the formula, and even @kbd{d B} to use Big display mode.
1306 % [calc-mode: justify: center]
1307 % [calc-mode: language: big]
1315 Calc has added annotations to the file to help it remember the modes
1316 that were used for this formula. They are formatted like comments
1317 in the @TeX{} typesetting language, just in case you are using @TeX{} or
1318 La@TeX{}. (In this example @TeX{} is not being used, so you might want
1319 to move these comments up to the top of the file or otherwise put them
1322 As an extra flourish, we can add an equation number using a
1323 righthand label: Type @kbd{d @} (1) @key{RET}}.
1327 % [calc-mode: justify: center]
1328 % [calc-mode: language: big]
1329 % [calc-mode: right-label: " (1)"]
1337 To leave Embedded mode, type @kbd{M-# e} again. The mode line
1338 and keyboard will revert to the way they were before. (If you have
1339 actually been trying this as you read along, you'll want to press
1340 @kbd{M-# 0} [with the digit zero] now to reset the modes you changed.)
1342 The related command @kbd{M-# w} operates on a single word, which
1343 generally means a single number, inside text. It uses any
1344 non-numeric characters rather than blank lines to delimit the
1345 formula it reads. Here's an example of its use:
1348 A slope of one-third corresponds to an angle of 1 degrees.
1351 Place the cursor on the @samp{1}, then type @kbd{M-# w} to enable
1352 Embedded mode on that number. Now type @kbd{3 /} (to get one-third),
1353 and @kbd{I T} (the Inverse Tangent converts a slope into an angle),
1354 then @w{@kbd{M-# w}} again to exit Embedded mode.
1357 A slope of one-third corresponds to an angle of 18.4349488229 degrees.
1360 @c [fix-ref Embedded Mode]
1361 @xref{Embedded Mode}, for full details.
1363 @node Other M-# Commands, , Embedded Mode Overview, Using Calc
1364 @subsection Other @kbd{M-#} Commands
1367 Two more Calc-related commands are @kbd{M-# g} and @kbd{M-# r},
1368 which ``grab'' data from a selected region of a buffer into the
1369 Calculator. The region is defined in the usual Emacs way, by
1370 a ``mark'' placed at one end of the region, and the Emacs
1371 cursor or ``point'' placed at the other.
1373 The @kbd{M-# g} command reads the region in the usual left-to-right,
1374 top-to-bottom order. The result is packaged into a Calc vector
1375 of numbers and placed on the stack. Calc (in its standard
1376 user interface) is then started. Type @kbd{v u} if you want
1377 to unpack this vector into separate numbers on the stack. Also,
1378 @kbd{C-u M-# g} interprets the region as a single number or
1381 The @kbd{M-# r} command reads a rectangle, with the point and
1382 mark defining opposite corners of the rectangle. The result
1383 is a matrix of numbers on the Calculator stack.
1385 Complementary to these is @kbd{M-# y}, which ``yanks'' the
1386 value at the top of the Calc stack back into an editing buffer.
1387 If you type @w{@kbd{M-# y}} while in such a buffer, the value is
1388 yanked at the current position. If you type @kbd{M-# y} while
1389 in the Calc buffer, Calc makes an educated guess as to which
1390 editing buffer you want to use. The Calc window does not have
1391 to be visible in order to use this command, as long as there
1392 is something on the Calc stack.
1394 Here, for reference, is the complete list of @kbd{M-#} commands.
1395 The shift, control, and meta keys are ignored for the keystroke
1396 following @kbd{M-#}.
1399 Commands for turning Calc on and off:
1403 Turn Calc on or off, employing the same user interface as last time.
1406 Turn Calc on or off using its standard bottom-of-the-screen
1407 interface. If Calc is already turned on but the cursor is not
1408 in the Calc window, move the cursor into the window.
1411 Same as @kbd{C}, but don't select the new Calc window. If
1412 Calc is already turned on and the cursor is in the Calc window,
1413 move it out of that window.
1416 Control whether @kbd{M-# c} and @kbd{M-# k} use the full screen.
1419 Use Quick mode for a single short calculation.
1422 Turn Calc Keypad mode on or off.
1425 Turn Calc Embedded mode on or off at the current formula.
1428 Turn Calc Embedded mode on or off, select the interesting part.
1431 Turn Calc Embedded mode on or off at the current word (number).
1434 Turn Calc on in a user-defined way, as defined by a @kbd{Z I} command.
1437 Quit Calc; turn off standard, Keypad, or Embedded mode if on.
1438 (This is like @kbd{q} or @key{OFF} inside of Calc.)
1445 Commands for moving data into and out of the Calculator:
1449 Grab the region into the Calculator as a vector.
1452 Grab the rectangular region into the Calculator as a matrix.
1455 Grab the rectangular region and compute the sums of its columns.
1458 Grab the rectangular region and compute the sums of its rows.
1461 Yank a value from the Calculator into the current editing buffer.
1468 Commands for use with Embedded mode:
1472 ``Activate'' the current buffer. Locate all formulas that
1473 contain @samp{:=} or @samp{=>} symbols and record their locations
1474 so that they can be updated automatically as variables are changed.
1477 Duplicate the current formula immediately below and select
1481 Insert a new formula at the current point.
1484 Move the cursor to the next active formula in the buffer.
1487 Move the cursor to the previous active formula in the buffer.
1490 Update (i.e., as if by the @kbd{=} key) the formula at the current point.
1493 Edit (as if by @code{calc-edit}) the formula at the current point.
1500 Miscellaneous commands:
1504 Run the Emacs Info system to read the Calc manual.
1505 (This is the same as @kbd{h i} inside of Calc.)
1508 Run the Emacs Info system to read the Calc Tutorial.
1511 Run the Emacs Info system to read the Calc Summary.
1514 Load Calc entirely into memory. (Normally the various parts
1515 are loaded only as they are needed.)
1518 Read a region of written keystroke names (like @kbd{C-n a b c @key{RET}})
1519 and record them as the current keyboard macro.
1522 (This is the ``zero'' digit key.) Reset the Calculator to
1523 its default state: Empty stack, and default mode settings.
1524 With any prefix argument, reset everything but the stack.
1527 @node History and Acknowledgements, , Using Calc, Getting Started
1528 @section History and Acknowledgements
1531 Calc was originally started as a two-week project to occupy a lull
1532 in the author's schedule. Basically, a friend asked if I remembered
1534 @texline @math{2^{32}}.
1535 @infoline @expr{2^32}.
1536 I didn't offhand, but I said, ``that's easy, just call up an
1537 @code{xcalc}.'' @code{Xcalc} duly reported that the answer to our
1538 question was @samp{4.294967e+09}---with no way to see the full ten
1539 digits even though we knew they were there in the program's memory! I
1540 was so annoyed, I vowed to write a calculator of my own, once and for
1543 I chose Emacs Lisp, a) because I had always been curious about it
1544 and b) because, being only a text editor extension language after
1545 all, Emacs Lisp would surely reach its limits long before the project
1546 got too far out of hand.
1548 To make a long story short, Emacs Lisp turned out to be a distressingly
1549 solid implementation of Lisp, and the humble task of calculating
1550 turned out to be more open-ended than one might have expected.
1552 Emacs Lisp doesn't have built-in floating point math, so it had to be
1553 simulated in software. In fact, Emacs integers will only comfortably
1554 fit six decimal digits or so---not enough for a decent calculator. So
1555 I had to write my own high-precision integer code as well, and once I had
1556 this I figured that arbitrary-size integers were just as easy as large
1557 integers. Arbitrary floating-point precision was the logical next step.
1558 Also, since the large integer arithmetic was there anyway it seemed only
1559 fair to give the user direct access to it, which in turn made it practical
1560 to support fractions as well as floats. All these features inspired me
1561 to look around for other data types that might be worth having.
1563 Around this time, my friend Rick Koshi showed me his nifty new HP-28
1564 calculator. It allowed the user to manipulate formulas as well as
1565 numerical quantities, and it could also operate on matrices. I
1566 decided that these would be good for Calc to have, too. And once
1567 things had gone this far, I figured I might as well take a look at
1568 serious algebra systems for further ideas. Since these systems did
1569 far more than I could ever hope to implement, I decided to focus on
1570 rewrite rules and other programming features so that users could
1571 implement what they needed for themselves.
1573 Rick complained that matrices were hard to read, so I put in code to
1574 format them in a 2D style. Once these routines were in place, Big mode
1575 was obligatory. Gee, what other language modes would be useful?
1577 Scott Hemphill and Allen Knutson, two friends with a strong mathematical
1578 bent, contributed ideas and algorithms for a number of Calc features
1579 including modulo forms, primality testing, and float-to-fraction conversion.
1581 Units were added at the eager insistence of Mass Sivilotti. Later,
1582 Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable
1583 expert assistance with the units table. As far as I can remember, the
1584 idea of using algebraic formulas and variables to represent units dates
1585 back to an ancient article in Byte magazine about muMath, an early
1586 algebra system for microcomputers.
1588 Many people have contributed to Calc by reporting bugs and suggesting
1589 features, large and small. A few deserve special mention: Tim Peters,
1590 who helped develop the ideas that led to the selection commands, rewrite
1591 rules, and many other algebra features;
1592 @texline Fran\c cois
1594 Pinard, who contributed an early prototype of the Calc Summary appendix
1595 as well as providing valuable suggestions in many other areas of Calc;
1596 Carl Witty, whose eagle eyes discovered many typographical and factual
1597 errors in the Calc manual; Tim Kay, who drove the development of
1598 Embedded mode; Ove Ewerlid, who made many suggestions relating to the
1599 algebra commands and contributed some code for polynomial operations;
1600 Randal Schwartz, who suggested the @code{calc-eval} function; Robert
1601 J. Chassell, who suggested the Calc Tutorial and exercises; and Juha
1602 Sarlin, who first worked out how to split Calc into quickly-loading
1603 parts. Bob Weiner helped immensely with the Lucid Emacs port.
1605 @cindex Bibliography
1606 @cindex Knuth, Art of Computer Programming
1607 @cindex Numerical Recipes
1608 @c Should these be expanded into more complete references?
1609 Among the books used in the development of Calc were Knuth's @emph{Art
1610 of Computer Programming} (especially volume II, @emph{Seminumerical
1611 Algorithms}); @emph{Numerical Recipes} by Press, Flannery, Teukolsky,
1612 and Vetterling; Bevington's @emph{Data Reduction and Error Analysis
1613 for the Physical Sciences}; @emph{Concrete Mathematics} by Graham,
1614 Knuth, and Patashnik; Steele's @emph{Common Lisp, the Language}; the
1615 @emph{CRC Standard Math Tables} (William H. Beyer, ed.); and
1616 Abramowitz and Stegun's venerable @emph{Handbook of Mathematical
1617 Functions}. Also, of course, Calc could not have been written without
1618 the excellent @emph{GNU Emacs Lisp Reference Manual}, by Bil Lewis and
1621 Final thanks go to Richard Stallman, without whose fine implementations
1622 of the Emacs editor, language, and environment, Calc would have been
1623 finished in two weeks.
1628 @c This node is accessed by the `M-# t' command.
1629 @node Interactive Tutorial, , , Top
1633 Some brief instructions on using the Emacs Info system for this tutorial:
1635 Press the space bar and Delete keys to go forward and backward in a
1636 section by screenfuls (or use the regular Emacs scrolling commands
1639 Press @kbd{n} or @kbd{p} to go to the Next or Previous section.
1640 If the section has a @dfn{menu}, press a digit key like @kbd{1}
1641 or @kbd{2} to go to a sub-section from the menu. Press @kbd{u} to
1642 go back up from a sub-section to the menu it is part of.
1644 Exercises in the tutorial all have cross-references to the
1645 appropriate page of the ``answers'' section. Press @kbd{f}, then
1646 the exercise number, to see the answer to an exercise. After
1647 you have followed a cross-reference, you can press the letter
1648 @kbd{l} to return to where you were before.
1650 You can press @kbd{?} at any time for a brief summary of Info commands.
1652 Press @kbd{1} now to enter the first section of the Tutorial.
1659 @node Tutorial, Introduction, Getting Started, Top
1663 This chapter explains how to use Calc and its many features, in
1664 a step-by-step, tutorial way. You are encouraged to run Calc and
1665 work along with the examples as you read (@pxref{Starting Calc}).
1666 If you are already familiar with advanced calculators, you may wish
1668 to skip on to the rest of this manual.
1670 @c to skip on to volume II of this manual, the @dfn{Calc Reference}.
1672 @c [fix-ref Embedded Mode]
1673 This tutorial describes the standard user interface of Calc only.
1674 The Quick mode and Keypad mode interfaces are fairly
1675 self-explanatory. @xref{Embedded Mode}, for a description of
1676 the Embedded mode interface.
1679 The easiest way to read this tutorial on-line is to have two windows on
1680 your Emacs screen, one with Calc and one with the Info system. (If you
1681 have a printed copy of the manual you can use that instead.) Press
1682 @kbd{M-# c} to turn Calc on or to switch into the Calc window, and
1683 press @kbd{M-# i} to start the Info system or to switch into its window.
1684 Or, you may prefer to use the tutorial in printed form.
1687 The easiest way to read this tutorial on-line is to have two windows on
1688 your Emacs screen, one with Calc and one with the Info system. (If you
1689 have a printed copy of the manual you can use that instead.) Press
1690 @kbd{M-# c} to turn Calc on or to switch into the Calc window, and
1691 press @kbd{M-# i} to start the Info system or to switch into its window.
1694 This tutorial is designed to be done in sequence. But the rest of this
1695 manual does not assume you have gone through the tutorial. The tutorial
1696 does not cover everything in the Calculator, but it touches on most
1700 You may wish to print out a copy of the Calc Summary and keep notes on
1701 it as you learn Calc. @xref{About This Manual}, to see how to make a
1702 printed summary. @xref{Summary}.
1705 The Calc Summary at the end of the reference manual includes some blank
1706 space for your own use. You may wish to keep notes there as you learn
1712 * Arithmetic Tutorial::
1713 * Vector/Matrix Tutorial::
1715 * Algebra Tutorial::
1716 * Programming Tutorial::
1718 * Answers to Exercises::
1721 @node Basic Tutorial, Arithmetic Tutorial, Tutorial, Tutorial
1722 @section Basic Tutorial
1725 In this section, we learn how RPN and algebraic-style calculations
1726 work, how to undo and redo an operation done by mistake, and how
1727 to control various modes of the Calculator.
1730 * RPN Tutorial:: Basic operations with the stack.
1731 * Algebraic Tutorial:: Algebraic entry; variables.
1732 * Undo Tutorial:: If you make a mistake: Undo and the trail.
1733 * Modes Tutorial:: Common mode-setting commands.
1736 @node RPN Tutorial, Algebraic Tutorial, Basic Tutorial, Basic Tutorial
1737 @subsection RPN Calculations and the Stack
1739 @cindex RPN notation
1742 Calc normally uses RPN notation. You may be familiar with the RPN
1743 system from Hewlett-Packard calculators, FORTH, or PostScript.
1744 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1749 Calc normally uses RPN notation. You may be familiar with the RPN
1750 system from Hewlett-Packard calculators, FORTH, or PostScript.
1751 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1755 The central component of an RPN calculator is the @dfn{stack}. A
1756 calculator stack is like a stack of dishes. New dishes (numbers) are
1757 added at the top of the stack, and numbers are normally only removed
1758 from the top of the stack.
1762 In an operation like @expr{2+3}, the 2 and 3 are called the @dfn{operands}
1763 and the @expr{+} is the @dfn{operator}. In an RPN calculator you always
1764 enter the operands first, then the operator. Each time you type a
1765 number, Calc adds or @dfn{pushes} it onto the top of the Stack.
1766 When you press an operator key like @kbd{+}, Calc @dfn{pops} the appropriate
1767 number of operands from the stack and pushes back the result.
1769 Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
1770 @kbd{2 @key{RET} 3 @key{RET} +}. (The @key{RET} key, Return, corresponds to
1771 the @key{ENTER} key on traditional RPN calculators.) Try this now if
1772 you wish; type @kbd{M-# c} to switch into the Calc window (you can type
1773 @kbd{M-# c} again or @kbd{M-# o} to switch back to the Tutorial window).
1774 The first four keystrokes ``push'' the numbers 2 and 3 onto the stack.
1775 The @kbd{+} key ``pops'' the top two numbers from the stack, adds them,
1776 and pushes the result (5) back onto the stack. Here's how the stack
1777 will look at various points throughout the calculation:
1785 M-# c 2 @key{RET} 3 @key{RET} + @key{DEL}
1789 The @samp{.} symbol is a marker that represents the top of the stack.
1790 Note that the ``top'' of the stack is really shown at the bottom of
1791 the Stack window. This may seem backwards, but it turns out to be
1792 less distracting in regular use.
1794 @cindex Stack levels
1795 @cindex Levels of stack
1796 The numbers @samp{1:} and @samp{2:} on the left are @dfn{stack level
1797 numbers}. Old RPN calculators always had four stack levels called
1798 @expr{x}, @expr{y}, @expr{z}, and @expr{t}. Calc's stack can grow
1799 as large as you like, so it uses numbers instead of letters. Some
1800 stack-manipulation commands accept a numeric argument that says
1801 which stack level to work on. Normal commands like @kbd{+} always
1802 work on the top few levels of the stack.
1804 @c [fix-ref Truncating the Stack]
1805 The Stack buffer is just an Emacs buffer, and you can move around in
1806 it using the regular Emacs motion commands. But no matter where the
1807 cursor is, even if you have scrolled the @samp{.} marker out of
1808 view, most Calc commands always move the cursor back down to level 1
1809 before doing anything. It is possible to move the @samp{.} marker
1810 upwards through the stack, temporarily ``hiding'' some numbers from
1811 commands like @kbd{+}. This is called @dfn{stack truncation} and
1812 we will not cover it in this tutorial; @pxref{Truncating the Stack},
1813 if you are interested.
1815 You don't really need the second @key{RET} in @kbd{2 @key{RET} 3
1816 @key{RET} +}. That's because if you type any operator name or
1817 other non-numeric key when you are entering a number, the Calculator
1818 automatically enters that number and then does the requested command.
1819 Thus @kbd{2 @key{RET} 3 +} will work just as well.
1821 Examples in this tutorial will often omit @key{RET} even when the
1822 stack displays shown would only happen if you did press @key{RET}:
1835 Here, after pressing @kbd{3} the stack would really show @samp{1: 2}
1836 with @samp{Calc:@: 3} in the minibuffer. In these situations, you can
1837 press the optional @key{RET} to see the stack as the figure shows.
1839 (@bullet{}) @strong{Exercise 1.} (This tutorial will include exercises
1840 at various points. Try them if you wish. Answers to all the exercises
1841 are located at the end of the Tutorial chapter. Each exercise will
1842 include a cross-reference to its particular answer. If you are
1843 reading with the Emacs Info system, press @kbd{f} and the
1844 exercise number to go to the answer, then the letter @kbd{l} to
1845 return to where you were.)
1848 Here's the first exercise: What will the keystrokes @kbd{1 @key{RET} 2
1849 @key{RET} 3 @key{RET} 4 + * -} compute? (@samp{*} is the symbol for
1850 multiplication.) Figure it out by hand, then try it with Calc to see
1851 if you're right. @xref{RPN Answer 1, 1}. (@bullet{})
1853 (@bullet{}) @strong{Exercise 2.} Compute
1854 @texline @math{(2\times4) + (7\times9.4) + {5\over4}}
1855 @infoline @expr{2*4 + 7*9.5 + 5/4}
1856 using the stack. @xref{RPN Answer 2, 2}. (@bullet{})
1858 The @key{DEL} key is called Backspace on some keyboards. It is
1859 whatever key you would use to correct a simple typing error when
1860 regularly using Emacs. The @key{DEL} key pops and throws away the
1861 top value on the stack. (You can still get that value back from
1862 the Trail if you should need it later on.) There are many places
1863 in this tutorial where we assume you have used @key{DEL} to erase the
1864 results of the previous example at the beginning of a new example.
1865 In the few places where it is really important to use @key{DEL} to
1866 clear away old results, the text will remind you to do so.
1868 (It won't hurt to let things accumulate on the stack, except that
1869 whenever you give a display-mode-changing command Calc will have to
1870 spend a long time reformatting such a large stack.)
1872 Since the @kbd{-} key is also an operator (it subtracts the top two
1873 stack elements), how does one enter a negative number? Calc uses
1874 the @kbd{_} (underscore) key to act like the minus sign in a number.
1875 So, typing @kbd{-5 @key{RET}} won't work because the @kbd{-} key
1876 will try to do a subtraction, but @kbd{_5 @key{RET}} works just fine.
1878 You can also press @kbd{n}, which means ``change sign.'' It changes
1879 the number at the top of the stack (or the number being entered)
1880 from positive to negative or vice-versa: @kbd{5 n @key{RET}}.
1882 @cindex Duplicating a stack entry
1883 If you press @key{RET} when you're not entering a number, the effect
1884 is to duplicate the top number on the stack. Consider this calculation:
1888 1: 3 2: 3 1: 9 2: 9 1: 81
1892 3 @key{RET} @key{RET} * @key{RET} *
1897 (Of course, an easier way to do this would be @kbd{3 @key{RET} 4 ^},
1898 to raise 3 to the fourth power.)
1900 The space-bar key (denoted @key{SPC} here) performs the same function
1901 as @key{RET}; you could replace all three occurrences of @key{RET} in
1902 the above example with @key{SPC} and the effect would be the same.
1904 @cindex Exchanging stack entries
1905 Another stack manipulation key is @key{TAB}. This exchanges the top
1906 two stack entries. Suppose you have computed @kbd{2 @key{RET} 3 +}
1907 to get 5, and then you realize what you really wanted to compute
1908 was @expr{20 / (2+3)}.
1912 1: 5 2: 5 2: 20 1: 4
1916 2 @key{RET} 3 + 20 @key{TAB} /
1921 Planning ahead, the calculation would have gone like this:
1925 1: 20 2: 20 3: 20 2: 20 1: 4
1930 20 @key{RET} 2 @key{RET} 3 + /
1934 A related stack command is @kbd{M-@key{TAB}} (hold @key{META} and type
1935 @key{TAB}). It rotates the top three elements of the stack upward,
1936 bringing the object in level 3 to the top.
1940 1: 10 2: 10 3: 10 3: 20 3: 30
1941 . 1: 20 2: 20 2: 30 2: 10
1945 10 @key{RET} 20 @key{RET} 30 @key{RET} M-@key{TAB} M-@key{TAB}
1949 (@bullet{}) @strong{Exercise 3.} Suppose the numbers 10, 20, and 30 are
1950 on the stack. Figure out how to add one to the number in level 2
1951 without affecting the rest of the stack. Also figure out how to add
1952 one to the number in level 3. @xref{RPN Answer 3, 3}. (@bullet{})
1954 Operations like @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, and @kbd{^} pop two
1955 arguments from the stack and push a result. Operations like @kbd{n} and
1956 @kbd{Q} (square root) pop a single number and push the result. You can
1957 think of them as simply operating on the top element of the stack.
1961 1: 3 1: 9 2: 9 1: 25 1: 5
1965 3 @key{RET} @key{RET} * 4 @key{RET} @key{RET} * + Q
1970 (Note that capital @kbd{Q} means to hold down the Shift key while
1971 typing @kbd{q}. Remember, plain unshifted @kbd{q} is the Quit command.)
1973 @cindex Pythagorean Theorem
1974 Here we've used the Pythagorean Theorem to determine the hypotenuse of a
1975 right triangle. Calc actually has a built-in command for that called
1976 @kbd{f h}, but let's suppose we can't remember the necessary keystrokes.
1977 We can still enter it by its full name using @kbd{M-x} notation:
1985 3 @key{RET} 4 @key{RET} M-x calc-hypot
1989 All Calculator commands begin with the word @samp{calc-}. Since it
1990 gets tiring to type this, Calc provides an @kbd{x} key which is just
1991 like the regular Emacs @kbd{M-x} key except that it types the @samp{calc-}
2000 3 @key{RET} 4 @key{RET} x hypot
2004 What happens if you take the square root of a negative number?
2008 1: 4 1: -4 1: (0, 2)
2016 The notation @expr{(a, b)} represents a complex number.
2017 Complex numbers are more traditionally written @expr{a + b i};
2018 Calc can display in this format, too, but for now we'll stick to the
2019 @expr{(a, b)} notation.
2021 If you don't know how complex numbers work, you can safely ignore this
2022 feature. Complex numbers only arise from operations that would be
2023 errors in a calculator that didn't have complex numbers. (For example,
2024 taking the square root or logarithm of a negative number produces a
2027 Complex numbers are entered in the notation shown. The @kbd{(} and
2028 @kbd{,} and @kbd{)} keys manipulate ``incomplete complex numbers.''
2032 1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3)
2040 You can perform calculations while entering parts of incomplete objects.
2041 However, an incomplete object cannot actually participate in a calculation:
2045 1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ...
2055 Adding 5 to an incomplete object makes no sense, so the last command
2056 produces an error message and leaves the stack the same.
2058 Incomplete objects can't participate in arithmetic, but they can be
2059 moved around by the regular stack commands.
2063 2: 2 3: 2 3: 3 1: ( ... 1: (2, 3)
2064 1: 3 2: 3 2: ( ... 2 .
2068 2 @key{RET} 3 @key{RET} ( M-@key{TAB} M-@key{TAB} )
2073 Note that the @kbd{,} (comma) key did not have to be used here.
2074 When you press @kbd{)} all the stack entries between the incomplete
2075 entry and the top are collected, so there's never really a reason
2076 to use the comma. It's up to you.
2078 (@bullet{}) @strong{Exercise 4.} To enter the complex number @expr{(2, 3)},
2079 your friend Joe typed @kbd{( 2 , @key{SPC} 3 )}. What happened?
2080 (Joe thought of a clever way to correct his mistake in only two
2081 keystrokes, but it didn't quite work. Try it to find out why.)
2082 @xref{RPN Answer 4, 4}. (@bullet{})
2084 Vectors are entered the same way as complex numbers, but with square
2085 brackets in place of parentheses. We'll meet vectors again later in
2088 Any Emacs command can be given a @dfn{numeric prefix argument} by
2089 typing a series of @key{META}-digits beforehand. If @key{META} is
2090 awkward for you, you can instead type @kbd{C-u} followed by the
2091 necessary digits. Numeric prefix arguments can be negative, as in
2092 @kbd{M-- M-3 M-5} or @w{@kbd{C-u - 3 5}}. Calc commands use numeric
2093 prefix arguments in a variety of ways. For example, a numeric prefix
2094 on the @kbd{+} operator adds any number of stack entries at once:
2098 1: 10 2: 10 3: 10 3: 10 1: 60
2099 . 1: 20 2: 20 2: 20 .
2103 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u 3 +
2107 For stack manipulation commands like @key{RET}, a positive numeric
2108 prefix argument operates on the top @var{n} stack entries at once. A
2109 negative argument operates on the entry in level @var{n} only. An
2110 argument of zero operates on the entire stack. In this example, we copy
2111 the second-to-top element of the stack:
2115 1: 10 2: 10 3: 10 3: 10 4: 10
2116 . 1: 20 2: 20 2: 20 3: 20
2121 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u -2 @key{RET}
2125 @cindex Clearing the stack
2126 @cindex Emptying the stack
2127 Another common idiom is @kbd{M-0 @key{DEL}}, which clears the stack.
2128 (The @kbd{M-0} numeric prefix tells @key{DEL} to operate on the
2131 @node Algebraic Tutorial, Undo Tutorial, RPN Tutorial, Basic Tutorial
2132 @subsection Algebraic-Style Calculations
2135 If you are not used to RPN notation, you may prefer to operate the
2136 Calculator in Algebraic mode, which is closer to the way
2137 non-RPN calculators work. In Algebraic mode, you enter formulas
2138 in traditional @expr{2+3} notation.
2140 You don't really need any special ``mode'' to enter algebraic formulas.
2141 You can enter a formula at any time by pressing the apostrophe (@kbd{'})
2142 key. Answer the prompt with the desired formula, then press @key{RET}.
2143 The formula is evaluated and the result is pushed onto the RPN stack.
2144 If you don't want to think in RPN at all, you can enter your whole
2145 computation as a formula, read the result from the stack, then press
2146 @key{DEL} to delete it from the stack.
2148 Try pressing the apostrophe key, then @kbd{2+3+4}, then @key{RET}.
2149 The result should be the number 9.
2151 Algebraic formulas use the operators @samp{+}, @samp{-}, @samp{*},
2152 @samp{/}, and @samp{^}. You can use parentheses to make the order
2153 of evaluation clear. In the absence of parentheses, @samp{^} is
2154 evaluated first, then @samp{*}, then @samp{/}, then finally
2155 @samp{+} and @samp{-}. For example, the expression
2158 2 + 3*4*5 / 6*7^8 - 9
2165 2 + ((3*4*5) / (6*(7^8)) - 9
2169 or, in large mathematical notation,
2184 $$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
2189 The result of this expression will be the number @mathit{-6.99999826533}.
2191 Calc's order of evaluation is the same as for most computer languages,
2192 except that @samp{*} binds more strongly than @samp{/}, as the above
2193 example shows. As in normal mathematical notation, the @samp{*} symbol
2194 can often be omitted: @samp{2 a} is the same as @samp{2*a}.
2196 Operators at the same level are evaluated from left to right, except
2197 that @samp{^} is evaluated from right to left. Thus, @samp{2-3-4} is
2198 equivalent to @samp{(2-3)-4} or @mathit{-5}, whereas @samp{2^3^4} is equivalent
2199 to @samp{2^(3^4)} (a very large integer; try it!).
2201 If you tire of typing the apostrophe all the time, there is
2202 Algebraic mode, where Calc automatically senses
2203 when you are about to type an algebraic expression. To enter this
2204 mode, press the two letters @w{@kbd{m a}}. (An @samp{Alg} indicator
2205 should appear in the Calc window's mode line.)
2207 Press @kbd{m a}, then @kbd{2+3+4} with no apostrophe, then @key{RET}.
2209 In Algebraic mode, when you press any key that would normally begin
2210 entering a number (such as a digit, a decimal point, or the @kbd{_}
2211 key), or if you press @kbd{(} or @kbd{[}, Calc automatically begins
2214 Functions which do not have operator symbols like @samp{+} and @samp{*}
2215 must be entered in formulas using function-call notation. For example,
2216 the function name corresponding to the square-root key @kbd{Q} is
2217 @code{sqrt}. To compute a square root in a formula, you would use
2218 the notation @samp{sqrt(@var{x})}.
2220 Press the apostrophe, then type @kbd{sqrt(5*2) - 3}. The result should
2221 be @expr{0.16227766017}.
2223 Note that if the formula begins with a function name, you need to use
2224 the apostrophe even if you are in Algebraic mode. If you type @kbd{arcsin}
2225 out of the blue, the @kbd{a r} will be taken as an Algebraic Rewrite
2226 command, and the @kbd{csin} will be taken as the name of the rewrite
2229 Some people prefer to enter complex numbers and vectors in algebraic
2230 form because they find RPN entry with incomplete objects to be too
2231 distracting, even though they otherwise use Calc as an RPN calculator.
2233 Still in Algebraic mode, type:
2237 1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1)
2238 . 1: (1, -2) . 1: 1 .
2241 (2,3) @key{RET} (1,-2) @key{RET} * 1 @key{RET} +
2245 Algebraic mode allows us to enter complex numbers without pressing
2246 an apostrophe first, but it also means we need to press @key{RET}
2247 after every entry, even for a simple number like @expr{1}.
2249 (You can type @kbd{C-u m a} to enable a special Incomplete Algebraic
2250 mode in which the @kbd{(} and @kbd{[} keys use algebraic entry even
2251 though regular numeric keys still use RPN numeric entry. There is also
2252 Total Algebraic mode, started by typing @kbd{m t}, in which all
2253 normal keys begin algebraic entry. You must then use the @key{META} key
2254 to type Calc commands: @kbd{M-m t} to get back out of Total Algebraic
2255 mode, @kbd{M-q} to quit, etc.)
2257 If you're still in Algebraic mode, press @kbd{m a} again to turn it off.
2259 Actual non-RPN calculators use a mixture of algebraic and RPN styles.
2260 In general, operators of two numbers (like @kbd{+} and @kbd{*})
2261 use algebraic form, but operators of one number (like @kbd{n} and @kbd{Q})
2262 use RPN form. Also, a non-RPN calculator allows you to see the
2263 intermediate results of a calculation as you go along. You can
2264 accomplish this in Calc by performing your calculation as a series
2265 of algebraic entries, using the @kbd{$} sign to tie them together.
2266 In an algebraic formula, @kbd{$} represents the number on the top
2267 of the stack. Here, we perform the calculation
2268 @texline @math{\sqrt{2\times4+1}},
2269 @infoline @expr{sqrt(2*4+1)},
2270 which on a traditional calculator would be done by pressing
2271 @kbd{2 * 4 + 1 =} and then the square-root key.
2278 ' 2*4 @key{RET} $+1 @key{RET} Q
2283 Notice that we didn't need to press an apostrophe for the @kbd{$+1},
2284 because the dollar sign always begins an algebraic entry.
2286 (@bullet{}) @strong{Exercise 1.} How could you get the same effect as
2287 pressing @kbd{Q} but using an algebraic entry instead? How about
2288 if the @kbd{Q} key on your keyboard were broken?
2289 @xref{Algebraic Answer 1, 1}. (@bullet{})
2291 The notations @kbd{$$}, @kbd{$$$}, and so on stand for higher stack
2292 entries. For example, @kbd{' $$+$ @key{RET}} is just like typing @kbd{+}.
2294 Algebraic formulas can include @dfn{variables}. To store in a
2295 variable, press @kbd{s s}, then type the variable name, then press
2296 @key{RET}. (There are actually two flavors of store command:
2297 @kbd{s s} stores a number in a variable but also leaves the number
2298 on the stack, while @w{@kbd{s t}} removes a number from the stack and
2299 stores it in the variable.) A variable name should consist of one
2300 or more letters or digits, beginning with a letter.
2304 1: 17 . 1: a + a^2 1: 306
2307 17 s t a @key{RET} ' a+a^2 @key{RET} =
2312 The @kbd{=} key @dfn{evaluates} a formula by replacing all its
2313 variables by the values that were stored in them.
2315 For RPN calculations, you can recall a variable's value on the
2316 stack either by entering its name as a formula and pressing @kbd{=},
2317 or by using the @kbd{s r} command.
2321 1: 17 2: 17 3: 17 2: 17 1: 306
2322 . 1: 17 2: 17 1: 289 .
2326 s r a @key{RET} ' a @key{RET} = 2 ^ +
2330 If you press a single digit for a variable name (as in @kbd{s t 3}, you
2331 get one of ten @dfn{quick variables} @code{q0} through @code{q9}.
2332 They are ``quick'' simply because you don't have to type the letter
2333 @code{q} or the @key{RET} after their names. In fact, you can type
2334 simply @kbd{s 3} as a shorthand for @kbd{s s 3}, and likewise for
2335 @kbd{t 3} and @w{@kbd{r 3}}.
2337 Any variables in an algebraic formula for which you have not stored
2338 values are left alone, even when you evaluate the formula.
2342 1: 2 a + 2 b 1: 34 + 2 b
2349 Calls to function names which are undefined in Calc are also left
2350 alone, as are calls for which the value is undefined.
2354 1: 2 + log10(0) + log10(x) + log10(5, 6) + foo(3)
2357 ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) @key{RET}
2362 In this example, the first call to @code{log10} works, but the other
2363 calls are not evaluated. In the second call, the logarithm is
2364 undefined for that value of the argument; in the third, the argument
2365 is symbolic, and in the fourth, there are too many arguments. In the
2366 fifth case, there is no function called @code{foo}. You will see a
2367 ``Wrong number of arguments'' message referring to @samp{log10(5,6)}.
2368 Press the @kbd{w} (``why'') key to see any other messages that may
2369 have arisen from the last calculation. In this case you will get
2370 ``logarithm of zero,'' then ``number expected: @code{x}''. Calc
2371 automatically displays the first message only if the message is
2372 sufficiently important; for example, Calc considers ``wrong number
2373 of arguments'' and ``logarithm of zero'' to be important enough to
2374 report automatically, while a message like ``number expected: @code{x}''
2375 will only show up if you explicitly press the @kbd{w} key.
2377 (@bullet{}) @strong{Exercise 2.} Joe entered the formula @samp{2 x y},
2378 stored 5 in @code{x}, pressed @kbd{=}, and got the expected result,
2379 @samp{10 y}. He then tried the same for the formula @samp{2 x (1+y)},
2380 expecting @samp{10 (1+y)}, but it didn't work. Why not?
2381 @xref{Algebraic Answer 2, 2}. (@bullet{})
2383 (@bullet{}) @strong{Exercise 3.} What result would you expect
2384 @kbd{1 @key{RET} 0 /} to give? What if you then type @kbd{0 *}?
2385 @xref{Algebraic Answer 3, 3}. (@bullet{})
2387 One interesting way to work with variables is to use the
2388 @dfn{evaluates-to} (@samp{=>}) operator. It works like this:
2389 Enter a formula algebraically in the usual way, but follow
2390 the formula with an @samp{=>} symbol. (There is also an @kbd{s =}
2391 command which builds an @samp{=>} formula using the stack.) On
2392 the stack, you will see two copies of the formula with an @samp{=>}
2393 between them. The lefthand formula is exactly like you typed it;
2394 the righthand formula has been evaluated as if by typing @kbd{=}.
2398 2: 2 + 3 => 5 2: 2 + 3 => 5
2399 1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b
2402 ' 2+3 => @key{RET} ' 2a+2b @key{RET} s = 10 s t a @key{RET}
2407 Notice that the instant we stored a new value in @code{a}, all
2408 @samp{=>} operators already on the stack that referred to @expr{a}
2409 were updated to use the new value. With @samp{=>}, you can push a
2410 set of formulas on the stack, then change the variables experimentally
2411 to see the effects on the formulas' values.
2413 You can also ``unstore'' a variable when you are through with it:
2418 1: 2 a + 2 b => 2 a + 2 b
2425 We will encounter formulas involving variables and functions again
2426 when we discuss the algebra and calculus features of the Calculator.
2428 @node Undo Tutorial, Modes Tutorial, Algebraic Tutorial, Basic Tutorial
2429 @subsection Undo and Redo
2432 If you make a mistake, you can usually correct it by pressing shift-@kbd{U},
2433 the ``undo'' command. First, clear the stack (@kbd{M-0 @key{DEL}}) and exit
2434 and restart Calc (@kbd{M-# M-# M-# M-#}) to make sure things start off
2435 with a clean slate. Now:
2439 1: 2 2: 2 1: 8 2: 2 1: 6
2447 You can undo any number of times. Calc keeps a complete record of
2448 all you have done since you last opened the Calc window. After the
2449 above example, you could type:
2461 You can also type @kbd{D} to ``redo'' a command that you have undone
2466 . 1: 2 2: 2 1: 6 1: 6
2475 It was not possible to redo past the @expr{6}, since that was placed there
2476 by something other than an undo command.
2479 You can think of undo and redo as a sort of ``time machine.'' Press
2480 @kbd{U} to go backward in time, @kbd{D} to go forward. If you go
2481 backward and do something (like @kbd{*}) then, as any science fiction
2482 reader knows, you have changed your future and you cannot go forward
2483 again. Thus, the inability to redo past the @expr{6} even though there
2484 was an earlier undo command.
2486 You can always recall an earlier result using the Trail. We've ignored
2487 the trail so far, but it has been faithfully recording everything we
2488 did since we loaded the Calculator. If the Trail is not displayed,
2489 press @kbd{t d} now to turn it on.
2491 Let's try grabbing an earlier result. The @expr{8} we computed was
2492 undone by a @kbd{U} command, and was lost even to Redo when we pressed
2493 @kbd{*}, but it's still there in the trail. There should be a little
2494 @samp{>} arrow (the @dfn{trail pointer}) resting on the last trail
2495 entry. If there isn't, press @kbd{t ]} to reset the trail pointer.
2496 Now, press @w{@kbd{t p}} to move the arrow onto the line containing
2497 @expr{8}, and press @w{@kbd{t y}} to ``yank'' that number back onto the
2500 If you press @kbd{t ]} again, you will see that even our Yank command
2501 went into the trail.
2503 Let's go further back in time. Earlier in the tutorial we computed
2504 a huge integer using the formula @samp{2^3^4}. We don't remember
2505 what it was, but the first digits were ``241''. Press @kbd{t r}
2506 (which stands for trail-search-reverse), then type @kbd{241}.
2507 The trail cursor will jump back to the next previous occurrence of
2508 the string ``241'' in the trail. This is just a regular Emacs
2509 incremental search; you can now press @kbd{C-s} or @kbd{C-r} to
2510 continue the search forwards or backwards as you like.
2512 To finish the search, press @key{RET}. This halts the incremental
2513 search and leaves the trail pointer at the thing we found. Now we
2514 can type @kbd{t y} to yank that number onto the stack. If we hadn't
2515 remembered the ``241'', we could simply have searched for @kbd{2^3^4},
2516 then pressed @kbd{@key{RET} t n} to halt and then move to the next item.
2518 You may have noticed that all the trail-related commands begin with
2519 the letter @kbd{t}. (The store-and-recall commands, on the other hand,
2520 all began with @kbd{s}.) Calc has so many commands that there aren't
2521 enough keys for all of them, so various commands are grouped into
2522 two-letter sequences where the first letter is called the @dfn{prefix}
2523 key. If you type a prefix key by accident, you can press @kbd{C-g}
2524 to cancel it. (In fact, you can press @kbd{C-g} to cancel almost
2525 anything in Emacs.) To get help on a prefix key, press that key
2526 followed by @kbd{?}. Some prefixes have several lines of help,
2527 so you need to press @kbd{?} repeatedly to see them all.
2528 You can also type @kbd{h h} to see all the help at once.
2530 Try pressing @kbd{t ?} now. You will see a line of the form,
2533 trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t-
2537 The word ``trail'' indicates that the @kbd{t} prefix key contains
2538 trail-related commands. Each entry on the line shows one command,
2539 with a single capital letter showing which letter you press to get
2540 that command. We have used @kbd{t n}, @kbd{t p}, @kbd{t ]}, and
2541 @kbd{t y} so far. The @samp{[MORE]} means you can press @kbd{?}
2542 again to see more @kbd{t}-prefix commands. Notice that the commands
2543 are roughly divided (by semicolons) into related groups.
2545 When you are in the help display for a prefix key, the prefix is
2546 still active. If you press another key, like @kbd{y} for example,
2547 it will be interpreted as a @kbd{t y} command. If all you wanted
2548 was to look at the help messages, press @kbd{C-g} afterwards to cancel
2551 One more way to correct an error is by editing the stack entries.
2552 The actual Stack buffer is marked read-only and must not be edited
2553 directly, but you can press @kbd{`} (the backquote or accent grave)
2554 to edit a stack entry.
2556 Try entering @samp{3.141439} now. If this is supposed to represent
2557 @cpi{}, it's got several errors. Press @kbd{`} to edit this number.
2558 Now use the normal Emacs cursor motion and editing keys to change
2559 the second 4 to a 5, and to transpose the 3 and the 9. When you
2560 press @key{RET}, the number on the stack will be replaced by your
2561 new number. This works for formulas, vectors, and all other types
2562 of values you can put on the stack. The @kbd{`} key also works
2563 during entry of a number or algebraic formula.
2565 @node Modes Tutorial, , Undo Tutorial, Basic Tutorial
2566 @subsection Mode-Setting Commands
2569 Calc has many types of @dfn{modes} that affect the way it interprets
2570 your commands or the way it displays data. We have already seen one
2571 mode, namely Algebraic mode. There are many others, too; we'll
2572 try some of the most common ones here.
2574 Perhaps the most fundamental mode in Calc is the current @dfn{precision}.
2575 Notice the @samp{12} on the Calc window's mode line:
2578 --%%-Calc: 12 Deg (Calculator)----All------
2582 Most of the symbols there are Emacs things you don't need to worry
2583 about, but the @samp{12} and the @samp{Deg} are mode indicators.
2584 The @samp{12} means that calculations should always be carried to
2585 12 significant figures. That is why, when we type @kbd{1 @key{RET} 7 /},
2586 we get @expr{0.142857142857} with exactly 12 digits, not counting
2587 leading and trailing zeros.
2589 You can set the precision to anything you like by pressing @kbd{p},
2590 then entering a suitable number. Try pressing @kbd{p 30 @key{RET}},
2591 then doing @kbd{1 @key{RET} 7 /} again:
2596 2: 0.142857142857142857142857142857
2601 Although the precision can be set arbitrarily high, Calc always
2602 has to have @emph{some} value for the current precision. After
2603 all, the true value @expr{1/7} is an infinitely repeating decimal;
2604 Calc has to stop somewhere.
2606 Of course, calculations are slower the more digits you request.
2607 Press @w{@kbd{p 12}} now to set the precision back down to the default.
2609 Calculations always use the current precision. For example, even
2610 though we have a 30-digit value for @expr{1/7} on the stack, if
2611 we use it in a calculation in 12-digit mode it will be rounded
2612 down to 12 digits before it is used. Try it; press @key{RET} to
2613 duplicate the number, then @w{@kbd{1 +}}. Notice that the @key{RET}
2614 key didn't round the number, because it doesn't do any calculation.
2615 But the instant we pressed @kbd{+}, the number was rounded down.
2620 2: 0.142857142857142857142857142857
2627 In fact, since we added a digit on the left, we had to lose one
2628 digit on the right from even the 12-digit value of @expr{1/7}.
2630 How did we get more than 12 digits when we computed @samp{2^3^4}? The
2631 answer is that Calc makes a distinction between @dfn{integers} and
2632 @dfn{floating-point} numbers, or @dfn{floats}. An integer is a number
2633 that does not contain a decimal point. There is no such thing as an
2634 ``infinitely repeating fraction integer,'' so Calc doesn't have to limit
2635 itself. If you asked for @samp{2^10000} (don't try this!), you would
2636 have to wait a long time but you would eventually get an exact answer.
2637 If you ask for @samp{2.^10000}, you will quickly get an answer which is
2638 correct only to 12 places. The decimal point tells Calc that it should
2639 use floating-point arithmetic to get the answer, not exact integer
2642 You can use the @kbd{F} (@code{calc-floor}) command to convert a
2643 floating-point value to an integer, and @kbd{c f} (@code{calc-float})
2644 to convert an integer to floating-point form.
2646 Let's try entering that last calculation:
2650 1: 2. 2: 2. 1: 1.99506311689e3010
2654 2.0 @key{RET} 10000 @key{RET} ^
2659 @cindex Scientific notation, entry of
2660 Notice the letter @samp{e} in there. It represents ``times ten to the
2661 power of,'' and is used by Calc automatically whenever writing the
2662 number out fully would introduce more extra zeros than you probably
2663 want to see. You can enter numbers in this notation, too.
2667 1: 2. 2: 2. 1: 1.99506311678e3010
2671 2.0 @key{RET} 1e4 @key{RET} ^
2675 @cindex Round-off errors
2677 Hey, the answer is different! Look closely at the middle columns
2678 of the two examples. In the first, the stack contained the
2679 exact integer @expr{10000}, but in the second it contained
2680 a floating-point value with a decimal point. When you raise a
2681 number to an integer power, Calc uses repeated squaring and
2682 multiplication to get the answer. When you use a floating-point
2683 power, Calc uses logarithms and exponentials. As you can see,
2684 a slight error crept in during one of these methods. Which
2685 one should we trust? Let's raise the precision a bit and find
2690 . 1: 2. 2: 2. 1: 1.995063116880828e3010
2694 p 16 @key{RET} 2. @key{RET} 1e4 ^ p 12 @key{RET}
2699 @cindex Guard digits
2700 Presumably, it doesn't matter whether we do this higher-precision
2701 calculation using an integer or floating-point power, since we
2702 have added enough ``guard digits'' to trust the first 12 digits
2703 no matter what. And the verdict is@dots{} Integer powers were more
2704 accurate; in fact, the result was only off by one unit in the
2707 @cindex Guard digits
2708 Calc does many of its internal calculations to a slightly higher
2709 precision, but it doesn't always bump the precision up enough.
2710 In each case, Calc added about two digits of precision during
2711 its calculation and then rounded back down to 12 digits
2712 afterward. In one case, it was enough; in the other, it
2713 wasn't. If you really need @var{x} digits of precision, it
2714 never hurts to do the calculation with a few extra guard digits.
2716 What if we want guard digits but don't want to look at them?
2717 We can set the @dfn{float format}. Calc supports four major
2718 formats for floating-point numbers, called @dfn{normal},
2719 @dfn{fixed-point}, @dfn{scientific notation}, and @dfn{engineering
2720 notation}. You get them by pressing @w{@kbd{d n}}, @kbd{d f},
2721 @kbd{d s}, and @kbd{d e}, respectively. In each case, you can
2722 supply a numeric prefix argument which says how many digits
2723 should be displayed. As an example, let's put a few numbers
2724 onto the stack and try some different display modes. First,
2725 use @kbd{M-0 @key{DEL}} to clear the stack, then enter the four
2730 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2731 3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000
2732 2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450
2733 1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345
2736 d n M-3 d n d s M-3 d s M-3 d f
2741 Notice that when we typed @kbd{M-3 d n}, the numbers were rounded down
2742 to three significant digits, but then when we typed @kbd{d s} all
2743 five significant figures reappeared. The float format does not
2744 affect how numbers are stored, it only affects how they are
2745 displayed. Only the current precision governs the actual rounding
2746 of numbers in the Calculator's memory.
2748 Engineering notation, not shown here, is like scientific notation
2749 except the exponent (the power-of-ten part) is always adjusted to be
2750 a multiple of three (as in ``kilo,'' ``micro,'' etc.). As a result
2751 there will be one, two, or three digits before the decimal point.
2753 Whenever you change a display-related mode, Calc redraws everything
2754 in the stack. This may be slow if there are many things on the stack,
2755 so Calc allows you to type shift-@kbd{H} before any mode command to
2756 prevent it from updating the stack. Anything Calc displays after the
2757 mode-changing command will appear in the new format.
2761 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2762 3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345.
2763 2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45
2764 1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345
2767 H d s @key{DEL} U @key{TAB} d @key{SPC} d n
2772 Here the @kbd{H d s} command changes to scientific notation but without
2773 updating the screen. Deleting the top stack entry and undoing it back
2774 causes it to show up in the new format; swapping the top two stack
2775 entries reformats both entries. The @kbd{d @key{SPC}} command refreshes the
2776 whole stack. The @kbd{d n} command changes back to the normal float
2777 format; since it doesn't have an @kbd{H} prefix, it also updates all
2778 the stack entries to be in @kbd{d n} format.
2780 Notice that the integer @expr{12345} was not affected by any
2781 of the float formats. Integers are integers, and are always
2784 @cindex Large numbers, readability
2785 Large integers have their own problems. Let's look back at
2786 the result of @kbd{2^3^4}.
2789 2417851639229258349412352
2793 Quick---how many digits does this have? Try typing @kbd{d g}:
2796 2,417,851,639,229,258,349,412,352
2800 Now how many digits does this have? It's much easier to tell!
2801 We can actually group digits into clumps of any size. Some
2802 people prefer @kbd{M-5 d g}:
2805 24178,51639,22925,83494,12352
2808 Let's see what happens to floating-point numbers when they are grouped.
2809 First, type @kbd{p 25 @key{RET}} to make sure we have enough precision
2810 to get ourselves into trouble. Now, type @kbd{1e13 /}:
2813 24,17851,63922.9258349412352
2817 The integer part is grouped but the fractional part isn't. Now try
2818 @kbd{M-- M-5 d g} (that's meta-minus-sign, meta-five):
2821 24,17851,63922.92583,49412,352
2824 If you find it hard to tell the decimal point from the commas, try
2825 changing the grouping character to a space with @kbd{d , @key{SPC}}:
2828 24 17851 63922.92583 49412 352
2831 Type @kbd{d , ,} to restore the normal grouping character, then
2832 @kbd{d g} again to turn grouping off. Also, press @kbd{p 12} to
2833 restore the default precision.
2835 Press @kbd{U} enough times to get the original big integer back.
2836 (Notice that @kbd{U} does not undo each mode-setting command; if
2837 you want to undo a mode-setting command, you have to do it yourself.)
2838 Now, type @kbd{d r 16 @key{RET}}:
2841 16#200000000000000000000
2845 The number is now displayed in @dfn{hexadecimal}, or ``base-16'' form.
2846 Suddenly it looks pretty simple; this should be no surprise, since we
2847 got this number by computing a power of two, and 16 is a power of 2.
2848 In fact, we can use @w{@kbd{d r 2 @key{RET}}} to see it in actual binary
2852 2#1000000000000000000000000000000000000000000000000000000 @dots{}
2856 We don't have enough space here to show all the zeros! They won't
2857 fit on a typical screen, either, so you will have to use horizontal
2858 scrolling to see them all. Press @kbd{<} and @kbd{>} to scroll the
2859 stack window left and right by half its width. Another way to view
2860 something large is to press @kbd{`} (back-quote) to edit the top of
2861 stack in a separate window. (Press @kbd{C-c C-c} when you are done.)
2863 You can enter non-decimal numbers using the @kbd{#} symbol, too.
2864 Let's see what the hexadecimal number @samp{5FE} looks like in
2865 binary. Type @kbd{16#5FE} (the letters can be typed in upper or
2866 lower case; they will always appear in upper case). It will also
2867 help to turn grouping on with @kbd{d g}:
2873 Notice that @kbd{d g} groups by fours by default if the display radix
2874 is binary or hexadecimal, but by threes if it is decimal, octal, or any
2877 Now let's see that number in decimal; type @kbd{d r 10}:
2883 Numbers are not @emph{stored} with any particular radix attached. They're
2884 just numbers; they can be entered in any radix, and are always displayed
2885 in whatever radix you've chosen with @kbd{d r}. The current radix applies
2886 to integers, fractions, and floats.
2888 @cindex Roundoff errors, in non-decimal numbers
2889 (@bullet{}) @strong{Exercise 1.} Your friend Joe tried to enter one-third
2890 as @samp{3#0.1} in @kbd{d r 3} mode with a precision of 12. He got
2891 @samp{3#0.0222222...} (with 25 2's) in the display. When he multiplied
2892 that by three, he got @samp{3#0.222222...} instead of the expected
2893 @samp{3#1}. Next, Joe entered @samp{3#0.2} and, to his great relief,
2894 saw @samp{3#0.2} on the screen. But when he typed @kbd{2 /}, he got
2895 @samp{3#0.10000001} (some zeros omitted). What's going on here?
2896 @xref{Modes Answer 1, 1}. (@bullet{})
2898 @cindex Scientific notation, in non-decimal numbers
2899 (@bullet{}) @strong{Exercise 2.} Scientific notation works in non-decimal
2900 modes in the natural way (the exponent is a power of the radix instead of
2901 a power of ten, although the exponent itself is always written in decimal).
2902 Thus @samp{8#1.23e3 = 8#1230.0}. Suppose we have the hexadecimal number
2903 @samp{f.e8f} times 16 to the 15th power: We write @samp{16#f.e8fe15}.
2904 What is wrong with this picture? What could we write instead that would
2905 work better? @xref{Modes Answer 2, 2}. (@bullet{})
2907 The @kbd{m} prefix key has another set of modes, relating to the way
2908 Calc interprets your inputs and does computations. Whereas @kbd{d}-prefix
2909 modes generally affect the way things look, @kbd{m}-prefix modes affect
2910 the way they are actually computed.
2912 The most popular @kbd{m}-prefix mode is the @dfn{angular mode}. Notice
2913 the @samp{Deg} indicator in the mode line. This means that if you use
2914 a command that interprets a number as an angle, it will assume the
2915 angle is measured in degrees. For example,
2919 1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5
2927 The shift-@kbd{S} command computes the sine of an angle. The sine
2929 @texline @math{\sqrt{2}/2};
2930 @infoline @expr{sqrt(2)/2};
2931 squaring this yields @expr{2/4 = 0.5}. However, there has been a slight
2932 roundoff error because the representation of
2933 @texline @math{\sqrt{2}/2}
2934 @infoline @expr{sqrt(2)/2}
2935 wasn't exact. The @kbd{c 1} command is a handy way to clean up numbers
2936 in this case; it temporarily reduces the precision by one digit while it
2937 re-rounds the number on the top of the stack.
2939 @cindex Roundoff errors, examples
2940 (@bullet{}) @strong{Exercise 3.} Your friend Joe computed the sine
2941 of 45 degrees as shown above, then, hoping to avoid an inexact
2942 result, he increased the precision to 16 digits before squaring.
2943 What happened? @xref{Modes Answer 3, 3}. (@bullet{})
2945 To do this calculation in radians, we would type @kbd{m r} first.
2946 (The indicator changes to @samp{Rad}.) 45 degrees corresponds to
2947 @cpiover{4} radians. To get @cpi{}, press the @kbd{P} key. (Once
2948 again, this is a shifted capital @kbd{P}. Remember, unshifted
2949 @kbd{p} sets the precision.)
2953 1: 3.14159265359 1: 0.785398163398 1: 0.707106781187
2960 Likewise, inverse trigonometric functions generate results in
2961 either radians or degrees, depending on the current angular mode.
2965 1: 0.707106781187 1: 0.785398163398 1: 45.
2968 .5 Q m r I S m d U I S
2973 Here we compute the Inverse Sine of
2974 @texline @math{\sqrt{0.5}},
2975 @infoline @expr{sqrt(0.5)},
2976 first in radians, then in degrees.
2978 Use @kbd{c d} and @kbd{c r} to convert a number from radians to degrees
2983 1: 45 1: 0.785398163397 1: 45.
2990 Another interesting mode is @dfn{Fraction mode}. Normally,
2991 dividing two integers produces a floating-point result if the
2992 quotient can't be expressed as an exact integer. Fraction mode
2993 causes integer division to produce a fraction, i.e., a rational
2998 2: 12 1: 1.33333333333 1: 4:3
3002 12 @key{RET} 9 / m f U / m f
3007 In the first case, we get an approximate floating-point result.
3008 In the second case, we get an exact fractional result (four-thirds).
3010 You can enter a fraction at any time using @kbd{:} notation.
3011 (Calc uses @kbd{:} instead of @kbd{/} as the fraction separator
3012 because @kbd{/} is already used to divide the top two stack
3013 elements.) Calculations involving fractions will always
3014 produce exact fractional results; Fraction mode only says
3015 what to do when dividing two integers.
3017 @cindex Fractions vs. floats
3018 @cindex Floats vs. fractions
3019 (@bullet{}) @strong{Exercise 4.} If fractional arithmetic is exact,
3020 why would you ever use floating-point numbers instead?
3021 @xref{Modes Answer 4, 4}. (@bullet{})
3023 Typing @kbd{m f} doesn't change any existing values in the stack.
3024 In the above example, we had to Undo the division and do it over
3025 again when we changed to Fraction mode. But if you use the
3026 evaluates-to operator you can get commands like @kbd{m f} to
3031 1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3
3034 ' 12/9 => @key{RET} p 4 @key{RET} m f
3039 In this example, the righthand side of the @samp{=>} operator
3040 on the stack is recomputed when we change the precision, then
3041 again when we change to Fraction mode. All @samp{=>} expressions
3042 on the stack are recomputed every time you change any mode that
3043 might affect their values.
3045 @node Arithmetic Tutorial, Vector/Matrix Tutorial, Basic Tutorial, Tutorial
3046 @section Arithmetic Tutorial
3049 In this section, we explore the arithmetic and scientific functions
3050 available in the Calculator.
3052 The standard arithmetic commands are @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/},
3053 and @kbd{^}. Each normally takes two numbers from the top of the stack
3054 and pushes back a result. The @kbd{n} and @kbd{&} keys perform
3055 change-sign and reciprocal operations, respectively.
3059 1: 5 1: 0.2 1: 5. 1: -5. 1: 5.
3066 @cindex Binary operators
3067 You can apply a ``binary operator'' like @kbd{+} across any number of
3068 stack entries by giving it a numeric prefix. You can also apply it
3069 pairwise to several stack elements along with the top one if you use
3074 3: 2 1: 9 3: 2 4: 2 3: 12
3075 2: 3 . 2: 3 3: 3 2: 13
3076 1: 4 1: 4 2: 4 1: 14
3080 2 @key{RET} 3 @key{RET} 4 M-3 + U 10 M-- M-3 +
3084 @cindex Unary operators
3085 You can apply a ``unary operator'' like @kbd{&} to the top @var{n}
3086 stack entries with a numeric prefix, too.
3091 2: 3 2: 0.333333333333 2: 3.
3095 2 @key{RET} 3 @key{RET} 4 M-3 & M-2 &
3099 Notice that the results here are left in floating-point form.
3100 We can convert them back to integers by pressing @kbd{F}, the
3101 ``floor'' function. This function rounds down to the next lower
3102 integer. There is also @kbd{R}, which rounds to the nearest
3120 Since dividing-and-flooring (i.e., ``integer quotient'') is such a
3121 common operation, Calc provides a special command for that purpose, the
3122 backslash @kbd{\}. Another common arithmetic operator is @kbd{%}, which
3123 computes the remainder that would arise from a @kbd{\} operation, i.e.,
3124 the ``modulo'' of two numbers. For example,
3128 2: 1234 1: 12 2: 1234 1: 34
3132 1234 @key{RET} 100 \ U %
3136 These commands actually work for any real numbers, not just integers.
3140 2: 3.1415 1: 3 2: 3.1415 1: 0.1415
3144 3.1415 @key{RET} 1 \ U %
3148 (@bullet{}) @strong{Exercise 1.} The @kbd{\} command would appear to be a
3149 frill, since you could always do the same thing with @kbd{/ F}. Think
3150 of a situation where this is not true---@kbd{/ F} would be inadequate.
3151 Now think of a way you could get around the problem if Calc didn't
3152 provide a @kbd{\} command. @xref{Arithmetic Answer 1, 1}. (@bullet{})
3154 We've already seen the @kbd{Q} (square root) and @kbd{S} (sine)
3155 commands. Other commands along those lines are @kbd{C} (cosine),
3156 @kbd{T} (tangent), @kbd{E} (@expr{e^x}) and @kbd{L} (natural
3157 logarithm). These can be modified by the @kbd{I} (inverse) and
3158 @kbd{H} (hyperbolic) prefix keys.
3160 Let's compute the sine and cosine of an angle, and verify the
3162 @texline @math{\sin^2x + \cos^2x = 1}.
3163 @infoline @expr{sin(x)^2 + cos(x)^2 = 1}.
3164 We'll arbitrarily pick @mathit{-64} degrees as a good value for @expr{x}.
3165 With the angular mode set to degrees (type @w{@kbd{m d}}), do:
3169 2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1.
3170 1: -64 1: -0.89879 1: -64 1: 0.43837 .
3173 64 n @key{RET} @key{RET} S @key{TAB} C f h
3178 (For brevity, we're showing only five digits of the results here.
3179 You can of course do these calculations to any precision you like.)
3181 Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum
3182 of squares, command.
3185 @texline @math{\displaystyle\tan x = {\sin x \over \cos x}}.
3186 @infoline @expr{tan(x) = sin(x) / cos(x)}.
3190 2: -0.89879 1: -2.0503 1: -64.
3198 A physical interpretation of this calculation is that if you move
3199 @expr{0.89879} units downward and @expr{0.43837} units to the right,
3200 your direction of motion is @mathit{-64} degrees from horizontal. Suppose
3201 we move in the opposite direction, up and to the left:
3205 2: -0.89879 2: 0.89879 1: -2.0503 1: -64.
3206 1: 0.43837 1: -0.43837 . .
3214 How can the angle be the same? The answer is that the @kbd{/} operation
3215 loses information about the signs of its inputs. Because the quotient
3216 is negative, we know exactly one of the inputs was negative, but we
3217 can't tell which one. There is an @kbd{f T} [@code{arctan2}] function which
3218 computes the inverse tangent of the quotient of a pair of numbers.
3219 Since you feed it the two original numbers, it has enough information
3220 to give you a full 360-degree answer.
3224 2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180.
3225 1: -0.43837 . 2: -0.89879 1: -64. .
3229 U U f T M-@key{RET} M-2 n f T -
3234 The resulting angles differ by 180 degrees; in other words, they
3235 point in opposite directions, just as we would expect.
3237 The @key{META}-@key{RET} we used in the third step is the
3238 ``last-arguments'' command. It is sort of like Undo, except that it
3239 restores the arguments of the last command to the stack without removing
3240 the command's result. It is useful in situations like this one,
3241 where we need to do several operations on the same inputs. We could
3242 have accomplished the same thing by using @kbd{M-2 @key{RET}} to duplicate
3243 the top two stack elements right after the @kbd{U U}, then a pair of
3244 @kbd{M-@key{TAB}} commands to cycle the 116 up around the duplicates.
3246 A similar identity is supposed to hold for hyperbolic sines and cosines,
3247 except that it is the @emph{difference}
3248 @texline @math{\cosh^2x - \sinh^2x}
3249 @infoline @expr{cosh(x)^2 - sinh(x)^2}
3250 that always equals one. Let's try to verify this identity.
3254 2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54
3255 1: -64 1: -3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54
3258 64 n @key{RET} @key{RET} H C 2 ^ @key{TAB} H S 2 ^
3263 @cindex Roundoff errors, examples
3264 Something's obviously wrong, because when we subtract these numbers
3265 the answer will clearly be zero! But if you think about it, if these
3266 numbers @emph{did} differ by one, it would be in the 55th decimal
3267 place. The difference we seek has been lost entirely to roundoff
3270 We could verify this hypothesis by doing the actual calculation with,
3271 say, 60 decimal places of precision. This will be slow, but not
3272 enormously so. Try it if you wish; sure enough, the answer is
3273 0.99999, reasonably close to 1.
3275 Of course, a more reasonable way to verify the identity is to use
3276 a more reasonable value for @expr{x}!
3278 @cindex Common logarithm
3279 Some Calculator commands use the Hyperbolic prefix for other purposes.
3280 The logarithm and exponential functions, for example, work to the base
3281 @expr{e} normally but use base-10 instead if you use the Hyperbolic
3286 1: 1000 1: 6.9077 1: 1000 1: 3
3294 First, we mistakenly compute a natural logarithm. Then we undo
3295 and compute a common logarithm instead.
3297 The @kbd{B} key computes a general base-@var{b} logarithm for any
3302 2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077
3303 1: 10 . . 1: 2.71828 .
3306 1000 @key{RET} 10 B H E H P B
3311 Here we first use @kbd{B} to compute the base-10 logarithm, then use
3312 the ``hyperbolic'' exponential as a cheap hack to recover the number
3313 1000, then use @kbd{B} again to compute the natural logarithm. Note
3314 that @kbd{P} with the hyperbolic prefix pushes the constant @expr{e}
3317 You may have noticed that both times we took the base-10 logarithm
3318 of 1000, we got an exact integer result. Calc always tries to give
3319 an exact rational result for calculations involving rational numbers
3320 where possible. But when we used @kbd{H E}, the result was a
3321 floating-point number for no apparent reason. In fact, if we had
3322 computed @kbd{10 @key{RET} 3 ^} we @emph{would} have gotten an
3323 exact integer 1000. But the @kbd{H E} command is rigged to generate
3324 a floating-point result all of the time so that @kbd{1000 H E} will
3325 not waste time computing a thousand-digit integer when all you
3326 probably wanted was @samp{1e1000}.
3328 (@bullet{}) @strong{Exercise 2.} Find a pair of integer inputs to
3329 the @kbd{B} command for which Calc could find an exact rational
3330 result but doesn't. @xref{Arithmetic Answer 2, 2}. (@bullet{})
3332 The Calculator also has a set of functions relating to combinatorics
3333 and statistics. You may be familiar with the @dfn{factorial} function,
3334 which computes the product of all the integers up to a given number.
3338 1: 100 1: 93326215443... 1: 100. 1: 9.3326e157
3346 Recall, the @kbd{c f} command converts the integer or fraction at the
3347 top of the stack to floating-point format. If you take the factorial
3348 of a floating-point number, you get a floating-point result
3349 accurate to the current precision. But if you give @kbd{!} an
3350 exact integer, you get an exact integer result (158 digits long
3353 If you take the factorial of a non-integer, Calc uses a generalized
3354 factorial function defined in terms of Euler's Gamma function
3355 @texline @math{\Gamma(n)}
3356 @infoline @expr{gamma(n)}
3357 (which is itself available as the @kbd{f g} command).
3361 3: 4. 3: 24. 1: 5.5 1: 52.342777847
3362 2: 4.5 2: 52.3427777847 . .
3366 M-3 ! M-0 @key{DEL} 5.5 f g
3371 Here we verify the identity
3372 @texline @math{n! = \Gamma(n+1)}.
3373 @infoline @expr{@var{n}!@: = gamma(@var{n}+1)}.
3375 The binomial coefficient @var{n}-choose-@var{m}
3376 @texline or @math{\displaystyle {n \choose m}}
3378 @texline @math{\displaystyle {n! \over m! \, (n-m)!}}
3379 @infoline @expr{n!@: / m!@: (n-m)!}
3380 for all reals @expr{n} and @expr{m}. The intermediate results in this
3381 formula can become quite large even if the final result is small; the
3382 @kbd{k c} command computes a binomial coefficient in a way that avoids
3383 large intermediate values.
3385 The @kbd{k} prefix key defines several common functions out of
3386 combinatorics and number theory. Here we compute the binomial
3387 coefficient 30-choose-20, then determine its prime factorization.
3391 2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29]
3395 30 @key{RET} 20 k c k f
3400 You can verify these prime factors by using @kbd{v u} to ``unpack''
3401 this vector into 8 separate stack entries, then @kbd{M-8 *} to
3402 multiply them back together. The result is the original number,
3406 Suppose a program you are writing needs a hash table with at least
3407 10000 entries. It's best to use a prime number as the actual size
3408 of a hash table. Calc can compute the next prime number after 10000:
3412 1: 10000 1: 10007 1: 9973
3420 Just for kicks we've also computed the next prime @emph{less} than
3423 @c [fix-ref Financial Functions]
3424 @xref{Financial Functions}, for a description of the Calculator
3425 commands that deal with business and financial calculations (functions
3426 like @code{pv}, @code{rate}, and @code{sln}).
3428 @c [fix-ref Binary Number Functions]
3429 @xref{Binary Functions}, to read about the commands for operating
3430 on binary numbers (like @code{and}, @code{xor}, and @code{lsh}).
3432 @node Vector/Matrix Tutorial, Types Tutorial, Arithmetic Tutorial, Tutorial
3433 @section Vector/Matrix Tutorial
3436 A @dfn{vector} is a list of numbers or other Calc data objects.
3437 Calc provides a large set of commands that operate on vectors. Some
3438 are familiar operations from vector analysis. Others simply treat
3439 a vector as a list of objects.
3442 * Vector Analysis Tutorial::
3447 @node Vector Analysis Tutorial, Matrix Tutorial, Vector/Matrix Tutorial, Vector/Matrix Tutorial
3448 @subsection Vector Analysis
3451 If you add two vectors, the result is a vector of the sums of the
3452 elements, taken pairwise.
3456 1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3]
3460 [1,2,3] s 1 [7 6 0] s 2 +
3465 Note that we can separate the vector elements with either commas or
3466 spaces. This is true whether we are using incomplete vectors or
3467 algebraic entry. The @kbd{s 1} and @kbd{s 2} commands save these
3468 vectors so we can easily reuse them later.
3470 If you multiply two vectors, the result is the sum of the products
3471 of the elements taken pairwise. This is called the @dfn{dot product}
3485 The dot product of two vectors is equal to the product of their
3486 lengths times the cosine of the angle between them. (Here the vector
3487 is interpreted as a line from the origin @expr{(0,0,0)} to the
3488 specified point in three-dimensional space.) The @kbd{A}
3489 (absolute value) command can be used to compute the length of a
3494 3: 19 3: 19 1: 0.550782 1: 56.579
3495 2: [1, 2, 3] 2: 3.741657 . .
3496 1: [7, 6, 0] 1: 9.219544
3499 M-@key{RET} M-2 A * / I C
3504 First we recall the arguments to the dot product command, then
3505 we compute the absolute values of the top two stack entries to
3506 obtain the lengths of the vectors, then we divide the dot product
3507 by the product of the lengths to get the cosine of the angle.
3508 The inverse cosine finds that the angle between the vectors
3509 is about 56 degrees.
3511 @cindex Cross product
3512 @cindex Perpendicular vectors
3513 The @dfn{cross product} of two vectors is a vector whose length
3514 is the product of the lengths of the inputs times the sine of the
3515 angle between them, and whose direction is perpendicular to both
3516 input vectors. Unlike the dot product, the cross product is
3517 defined only for three-dimensional vectors. Let's double-check
3518 our computation of the angle using the cross product.
3522 2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579
3523 1: [7, 6, 0] 2: [1, 2, 3] . .
3527 r 1 r 2 V C s 3 M-@key{RET} M-2 A * / A I S
3532 First we recall the original vectors and compute their cross product,
3533 which we also store for later reference. Now we divide the vector
3534 by the product of the lengths of the original vectors. The length of
3535 this vector should be the sine of the angle; sure enough, it is!
3537 @c [fix-ref General Mode Commands]
3538 Vector-related commands generally begin with the @kbd{v} prefix key.
3539 Some are uppercase letters and some are lowercase. To make it easier
3540 to type these commands, the shift-@kbd{V} prefix key acts the same as
3541 the @kbd{v} key. (@xref{General Mode Commands}, for a way to make all
3542 prefix keys have this property.)
3544 If we take the dot product of two perpendicular vectors we expect
3545 to get zero, since the cosine of 90 degrees is zero. Let's check
3546 that the cross product is indeed perpendicular to both inputs:
3550 2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0
3551 1: [-18, 21, -8] . 1: [-18, 21, -8] .
3554 r 1 r 3 * @key{DEL} r 2 r 3 *
3558 @cindex Normalizing a vector
3559 @cindex Unit vectors
3560 (@bullet{}) @strong{Exercise 1.} Given a vector on the top of the
3561 stack, what keystrokes would you use to @dfn{normalize} the
3562 vector, i.e., to reduce its length to one without changing its
3563 direction? @xref{Vector Answer 1, 1}. (@bullet{})
3565 (@bullet{}) @strong{Exercise 2.} Suppose a certain particle can be
3566 at any of several positions along a ruler. You have a list of
3567 those positions in the form of a vector, and another list of the
3568 probabilities for the particle to be at the corresponding positions.
3569 Find the average position of the particle.
3570 @xref{Vector Answer 2, 2}. (@bullet{})
3572 @node Matrix Tutorial, List Tutorial, Vector Analysis Tutorial, Vector/Matrix Tutorial
3573 @subsection Matrices
3576 A @dfn{matrix} is just a vector of vectors, all the same length.
3577 This means you can enter a matrix using nested brackets. You can
3578 also use the semicolon character to enter a matrix. We'll show
3583 1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3584 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3587 [[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6] @key{RET}
3592 We'll be using this matrix again, so type @kbd{s 4} to save it now.
3594 Note that semicolons work with incomplete vectors, but they work
3595 better in algebraic entry. That's why we use the apostrophe in
3598 When two matrices are multiplied, the lefthand matrix must have
3599 the same number of columns as the righthand matrix has rows.
3600 Row @expr{i}, column @expr{j} of the result is effectively the
3601 dot product of row @expr{i} of the left matrix by column @expr{j}
3602 of the right matrix.
3604 If we try to duplicate this matrix and multiply it by itself,
3605 the dimensions are wrong and the multiplication cannot take place:
3609 1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ]
3610 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3618 Though rather hard to read, this is a formula which shows the product
3619 of two matrices. The @samp{*} function, having invalid arguments, has
3620 been left in symbolic form.
3622 We can multiply the matrices if we @dfn{transpose} one of them first.
3626 2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ]
3627 [ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ]
3628 1: [ [ 1, 4 ] . [ 27, 36, 45 ] ]
3633 U v t * U @key{TAB} *
3637 Matrix multiplication is not commutative; indeed, switching the
3638 order of the operands can even change the dimensions of the result
3639 matrix, as happened here!
3641 If you multiply a plain vector by a matrix, it is treated as a
3642 single row or column depending on which side of the matrix it is
3643 on. The result is a plain vector which should also be interpreted
3644 as a row or column as appropriate.
3648 2: [ [ 1, 2, 3 ] 1: [14, 32]
3657 Multiplying in the other order wouldn't work because the number of
3658 rows in the matrix is different from the number of elements in the
3661 (@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows
3663 @texline @math{2\times3}
3665 matrix to get @expr{[6, 15]}. Now use @samp{*} to sum along the columns
3666 to get @expr{[5, 7, 9]}.
3667 @xref{Matrix Answer 1, 1}. (@bullet{})
3669 @cindex Identity matrix
3670 An @dfn{identity matrix} is a square matrix with ones along the
3671 diagonal and zeros elsewhere. It has the property that multiplication
3672 by an identity matrix, on the left or on the right, always produces
3673 the original matrix.
3677 1: [ [ 1, 2, 3 ] 2: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3678 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3679 . 1: [ [ 1, 0, 0 ] .
3684 r 4 v i 3 @key{RET} *
3688 If a matrix is square, it is often possible to find its @dfn{inverse},
3689 that is, a matrix which, when multiplied by the original matrix, yields
3690 an identity matrix. The @kbd{&} (reciprocal) key also computes the
3691 inverse of a matrix.
3695 1: [ [ 1, 2, 3 ] 1: [ [ -2.4, 1.2, -0.2 ]
3696 [ 4, 5, 6 ] [ 2.8, -1.4, 0.4 ]
3697 [ 7, 6, 0 ] ] [ -0.73333, 0.53333, -0.2 ] ]
3705 The vertical bar @kbd{|} @dfn{concatenates} numbers, vectors, and
3706 matrices together. Here we have used it to add a new row onto
3707 our matrix to make it square.
3709 We can multiply these two matrices in either order to get an identity.
3713 1: [ [ 1., 0., 0. ] 1: [ [ 1., 0., 0. ]
3714 [ 0., 1., 0. ] [ 0., 1., 0. ]
3715 [ 0., 0., 1. ] ] [ 0., 0., 1. ] ]
3718 M-@key{RET} * U @key{TAB} *
3722 @cindex Systems of linear equations
3723 @cindex Linear equations, systems of
3724 Matrix inverses are related to systems of linear equations in algebra.
3725 Suppose we had the following set of equations:
3739 $$ \openup1\jot \tabskip=0pt plus1fil
3740 \halign to\displaywidth{\tabskip=0pt
3741 $\hfil#$&$\hfil{}#{}$&
3742 $\hfil#$&$\hfil{}#{}$&
3743 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3752 This can be cast into the matrix equation,
3757 [ [ 1, 2, 3 ] [ [ a ] [ [ 6 ]
3758 [ 4, 5, 6 ] * [ b ] = [ 2 ]
3759 [ 7, 6, 0 ] ] [ c ] ] [ 3 ] ]
3766 $$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
3768 \pmatrix{ a \cr b \cr c } = \pmatrix{ 6 \cr 2 \cr 3 }
3773 We can solve this system of equations by multiplying both sides by the
3774 inverse of the matrix. Calc can do this all in one step:
3778 2: [6, 2, 3] 1: [-12.6, 15.2, -3.93333]
3789 The result is the @expr{[a, b, c]} vector that solves the equations.
3790 (Dividing by a square matrix is equivalent to multiplying by its
3793 Let's verify this solution:
3797 2: [ [ 1, 2, 3 ] 1: [6., 2., 3.]
3800 1: [-12.6, 15.2, -3.93333]
3808 Note that we had to be careful about the order in which we multiplied
3809 the matrix and vector. If we multiplied in the other order, Calc would
3810 assume the vector was a row vector in order to make the dimensions
3811 come out right, and the answer would be incorrect. If you
3812 don't feel safe letting Calc take either interpretation of your
3813 vectors, use explicit
3814 @texline @math{N\times1}
3817 @texline @math{1\times N}
3819 matrices instead. In this case, you would enter the original column
3820 vector as @samp{[[6], [2], [3]]} or @samp{[6; 2; 3]}.
3822 (@bullet{}) @strong{Exercise 2.} Algebraic entry allows you to make
3823 vectors and matrices that include variables. Solve the following
3824 system of equations to get expressions for @expr{x} and @expr{y}
3825 in terms of @expr{a} and @expr{b}.
3838 $$ \eqalign{ x &+ a y = 6 \cr
3845 @xref{Matrix Answer 2, 2}. (@bullet{})
3847 @cindex Least-squares for over-determined systems
3848 @cindex Over-determined systems of equations
3849 (@bullet{}) @strong{Exercise 3.} A system of equations is ``over-determined''
3850 if it has more equations than variables. It is often the case that
3851 there are no values for the variables that will satisfy all the
3852 equations at once, but it is still useful to find a set of values
3853 which ``nearly'' satisfy all the equations. In terms of matrix equations,
3854 you can't solve @expr{A X = B} directly because the matrix @expr{A}
3855 is not square for an over-determined system. Matrix inversion works
3856 only for square matrices. One common trick is to multiply both sides
3857 on the left by the transpose of @expr{A}:
3859 @samp{trn(A)*A*X = trn(A)*B}.
3863 $A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
3866 @texline @math{A^T A}
3867 @infoline @expr{trn(A)*A}
3868 is a square matrix so a solution is possible. It turns out that the
3869 @expr{X} vector you compute in this way will be a ``least-squares''
3870 solution, which can be regarded as the ``closest'' solution to the set
3871 of equations. Use Calc to solve the following over-determined
3887 $$ \openup1\jot \tabskip=0pt plus1fil
3888 \halign to\displaywidth{\tabskip=0pt
3889 $\hfil#$&$\hfil{}#{}$&
3890 $\hfil#$&$\hfil{}#{}$&
3891 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3895 2a&+&4b&+&6c&=11 \cr}
3901 @xref{Matrix Answer 3, 3}. (@bullet{})
3903 @node List Tutorial, , Matrix Tutorial, Vector/Matrix Tutorial
3904 @subsection Vectors as Lists
3908 Although Calc has a number of features for manipulating vectors and
3909 matrices as mathematical objects, you can also treat vectors as
3910 simple lists of values. For example, we saw that the @kbd{k f}
3911 command returns a vector which is a list of the prime factors of a
3914 You can pack and unpack stack entries into vectors:
3918 3: 10 1: [10, 20, 30] 3: 10
3927 You can also build vectors out of consecutive integers, or out
3928 of many copies of a given value:
3932 1: [1, 2, 3, 4] 2: [1, 2, 3, 4] 2: [1, 2, 3, 4]
3933 . 1: 17 1: [17, 17, 17, 17]
3936 v x 4 @key{RET} 17 v b 4 @key{RET}
3940 You can apply an operator to every element of a vector using the
3945 1: [17, 34, 51, 68] 1: [289, 1156, 2601, 4624] 1: [17, 34, 51, 68]
3953 In the first step, we multiply the vector of integers by the vector
3954 of 17's elementwise. In the second step, we raise each element to
3955 the power two. (The general rule is that both operands must be
3956 vectors of the same length, or else one must be a vector and the
3957 other a plain number.) In the final step, we take the square root
3960 (@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two
3962 @texline @math{2^{-4}}
3963 @infoline @expr{2^-4}
3964 to @expr{2^4}. @xref{List Answer 1, 1}. (@bullet{})
3966 You can also @dfn{reduce} a binary operator across a vector.
3967 For example, reducing @samp{*} computes the product of all the
3968 elements in the vector:
3972 1: 123123 1: [3, 7, 11, 13, 41] 1: 123123
3980 In this example, we decompose 123123 into its prime factors, then
3981 multiply those factors together again to yield the original number.
3983 We could compute a dot product ``by hand'' using mapping and
3988 2: [1, 2, 3] 1: [7, 12, 0] 1: 19
3997 Recalling two vectors from the previous section, we compute the
3998 sum of pairwise products of the elements to get the same answer
3999 for the dot product as before.
4001 A slight variant of vector reduction is the @dfn{accumulate} operation,
4002 @kbd{V U}. This produces a vector of the intermediate results from
4003 a corresponding reduction. Here we compute a table of factorials:
4007 1: [1, 2, 3, 4, 5, 6] 1: [1, 2, 6, 24, 120, 720]
4010 v x 6 @key{RET} V U *
4014 Calc allows vectors to grow as large as you like, although it gets
4015 rather slow if vectors have more than about a hundred elements.
4016 Actually, most of the time is spent formatting these large vectors
4017 for display, not calculating on them. Try the following experiment
4018 (if your computer is very fast you may need to substitute a larger
4023 1: [1, 2, 3, 4, ... 1: [2, 3, 4, 5, ...
4026 v x 500 @key{RET} 1 V M +
4030 Now press @kbd{v .} (the letter @kbd{v}, then a period) and try the
4031 experiment again. In @kbd{v .} mode, long vectors are displayed
4032 ``abbreviated'' like this:
4036 1: [1, 2, 3, ..., 500] 1: [2, 3, 4, ..., 501]
4039 v x 500 @key{RET} 1 V M +
4044 (where now the @samp{...} is actually part of the Calc display).
4045 You will find both operations are now much faster. But notice that
4046 even in @w{@kbd{v .}} mode, the full vectors are still shown in the Trail.
4047 Type @w{@kbd{t .}} to cause the trail to abbreviate as well, and try the
4048 experiment one more time. Operations on long vectors are now quite
4049 fast! (But of course if you use @kbd{t .} you will lose the ability
4050 to get old vectors back using the @kbd{t y} command.)
4052 An easy way to view a full vector when @kbd{v .} mode is active is
4053 to press @kbd{`} (back-quote) to edit the vector; editing always works
4054 with the full, unabbreviated value.
4056 @cindex Least-squares for fitting a straight line
4057 @cindex Fitting data to a line
4058 @cindex Line, fitting data to
4059 @cindex Data, extracting from buffers
4060 @cindex Columns of data, extracting
4061 As a larger example, let's try to fit a straight line to some data,
4062 using the method of least squares. (Calc has a built-in command for
4063 least-squares curve fitting, but we'll do it by hand here just to
4064 practice working with vectors.) Suppose we have the following list
4065 of values in a file we have loaded into Emacs:
4092 If you are reading this tutorial in printed form, you will find it
4093 easiest to press @kbd{M-# i} to enter the on-line Info version of
4094 the manual and find this table there. (Press @kbd{g}, then type
4095 @kbd{List Tutorial}, to jump straight to this section.)
4097 Position the cursor at the upper-left corner of this table, just
4098 to the left of the @expr{1.34}. Press @kbd{C-@@} to set the mark.
4099 (On your system this may be @kbd{C-2}, @kbd{C-@key{SPC}}, or @kbd{NUL}.)
4100 Now position the cursor to the lower-right, just after the @expr{1.354}.
4101 You have now defined this region as an Emacs ``rectangle.'' Still
4102 in the Info buffer, type @kbd{M-# r}. This command
4103 (@code{calc-grab-rectangle}) will pop you back into the Calculator, with
4104 the contents of the rectangle you specified in the form of a matrix.
4108 1: [ [ 1.34, 0.234 ]
4115 (You may wish to use @kbd{v .} mode to abbreviate the display of this
4118 We want to treat this as a pair of lists. The first step is to
4119 transpose this matrix into a pair of rows. Remember, a matrix is
4120 just a vector of vectors. So we can unpack the matrix into a pair
4121 of row vectors on the stack.
4125 1: [ [ 1.34, 1.41, 1.49, ... ] 2: [1.34, 1.41, 1.49, ... ]
4126 [ 0.234, 0.298, 0.402, ... ] ] 1: [0.234, 0.298, 0.402, ... ]
4134 Let's store these in quick variables 1 and 2, respectively.
4138 1: [1.34, 1.41, 1.49, ... ] .
4146 (Recall that @kbd{t 2} is a variant of @kbd{s 2} that removes the
4147 stored value from the stack.)
4149 In a least squares fit, the slope @expr{m} is given by the formula
4153 m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
4159 $$ m = {N \sum x y - \sum x \sum y \over
4160 N \sum x^2 - \left( \sum x \right)^2} $$
4166 @texline @math{\sum x}
4167 @infoline @expr{sum(x)}
4168 represents the sum of all the values of @expr{x}. While there is an
4169 actual @code{sum} function in Calc, it's easier to sum a vector using a
4170 simple reduction. First, let's compute the four different sums that
4178 r 1 V R + t 3 r 1 2 V M ^ V R + t 4
4185 1: 13.613 1: 33.36554
4188 r 2 V R + t 5 r 1 r 2 V M * V R + t 6
4194 These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
4195 respectively. (We could have used @kbd{*} to compute @samp{sum(x^2)} and
4200 These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
4201 respectively. (We could have used \kbd{*} to compute $\sum x^2$ and
4205 Finally, we also need @expr{N}, the number of data points. This is just
4206 the length of either of our lists.
4218 (That's @kbd{v} followed by a lower-case @kbd{l}.)
4220 Now we grind through the formula:
4224 1: 633.94526 2: 633.94526 1: 67.23607
4228 r 7 r 6 * r 3 r 5 * -
4235 2: 67.23607 3: 67.23607 2: 67.23607 1: 0.52141679
4236 1: 1862.0057 2: 1862.0057 1: 128.9488 .
4240 r 7 r 4 * r 3 2 ^ - / t 8
4244 That gives us the slope @expr{m}. The y-intercept @expr{b} can now
4245 be found with the simple formula,
4249 b = (sum(y) - m sum(x)) / N
4255 $$ b = {\sum y - m \sum x \over N} $$
4262 1: 13.613 2: 13.613 1: -8.09358 1: -0.425978
4266 r 5 r 8 r 3 * - r 7 / t 9
4270 Let's ``plot'' this straight line approximation,
4271 @texline @math{y \approx m x + b},
4272 @infoline @expr{m x + b},
4273 and compare it with the original data.
4277 1: [0.699, 0.735, ... ] 1: [0.273, 0.309, ... ]
4285 Notice that multiplying a vector by a constant, and adding a constant
4286 to a vector, can be done without mapping commands since these are
4287 common operations from vector algebra. As far as Calc is concerned,
4288 we've just been doing geometry in 19-dimensional space!
4290 We can subtract this vector from our original @expr{y} vector to get
4291 a feel for the error of our fit. Let's find the maximum error:
4295 1: [0.0387, 0.0112, ... ] 1: [0.0387, 0.0112, ... ] 1: 0.0897
4303 First we compute a vector of differences, then we take the absolute
4304 values of these differences, then we reduce the @code{max} function
4305 across the vector. (The @code{max} function is on the two-key sequence
4306 @kbd{f x}; because it is so common to use @code{max} in a vector
4307 operation, the letters @kbd{X} and @kbd{N} are also accepted for
4308 @code{max} and @code{min} in this context. In general, you answer
4309 the @kbd{V M} or @kbd{V R} prompt with the actual key sequence that
4310 invokes the function you want. You could have typed @kbd{V R f x} or
4311 even @kbd{V R x max @key{RET}} if you had preferred.)
4313 If your system has the GNUPLOT program, you can see graphs of your
4314 data and your straight line to see how well they match. (If you have
4315 GNUPLOT 3.0, the following instructions will work regardless of the
4316 kind of display you have. Some GNUPLOT 2.0, non-X-windows systems
4317 may require additional steps to view the graphs.)
4319 Let's start by plotting the original data. Recall the ``@var{x}'' and ``@var{y}''
4320 vectors onto the stack and press @kbd{g f}. This ``fast'' graphing
4321 command does everything you need to do for simple, straightforward
4326 2: [1.34, 1.41, 1.49, ... ]
4327 1: [0.234, 0.298, 0.402, ... ]
4334 If all goes well, you will shortly get a new window containing a graph
4335 of the data. (If not, contact your GNUPLOT or Calc installer to find
4336 out what went wrong.) In the X window system, this will be a separate
4337 graphics window. For other kinds of displays, the default is to
4338 display the graph in Emacs itself using rough character graphics.
4339 Press @kbd{q} when you are done viewing the character graphics.
4341 Next, let's add the line we got from our least-squares fit.
4343 (If you are reading this tutorial on-line while running Calc, typing
4344 @kbd{g a} may cause the tutorial to disappear from its window and be
4345 replaced by a buffer named @samp{*Gnuplot Commands*}. The tutorial
4346 will reappear when you terminate GNUPLOT by typing @kbd{g q}.)
4351 2: [1.34, 1.41, 1.49, ... ]
4352 1: [0.273, 0.309, 0.351, ... ]
4355 @key{DEL} r 0 g a g p
4359 It's not very useful to get symbols to mark the data points on this
4360 second curve; you can type @kbd{g S g p} to remove them. Type @kbd{g q}
4361 when you are done to remove the X graphics window and terminate GNUPLOT.
4363 (@bullet{}) @strong{Exercise 2.} An earlier exercise showed how to do
4364 least squares fitting to a general system of equations. Our 19 data
4365 points are really 19 equations of the form @expr{y_i = m x_i + b} for
4366 different pairs of @expr{(x_i,y_i)}. Use the matrix-transpose method
4367 to solve for @expr{m} and @expr{b}, duplicating the above result.
4368 @xref{List Answer 2, 2}. (@bullet{})
4370 @cindex Geometric mean
4371 (@bullet{}) @strong{Exercise 3.} If the input data do not form a
4372 rectangle, you can use @w{@kbd{M-# g}} (@code{calc-grab-region})
4373 to grab the data the way Emacs normally works with regions---it reads
4374 left-to-right, top-to-bottom, treating line breaks the same as spaces.
4375 Use this command to find the geometric mean of the following numbers.
4376 (The geometric mean is the @var{n}th root of the product of @var{n} numbers.)
4385 The @kbd{M-# g} command accepts numbers separated by spaces or commas,
4386 with or without surrounding vector brackets.
4387 @xref{List Answer 3, 3}. (@bullet{})
4390 As another example, a theorem about binomial coefficients tells
4391 us that the alternating sum of binomial coefficients
4392 @var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so
4393 on up to @var{n}-choose-@var{n},
4394 always comes out to zero. Let's verify this
4398 As another example, a theorem about binomial coefficients tells
4399 us that the alternating sum of binomial coefficients
4400 ${n \choose 0} - {n \choose 1} + {n \choose 2} - \cdots \pm {n \choose n}$
4401 always comes out to zero. Let's verify this
4407 1: [1, 2, 3, 4, 5, 6, 7] 1: [0, 1, 2, 3, 4, 5, 6]
4417 1: [1, -6, 15, -20, 15, -6, 1] 1: 0
4420 V M ' (-1)^$ choose(6,$) @key{RET} V R +
4424 The @kbd{V M '} command prompts you to enter any algebraic expression
4425 to define the function to map over the vector. The symbol @samp{$}
4426 inside this expression represents the argument to the function.
4427 The Calculator applies this formula to each element of the vector,
4428 substituting each element's value for the @samp{$} sign(s) in turn.
4430 To define a two-argument function, use @samp{$$} for the first
4431 argument and @samp{$} for the second: @kbd{V M ' $$-$ @key{RET}} is
4432 equivalent to @kbd{V M -}. This is analogous to regular algebraic
4433 entry, where @samp{$$} would refer to the next-to-top stack entry
4434 and @samp{$} would refer to the top stack entry, and @kbd{' $$-$ @key{RET}}
4435 would act exactly like @kbd{-}.
4437 Notice that the @kbd{V M '} command has recorded two things in the
4438 trail: The result, as usual, and also a funny-looking thing marked
4439 @samp{oper} that represents the operator function you typed in.
4440 The function is enclosed in @samp{< >} brackets, and the argument is
4441 denoted by a @samp{#} sign. If there were several arguments, they
4442 would be shown as @samp{#1}, @samp{#2}, and so on. (For example,
4443 @kbd{V M ' $$-$} will put the function @samp{<#1 - #2>} on the
4444 trail.) This object is a ``nameless function''; you can use nameless
4445 @w{@samp{< >}} notation to answer the @kbd{V M '} prompt if you like.
4446 Nameless function notation has the interesting, occasionally useful
4447 property that a nameless function is not actually evaluated until
4448 it is used. For example, @kbd{V M ' $+random(2.0)} evaluates
4449 @samp{random(2.0)} once and adds that random number to all elements
4450 of the vector, but @kbd{V M ' <#+random(2.0)>} evaluates the
4451 @samp{random(2.0)} separately for each vector element.
4453 Another group of operators that are often useful with @kbd{V M} are
4454 the relational operators: @kbd{a =}, for example, compares two numbers
4455 and gives the result 1 if they are equal, or 0 if not. Similarly,
4456 @w{@kbd{a <}} checks for one number being less than another.
4458 Other useful vector operations include @kbd{v v}, to reverse a
4459 vector end-for-end; @kbd{V S}, to sort the elements of a vector
4460 into increasing order; and @kbd{v r} and @w{@kbd{v c}}, to extract
4461 one row or column of a matrix, or (in both cases) to extract one
4462 element of a plain vector. With a negative argument, @kbd{v r}
4463 and @kbd{v c} instead delete one row, column, or vector element.
4465 @cindex Divisor functions
4466 (@bullet{}) @strong{Exercise 4.} The @expr{k}th @dfn{divisor function}
4470 is the sum of the @expr{k}th powers of all the divisors of an
4471 integer @expr{n}. Figure out a method for computing the divisor
4472 function for reasonably small values of @expr{n}. As a test,
4473 the 0th and 1st divisor functions of 30 are 8 and 72, respectively.
4474 @xref{List Answer 4, 4}. (@bullet{})
4476 @cindex Square-free numbers
4477 @cindex Duplicate values in a list
4478 (@bullet{}) @strong{Exercise 5.} The @kbd{k f} command produces a
4479 list of prime factors for a number. Sometimes it is important to
4480 know that a number is @dfn{square-free}, i.e., that no prime occurs
4481 more than once in its list of prime factors. Find a sequence of
4482 keystrokes to tell if a number is square-free; your method should
4483 leave 1 on the stack if it is, or 0 if it isn't.
4484 @xref{List Answer 5, 5}. (@bullet{})
4486 @cindex Triangular lists
4487 (@bullet{}) @strong{Exercise 6.} Build a list of lists that looks
4488 like the following diagram. (You may wish to use the @kbd{v /}
4489 command to enable multi-line display of vectors.)
4498 [1, 2, 3, 4, 5, 6] ]
4503 @xref{List Answer 6, 6}. (@bullet{})
4505 (@bullet{}) @strong{Exercise 7.} Build the following list of lists.
4513 [10, 11, 12, 13, 14],
4514 [15, 16, 17, 18, 19, 20] ]
4519 @xref{List Answer 7, 7}. (@bullet{})
4521 @cindex Maximizing a function over a list of values
4522 @c [fix-ref Numerical Solutions]
4523 (@bullet{}) @strong{Exercise 8.} Compute a list of values of Bessel's
4524 @texline @math{J_1(x)}
4526 function @samp{besJ(1,x)} for @expr{x} from 0 to 5 in steps of 0.25.
4527 Find the value of @expr{x} (from among the above set of values) for
4528 which @samp{besJ(1,x)} is a maximum. Use an ``automatic'' method,
4529 i.e., just reading along the list by hand to find the largest value
4530 is not allowed! (There is an @kbd{a X} command which does this kind
4531 of thing automatically; @pxref{Numerical Solutions}.)
4532 @xref{List Answer 8, 8}. (@bullet{})
4534 @cindex Digits, vectors of
4535 (@bullet{}) @strong{Exercise 9.} You are given an integer in the range
4536 @texline @math{0 \le N < 10^m}
4537 @infoline @expr{0 <= N < 10^m}
4538 for @expr{m=12} (i.e., an integer of less than
4539 twelve digits). Convert this integer into a vector of @expr{m}
4540 digits, each in the range from 0 to 9. In vector-of-digits notation,
4541 add one to this integer to produce a vector of @expr{m+1} digits
4542 (since there could be a carry out of the most significant digit).
4543 Convert this vector back into a regular integer. A good integer
4544 to try is 25129925999. @xref{List Answer 9, 9}. (@bullet{})
4546 (@bullet{}) @strong{Exercise 10.} Your friend Joe tried to use
4547 @kbd{V R a =} to test if all numbers in a list were equal. What
4548 happened? How would you do this test? @xref{List Answer 10, 10}. (@bullet{})
4550 (@bullet{}) @strong{Exercise 11.} The area of a circle of radius one
4551 is @cpi{}. The area of the
4552 @texline @math{2\times2}
4554 square that encloses that circle is 4. So if we throw @var{n} darts at
4555 random points in the square, about @cpiover{4} of them will land inside
4556 the circle. This gives us an entertaining way to estimate the value of
4557 @cpi{}. The @w{@kbd{k r}}
4558 command picks a random number between zero and the value on the stack.
4559 We could get a random floating-point number between @mathit{-1} and 1 by typing
4560 @w{@kbd{2.0 k r 1 -}}. Build a vector of 100 random @expr{(x,y)} points in
4561 this square, then use vector mapping and reduction to count how many
4562 points lie inside the unit circle. Hint: Use the @kbd{v b} command.
4563 @xref{List Answer 11, 11}. (@bullet{})
4565 @cindex Matchstick problem
4566 (@bullet{}) @strong{Exercise 12.} The @dfn{matchstick problem} provides
4567 another way to calculate @cpi{}. Say you have an infinite field
4568 of vertical lines with a spacing of one inch. Toss a one-inch matchstick
4569 onto the field. The probability that the matchstick will land crossing
4570 a line turns out to be
4571 @texline @math{2/\pi}.
4572 @infoline @expr{2/pi}.
4573 Toss 100 matchsticks to estimate @cpi{}. (If you want still more fun,
4574 the probability that the GCD (@w{@kbd{k g}}) of two large integers is
4576 @texline @math{6/\pi^2}.
4577 @infoline @expr{6/pi^2}.
4578 That provides yet another way to estimate @cpi{}.)
4579 @xref{List Answer 12, 12}. (@bullet{})
4581 (@bullet{}) @strong{Exercise 13.} An algebraic entry of a string in
4582 double-quote marks, @samp{"hello"}, creates a vector of the numerical
4583 (ASCII) codes of the characters (here, @expr{[104, 101, 108, 108, 111]}).
4584 Sometimes it is convenient to compute a @dfn{hash code} of a string,
4585 which is just an integer that represents the value of that string.
4586 Two equal strings have the same hash code; two different strings
4587 @dfn{probably} have different hash codes. (For example, Calc has
4588 over 400 function names, but Emacs can quickly find the definition for
4589 any given name because it has sorted the functions into ``buckets'' by
4590 their hash codes. Sometimes a few names will hash into the same bucket,
4591 but it is easier to search among a few names than among all the names.)
4592 One popular hash function is computed as follows: First set @expr{h = 0}.
4593 Then, for each character from the string in turn, set @expr{h = 3h + c_i}
4594 where @expr{c_i} is the character's ASCII code. If we have 511 buckets,
4595 we then take the hash code modulo 511 to get the bucket number. Develop a
4596 simple command or commands for converting string vectors into hash codes.
4597 The hash code for @samp{"Testing, 1, 2, 3"} is 1960915098, which modulo
4598 511 is 121. @xref{List Answer 13, 13}. (@bullet{})
4600 (@bullet{}) @strong{Exercise 14.} The @kbd{H V R} and @kbd{H V U}
4601 commands do nested function evaluations. @kbd{H V U} takes a starting
4602 value and a number of steps @var{n} from the stack; it then applies the
4603 function you give to the starting value 0, 1, 2, up to @var{n} times
4604 and returns a vector of the results. Use this command to create a
4605 ``random walk'' of 50 steps. Start with the two-dimensional point
4606 @expr{(0,0)}; then take one step a random distance between @mathit{-1} and 1
4607 in both @expr{x} and @expr{y}; then take another step, and so on. Use the
4608 @kbd{g f} command to display this random walk. Now modify your random
4609 walk to walk a unit distance, but in a random direction, at each step.
4610 (Hint: The @code{sincos} function returns a vector of the cosine and
4611 sine of an angle.) @xref{List Answer 14, 14}. (@bullet{})
4613 @node Types Tutorial, Algebra Tutorial, Vector/Matrix Tutorial, Tutorial
4614 @section Types Tutorial
4617 Calc understands a variety of data types as well as simple numbers.
4618 In this section, we'll experiment with each of these types in turn.
4620 The numbers we've been using so far have mainly been either @dfn{integers}
4621 or @dfn{floats}. We saw that floats are usually a good approximation to
4622 the mathematical concept of real numbers, but they are only approximations
4623 and are susceptible to roundoff error. Calc also supports @dfn{fractions},
4624 which can exactly represent any rational number.
4628 1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414
4632 10 ! 49 @key{RET} : 2 + &
4637 The @kbd{:} command divides two integers to get a fraction; @kbd{/}
4638 would normally divide integers to get a floating-point result.
4639 Notice we had to type @key{RET} between the @kbd{49} and the @kbd{:}
4640 since the @kbd{:} would otherwise be interpreted as part of a
4641 fraction beginning with 49.
4643 You can convert between floating-point and fractional format using
4644 @kbd{c f} and @kbd{c F}:
4648 1: 1.35027217629e-5 1: 7:518414
4655 The @kbd{c F} command replaces a floating-point number with the
4656 ``simplest'' fraction whose floating-point representation is the
4657 same, to within the current precision.
4661 1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113
4664 P c F @key{DEL} p 5 @key{RET} P c F
4668 (@bullet{}) @strong{Exercise 1.} A calculation has produced the
4669 result 1.26508260337. You suspect it is the square root of the
4670 product of @cpi{} and some rational number. Is it? (Be sure
4671 to allow for roundoff error!) @xref{Types Answer 1, 1}. (@bullet{})
4673 @dfn{Complex numbers} can be stored in both rectangular and polar form.
4677 1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.)
4685 The square root of @mathit{-9} is by default rendered in rectangular form
4686 (@w{@expr{0 + 3i}}), but we can convert it to polar form (3 with a
4687 phase angle of 90 degrees). All the usual arithmetic and scientific
4688 operations are defined on both types of complex numbers.
4690 Another generalized kind of number is @dfn{infinity}. Infinity
4691 isn't really a number, but it can sometimes be treated like one.
4692 Calc uses the symbol @code{inf} to represent positive infinity,
4693 i.e., a value greater than any real number. Naturally, you can
4694 also write @samp{-inf} for minus infinity, a value less than any
4695 real number. The word @code{inf} can only be input using
4700 2: inf 2: -inf 2: -inf 2: -inf 1: nan
4701 1: -17 1: -inf 1: -inf 1: inf .
4704 ' inf @key{RET} 17 n * @key{RET} 72 + A +
4709 Since infinity is infinitely large, multiplying it by any finite
4710 number (like @mathit{-17}) has no effect, except that since @mathit{-17}
4711 is negative, it changes a plus infinity to a minus infinity.
4712 (``A huge positive number, multiplied by @mathit{-17}, yields a huge
4713 negative number.'') Adding any finite number to infinity also
4714 leaves it unchanged. Taking an absolute value gives us plus
4715 infinity again. Finally, we add this plus infinity to the minus
4716 infinity we had earlier. If you work it out, you might expect
4717 the answer to be @mathit{-72} for this. But the 72 has been completely
4718 lost next to the infinities; by the time we compute @w{@samp{inf - inf}}
4719 the finite difference between them, if any, is undetectable.
4720 So we say the result is @dfn{indeterminate}, which Calc writes
4721 with the symbol @code{nan} (for Not A Number).
4723 Dividing by zero is normally treated as an error, but you can get
4724 Calc to write an answer in terms of infinity by pressing @kbd{m i}
4725 to turn on Infinite mode.
4729 3: nan 2: nan 2: nan 2: nan 1: nan
4730 2: 1 1: 1 / 0 1: uinf 1: uinf .
4734 1 @key{RET} 0 / m i U / 17 n * +
4739 Dividing by zero normally is left unevaluated, but after @kbd{m i}
4740 it instead gives an infinite result. The answer is actually
4741 @code{uinf}, ``undirected infinity.'' If you look at a graph of
4742 @expr{1 / x} around @w{@expr{x = 0}}, you'll see that it goes toward
4743 plus infinity as you approach zero from above, but toward minus
4744 infinity as you approach from below. Since we said only @expr{1 / 0},
4745 Calc knows that the answer is infinite but not in which direction.
4746 That's what @code{uinf} means. Notice that multiplying @code{uinf}
4747 by a negative number still leaves plain @code{uinf}; there's no
4748 point in saying @samp{-uinf} because the sign of @code{uinf} is
4749 unknown anyway. Finally, we add @code{uinf} to our @code{nan},
4750 yielding @code{nan} again. It's easy to see that, because
4751 @code{nan} means ``totally unknown'' while @code{uinf} means
4752 ``unknown sign but known to be infinite,'' the more mysterious
4753 @code{nan} wins out when it is combined with @code{uinf}, or, for
4754 that matter, with anything else.
4756 (@bullet{}) @strong{Exercise 2.} Predict what Calc will answer
4757 for each of these formulas: @samp{inf / inf}, @samp{exp(inf)},
4758 @samp{exp(-inf)}, @samp{sqrt(-inf)}, @samp{sqrt(uinf)},
4759 @samp{abs(uinf)}, @samp{ln(0)}.
4760 @xref{Types Answer 2, 2}. (@bullet{})
4762 (@bullet{}) @strong{Exercise 3.} We saw that @samp{inf - inf = nan},
4763 which stands for an unknown value. Can @code{nan} stand for
4764 a complex number? Can it stand for infinity?
4765 @xref{Types Answer 3, 3}. (@bullet{})
4767 @dfn{HMS forms} represent a value in terms of hours, minutes, and
4772 1: 2@@ 30' 0" 1: 3@@ 30' 0" 2: 3@@ 30' 0" 1: 2.
4773 . . 1: 1@@ 45' 0." .
4776 2@@ 30' @key{RET} 1 + @key{RET} 2 / /
4780 HMS forms can also be used to hold angles in degrees, minutes, and
4785 1: 0.5 1: 26.56505 1: 26@@ 33' 54.18" 1: 0.44721
4793 First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
4794 form, then we take the sine of that angle. Note that the trigonometric
4795 functions will accept HMS forms directly as input.
4798 (@bullet{}) @strong{Exercise 4.} The Beatles' @emph{Abbey Road} is
4799 47 minutes and 26 seconds long, and contains 17 songs. What is the
4800 average length of a song on @emph{Abbey Road}? If the Extended Disco
4801 Version of @emph{Abbey Road} added 20 seconds to the length of each
4802 song, how long would the album be? @xref{Types Answer 4, 4}. (@bullet{})
4804 A @dfn{date form} represents a date, or a date and time. Dates must
4805 be entered using algebraic entry. Date forms are surrounded by
4806 @samp{< >} symbols; most standard formats for dates are recognized.
4810 2: <Sun Jan 13, 1991> 1: 2.25
4811 1: <6:00pm Thu Jan 10, 1991> .
4814 ' <13 Jan 1991>, <1/10/91, 6pm> @key{RET} -
4819 In this example, we enter two dates, then subtract to find the
4820 number of days between them. It is also possible to add an
4821 HMS form or a number (of days) to a date form to get another
4826 1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991>
4833 @c [fix-ref Date Arithmetic]
4835 The @kbd{t N} (``now'') command pushes the current date and time on the
4836 stack; then we add two days, ten hours and five minutes to the date and
4837 time. Other date-and-time related commands include @kbd{t J}, which
4838 does Julian day conversions, @kbd{t W}, which finds the beginning of
4839 the week in which a date form lies, and @kbd{t I}, which increments a
4840 date by one or several months. @xref{Date Arithmetic}, for more.
4842 (@bullet{}) @strong{Exercise 5.} How many days until the next
4843 Friday the 13th? @xref{Types Answer 5, 5}. (@bullet{})
4845 (@bullet{}) @strong{Exercise 6.} How many leap years will there be
4846 between now and the year 10001 A.D.? @xref{Types Answer 6, 6}. (@bullet{})
4848 @cindex Slope and angle of a line
4849 @cindex Angle and slope of a line
4850 An @dfn{error form} represents a mean value with an attached standard
4851 deviation, or error estimate. Suppose our measurements indicate that
4852 a certain telephone pole is about 30 meters away, with an estimated
4853 error of 1 meter, and 8 meters tall, with an estimated error of 0.2
4854 meters. What is the slope of a line from here to the top of the
4855 pole, and what is the equivalent angle in degrees?
4859 1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594
4863 8 p .2 @key{RET} 30 p 1 / I T
4868 This means that the angle is about 15 degrees, and, assuming our
4869 original error estimates were valid standard deviations, there is about
4870 a 60% chance that the result is correct within 0.59 degrees.
4872 @cindex Torus, volume of
4873 (@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is
4874 @texline @math{2 \pi^2 R r^2}
4875 @infoline @w{@expr{2 pi^2 R r^2}}
4876 where @expr{R} is the radius of the circle that
4877 defines the center of the tube and @expr{r} is the radius of the tube
4878 itself. Suppose @expr{R} is 20 cm and @expr{r} is 4 cm, each known to
4879 within 5 percent. What is the volume and the relative uncertainty of
4880 the volume? @xref{Types Answer 7, 7}. (@bullet{})
4882 An @dfn{interval form} represents a range of values. While an
4883 error form is best for making statistical estimates, intervals give
4884 you exact bounds on an answer. Suppose we additionally know that
4885 our telephone pole is definitely between 28 and 31 meters away,
4886 and that it is between 7.7 and 8.1 meters tall.
4890 1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1]
4894 [ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T
4899 If our bounds were correct, then the angle to the top of the pole
4900 is sure to lie in the range shown.
4902 The square brackets around these intervals indicate that the endpoints
4903 themselves are allowable values. In other words, the distance to the
4904 telephone pole is between 28 and 31, @emph{inclusive}. You can also
4905 make an interval that is exclusive of its endpoints by writing
4906 parentheses instead of square brackets. You can even make an interval
4907 which is inclusive (``closed'') on one end and exclusive (``open'') on
4912 1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3)
4916 [ 1 .. 10 ) & [ 2 .. 3 ) *
4921 The Calculator automatically keeps track of which end values should
4922 be open and which should be closed. You can also make infinite or
4923 semi-infinite intervals by using @samp{-inf} or @samp{inf} for one
4926 (@bullet{}) @strong{Exercise 8.} What answer would you expect from
4927 @samp{@w{1 /} @w{(0 .. 10)}}? What about @samp{@w{1 /} @w{(-10 .. 0)}}? What
4928 about @samp{@w{1 /} @w{[0 .. 10]}} (where the interval actually includes
4929 zero)? What about @samp{@w{1 /} @w{(-10 .. 10)}}?
4930 @xref{Types Answer 8, 8}. (@bullet{})
4932 (@bullet{}) @strong{Exercise 9.} Two easy ways of squaring a number
4933 are @kbd{@key{RET} *} and @w{@kbd{2 ^}}. Normally these produce the same
4934 answer. Would you expect this still to hold true for interval forms?
4935 If not, which of these will result in a larger interval?
4936 @xref{Types Answer 9, 9}. (@bullet{})
4938 A @dfn{modulo form} is used for performing arithmetic modulo @var{m}.
4939 For example, arithmetic involving time is generally done modulo 12
4944 1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24
4947 17 M 24 @key{RET} 10 + n 5 /
4952 In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
4953 new number which, when multiplied by 5 modulo 24, produces the original
4954 number, 21. If @var{m} is prime and the divisor is not a multiple of
4955 @var{m}, it is always possible to find such a number. For non-prime
4956 @var{m} like 24, it is only sometimes possible.
4960 1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16
4963 10 M 24 @key{RET} 100 ^ 10 @key{RET} 100 ^ 24 %
4968 These two calculations get the same answer, but the first one is
4969 much more efficient because it avoids the huge intermediate value
4970 that arises in the second one.
4972 @cindex Fermat, primality test of
4973 (@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat
4975 @texline @w{@math{x^{n-1} \bmod n = 1}}
4976 @infoline @expr{x^(n-1) mod n = 1}
4977 if @expr{n} is a prime number and @expr{x} is an integer less than
4978 @expr{n}. If @expr{n} is @emph{not} a prime number, this will
4979 @emph{not} be true for most values of @expr{x}. Thus we can test
4980 informally if a number is prime by trying this formula for several
4981 values of @expr{x}. Use this test to tell whether the following numbers
4982 are prime: 811749613, 15485863. @xref{Types Answer 10, 10}. (@bullet{})
4984 It is possible to use HMS forms as parts of error forms, intervals,
4985 modulo forms, or as the phase part of a polar complex number.
4986 For example, the @code{calc-time} command pushes the current time
4987 of day on the stack as an HMS/modulo form.
4991 1: 17@@ 34' 45" mod 24@@ 0' 0" 1: 6@@ 22' 15" mod 24@@ 0' 0"
4999 This calculation tells me it is six hours and 22 minutes until midnight.
5001 (@bullet{}) @strong{Exercise 11.} A rule of thumb is that one year
5003 @texline @math{\pi \times 10^7}
5004 @infoline @w{@expr{pi * 10^7}}
5005 seconds. What time will it be that many seconds from right now?
5006 @xref{Types Answer 11, 11}. (@bullet{})
5008 (@bullet{}) @strong{Exercise 12.} You are preparing to order packaging
5009 for the CD release of the Extended Disco Version of @emph{Abbey Road}.
5010 You are told that the songs will actually be anywhere from 20 to 60
5011 seconds longer than the originals. One CD can hold about 75 minutes
5012 of music. Should you order single or double packages?
5013 @xref{Types Answer 12, 12}. (@bullet{})
5015 Another kind of data the Calculator can manipulate is numbers with
5016 @dfn{units}. This isn't strictly a new data type; it's simply an
5017 application of algebraic expressions, where we use variables with
5018 suggestive names like @samp{cm} and @samp{in} to represent units
5019 like centimeters and inches.
5023 1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m
5026 ' 2in @key{RET} u c cm @key{RET} u c fath @key{RET} u b
5031 We enter the quantity ``2 inches'' (actually an algebraic expression
5032 which means two times the variable @samp{in}), then we convert it
5033 first to centimeters, then to fathoms, then finally to ``base'' units,
5034 which in this case means meters.
5038 1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm
5041 ' 9 acre @key{RET} Q u s ' $+30 cm @key{RET}
5048 1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2
5056 Since units expressions are really just formulas, taking the square
5057 root of @samp{acre} is undefined. After all, @code{acre} might be an
5058 algebraic variable that you will someday assign a value. We use the
5059 ``units-simplify'' command to simplify the expression with variables
5060 being interpreted as unit names.
5062 In the final step, we have converted not to a particular unit, but to a
5063 units system. The ``cgs'' system uses centimeters instead of meters
5064 as its standard unit of length.
5066 There is a wide variety of units defined in the Calculator.
5070 1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c
5073 ' 55 mph @key{RET} u c kph @key{RET} u c km/hr @key{RET} u c c @key{RET}
5078 We express a speed first in miles per hour, then in kilometers per
5079 hour, then again using a slightly more explicit notation, then
5080 finally in terms of fractions of the speed of light.
5082 Temperature conversions are a bit more tricky. There are two ways to
5083 interpret ``20 degrees Fahrenheit''---it could mean an actual
5084 temperature, or it could mean a change in temperature. For normal
5085 units there is no difference, but temperature units have an offset
5086 as well as a scale factor and so there must be two explicit commands
5091 1: 20 degF 1: 11.1111 degC 1: -20:3 degC 1: -6.666 degC
5094 ' 20 degF @key{RET} u c degC @key{RET} U u t degC @key{RET} c f
5099 First we convert a change of 20 degrees Fahrenheit into an equivalent
5100 change in degrees Celsius (or Centigrade). Then, we convert the
5101 absolute temperature 20 degrees Fahrenheit into Celsius. Since
5102 this comes out as an exact fraction, we then convert to floating-point
5103 for easier comparison with the other result.
5105 For simple unit conversions, you can put a plain number on the stack.
5106 Then @kbd{u c} and @kbd{u t} will prompt for both old and new units.
5107 When you use this method, you're responsible for remembering which
5108 numbers are in which units:
5112 1: 55 1: 88.5139 1: 8.201407e-8
5115 55 u c mph @key{RET} kph @key{RET} u c km/hr @key{RET} c @key{RET}
5119 To see a complete list of built-in units, type @kbd{u v}. Press
5120 @w{@kbd{M-# c}} again to re-enter the Calculator when you're done looking
5123 (@bullet{}) @strong{Exercise 13.} How many seconds are there really
5124 in a year? @xref{Types Answer 13, 13}. (@bullet{})
5126 @cindex Speed of light
5127 (@bullet{}) @strong{Exercise 14.} Supercomputer designs are limited by
5128 the speed of light (and of electricity, which is nearly as fast).
5129 Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its
5130 cabinet is one meter across. Is speed of light going to be a
5131 significant factor in its design? @xref{Types Answer 14, 14}. (@bullet{})
5133 (@bullet{}) @strong{Exercise 15.} Sam the Slug normally travels about
5134 five yards in an hour. He has obtained a supply of Power Pills; each
5135 Power Pill he eats doubles his speed. How many Power Pills can he
5136 swallow and still travel legally on most US highways?
5137 @xref{Types Answer 15, 15}. (@bullet{})
5139 @node Algebra Tutorial, Programming Tutorial, Types Tutorial, Tutorial
5140 @section Algebra and Calculus Tutorial
5143 This section shows how to use Calc's algebra facilities to solve
5144 equations, do simple calculus problems, and manipulate algebraic
5148 * Basic Algebra Tutorial::
5149 * Rewrites Tutorial::
5152 @node Basic Algebra Tutorial, Rewrites Tutorial, Algebra Tutorial, Algebra Tutorial
5153 @subsection Basic Algebra
5156 If you enter a formula in Algebraic mode that refers to variables,
5157 the formula itself is pushed onto the stack. You can manipulate
5158 formulas as regular data objects.
5162 1: 2 x^2 - 6 1: 6 - 2 x^2 1: (6 - 2 x^2) (3 x^2 + y)
5165 ' 2x^2-6 @key{RET} n ' 3x^2+y @key{RET} *
5169 (@bullet{}) @strong{Exercise 1.} Do @kbd{' x @key{RET} Q 2 ^} and
5170 @kbd{' x @key{RET} 2 ^ Q} both wind up with the same result (@samp{x})?
5171 Why or why not? @xref{Algebra Answer 1, 1}. (@bullet{})
5173 There are also commands for doing common algebraic operations on
5174 formulas. Continuing with the formula from the last example,
5178 1: 18 x^2 + 6 y - 6 x^4 - 2 x^2 y 1: (18 - 2 y) x^2 - 6 x^4 + 6 y
5186 First we ``expand'' using the distributive law, then we ``collect''
5187 terms involving like powers of @expr{x}.
5189 Let's find the value of this expression when @expr{x} is 2 and @expr{y}
5194 1: 17 x^2 - 6 x^4 + 3 1: -25
5197 1:2 s l y @key{RET} 2 s l x @key{RET}
5202 The @kbd{s l} command means ``let''; it takes a number from the top of
5203 the stack and temporarily assigns it as the value of the variable
5204 you specify. It then evaluates (as if by the @kbd{=} key) the
5205 next expression on the stack. After this command, the variable goes
5206 back to its original value, if any.
5208 (An earlier exercise in this tutorial involved storing a value in the
5209 variable @code{x}; if this value is still there, you will have to
5210 unstore it with @kbd{s u x @key{RET}} before the above example will work
5213 @cindex Maximum of a function using Calculus
5214 Let's find the maximum value of our original expression when @expr{y}
5215 is one-half and @expr{x} ranges over all possible values. We can
5216 do this by taking the derivative with respect to @expr{x} and examining
5217 values of @expr{x} for which the derivative is zero. If the second
5218 derivative of the function at that value of @expr{x} is negative,
5219 the function has a local maximum there.
5223 1: 17 x^2 - 6 x^4 + 3 1: 34 x - 24 x^3
5226 U @key{DEL} s 1 a d x @key{RET} s 2
5231 Well, the derivative is clearly zero when @expr{x} is zero. To find
5232 the other root(s), let's divide through by @expr{x} and then solve:
5236 1: (34 x - 24 x^3) / x 1: 34 x / x - 24 x^3 / x 1: 34 - 24 x^2
5239 ' x @key{RET} / a x a s
5246 1: 34 - 24 x^2 = 0 1: x = 1.19023
5249 0 a = s 3 a S x @key{RET}
5254 Notice the use of @kbd{a s} to ``simplify'' the formula. When the
5255 default algebraic simplifications don't do enough, you can use
5256 @kbd{a s} to tell Calc to spend more time on the job.
5258 Now we compute the second derivative and plug in our values of @expr{x}:
5262 1: 1.19023 2: 1.19023 2: 1.19023
5263 . 1: 34 x - 24 x^3 1: 34 - 72 x^2
5266 a . r 2 a d x @key{RET} s 4
5271 (The @kbd{a .} command extracts just the righthand side of an equation.
5272 Another method would have been to use @kbd{v u} to unpack the equation
5273 @w{@samp{x = 1.19}} to @samp{x} and @samp{1.19}, then use @kbd{M-- M-2 @key{DEL}}
5274 to delete the @samp{x}.)
5278 2: 34 - 72 x^2 1: -68. 2: 34 - 72 x^2 1: 34
5282 @key{TAB} s l x @key{RET} U @key{DEL} 0 s l x @key{RET}
5287 The first of these second derivatives is negative, so we know the function
5288 has a maximum value at @expr{x = 1.19023}. (The function also has a
5289 local @emph{minimum} at @expr{x = 0}.)
5291 When we solved for @expr{x}, we got only one value even though
5292 @expr{34 - 24 x^2 = 0} is a quadratic equation that ought to have
5293 two solutions. The reason is that @w{@kbd{a S}} normally returns a
5294 single ``principal'' solution. If it needs to come up with an
5295 arbitrary sign (as occurs in the quadratic formula) it picks @expr{+}.
5296 If it needs an arbitrary integer, it picks zero. We can get a full
5297 solution by pressing @kbd{H} (the Hyperbolic flag) before @kbd{a S}.
5301 1: 34 - 24 x^2 = 0 1: x = 1.19023 s1 1: x = -1.19023
5304 r 3 H a S x @key{RET} s 5 1 n s l s1 @key{RET}
5309 Calc has invented the variable @samp{s1} to represent an unknown sign;
5310 it is supposed to be either @mathit{+1} or @mathit{-1}. Here we have used
5311 the ``let'' command to evaluate the expression when the sign is negative.
5312 If we plugged this into our second derivative we would get the same,
5313 negative, answer, so @expr{x = -1.19023} is also a maximum.
5315 To find the actual maximum value, we must plug our two values of @expr{x}
5316 into the original formula.
5320 2: 17 x^2 - 6 x^4 + 3 1: 24.08333 s1^2 - 12.04166 s1^4 + 3
5324 r 1 r 5 s l @key{RET}
5329 (Here we see another way to use @kbd{s l}; if its input is an equation
5330 with a variable on the lefthand side, then @kbd{s l} treats the equation
5331 like an assignment to that variable if you don't give a variable name.)
5333 It's clear that this will have the same value for either sign of
5334 @code{s1}, but let's work it out anyway, just for the exercise:
5338 2: [-1, 1] 1: [15.04166, 15.04166]
5339 1: 24.08333 s1^2 ... .
5342 [ 1 n , 1 ] @key{TAB} V M $ @key{RET}
5347 Here we have used a vector mapping operation to evaluate the function
5348 at several values of @samp{s1} at once. @kbd{V M $} is like @kbd{V M '}
5349 except that it takes the formula from the top of the stack. The
5350 formula is interpreted as a function to apply across the vector at the
5351 next-to-top stack level. Since a formula on the stack can't contain
5352 @samp{$} signs, Calc assumes the variables in the formula stand for
5353 different arguments. It prompts you for an @dfn{argument list}, giving
5354 the list of all variables in the formula in alphabetical order as the
5355 default list. In this case the default is @samp{(s1)}, which is just
5356 what we want so we simply press @key{RET} at the prompt.
5358 If there had been several different values, we could have used
5359 @w{@kbd{V R X}} to find the global maximum.
5361 Calc has a built-in @kbd{a P} command that solves an equation using
5362 @w{@kbd{H a S}} and returns a vector of all the solutions. It simply
5363 automates the job we just did by hand. Applied to our original
5364 cubic polynomial, it would produce the vector of solutions
5365 @expr{[1.19023, -1.19023, 0]}. (There is also an @kbd{a X} command
5366 which finds a local maximum of a function. It uses a numerical search
5367 method rather than examining the derivatives, and thus requires you
5368 to provide some kind of initial guess to show it where to look.)
5370 (@bullet{}) @strong{Exercise 2.} Given a vector of the roots of a
5371 polynomial (such as the output of an @kbd{a P} command), what
5372 sequence of commands would you use to reconstruct the original
5373 polynomial? (The answer will be unique to within a constant
5374 multiple; choose the solution where the leading coefficient is one.)
5375 @xref{Algebra Answer 2, 2}. (@bullet{})
5377 The @kbd{m s} command enables Symbolic mode, in which formulas
5378 like @samp{sqrt(5)} that can't be evaluated exactly are left in
5379 symbolic form rather than giving a floating-point approximate answer.
5380 Fraction mode (@kbd{m f}) is also useful when doing algebra.
5384 2: 34 x - 24 x^3 2: 34 x - 24 x^3
5385 1: 34 x - 24 x^3 1: [sqrt(51) / 6, sqrt(51) / -6, 0]
5388 r 2 @key{RET} m s m f a P x @key{RET}
5392 One more mode that makes reading formulas easier is Big mode.
5401 1: [-----, -----, 0]
5410 Here things like powers, square roots, and quotients and fractions
5411 are displayed in a two-dimensional pictorial form. Calc has other
5412 language modes as well, such as C mode, FORTRAN mode, @TeX{} mode
5417 2: 34*x - 24*pow(x, 3) 2: 34*x - 24*x**3
5418 1: @{sqrt(51) / 6, sqrt(51) / -6, 0@} 1: /sqrt(51) / 6, sqrt(51) / -6, 0/
5429 2: [@{\sqrt@{51@} \over 6@}, @{\sqrt@{51@} \over -6@}, 0]
5430 1: @{2 \over 3@} \sqrt@{5@}
5433 d T ' 2 \sqrt@{5@} \over 3 @key{RET}
5438 As you can see, language modes affect both entry and display of
5439 formulas. They affect such things as the names used for built-in
5440 functions, the set of arithmetic operators and their precedences,
5441 and notations for vectors and matrices.
5443 Notice that @samp{sqrt(51)} may cause problems with older
5444 implementations of C and FORTRAN, which would require something more
5445 like @samp{sqrt(51.0)}. It is always wise to check over the formulas
5446 produced by the various language modes to make sure they are fully
5449 Type @kbd{m s}, @kbd{m f}, and @kbd{d N} to reset these modes. (You
5450 may prefer to remain in Big mode, but all the examples in the tutorial
5451 are shown in normal mode.)
5453 @cindex Area under a curve
5454 What is the area under the portion of this curve from @expr{x = 1} to @expr{2}?
5455 This is simply the integral of the function:
5459 1: 17 x^2 - 6 x^4 + 3 1: 5.6666 x^3 - 1.2 x^5 + 3 x
5467 We want to evaluate this at our two values for @expr{x} and subtract.
5468 One way to do it is again with vector mapping and reduction:
5472 2: [2, 1] 1: [12.93333, 7.46666] 1: 5.46666
5473 1: 5.6666 x^3 ... . .
5475 [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5479 (@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @expr{y}
5481 @texline @math{x \sin \pi x}
5482 @infoline @w{@expr{x sin(pi x)}}
5483 (where the sine is calculated in radians). Find the values of the
5484 integral for integers @expr{y} from 1 to 5. @xref{Algebra Answer 3,
5487 Calc's integrator can do many simple integrals symbolically, but many
5488 others are beyond its capabilities. Suppose we wish to find the area
5490 @texline @math{\sin x \ln x}
5491 @infoline @expr{sin(x) ln(x)}
5492 over the same range of @expr{x}. If you entered this formula and typed
5493 @kbd{a i x @key{RET}} (don't bother to try this), Calc would work for a
5494 long time but would be unable to find a solution. In fact, there is no
5495 closed-form solution to this integral. Now what do we do?
5497 @cindex Integration, numerical
5498 @cindex Numerical integration
5499 One approach would be to do the integral numerically. It is not hard
5500 to do this by hand using vector mapping and reduction. It is rather
5501 slow, though, since the sine and logarithm functions take a long time.
5502 We can save some time by reducing the working precision.
5506 3: 10 1: [1, 1.1, 1.2, ... , 1.8, 1.9]
5511 10 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
5516 (Note that we have used the extended version of @kbd{v x}; we could
5517 also have used plain @kbd{v x} as follows: @kbd{v x 10 @key{RET} 9 + .1 *}.)
5521 2: [1, 1.1, ... ] 1: [0., 0.084941, 0.16993, ... ]
5525 ' sin(x) ln(x) @key{RET} s 1 m r p 5 @key{RET} V M $ @key{RET}
5540 (If you got wildly different results, did you remember to switch
5543 Here we have divided the curve into ten segments of equal width;
5544 approximating these segments as rectangular boxes (i.e., assuming
5545 the curve is nearly flat at that resolution), we compute the areas
5546 of the boxes (height times width), then sum the areas. (It is
5547 faster to sum first, then multiply by the width, since the width
5548 is the same for every box.)
5550 The true value of this integral turns out to be about 0.374, so
5551 we're not doing too well. Let's try another approach.
5555 1: sin(x) ln(x) 1: 0.84147 x - 0.84147 + 0.11957 (x - 1)^2 - ...
5558 r 1 a t x=1 @key{RET} 4 @key{RET}
5563 Here we have computed the Taylor series expansion of the function
5564 about the point @expr{x=1}. We can now integrate this polynomial
5565 approximation, since polynomials are easy to integrate.
5569 1: 0.42074 x^2 + ... 1: [-0.0446, -0.42073] 1: 0.3761
5572 a i x @key{RET} [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5577 Better! By increasing the precision and/or asking for more terms
5578 in the Taylor series, we can get a result as accurate as we like.
5579 (Taylor series converge better away from singularities in the
5580 function such as the one at @code{ln(0)}, so it would also help to
5581 expand the series about the points @expr{x=2} or @expr{x=1.5} instead
5584 @cindex Simpson's rule
5585 @cindex Integration by Simpson's rule
5586 (@bullet{}) @strong{Exercise 4.} Our first method approximated the
5587 curve by stairsteps of width 0.1; the total area was then the sum
5588 of the areas of the rectangles under these stairsteps. Our second
5589 method approximated the function by a polynomial, which turned out
5590 to be a better approximation than stairsteps. A third method is
5591 @dfn{Simpson's rule}, which is like the stairstep method except
5592 that the steps are not required to be flat. Simpson's rule boils
5593 down to the formula,
5597 (h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
5598 + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
5605 \qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
5606 \hfill \cr \hfill {} + 2 f(a+(n-2)h) + 4 f(a+(n-1)h) + f(a+n h)) \qquad
5612 where @expr{n} (which must be even) is the number of slices and @expr{h}
5613 is the width of each slice. These are 10 and 0.1 in our example.
5614 For reference, here is the corresponding formula for the stairstep
5619 h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
5620 + f(a+(n-2)*h) + f(a+(n-1)*h))
5626 $$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
5627 + f(a+(n-2)h) + f(a+(n-1)h)) $$
5631 Compute the integral from 1 to 2 of
5632 @texline @math{\sin x \ln x}
5633 @infoline @expr{sin(x) ln(x)}
5634 using Simpson's rule with 10 slices.
5635 @xref{Algebra Answer 4, 4}. (@bullet{})
5637 Calc has a built-in @kbd{a I} command for doing numerical integration.
5638 It uses @dfn{Romberg's method}, which is a more sophisticated cousin
5639 of Simpson's rule. In particular, it knows how to keep refining the
5640 result until the current precision is satisfied.
5642 @c [fix-ref Selecting Sub-Formulas]
5643 Aside from the commands we've seen so far, Calc also provides a
5644 large set of commands for operating on parts of formulas. You
5645 indicate the desired sub-formula by placing the cursor on any part
5646 of the formula before giving a @dfn{selection} command. Selections won't
5647 be covered in the tutorial; @pxref{Selecting Subformulas}, for
5648 details and examples.
5650 @c hard exercise: simplify (2^(n r) - 2^(r*(n - 1))) / (2^r - 1) 2^(n - 1)
5651 @c to 2^((n-1)*(r-1)).
5653 @node Rewrites Tutorial, , Basic Algebra Tutorial, Algebra Tutorial
5654 @subsection Rewrite Rules
5657 No matter how many built-in commands Calc provided for doing algebra,
5658 there would always be something you wanted to do that Calc didn't have
5659 in its repertoire. So Calc also provides a @dfn{rewrite rule} system
5660 that you can use to define your own algebraic manipulations.
5662 Suppose we want to simplify this trigonometric formula:
5666 1: 1 / cos(x) - sin(x) tan(x)
5669 ' 1/cos(x) - sin(x) tan(x) @key{RET} s 1
5674 If we were simplifying this by hand, we'd probably replace the
5675 @samp{tan} with a @samp{sin/cos} first, then combine over a common
5676 denominator. There is no Calc command to do the former; the @kbd{a n}
5677 algebra command will do the latter but we'll do both with rewrite
5678 rules just for practice.
5680 Rewrite rules are written with the @samp{:=} symbol.
5684 1: 1 / cos(x) - sin(x)^2 / cos(x)
5687 a r tan(a) := sin(a)/cos(a) @key{RET}
5692 (The ``assignment operator'' @samp{:=} has several uses in Calc. All
5693 by itself the formula @samp{tan(a) := sin(a)/cos(a)} doesn't do anything,
5694 but when it is given to the @kbd{a r} command, that command interprets
5695 it as a rewrite rule.)
5697 The lefthand side, @samp{tan(a)}, is called the @dfn{pattern} of the
5698 rewrite rule. Calc searches the formula on the stack for parts that
5699 match the pattern. Variables in a rewrite pattern are called
5700 @dfn{meta-variables}, and when matching the pattern each meta-variable
5701 can match any sub-formula. Here, the meta-variable @samp{a} matched
5702 the actual variable @samp{x}.
5704 When the pattern part of a rewrite rule matches a part of the formula,
5705 that part is replaced by the righthand side with all the meta-variables
5706 substituted with the things they matched. So the result is
5707 @samp{sin(x) / cos(x)}. Calc's normal algebraic simplifications then
5708 mix this in with the rest of the original formula.
5710 To merge over a common denominator, we can use another simple rule:
5714 1: (1 - sin(x)^2) / cos(x)
5717 a r a/x + b/x := (a+b)/x @key{RET}
5721 This rule points out several interesting features of rewrite patterns.
5722 First, if a meta-variable appears several times in a pattern, it must
5723 match the same thing everywhere. This rule detects common denominators
5724 because the same meta-variable @samp{x} is used in both of the
5727 Second, meta-variable names are independent from variables in the
5728 target formula. Notice that the meta-variable @samp{x} here matches
5729 the subformula @samp{cos(x)}; Calc never confuses the two meanings of
5732 And third, rewrite patterns know a little bit about the algebraic
5733 properties of formulas. The pattern called for a sum of two quotients;
5734 Calc was able to match a difference of two quotients by matching
5735 @samp{a = 1}, @samp{b = -sin(x)^2}, and @samp{x = cos(x)}.
5737 @c [fix-ref Algebraic Properties of Rewrite Rules]
5738 We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
5739 the rule. It would have worked just the same in all cases. (If we
5740 really wanted the rule to apply only to @samp{+} or only to @samp{-},
5741 we could have used the @code{plain} symbol. @xref{Algebraic Properties
5742 of Rewrite Rules}, for some examples of this.)
5744 One more rewrite will complete the job. We want to use the identity
5745 @samp{sin(x)^2 + cos(x)^2 = 1}, but of course we must first rearrange
5746 the identity in a way that matches our formula. The obvious rule
5747 would be @samp{@w{1 - sin(x)^2} := cos(x)^2}, but a little thought shows
5748 that the rule @samp{sin(x)^2 := 1 - cos(x)^2} will also work. The
5749 latter rule has a more general pattern so it will work in many other
5754 1: (1 + cos(x)^2 - 1) / cos(x) 1: cos(x)
5757 a r sin(x)^2 := 1 - cos(x)^2 @key{RET} a s
5761 You may ask, what's the point of using the most general rule if you
5762 have to type it in every time anyway? The answer is that Calc allows
5763 you to store a rewrite rule in a variable, then give the variable
5764 name in the @kbd{a r} command. In fact, this is the preferred way to
5765 use rewrites. For one, if you need a rule once you'll most likely
5766 need it again later. Also, if the rule doesn't work quite right you
5767 can simply Undo, edit the variable, and run the rule again without
5768 having to retype it.
5772 ' tan(x) := sin(x)/cos(x) @key{RET} s t tsc @key{RET}
5773 ' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET}
5774 ' sin(x)^2 := 1 - cos(x)^2 @key{RET} s t sinsqr @key{RET}
5776 1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
5779 r 1 a r tsc @key{RET} a r merge @key{RET} a r sinsqr @key{RET} a s
5783 To edit a variable, type @kbd{s e} and the variable name, use regular
5784 Emacs editing commands as necessary, then type @kbd{C-c C-c} to store
5785 the edited value back into the variable.
5786 You can also use @w{@kbd{s e}} to create a new variable if you wish.
5788 Notice that the first time you use each rule, Calc puts up a ``compiling''
5789 message briefly. The pattern matcher converts rules into a special
5790 optimized pattern-matching language rather than using them directly.
5791 This allows @kbd{a r} to apply even rather complicated rules very
5792 efficiently. If the rule is stored in a variable, Calc compiles it
5793 only once and stores the compiled form along with the variable. That's
5794 another good reason to store your rules in variables rather than
5795 entering them on the fly.
5797 (@bullet{}) @strong{Exercise 1.} Type @kbd{m s} to get Symbolic
5798 mode, then enter the formula @samp{@w{(2 + sqrt(2))} / @w{(1 + sqrt(2))}}.
5799 Using a rewrite rule, simplify this formula by multiplying the top and
5800 bottom by the conjugate @w{@samp{1 - sqrt(2)}}. The result will have
5801 to be expanded by the distributive law; do this with another
5802 rewrite. @xref{Rewrites Answer 1, 1}. (@bullet{})
5804 The @kbd{a r} command can also accept a vector of rewrite rules, or
5805 a variable containing a vector of rules.
5809 1: [tsc, merge, sinsqr] 1: [tan(x) := sin(x) / cos(x), ... ]
5812 ' [tsc,merge,sinsqr] @key{RET} =
5819 1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
5822 s t trig @key{RET} r 1 a r trig @key{RET} a s
5826 @c [fix-ref Nested Formulas with Rewrite Rules]
5827 Calc tries all the rules you give against all parts of the formula,
5828 repeating until no further change is possible. (The exact order in
5829 which things are tried is rather complex, but for simple rules like
5830 the ones we've used here the order doesn't really matter.
5831 @xref{Nested Formulas with Rewrite Rules}.)
5833 Calc actually repeats only up to 100 times, just in case your rule set
5834 has gotten into an infinite loop. You can give a numeric prefix argument
5835 to @kbd{a r} to specify any limit. In particular, @kbd{M-1 a r} does
5836 only one rewrite at a time.
5840 1: 1 / cos(x) - sin(x)^2 / cos(x) 1: (1 - sin(x)^2) / cos(x)
5843 r 1 M-1 a r trig @key{RET} M-1 a r trig @key{RET}
5847 You can type @kbd{M-0 a r} if you want no limit at all on the number
5848 of rewrites that occur.
5850 Rewrite rules can also be @dfn{conditional}. Simply follow the rule
5851 with a @samp{::} symbol and the desired condition. For example,
5855 1: exp(2 pi i) + exp(3 pi i) + exp(4 pi i)
5858 ' exp(2 pi i) + exp(3 pi i) + exp(4 pi i) @key{RET}
5865 1: 1 + exp(3 pi i) + 1
5868 a r exp(k pi i) := 1 :: k % 2 = 0 @key{RET}
5873 (Recall, @samp{k % 2} is the remainder from dividing @samp{k} by 2,
5874 which will be zero only when @samp{k} is an even integer.)
5876 An interesting point is that the variables @samp{pi} and @samp{i}
5877 were matched literally rather than acting as meta-variables.
5878 This is because they are special-constant variables. The special
5879 constants @samp{e}, @samp{phi}, and so on also match literally.
5880 A common error with rewrite
5881 rules is to write, say, @samp{f(a,b,c,d,e) := g(a+b+c+d+e)}, expecting
5882 to match any @samp{f} with five arguments but in fact matching
5883 only when the fifth argument is literally @samp{e}!
5885 @cindex Fibonacci numbers
5890 Rewrite rules provide an interesting way to define your own functions.
5891 Suppose we want to define @samp{fib(n)} to produce the @var{n}th
5892 Fibonacci number. The first two Fibonacci numbers are each 1;
5893 later numbers are formed by summing the two preceding numbers in
5894 the sequence. This is easy to express in a set of three rules:
5898 ' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] @key{RET} s t fib
5903 ' fib(7) @key{RET} a r fib @key{RET}
5907 One thing that is guaranteed about the order that rewrites are tried
5908 is that, for any given subformula, earlier rules in the rule set will
5909 be tried for that subformula before later ones. So even though the
5910 first and third rules both match @samp{fib(1)}, we know the first will
5911 be used preferentially.
5913 This rule set has one dangerous bug: Suppose we apply it to the
5914 formula @samp{fib(x)}? (Don't actually try this.) The third rule
5915 will match @samp{fib(x)} and replace it with @w{@samp{fib(x-1) + fib(x-2)}}.
5916 Each of these will then be replaced to get @samp{fib(x-2) + 2 fib(x-3) +
5917 fib(x-4)}, and so on, expanding forever. What we really want is to apply
5918 the third rule only when @samp{n} is an integer greater than two. Type
5919 @w{@kbd{s e fib @key{RET}}}, then edit the third rule to:
5922 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2
5930 1: fib(6) + fib(x) + fib(0) 1: 8 + fib(x) + fib(0)
5933 ' fib(6)+fib(x)+fib(0) @key{RET} a r fib @key{RET}
5938 We've created a new function, @code{fib}, and a new command,
5939 @w{@kbd{a r fib @key{RET}}}, which means ``evaluate all @code{fib} calls in
5940 this formula.'' To make things easier still, we can tell Calc to
5941 apply these rules automatically by storing them in the special
5942 variable @code{EvalRules}.
5946 1: [fib(1) := ...] . 1: [8, 13]
5949 s r fib @key{RET} s t EvalRules @key{RET} ' [fib(6), fib(7)] @key{RET}
5953 It turns out that this rule set has the problem that it does far
5954 more work than it needs to when @samp{n} is large. Consider the
5955 first few steps of the computation of @samp{fib(6)}:
5961 fib(4) + fib(3) + fib(3) + fib(2) =
5962 fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...
5967 Note that @samp{fib(3)} appears three times here. Unless Calc's
5968 algebraic simplifier notices the multiple @samp{fib(3)}s and combines
5969 them (and, as it happens, it doesn't), this rule set does lots of
5970 needless recomputation. To cure the problem, type @code{s e EvalRules}
5971 to edit the rules (or just @kbd{s E}, a shorthand command for editing
5972 @code{EvalRules}) and add another condition:
5975 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember
5979 If a @samp{:: remember} condition appears anywhere in a rule, then if
5980 that rule succeeds Calc will add another rule that describes that match
5981 to the front of the rule set. (Remembering works in any rule set, but
5982 for technical reasons it is most effective in @code{EvalRules}.) For
5983 example, if the rule rewrites @samp{fib(7)} to something that evaluates
5984 to 13, then the rule @samp{fib(7) := 13} will be added to the rule set.
5986 Type @kbd{' fib(8) @key{RET}} to compute the eighth Fibonacci number, then
5987 type @kbd{s E} again to see what has happened to the rule set.
5989 With the @code{remember} feature, our rule set can now compute
5990 @samp{fib(@var{n})} in just @var{n} steps. In the process it builds
5991 up a table of all Fibonacci numbers up to @var{n}. After we have
5992 computed the result for a particular @var{n}, we can get it back
5993 (and the results for all smaller @var{n}) later in just one step.
5995 All Calc operations will run somewhat slower whenever @code{EvalRules}
5996 contains any rules. You should type @kbd{s u EvalRules @key{RET}} now to
5997 un-store the variable.
5999 (@bullet{}) @strong{Exercise 2.} Sometimes it is possible to reformulate
6000 a problem to reduce the amount of recursion necessary to solve it.
6001 Create a rule that, in about @var{n} simple steps and without recourse
6002 to the @code{remember} option, replaces @samp{fib(@var{n}, 1, 1)} with
6003 @samp{fib(1, @var{x}, @var{y})} where @var{x} and @var{y} are the
6004 @var{n}th and @var{n+1}st Fibonacci numbers, respectively. This rule is
6005 rather clunky to use, so add a couple more rules to make the ``user
6006 interface'' the same as for our first version: enter @samp{fib(@var{n})},
6007 get back a plain number. @xref{Rewrites Answer 2, 2}. (@bullet{})
6009 There are many more things that rewrites can do. For example, there
6010 are @samp{&&&} and @samp{|||} pattern operators that create ``and''
6011 and ``or'' combinations of rules. As one really simple example, we
6012 could combine our first two Fibonacci rules thusly:
6015 [fib(1 ||| 2) := 1, fib(n) := ... ]
6019 That means ``@code{fib} of something matching either 1 or 2 rewrites
6022 You can also make meta-variables optional by enclosing them in @code{opt}.
6023 For example, the pattern @samp{a + b x} matches @samp{2 + 3 x} but not
6024 @samp{2 + x} or @samp{3 x} or @samp{x}. The pattern @samp{opt(a) + opt(b) x}
6025 matches all of these forms, filling in a default of zero for @samp{a}
6026 and one for @samp{b}.
6028 (@bullet{}) @strong{Exercise 3.} Your friend Joe had @samp{2 + 3 x}
6029 on the stack and tried to use the rule
6030 @samp{opt(a) + opt(b) x := f(a, b, x)}. What happened?
6031 @xref{Rewrites Answer 3, 3}. (@bullet{})
6033 (@bullet{}) @strong{Exercise 4.} Starting with a positive integer @expr{a},
6034 divide @expr{a} by two if it is even, otherwise compute @expr{3 a + 1}.
6035 Now repeat this step over and over. A famous unproved conjecture
6036 is that for any starting @expr{a}, the sequence always eventually
6037 reaches 1. Given the formula @samp{seq(@var{a}, 0)}, write a set of
6038 rules that convert this into @samp{seq(1, @var{n})} where @var{n}
6039 is the number of steps it took the sequence to reach the value 1.
6040 Now enhance the rules to accept @samp{seq(@var{a})} as a starting
6041 configuration, and to stop with just the number @var{n} by itself.
6042 Now make the result be a vector of values in the sequence, from @var{a}
6043 to 1. (The formula @samp{@var{x}|@var{y}} appends the vectors @var{x}
6044 and @var{y}.) For example, rewriting @samp{seq(6)} should yield the
6045 vector @expr{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
6046 @xref{Rewrites Answer 4, 4}. (@bullet{})
6048 (@bullet{}) @strong{Exercise 5.} Define, using rewrite rules, a function
6049 @samp{nterms(@var{x})} that returns the number of terms in the sum
6050 @var{x}, or 1 if @var{x} is not a sum. (A @dfn{sum} for our purposes
6051 is one or more non-sum terms separated by @samp{+} or @samp{-} signs,
6052 so that @expr{2 - 3 (x + y) + x y} is a sum of three terms.)
6053 @xref{Rewrites Answer 5, 5}. (@bullet{})
6055 (@bullet{}) @strong{Exercise 6.} A Taylor series for a function is an
6056 infinite series that exactly equals the value of that function at
6057 values of @expr{x} near zero.
6061 cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
6067 $$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
6071 The @kbd{a t} command produces a @dfn{truncated Taylor series} which
6072 is obtained by dropping all the terms higher than, say, @expr{x^2}.
6073 Calc represents the truncated Taylor series as a polynomial in @expr{x}.
6074 Mathematicians often write a truncated series using a ``big-O'' notation
6075 that records what was the lowest term that was truncated.
6079 cos(x) = 1 - x^2 / 2! + O(x^3)
6085 $$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
6090 The meaning of @expr{O(x^3)} is ``a quantity which is negligibly small
6091 if @expr{x^3} is considered negligibly small as @expr{x} goes to zero.''
6093 The exercise is to create rewrite rules that simplify sums and products of
6094 power series represented as @samp{@var{polynomial} + O(@var{var}^@var{n})}.
6095 For example, given @samp{1 - x^2 / 2 + O(x^3)} and @samp{x - x^3 / 6 + O(x^4)}
6096 on the stack, we want to be able to type @kbd{*} and get the result
6097 @samp{x - 2:3 x^3 + O(x^4)}. Don't worry if the terms of the sum are
6098 rearranged or if @kbd{a s} needs to be typed after rewriting. (This one
6099 is rather tricky; the solution at the end of this chapter uses 6 rewrite
6100 rules. Hint: The @samp{constant(x)} condition tests whether @samp{x} is
6101 a number.) @xref{Rewrites Answer 6, 6}. (@bullet{})
6103 Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}.
6104 What happens? (Be sure to remove this rule afterward, or you might get
6105 a nasty surprise when you use Calc to balance your checkbook!)
6107 @xref{Rewrite Rules}, for the whole story on rewrite rules.
6109 @node Programming Tutorial, Answers to Exercises, Algebra Tutorial, Tutorial
6110 @section Programming Tutorial
6113 The Calculator is written entirely in Emacs Lisp, a highly extensible
6114 language. If you know Lisp, you can program the Calculator to do
6115 anything you like. Rewrite rules also work as a powerful programming
6116 system. But Lisp and rewrite rules take a while to master, and often
6117 all you want to do is define a new function or repeat a command a few
6118 times. Calc has features that allow you to do these things easily.
6120 One very limited form of programming is defining your own functions.
6121 Calc's @kbd{Z F} command allows you to define a function name and
6122 key sequence to correspond to any formula. Programming commands use
6123 the shift-@kbd{Z} prefix; the user commands they create use the lower
6124 case @kbd{z} prefix.
6128 1: 1 + x + x^2 / 2 + x^3 / 6 1: 1 + x + x^2 / 2 + x^3 / 6
6131 ' 1 + x + x^2/2! + x^3/3! @key{RET} Z F e myexp @key{RET} @key{RET} @key{RET} y
6135 This polynomial is a Taylor series approximation to @samp{exp(x)}.
6136 The @kbd{Z F} command asks a number of questions. The above answers
6137 say that the key sequence for our function should be @kbd{z e}; the
6138 @kbd{M-x} equivalent should be @code{calc-myexp}; the name of the
6139 function in algebraic formulas should also be @code{myexp}; the
6140 default argument list @samp{(x)} is acceptable; and finally @kbd{y}
6141 answers the question ``leave it in symbolic form for non-constant
6146 1: 1.3495 2: 1.3495 3: 1.3495
6147 . 1: 1.34986 2: 1.34986
6151 .3 z e .3 E ' a+1 @key{RET} z e
6156 First we call our new @code{exp} approximation with 0.3 as an
6157 argument, and compare it with the true @code{exp} function. Then
6158 we note that, as requested, if we try to give @kbd{z e} an
6159 argument that isn't a plain number, it leaves the @code{myexp}
6160 function call in symbolic form. If we had answered @kbd{n} to the
6161 final question, @samp{myexp(a + 1)} would have evaluated by plugging
6162 in @samp{a + 1} for @samp{x} in the defining formula.
6164 @cindex Sine integral Si(x)
6169 (@bullet{}) @strong{Exercise 1.} The ``sine integral'' function
6170 @texline @math{{\rm Si}(x)}
6171 @infoline @expr{Si(x)}
6172 is defined as the integral of @samp{sin(t)/t} for
6173 @expr{t = 0} to @expr{x} in radians. (It was invented because this
6174 integral has no solution in terms of basic functions; if you give it
6175 to Calc's @kbd{a i} command, it will ponder it for a long time and then
6176 give up.) We can use the numerical integration command, however,
6177 which in algebraic notation is written like @samp{ninteg(f(t), t, 0, x)}
6178 with any integrand @samp{f(t)}. Define a @kbd{z s} command and
6179 @code{Si} function that implement this. You will need to edit the
6180 default argument list a bit. As a test, @samp{Si(1)} should return
6181 0.946083. (If you don't get this answer, you might want to check that
6182 Calc is in Radians mode. Also, @code{ninteg} will run a lot faster if
6183 you reduce the precision to, say, six digits beforehand.)
6184 @xref{Programming Answer 1, 1}. (@bullet{})
6186 The simplest way to do real ``programming'' of Emacs is to define a
6187 @dfn{keyboard macro}. A keyboard macro is simply a sequence of
6188 keystrokes which Emacs has stored away and can play back on demand.
6189 For example, if you find yourself typing @kbd{H a S x @key{RET}} often,
6190 you may wish to program a keyboard macro to type this for you.
6194 1: y = sqrt(x) 1: x = y^2
6197 ' y=sqrt(x) @key{RET} C-x ( H a S x @key{RET} C-x )
6199 1: y = cos(x) 1: x = s1 arccos(y) + 2 pi n1
6202 ' y=cos(x) @key{RET} X
6207 When you type @kbd{C-x (}, Emacs begins recording. But it is also
6208 still ready to execute your keystrokes, so you're really ``training''
6209 Emacs by walking it through the procedure once. When you type
6210 @w{@kbd{C-x )}}, the macro is recorded. You can now type @kbd{X} to
6211 re-execute the same keystrokes.
6213 You can give a name to your macro by typing @kbd{Z K}.
6217 1: . 1: y = x^4 1: x = s2 sqrt(s1 sqrt(y))
6220 Z K x @key{RET} ' y=x^4 @key{RET} z x
6225 Notice that we use shift-@kbd{Z} to define the command, and lower-case
6226 @kbd{z} to call it up.
6228 Keyboard macros can call other macros.
6232 1: abs(x) 1: x = s1 y 1: 2 / x 1: x = 2 / y
6235 ' abs(x) @key{RET} C-x ( ' y @key{RET} a = z x C-x ) ' 2/x @key{RET} X
6239 (@bullet{}) @strong{Exercise 2.} Define a keyboard macro to negate
6240 the item in level 3 of the stack, without disturbing the rest of
6241 the stack. @xref{Programming Answer 2, 2}. (@bullet{})
6243 (@bullet{}) @strong{Exercise 3.} Define keyboard macros to compute
6244 the following functions:
6249 @texline @math{\displaystyle{\sin x \over x}},
6250 @infoline @expr{sin(x) / x},
6251 where @expr{x} is the number on the top of the stack.
6254 Compute the base-@expr{b} logarithm, just like the @kbd{B} key except
6255 the arguments are taken in the opposite order.
6258 Produce a vector of integers from 1 to the integer on the top of
6262 @xref{Programming Answer 3, 3}. (@bullet{})
6264 (@bullet{}) @strong{Exercise 4.} Define a keyboard macro to compute
6265 the average (mean) value of a list of numbers.
6266 @xref{Programming Answer 4, 4}. (@bullet{})
6268 In many programs, some of the steps must execute several times.
6269 Calc has @dfn{looping} commands that allow this. Loops are useful
6270 inside keyboard macros, but actually work at any time.
6274 1: x^6 2: x^6 1: 360 x^2
6278 ' x^6 @key{RET} 4 Z < a d x @key{RET} Z >
6283 Here we have computed the fourth derivative of @expr{x^6} by
6284 enclosing a derivative command in a ``repeat loop'' structure.
6285 This structure pops a repeat count from the stack, then
6286 executes the body of the loop that many times.
6288 If you make a mistake while entering the body of the loop,
6289 type @w{@kbd{Z C-g}} to cancel the loop command.
6291 @cindex Fibonacci numbers
6292 Here's another example:
6301 1 @key{RET} @key{RET} 20 Z < @key{TAB} C-j + Z >
6306 The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
6307 numbers, respectively. (To see what's going on, try a few repetitions
6308 of the loop body by hand; @kbd{C-j}, also on the Line-Feed or @key{LFD}
6309 key if you have one, makes a copy of the number in level 2.)
6311 @cindex Golden ratio
6312 @cindex Phi, golden ratio
6313 A fascinating property of the Fibonacci numbers is that the @expr{n}th
6314 Fibonacci number can be found directly by computing
6315 @texline @math{\phi^n / \sqrt{5}}
6316 @infoline @expr{phi^n / sqrt(5)}
6317 and then rounding to the nearest integer, where
6318 @texline @math{\phi} (``phi''),
6319 @infoline @expr{phi},
6320 the ``golden ratio,'' is
6321 @texline @math{(1 + \sqrt{5}) / 2}.
6322 @infoline @expr{(1 + sqrt(5)) / 2}.
6323 (For convenience, this constant is available from the @code{phi}
6324 variable, or the @kbd{I H P} command.)
6328 1: 1.61803 1: 24476.0000409 1: 10945.9999817 1: 10946
6335 @cindex Continued fractions
6336 (@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction}
6338 @texline @math{\phi}
6339 @infoline @expr{phi}
6341 @texline @math{1 + 1/(1 + 1/(1 + 1/( \ldots )))}.
6342 @infoline @expr{1 + 1/(1 + 1/(1 + 1/( ...@: )))}.
6343 We can compute an approximate value by carrying this however far
6344 and then replacing the innermost
6345 @texline @math{1/( \ldots )}
6346 @infoline @expr{1/( ...@: )}
6348 @texline @math{\phi}
6349 @infoline @expr{phi}
6350 using a twenty-term continued fraction.
6351 @xref{Programming Answer 5, 5}. (@bullet{})
6353 (@bullet{}) @strong{Exercise 6.} Linear recurrences like the one for
6354 Fibonacci numbers can be expressed in terms of matrices. Given a
6355 vector @w{@expr{[a, b]}} determine a matrix which, when multiplied by this
6356 vector, produces the vector @expr{[b, c]}, where @expr{a}, @expr{b} and
6357 @expr{c} are three successive Fibonacci numbers. Now write a program
6358 that, given an integer @expr{n}, computes the @expr{n}th Fibonacci number
6359 using matrix arithmetic. @xref{Programming Answer 6, 6}. (@bullet{})
6361 @cindex Harmonic numbers
6362 A more sophisticated kind of loop is the @dfn{for} loop. Suppose
6363 we wish to compute the 20th ``harmonic'' number, which is equal to
6364 the sum of the reciprocals of the integers from 1 to 20.
6373 0 @key{RET} 1 @key{RET} 20 Z ( & + 1 Z )
6378 The ``for'' loop pops two numbers, the lower and upper limits, then
6379 repeats the body of the loop as an internal counter increases from
6380 the lower limit to the upper one. Just before executing the loop
6381 body, it pushes the current loop counter. When the loop body
6382 finishes, it pops the ``step,'' i.e., the amount by which to
6383 increment the loop counter. As you can see, our loop always
6386 This harmonic number function uses the stack to hold the running
6387 total as well as for the various loop housekeeping functions. If
6388 you find this disorienting, you can sum in a variable instead:
6392 1: 0 2: 1 . 1: 3.597739
6396 0 t 7 1 @key{RET} 20 Z ( & s + 7 1 Z ) r 7
6401 The @kbd{s +} command adds the top-of-stack into the value in a
6402 variable (and removes that value from the stack).
6404 It's worth noting that many jobs that call for a ``for'' loop can
6405 also be done more easily by Calc's high-level operations. Two
6406 other ways to compute harmonic numbers are to use vector mapping
6407 and reduction (@kbd{v x 20}, then @w{@kbd{V M &}}, then @kbd{V R +}),
6408 or to use the summation command @kbd{a +}. Both of these are
6409 probably easier than using loops. However, there are some
6410 situations where loops really are the way to go:
6412 (@bullet{}) @strong{Exercise 7.} Use a ``for'' loop to find the first
6413 harmonic number which is greater than 4.0.
6414 @xref{Programming Answer 7, 7}. (@bullet{})
6416 Of course, if we're going to be using variables in our programs,
6417 we have to worry about the programs clobbering values that the
6418 caller was keeping in those same variables. This is easy to
6423 . 1: 0.6667 1: 0.6667 3: 0.6667
6428 Z ` p 4 @key{RET} 2 @key{RET} 3 / s 7 s s a @key{RET} Z ' r 7 s r a @key{RET}
6433 When we type @kbd{Z `} (that's a back-quote character), Calc saves
6434 its mode settings and the contents of the ten ``quick variables''
6435 for later reference. When we type @kbd{Z '} (that's an apostrophe
6436 now), Calc restores those saved values. Thus the @kbd{p 4} and
6437 @kbd{s 7} commands have no effect outside this sequence. Wrapping
6438 this around the body of a keyboard macro ensures that it doesn't
6439 interfere with what the user of the macro was doing. Notice that
6440 the contents of the stack, and the values of named variables,
6441 survive past the @kbd{Z '} command.
6443 @cindex Bernoulli numbers, approximate
6444 The @dfn{Bernoulli numbers} are a sequence with the interesting
6445 property that all of the odd Bernoulli numbers are zero, and the
6446 even ones, while difficult to compute, can be roughly approximated
6448 @texline @math{\displaystyle{2 n! \over (2 \pi)^n}}.
6449 @infoline @expr{2 n!@: / (2 pi)^n}.
6450 Let's write a keyboard macro to compute (approximate) Bernoulli numbers.
6451 (Calc has a command, @kbd{k b}, to compute exact Bernoulli numbers, but
6452 this command is very slow for large @expr{n} since the higher Bernoulli
6453 numbers are very large fractions.)
6460 10 C-x ( @key{RET} 2 % Z [ @key{DEL} 0 Z : ' 2 $! / (2 pi)^$ @key{RET} = Z ] C-x )
6465 You can read @kbd{Z [} as ``then,'' @kbd{Z :} as ``else,'' and
6466 @kbd{Z ]} as ``end-if.'' There is no need for an explicit ``if''
6467 command. For the purposes of @w{@kbd{Z [}}, the condition is ``true''
6468 if the value it pops from the stack is a nonzero number, or ``false''
6469 if it pops zero or something that is not a number (like a formula).
6470 Here we take our integer argument modulo 2; this will be nonzero
6471 if we're asking for an odd Bernoulli number.
6473 The actual tenth Bernoulli number is @expr{5/66}.
6477 3: 0.0756823 1: 0 1: 0.25305 1: 0 1: 1.16659
6482 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
6486 Just to exercise loops a bit more, let's compute a table of even
6491 3: [] 1: [0.10132, 0.03079, 0.02340, 0.033197, ...]
6496 [ ] 2 @key{RET} 30 Z ( X | 2 Z )
6501 The vertical-bar @kbd{|} is the vector-concatenation command. When
6502 we execute it, the list we are building will be in stack level 2
6503 (initially this is an empty list), and the next Bernoulli number
6504 will be in level 1. The effect is to append the Bernoulli number
6505 onto the end of the list. (To create a table of exact fractional
6506 Bernoulli numbers, just replace @kbd{X} with @kbd{k b} in the above
6507 sequence of keystrokes.)
6509 With loops and conditionals, you can program essentially anything
6510 in Calc. One other command that makes looping easier is @kbd{Z /},
6511 which takes a condition from the stack and breaks out of the enclosing
6512 loop if the condition is true (non-zero). You can use this to make
6513 ``while'' and ``until'' style loops.
6515 If you make a mistake when entering a keyboard macro, you can edit
6516 it using @kbd{Z E}. First, you must attach it to a key with @kbd{Z K}.
6517 One technique is to enter a throwaway dummy definition for the macro,
6518 then enter the real one in the edit command.
6522 1: 3 1: 3 Calc Macro Edit Mode.
6523 . . Original keys: 1 <return> 2 +
6530 C-x ( 1 @key{RET} 2 + C-x ) Z K h @key{RET} Z E h
6535 A keyboard macro is stored as a pure keystroke sequence. The
6536 @file{edmacro} package (invoked by @kbd{Z E}) scans along the
6537 macro and tries to decode it back into human-readable steps.
6538 Descriptions of the keystrokes are given as comments, which begin with
6539 @samp{;;}, and which are ignored when the edited macro is saved.
6540 Spaces and line breaks are also ignored when the edited macro is saved.
6541 To enter a space into the macro, type @code{SPC}. All the special
6542 characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC}, @code{DEL},
6543 and @code{NUL} must be written in all uppercase, as must the prefixes
6544 @code{C-} and @code{M-}.
6546 Let's edit in a new definition, for computing harmonic numbers.
6547 First, erase the four lines of the old definition. Then, type
6548 in the new definition (or use Emacs @kbd{M-w} and @kbd{C-y} commands
6549 to copy it from this page of the Info file; you can of course skip
6550 typing the comments, which begin with @samp{;;}).
6553 Z` ;; calc-kbd-push (Save local values)
6554 0 ;; calc digits (Push a zero onto the stack)
6555 st ;; calc-store-into (Store it in the following variable)
6556 1 ;; calc quick variable (Quick variable q1)
6557 1 ;; calc digits (Initial value for the loop)
6558 TAB ;; calc-roll-down (Swap initial and final)
6559 Z( ;; calc-kbd-for (Begin the "for" loop)
6560 & ;; calc-inv (Take the reciprocal)
6561 s+ ;; calc-store-plus (Add to the following variable)
6562 1 ;; calc quick variable (Quick variable q1)
6563 1 ;; calc digits (The loop step is 1)
6564 Z) ;; calc-kbd-end-for (End the "for" loop)
6565 sr ;; calc-recall (Recall the final accumulated value)
6566 1 ;; calc quick variable (Quick variable q1)
6567 Z' ;; calc-kbd-pop (Restore values)
6571 Press @kbd{C-c C-c} to finish editing and return to the Calculator.
6582 The @file{edmacro} package defines a handy @code{read-kbd-macro} command
6583 which reads the current region of the current buffer as a sequence of
6584 keystroke names, and defines that sequence on the @kbd{X}
6585 (and @kbd{C-x e}) key. Because this is so useful, Calc puts this
6586 command on the @kbd{M-# m} key. Try reading in this macro in the
6587 following form: Press @kbd{C-@@} (or @kbd{C-@key{SPC}}) at
6588 one end of the text below, then type @kbd{M-# m} at the other.
6600 (@bullet{}) @strong{Exercise 8.} A general algorithm for solving
6601 equations numerically is @dfn{Newton's Method}. Given the equation
6602 @expr{f(x) = 0} for any function @expr{f}, and an initial guess
6603 @expr{x_0} which is reasonably close to the desired solution, apply
6604 this formula over and over:
6608 new_x = x - f(x)/f'(x)
6613 $$ x_{\rm new} = x - {f(x) \over f'(x)} $$
6618 where @expr{f'(x)} is the derivative of @expr{f}. The @expr{x}
6619 values will quickly converge to a solution, i.e., eventually
6620 @texline @math{x_{\rm new}}
6621 @infoline @expr{new_x}
6622 and @expr{x} will be equal to within the limits
6623 of the current precision. Write a program which takes a formula
6624 involving the variable @expr{x}, and an initial guess @expr{x_0},
6625 on the stack, and produces a value of @expr{x} for which the formula
6626 is zero. Use it to find a solution of
6627 @texline @math{\sin(\cos x) = 0.5}
6628 @infoline @expr{sin(cos(x)) = 0.5}
6629 near @expr{x = 4.5}. (Use angles measured in radians.) Note that
6630 the built-in @w{@kbd{a R}} (@code{calc-find-root}) command uses Newton's
6631 method when it is able. @xref{Programming Answer 8, 8}. (@bullet{})
6633 @cindex Digamma function
6634 @cindex Gamma constant, Euler's
6635 @cindex Euler's gamma constant
6636 (@bullet{}) @strong{Exercise 9.} The @dfn{digamma} function
6637 @texline @math{\psi(z) (``psi'')}
6638 @infoline @expr{psi(z)}
6639 is defined as the derivative of
6640 @texline @math{\ln \Gamma(z)}.
6641 @infoline @expr{ln(gamma(z))}.
6642 For large values of @expr{z}, it can be approximated by the infinite sum
6646 psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
6651 $$ \psi(z) \approx \ln z - {1\over2z} -
6652 \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}}
6659 @texline @math{\sum}
6660 @infoline @expr{sum}
6661 represents the sum over @expr{n} from 1 to infinity
6662 (or to some limit high enough to give the desired accuracy), and
6663 the @code{bern} function produces (exact) Bernoulli numbers.
6664 While this sum is not guaranteed to converge, in practice it is safe.
6665 An interesting mathematical constant is Euler's gamma, which is equal
6666 to about 0.5772. One way to compute it is by the formula,
6667 @texline @math{\gamma = -\psi(1)}.
6668 @infoline @expr{gamma = -psi(1)}.
6669 Unfortunately, 1 isn't a large enough argument
6670 for the above formula to work (5 is a much safer value for @expr{z}).
6671 Fortunately, we can compute
6672 @texline @math{\psi(1)}
6673 @infoline @expr{psi(1)}
6675 @texline @math{\psi(5)}
6676 @infoline @expr{psi(5)}
6677 using the recurrence
6678 @texline @math{\psi(z+1) = \psi(z) + {1 \over z}}.
6679 @infoline @expr{psi(z+1) = psi(z) + 1/z}.
6680 Your task: Develop a program to compute
6681 @texline @math{\psi(z)};
6682 @infoline @expr{psi(z)};
6683 it should ``pump up'' @expr{z}
6684 if necessary to be greater than 5, then use the above summation
6685 formula. Use looping commands to compute the sum. Use your function
6687 @texline @math{\gamma}
6688 @infoline @expr{gamma}
6689 to twelve decimal places. (Calc has a built-in command
6690 for Euler's constant, @kbd{I P}, which you can use to check your answer.)
6691 @xref{Programming Answer 9, 9}. (@bullet{})
6693 @cindex Polynomial, list of coefficients
6694 (@bullet{}) @strong{Exercise 10.} Given a polynomial in @expr{x} and
6695 a number @expr{m} on the stack, where the polynomial is of degree
6696 @expr{m} or less (i.e., does not have any terms higher than @expr{x^m}),
6697 write a program to convert the polynomial into a list-of-coefficients
6698 notation. For example, @expr{5 x^4 + (x + 1)^2} with @expr{m = 6}
6699 should produce the list @expr{[1, 2, 1, 0, 5, 0, 0]}. Also develop
6700 a way to convert from this form back to the standard algebraic form.
6701 @xref{Programming Answer 10, 10}. (@bullet{})
6704 (@bullet{}) @strong{Exercise 11.} The @dfn{Stirling numbers of the
6705 first kind} are defined by the recurrences,
6709 s(n,n) = 1 for n >= 0,
6710 s(n,0) = 0 for n > 0,
6711 s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1.
6717 $$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr
6718 s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr
6719 s(n+1,m) &= s(n,m-1) - n \, s(n,m) \qquad
6720 \hbox{for } n \ge m \ge 1.}
6724 (These numbers are also sometimes written $\displaystyle{n \brack m}$.)
6727 This can be implemented using a @dfn{recursive} program in Calc; the
6728 program must invoke itself in order to calculate the two righthand
6729 terms in the general formula. Since it always invokes itself with
6730 ``simpler'' arguments, it's easy to see that it must eventually finish
6731 the computation. Recursion is a little difficult with Emacs keyboard
6732 macros since the macro is executed before its definition is complete.
6733 So here's the recommended strategy: Create a ``dummy macro'' and assign
6734 it to a key with, e.g., @kbd{Z K s}. Now enter the true definition,
6735 using the @kbd{z s} command to call itself recursively, then assign it
6736 to the same key with @kbd{Z K s}. Now the @kbd{z s} command will run
6737 the complete recursive program. (Another way is to use @w{@kbd{Z E}}
6738 or @kbd{M-# m} (@code{read-kbd-macro}) to read the whole macro at once,
6739 thus avoiding the ``training'' phase.) The task: Write a program
6740 that computes Stirling numbers of the first kind, given @expr{n} and
6741 @expr{m} on the stack. Test it with @emph{small} inputs like
6742 @expr{s(4,2)}. (There is a built-in command for Stirling numbers,
6743 @kbd{k s}, which you can use to check your answers.)
6744 @xref{Programming Answer 11, 11}. (@bullet{})
6746 The programming commands we've seen in this part of the tutorial
6747 are low-level, general-purpose operations. Often you will find
6748 that a higher-level function, such as vector mapping or rewrite
6749 rules, will do the job much more easily than a detailed, step-by-step
6752 (@bullet{}) @strong{Exercise 12.} Write another program for
6753 computing Stirling numbers of the first kind, this time using
6754 rewrite rules. Once again, @expr{n} and @expr{m} should be taken
6755 from the stack. @xref{Programming Answer 12, 12}. (@bullet{})
6760 This ends the tutorial section of the Calc manual. Now you know enough
6761 about Calc to use it effectively for many kinds of calculations. But
6762 Calc has many features that were not even touched upon in this tutorial.
6764 The rest of this manual tells the whole story.
6766 @c Volume II of this manual, the @dfn{Calc Reference}, tells the whole story.
6769 @node Answers to Exercises, , Programming Tutorial, Tutorial
6770 @section Answers to Exercises
6773 This section includes answers to all the exercises in the Calc tutorial.
6776 * RPN Answer 1:: 1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -
6777 * RPN Answer 2:: 2*4 + 7*9.5 + 5/4
6778 * RPN Answer 3:: Operating on levels 2 and 3
6779 * RPN Answer 4:: Joe's complex problems
6780 * Algebraic Answer 1:: Simulating Q command
6781 * Algebraic Answer 2:: Joe's algebraic woes
6782 * Algebraic Answer 3:: 1 / 0
6783 * Modes Answer 1:: 3#0.1 = 3#0.0222222?
6784 * Modes Answer 2:: 16#f.e8fe15
6785 * Modes Answer 3:: Joe's rounding bug
6786 * Modes Answer 4:: Why floating point?
6787 * Arithmetic Answer 1:: Why the \ command?
6788 * Arithmetic Answer 2:: Tripping up the B command
6789 * Vector Answer 1:: Normalizing a vector
6790 * Vector Answer 2:: Average position
6791 * Matrix Answer 1:: Row and column sums
6792 * Matrix Answer 2:: Symbolic system of equations
6793 * Matrix Answer 3:: Over-determined system
6794 * List Answer 1:: Powers of two
6795 * List Answer 2:: Least-squares fit with matrices
6796 * List Answer 3:: Geometric mean
6797 * List Answer 4:: Divisor function
6798 * List Answer 5:: Duplicate factors
6799 * List Answer 6:: Triangular list
6800 * List Answer 7:: Another triangular list
6801 * List Answer 8:: Maximum of Bessel function
6802 * List Answer 9:: Integers the hard way
6803 * List Answer 10:: All elements equal
6804 * List Answer 11:: Estimating pi with darts
6805 * List Answer 12:: Estimating pi with matchsticks
6806 * List Answer 13:: Hash codes
6807 * List Answer 14:: Random walk
6808 * Types Answer 1:: Square root of pi times rational
6809 * Types Answer 2:: Infinities
6810 * Types Answer 3:: What can "nan" be?
6811 * Types Answer 4:: Abbey Road
6812 * Types Answer 5:: Friday the 13th
6813 * Types Answer 6:: Leap years
6814 * Types Answer 7:: Erroneous donut
6815 * Types Answer 8:: Dividing intervals
6816 * Types Answer 9:: Squaring intervals
6817 * Types Answer 10:: Fermat's primality test
6818 * Types Answer 11:: pi * 10^7 seconds
6819 * Types Answer 12:: Abbey Road on CD
6820 * Types Answer 13:: Not quite pi * 10^7 seconds
6821 * Types Answer 14:: Supercomputers and c
6822 * Types Answer 15:: Sam the Slug
6823 * Algebra Answer 1:: Squares and square roots
6824 * Algebra Answer 2:: Building polynomial from roots
6825 * Algebra Answer 3:: Integral of x sin(pi x)
6826 * Algebra Answer 4:: Simpson's rule
6827 * Rewrites Answer 1:: Multiplying by conjugate
6828 * Rewrites Answer 2:: Alternative fib rule
6829 * Rewrites Answer 3:: Rewriting opt(a) + opt(b) x
6830 * Rewrites Answer 4:: Sequence of integers
6831 * Rewrites Answer 5:: Number of terms in sum
6832 * Rewrites Answer 6:: Truncated Taylor series
6833 * Programming Answer 1:: Fresnel's C(x)
6834 * Programming Answer 2:: Negate third stack element
6835 * Programming Answer 3:: Compute sin(x) / x, etc.
6836 * Programming Answer 4:: Average value of a list
6837 * Programming Answer 5:: Continued fraction phi
6838 * Programming Answer 6:: Matrix Fibonacci numbers
6839 * Programming Answer 7:: Harmonic number greater than 4
6840 * Programming Answer 8:: Newton's method
6841 * Programming Answer 9:: Digamma function
6842 * Programming Answer 10:: Unpacking a polynomial
6843 * Programming Answer 11:: Recursive Stirling numbers
6844 * Programming Answer 12:: Stirling numbers with rewrites
6847 @c The following kludgery prevents the individual answers from
6848 @c being entered on the table of contents.
6850 \global\let\oldwrite=\write
6851 \gdef\skipwrite#1#2{\let\write=\oldwrite}
6852 \global\let\oldchapternofonts=\chapternofonts
6853 \gdef\chapternofonts{\let\write=\skipwrite\oldchapternofonts}
6856 @node RPN Answer 1, RPN Answer 2, Answers to Exercises, Answers to Exercises
6857 @subsection RPN Tutorial Exercise 1
6860 @kbd{1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -}
6863 @texline @math{1 - (2 \times (3 + 4)) = -13}.
6864 @infoline @expr{1 - (2 * (3 + 4)) = -13}.
6866 @node RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises
6867 @subsection RPN Tutorial Exercise 2
6870 @texline @math{2\times4 + 7\times9.5 + {5\over4} = 75.75}
6871 @infoline @expr{2*4 + 7*9.5 + 5/4 = 75.75}
6873 After computing the intermediate term
6874 @texline @math{2\times4 = 8},
6875 @infoline @expr{2*4 = 8},
6876 you can leave that result on the stack while you compute the second
6877 term. With both of these results waiting on the stack you can then
6878 compute the final term, then press @kbd{+ +} to add everything up.
6887 2 @key{RET} 4 * 7 @key{RET} 9.5 *
6894 4: 8 3: 8 2: 8 1: 75.75
6895 3: 66.5 2: 66.5 1: 67.75 .
6904 Alternatively, you could add the first two terms before going on
6905 with the third term.
6909 2: 8 1: 74.5 3: 74.5 2: 74.5 1: 75.75
6910 1: 66.5 . 2: 5 1: 1.25 .
6914 ... + 5 @key{RET} 4 / +
6918 On an old-style RPN calculator this second method would have the
6919 advantage of using only three stack levels. But since Calc's stack
6920 can grow arbitrarily large this isn't really an issue. Which method
6921 you choose is purely a matter of taste.
6923 @node RPN Answer 3, RPN Answer 4, RPN Answer 2, Answers to Exercises
6924 @subsection RPN Tutorial Exercise 3
6927 The @key{TAB} key provides a way to operate on the number in level 2.
6931 3: 10 3: 10 4: 10 3: 10 3: 10
6932 2: 20 2: 30 3: 30 2: 30 2: 21
6933 1: 30 1: 20 2: 20 1: 21 1: 30
6937 @key{TAB} 1 + @key{TAB}
6941 Similarly, @kbd{M-@key{TAB}} gives you access to the number in level 3.
6945 3: 10 3: 21 3: 21 3: 30 3: 11
6946 2: 21 2: 30 2: 30 2: 11 2: 21
6947 1: 30 1: 10 1: 11 1: 21 1: 30
6950 M-@key{TAB} 1 + M-@key{TAB} M-@key{TAB}
6954 @node RPN Answer 4, Algebraic Answer 1, RPN Answer 3, Answers to Exercises
6955 @subsection RPN Tutorial Exercise 4
6958 Either @kbd{( 2 , 3 )} or @kbd{( 2 @key{SPC} 3 )} would have worked,
6959 but using both the comma and the space at once yields:
6963 1: ( ... 2: ( ... 1: (2, ... 2: (2, ... 2: (2, ...
6964 . 1: 2 . 1: (2, ... 1: (2, 3)
6971 Joe probably tried to type @kbd{@key{TAB} @key{DEL}} to swap the
6972 extra incomplete object to the top of the stack and delete it.
6973 But a feature of Calc is that @key{DEL} on an incomplete object
6974 deletes just one component out of that object, so he had to press
6975 @key{DEL} twice to finish the job.
6979 2: (2, ... 2: (2, 3) 2: (2, 3) 1: (2, 3)
6980 1: (2, 3) 1: (2, ... 1: ( ... .
6983 @key{TAB} @key{DEL} @key{DEL}
6987 (As it turns out, deleting the second-to-top stack entry happens often
6988 enough that Calc provides a special key, @kbd{M-@key{DEL}}, to do just that.
6989 @kbd{M-@key{DEL}} is just like @kbd{@key{TAB} @key{DEL}}, except that it doesn't exhibit
6990 the ``feature'' that tripped poor Joe.)
6992 @node Algebraic Answer 1, Algebraic Answer 2, RPN Answer 4, Answers to Exercises
6993 @subsection Algebraic Entry Tutorial Exercise 1
6996 Type @kbd{' sqrt($) @key{RET}}.
6998 If the @kbd{Q} key is broken, you could use @kbd{' $^0.5 @key{RET}}.
6999 Or, RPN style, @kbd{0.5 ^}.
7001 (Actually, @samp{$^1:2}, using the fraction one-half as the power, is
7002 a closer equivalent, since @samp{9^0.5} yields @expr{3.0} whereas
7003 @samp{sqrt(9)} and @samp{9^1:2} yield the exact integer @expr{3}.)
7005 @node Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1, Answers to Exercises
7006 @subsection Algebraic Entry Tutorial Exercise 2
7009 In the formula @samp{2 x (1+y)}, @samp{x} was interpreted as a function
7010 name with @samp{1+y} as its argument. Assigning a value to a variable
7011 has no relation to a function by the same name. Joe needed to use an
7012 explicit @samp{*} symbol here: @samp{2 x*(1+y)}.
7014 @node Algebraic Answer 3, Modes Answer 1, Algebraic Answer 2, Answers to Exercises
7015 @subsection Algebraic Entry Tutorial Exercise 3
7018 The result from @kbd{1 @key{RET} 0 /} will be the formula @expr{1 / 0}.
7019 The ``function'' @samp{/} cannot be evaluated when its second argument
7020 is zero, so it is left in symbolic form. When you now type @kbd{0 *},
7021 the result will be zero because Calc uses the general rule that ``zero
7022 times anything is zero.''
7024 @c [fix-ref Infinities]
7025 The @kbd{m i} command enables an @dfn{Infinite mode} in which @expr{1 / 0}
7026 results in a special symbol that represents ``infinity.'' If you
7027 multiply infinity by zero, Calc uses another special new symbol to
7028 show that the answer is ``indeterminate.'' @xref{Infinities}, for
7029 further discussion of infinite and indeterminate values.
7031 @node Modes Answer 1, Modes Answer 2, Algebraic Answer 3, Answers to Exercises
7032 @subsection Modes Tutorial Exercise 1
7035 Calc always stores its numbers in decimal, so even though one-third has
7036 an exact base-3 representation (@samp{3#0.1}), it is still stored as
7037 0.3333333 (chopped off after 12 or however many decimal digits) inside
7038 the calculator's memory. When this inexact number is converted back
7039 to base 3 for display, it may still be slightly inexact. When we
7040 multiply this number by 3, we get 0.999999, also an inexact value.
7042 When Calc displays a number in base 3, it has to decide how many digits
7043 to show. If the current precision is 12 (decimal) digits, that corresponds
7044 to @samp{12 / log10(3) = 25.15} base-3 digits. Because 25.15 is not an
7045 exact integer, Calc shows only 25 digits, with the result that stored
7046 numbers carry a little bit of extra information that may not show up on
7047 the screen. When Joe entered @samp{3#0.2}, the stored number 0.666666
7048 happened to round to a pleasing value when it lost that last 0.15 of a
7049 digit, but it was still inexact in Calc's memory. When he divided by 2,
7050 he still got the dreaded inexact value 0.333333. (Actually, he divided
7051 0.666667 by 2 to get 0.333334, which is why he got something a little
7052 higher than @code{3#0.1} instead of a little lower.)
7054 If Joe didn't want to be bothered with all this, he could have typed
7055 @kbd{M-24 d n} to display with one less digit than the default. (If
7056 you give @kbd{d n} a negative argument, it uses default-minus-that,
7057 so @kbd{M-- d n} would be an easier way to get the same effect.) Those
7058 inexact results would still be lurking there, but they would now be
7059 rounded to nice, natural-looking values for display purposes. (Remember,
7060 @samp{0.022222} in base 3 is like @samp{0.099999} in base 10; rounding
7061 off one digit will round the number up to @samp{0.1}.) Depending on the
7062 nature of your work, this hiding of the inexactness may be a benefit or
7063 a danger. With the @kbd{d n} command, Calc gives you the choice.
7065 Incidentally, another consequence of all this is that if you type
7066 @kbd{M-30 d n} to display more digits than are ``really there,''
7067 you'll see garbage digits at the end of the number. (In decimal
7068 display mode, with decimally-stored numbers, these garbage digits are
7069 always zero so they vanish and you don't notice them.) Because Calc
7070 rounds off that 0.15 digit, there is the danger that two numbers could
7071 be slightly different internally but still look the same. If you feel
7072 uneasy about this, set the @kbd{d n} precision to be a little higher
7073 than normal; you'll get ugly garbage digits, but you'll always be able
7074 to tell two distinct numbers apart.
7076 An interesting side note is that most computers store their
7077 floating-point numbers in binary, and convert to decimal for display.
7078 Thus everyday programs have the same problem: Decimal 0.1 cannot be
7079 represented exactly in binary (try it: @kbd{0.1 d 2}), so @samp{0.1 * 10}
7080 comes out as an inexact approximation to 1 on some machines (though
7081 they generally arrange to hide it from you by rounding off one digit as
7082 we did above). Because Calc works in decimal instead of binary, you can
7083 be sure that numbers that look exact @emph{are} exact as long as you stay
7084 in decimal display mode.
7086 It's not hard to show that any number that can be represented exactly
7087 in binary, octal, or hexadecimal is also exact in decimal, so the kinds
7088 of problems we saw in this exercise are likely to be severe only when
7089 you use a relatively unusual radix like 3.
7091 @node Modes Answer 2, Modes Answer 3, Modes Answer 1, Answers to Exercises
7092 @subsection Modes Tutorial Exercise 2
7094 If the radix is 15 or higher, we can't use the letter @samp{e} to mark
7095 the exponent because @samp{e} is interpreted as a digit. When Calc
7096 needs to display scientific notation in a high radix, it writes
7097 @samp{16#F.E8F*16.^15}. You can enter a number like this as an
7098 algebraic entry. Also, pressing @kbd{e} without any digits before it
7099 normally types @kbd{1e}, but in a high radix it types @kbd{16.^} and
7100 puts you in algebraic entry: @kbd{16#f.e8f @key{RET} e 15 @key{RET} *} is another
7101 way to enter this number.
7103 The reason Calc puts a decimal point in the @samp{16.^} is to prevent
7104 huge integers from being generated if the exponent is large (consider
7105 @samp{16#1.23*16^1000}, where we compute @samp{16^1000} as a giant
7106 exact integer and then throw away most of the digits when we multiply
7107 it by the floating-point @samp{16#1.23}). While this wouldn't normally
7108 matter for display purposes, it could give you a nasty surprise if you
7109 copied that number into a file and later moved it back into Calc.
7111 @node Modes Answer 3, Modes Answer 4, Modes Answer 2, Answers to Exercises
7112 @subsection Modes Tutorial Exercise 3
7115 The answer he got was @expr{0.5000000000006399}.
7117 The problem is not that the square operation is inexact, but that the
7118 sine of 45 that was already on the stack was accurate to only 12 places.
7119 Arbitrary-precision calculations still only give answers as good as
7122 The real problem is that there is no 12-digit number which, when
7123 squared, comes out to 0.5 exactly. The @kbd{f [} and @kbd{f ]}
7124 commands decrease or increase a number by one unit in the last
7125 place (according to the current precision). They are useful for
7126 determining facts like this.
7130 1: 0.707106781187 1: 0.500000000001
7140 1: 0.707106781187 1: 0.707106781186 1: 0.499999999999
7147 A high-precision calculation must be carried out in high precision
7148 all the way. The only number in the original problem which was known
7149 exactly was the quantity 45 degrees, so the precision must be raised
7150 before anything is done after the number 45 has been entered in order
7151 for the higher precision to be meaningful.
7153 @node Modes Answer 4, Arithmetic Answer 1, Modes Answer 3, Answers to Exercises
7154 @subsection Modes Tutorial Exercise 4
7157 Many calculations involve real-world quantities, like the width and
7158 height of a piece of wood or the volume of a jar. Such quantities
7159 can't be measured exactly anyway, and if the data that is input to
7160 a calculation is inexact, doing exact arithmetic on it is a waste
7163 Fractions become unwieldy after too many calculations have been
7164 done with them. For example, the sum of the reciprocals of the
7165 integers from 1 to 10 is 7381:2520. The sum from 1 to 30 is
7166 9304682830147:2329089562800. After a point it will take a long
7167 time to add even one more term to this sum, but a floating-point
7168 calculation of the sum will not have this problem.
7170 Also, rational numbers cannot express the results of all calculations.
7171 There is no fractional form for the square root of two, so if you type
7172 @w{@kbd{2 Q}}, Calc has no choice but to give you a floating-point answer.
7174 @node Arithmetic Answer 1, Arithmetic Answer 2, Modes Answer 4, Answers to Exercises
7175 @subsection Arithmetic Tutorial Exercise 1
7178 Dividing two integers that are larger than the current precision may
7179 give a floating-point result that is inaccurate even when rounded
7180 down to an integer. Consider @expr{123456789 / 2} when the current
7181 precision is 6 digits. The true answer is @expr{61728394.5}, but
7182 with a precision of 6 this will be rounded to
7183 @texline @math{12345700.0/2.0 = 61728500.0}.
7184 @infoline @expr{12345700.@: / 2.@: = 61728500.}.
7185 The result, when converted to an integer, will be off by 106.
7187 Here are two solutions: Raise the precision enough that the
7188 floating-point round-off error is strictly to the right of the
7189 decimal point. Or, convert to Fraction mode so that @expr{123456789 / 2}
7190 produces the exact fraction @expr{123456789:2}, which can be rounded
7191 down by the @kbd{F} command without ever switching to floating-point
7194 @node Arithmetic Answer 2, Vector Answer 1, Arithmetic Answer 1, Answers to Exercises
7195 @subsection Arithmetic Tutorial Exercise 2
7198 @kbd{27 @key{RET} 9 B} could give the exact result @expr{3:2}, but it
7199 does a floating-point calculation instead and produces @expr{1.5}.
7201 Calc will find an exact result for a logarithm if the result is an integer
7202 or (when in Fraction mode) the reciprocal of an integer. But there is
7203 no efficient way to search the space of all possible rational numbers
7204 for an exact answer, so Calc doesn't try.
7206 @node Vector Answer 1, Vector Answer 2, Arithmetic Answer 2, Answers to Exercises
7207 @subsection Vector Tutorial Exercise 1
7210 Duplicate the vector, compute its length, then divide the vector
7211 by its length: @kbd{@key{RET} A /}.
7215 1: [1, 2, 3] 2: [1, 2, 3] 1: [0.27, 0.53, 0.80] 1: 1.
7216 . 1: 3.74165738677 . .
7223 The final @kbd{A} command shows that the normalized vector does
7224 indeed have unit length.
7226 @node Vector Answer 2, Matrix Answer 1, Vector Answer 1, Answers to Exercises
7227 @subsection Vector Tutorial Exercise 2
7230 The average position is equal to the sum of the products of the
7231 positions times their corresponding probabilities. This is the
7232 definition of the dot product operation. So all you need to do
7233 is to put the two vectors on the stack and press @kbd{*}.
7235 @node Matrix Answer 1, Matrix Answer 2, Vector Answer 2, Answers to Exercises
7236 @subsection Matrix Tutorial Exercise 1
7239 The trick is to multiply by a vector of ones. Use @kbd{r 4 [1 1 1] *} to
7240 get the row sum. Similarly, use @kbd{[1 1] r 4 *} to get the column sum.
7242 @node Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to Exercises
7243 @subsection Matrix Tutorial Exercise 2
7256 $$ \eqalign{ x &+ a y = 6 \cr
7262 Just enter the righthand side vector, then divide by the lefthand side
7267 1: [6, 10] 2: [6, 10] 1: [6 - 4 a / (b - a), 4 / (b - a) ]
7272 ' [6 10] @key{RET} ' [1 a; 1 b] @key{RET} /
7276 This can be made more readable using @kbd{d B} to enable Big display
7282 1: [6 - -----, -----]
7287 Type @kbd{d N} to return to Normal display mode afterwards.
7289 @node Matrix Answer 3, List Answer 1, Matrix Answer 2, Answers to Exercises
7290 @subsection Matrix Tutorial Exercise 3
7294 @texline @math{A^T A \, X = A^T B},
7295 @infoline @expr{trn(A) * A * X = trn(A) * B},
7297 @texline @math{A' = A^T A}
7298 @infoline @expr{A2 = trn(A) * A}
7300 @texline @math{B' = A^T B};
7301 @infoline @expr{B2 = trn(A) * B};
7302 now, we have a system
7303 @texline @math{A' X = B'}
7304 @infoline @expr{A2 * X = B2}
7305 which we can solve using Calc's @samp{/} command.
7320 $$ \openup1\jot \tabskip=0pt plus1fil
7321 \halign to\displaywidth{\tabskip=0pt
7322 $\hfil#$&$\hfil{}#{}$&
7323 $\hfil#$&$\hfil{}#{}$&
7324 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
7328 2a&+&4b&+&6c&=11 \cr}
7333 The first step is to enter the coefficient matrix. We'll store it in
7334 quick variable number 7 for later reference. Next, we compute the
7341 1: [ [ 1, 2, 3 ] 2: [ [ 1, 4, 7, 2 ] 1: [57, 84, 96]
7342 [ 4, 5, 6 ] [ 2, 5, 6, 4 ] .
7343 [ 7, 6, 0 ] [ 3, 6, 0, 6 ] ]
7344 [ 2, 4, 6 ] ] 1: [6, 2, 3, 11]
7347 ' [1 2 3; 4 5 6; 7 6 0; 2 4 6] @key{RET} s 7 v t [6 2 3 11] *
7352 Now we compute the matrix
7359 2: [57, 84, 96] 1: [-11.64, 14.08, -3.64]
7360 1: [ [ 70, 72, 39 ] .
7370 (The actual computed answer will be slightly inexact due to
7373 Notice that the answers are similar to those for the
7374 @texline @math{3\times3}
7376 system solved in the text. That's because the fourth equation that was
7377 added to the system is almost identical to the first one multiplied
7378 by two. (If it were identical, we would have gotten the exact same
7380 @texline @math{4\times3}
7382 system would be equivalent to the original
7383 @texline @math{3\times3}
7387 Since the first and fourth equations aren't quite equivalent, they
7388 can't both be satisfied at once. Let's plug our answers back into
7389 the original system of equations to see how well they match.
7393 2: [-11.64, 14.08, -3.64] 1: [5.6, 2., 3., 11.2]
7405 This is reasonably close to our original @expr{B} vector,
7406 @expr{[6, 2, 3, 11]}.
7408 @node List Answer 1, List Answer 2, Matrix Answer 3, Answers to Exercises
7409 @subsection List Tutorial Exercise 1
7412 We can use @kbd{v x} to build a vector of integers. This needs to be
7413 adjusted to get the range of integers we desire. Mapping @samp{-}
7414 across the vector will accomplish this, although it turns out the
7415 plain @samp{-} key will work just as well.
7420 1: [1, 2, 3, 4, 5, 6, 7, 8, 9] 1: [-4, -3, -2, -1, 0, 1, 2, 3, 4]
7423 2 v x 9 @key{RET} 5 V M - or 5 -
7428 Now we use @kbd{V M ^} to map the exponentiation operator across the
7433 1: [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
7440 @node List Answer 2, List Answer 3, List Answer 1, Answers to Exercises
7441 @subsection List Tutorial Exercise 2
7444 Given @expr{x} and @expr{y} vectors in quick variables 1 and 2 as before,
7445 the first job is to form the matrix that describes the problem.
7455 $$ m \times x + b \times 1 = y $$
7460 @texline @math{19\times2}
7462 matrix with our @expr{x} vector as one column and
7463 ones as the other column. So, first we build the column of ones, then
7464 we combine the two columns to form our @expr{A} matrix.
7468 2: [1.34, 1.41, 1.49, ... ] 1: [ [ 1.34, 1 ]
7469 1: [1, 1, 1, ...] [ 1.41, 1 ]
7473 r 1 1 v b 19 @key{RET} M-2 v p v t s 3
7479 @texline @math{A^T y}
7480 @infoline @expr{trn(A) * y}
7482 @texline @math{A^T A}
7483 @infoline @expr{trn(A) * A}
7488 1: [33.36554, 13.613] 2: [33.36554, 13.613]
7489 . 1: [ [ 98.0003, 41.63 ]
7493 v t r 2 * r 3 v t r 3 *
7498 (Hey, those numbers look familiar!)
7502 1: [0.52141679, -0.425978]
7509 Since we were solving equations of the form
7510 @texline @math{m \times x + b \times 1 = y},
7511 @infoline @expr{m*x + b*1 = y},
7512 these numbers should be @expr{m} and @expr{b}, respectively. Sure
7513 enough, they agree exactly with the result computed using @kbd{V M} and
7516 The moral of this story: @kbd{V M} and @kbd{V R} will probably solve
7517 your problem, but there is often an easier way using the higher-level
7518 arithmetic functions!
7520 @c [fix-ref Curve Fitting]
7521 In fact, there is a built-in @kbd{a F} command that does least-squares
7522 fits. @xref{Curve Fitting}.
7524 @node List Answer 3, List Answer 4, List Answer 2, Answers to Exercises
7525 @subsection List Tutorial Exercise 3
7528 Move to one end of the list and press @kbd{C-@@} (or @kbd{C-@key{SPC}} or
7529 whatever) to set the mark, then move to the other end of the list
7530 and type @w{@kbd{M-# g}}.
7534 1: [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5]
7539 To make things interesting, let's assume we don't know at a glance
7540 how many numbers are in this list. Then we could type:
7544 2: [2.3, 6, 22, ... ] 2: [2.3, 6, 22, ... ]
7545 1: [2.3, 6, 22, ... ] 1: 126356422.5
7555 2: 126356422.5 2: 126356422.5 1: 7.94652913734
7556 1: [2.3, 6, 22, ... ] 1: 9 .
7564 (The @kbd{I ^} command computes the @var{n}th root of a number.
7565 You could also type @kbd{& ^} to take the reciprocal of 9 and
7566 then raise the number to that power.)
7568 @node List Answer 4, List Answer 5, List Answer 3, Answers to Exercises
7569 @subsection List Tutorial Exercise 4
7572 A number @expr{j} is a divisor of @expr{n} if
7573 @texline @math{n \mathbin{\hbox{\code{\%}}} j = 0}.
7574 @infoline @samp{n % j = 0}.
7575 The first step is to get a vector that identifies the divisors.
7579 2: 30 2: [0, 0, 0, 2, ...] 1: [1, 1, 1, 0, ...]
7580 1: [1, 2, 3, 4, ...] 1: 0 .
7583 30 @key{RET} v x 30 @key{RET} s 1 V M % 0 V M a = s 2
7588 This vector has 1's marking divisors of 30 and 0's marking non-divisors.
7590 The zeroth divisor function is just the total number of divisors.
7591 The first divisor function is the sum of the divisors.
7596 2: [1, 2, 3, 4, ...] 1: [1, 2, 3, 0, ...] 1: 72
7597 1: [1, 1, 1, 0, ...] . .
7600 V R + r 1 r 2 V M * V R +
7605 Once again, the last two steps just compute a dot product for which
7606 a simple @kbd{*} would have worked equally well.
7608 @node List Answer 5, List Answer 6, List Answer 4, Answers to Exercises
7609 @subsection List Tutorial Exercise 5
7612 The obvious first step is to obtain the list of factors with @kbd{k f}.
7613 This list will always be in sorted order, so if there are duplicates
7614 they will be right next to each other. A suitable method is to compare
7615 the list with a copy of itself shifted over by one.
7619 1: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19, 0]
7620 . 1: [3, 7, 7, 7, 19, 0] 1: [0, 3, 7, 7, 7, 19]
7623 19551 k f @key{RET} 0 | @key{TAB} 0 @key{TAB} |
7630 1: [0, 0, 1, 1, 0, 0] 1: 2 1: 0
7638 Note that we have to arrange for both vectors to have the same length
7639 so that the mapping operation works; no prime factor will ever be
7640 zero, so adding zeros on the left and right is safe. From then on
7641 the job is pretty straightforward.
7643 Incidentally, Calc provides the
7644 @texline @dfn{M@"obius} @math{\mu}
7645 @infoline @dfn{Moebius mu}
7646 function which is zero if and only if its argument is square-free. It
7647 would be a much more convenient way to do the above test in practice.
7649 @node List Answer 6, List Answer 7, List Answer 5, Answers to Exercises
7650 @subsection List Tutorial Exercise 6
7653 First use @kbd{v x 6 @key{RET}} to get a list of integers, then @kbd{V M v x}
7654 to get a list of lists of integers!
7656 @node List Answer 7, List Answer 8, List Answer 6, Answers to Exercises
7657 @subsection List Tutorial Exercise 7
7660 Here's one solution. First, compute the triangular list from the previous
7661 exercise and type @kbd{1 -} to subtract one from all the elements.
7674 The numbers down the lefthand edge of the list we desire are called
7675 the ``triangular numbers'' (now you know why!). The @expr{n}th
7676 triangular number is the sum of the integers from 1 to @expr{n}, and
7677 can be computed directly by the formula
7678 @texline @math{n (n+1) \over 2}.
7679 @infoline @expr{n * (n+1) / 2}.
7683 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7684 1: [0, 1, 2, 3, 4, 5] 1: [0, 1, 3, 6, 10, 15]
7687 v x 6 @key{RET} 1 - V M ' $ ($+1)/2 @key{RET}
7692 Adding this list to the above list of lists produces the desired
7701 [10, 11, 12, 13, 14],
7702 [15, 16, 17, 18, 19, 20] ]
7709 If we did not know the formula for triangular numbers, we could have
7710 computed them using a @kbd{V U +} command. We could also have
7711 gotten them the hard way by mapping a reduction across the original
7716 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7717 1: [ [0], [0, 1], ... ] 1: [0, 1, 3, 6, 10, 15]
7725 (This means ``map a @kbd{V R +} command across the vector,'' and
7726 since each element of the main vector is itself a small vector,
7727 @kbd{V R +} computes the sum of its elements.)
7729 @node List Answer 8, List Answer 9, List Answer 7, Answers to Exercises
7730 @subsection List Tutorial Exercise 8
7733 The first step is to build a list of values of @expr{x}.
7737 1: [1, 2, 3, ..., 21] 1: [0, 1, 2, ..., 20] 1: [0, 0.25, 0.5, ..., 5]
7740 v x 21 @key{RET} 1 - 4 / s 1
7744 Next, we compute the Bessel function values.
7748 1: [0., 0.124, 0.242, ..., -0.328]
7751 V M ' besJ(1,$) @key{RET}
7756 (Another way to do this would be @kbd{1 @key{TAB} V M f j}.)
7758 A way to isolate the maximum value is to compute the maximum using
7759 @kbd{V R X}, then compare all the Bessel values with that maximum.
7763 2: [0., 0.124, 0.242, ... ] 1: [0, 0, 0, ... ] 2: [0, 0, 0, ... ]
7767 @key{RET} V R X V M a = @key{RET} V R + @key{DEL}
7772 It's a good idea to verify, as in the last step above, that only
7773 one value is equal to the maximum. (After all, a plot of
7774 @texline @math{\sin x}
7775 @infoline @expr{sin(x)}
7776 might have many points all equal to the maximum value, 1.)
7778 The vector we have now has a single 1 in the position that indicates
7779 the maximum value of @expr{x}. Now it is a simple matter to convert
7780 this back into the corresponding value itself.
7784 2: [0, 0, 0, ... ] 1: [0, 0., 0., ... ] 1: 1.75
7785 1: [0, 0.25, 0.5, ... ] . .
7792 If @kbd{a =} had produced more than one @expr{1} value, this method
7793 would have given the sum of all maximum @expr{x} values; not very
7794 useful! In this case we could have used @kbd{v m} (@code{calc-mask-vector})
7795 instead. This command deletes all elements of a ``data'' vector that
7796 correspond to zeros in a ``mask'' vector, leaving us with, in this
7797 example, a vector of maximum @expr{x} values.
7799 The built-in @kbd{a X} command maximizes a function using more
7800 efficient methods. Just for illustration, let's use @kbd{a X}
7801 to maximize @samp{besJ(1,x)} over this same interval.
7805 2: besJ(1, x) 1: [1.84115, 0.581865]
7809 ' besJ(1,x), [0..5] @key{RET} a X x @key{RET}
7814 The output from @kbd{a X} is a vector containing the value of @expr{x}
7815 that maximizes the function, and the function's value at that maximum.
7816 As you can see, our simple search got quite close to the right answer.
7818 @node List Answer 9, List Answer 10, List Answer 8, Answers to Exercises
7819 @subsection List Tutorial Exercise 9
7822 Step one is to convert our integer into vector notation.
7826 1: 25129925999 3: 25129925999
7828 1: [11, 10, 9, ..., 1, 0]
7831 25129925999 @key{RET} 10 @key{RET} 12 @key{RET} v x 12 @key{RET} -
7838 1: 25129925999 1: [0, 2, 25, 251, 2512, ... ]
7839 2: [100000000000, ... ] .
7847 (Recall, the @kbd{\} command computes an integer quotient.)
7851 1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
7858 Next we must increment this number. This involves adding one to
7859 the last digit, plus handling carries. There is a carry to the
7860 left out of a digit if that digit is a nine and all the digits to
7861 the right of it are nines.
7865 1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ]
7875 1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
7883 Accumulating @kbd{*} across a vector of ones and zeros will preserve
7884 only the initial run of ones. These are the carries into all digits
7885 except the rightmost digit. Concatenating a one on the right takes
7886 care of aligning the carries properly, and also adding one to the
7891 2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
7892 1: [0, 0, 2, 5, ... ] .
7895 0 r 2 | V M + 10 V M %
7900 Here we have concatenated 0 to the @emph{left} of the original number;
7901 this takes care of shifting the carries by one with respect to the
7902 digits that generated them.
7904 Finally, we must convert this list back into an integer.
7908 3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ]
7909 2: 1000000000000 1: [1000000000000, 100000000000, ... ]
7910 1: [100000000000, ... ] .
7913 10 @key{RET} 12 ^ r 1 |
7920 1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000
7928 Another way to do this final step would be to reduce the formula
7929 @w{@samp{10 $$ + $}} across the vector of digits.
7933 1: [0, 0, 2, 5, ... ] 1: 25129926000
7936 V R ' 10 $$ + $ @key{RET}
7940 @node List Answer 10, List Answer 11, List Answer 9, Answers to Exercises
7941 @subsection List Tutorial Exercise 10
7944 For the list @expr{[a, b, c, d]}, the result is @expr{((a = b) = c) = d},
7945 which will compare @expr{a} and @expr{b} to produce a 1 or 0, which is
7946 then compared with @expr{c} to produce another 1 or 0, which is then
7947 compared with @expr{d}. This is not at all what Joe wanted.
7949 Here's a more correct method:
7953 1: [7, 7, 7, 8, 7] 2: [7, 7, 7, 8, 7]
7957 ' [7,7,7,8,7] @key{RET} @key{RET} v r 1 @key{RET}
7964 1: [1, 1, 1, 0, 1] 1: 0
7971 @node List Answer 11, List Answer 12, List Answer 10, Answers to Exercises
7972 @subsection List Tutorial Exercise 11
7975 The circle of unit radius consists of those points @expr{(x,y)} for which
7976 @expr{x^2 + y^2 < 1}. We start by generating a vector of @expr{x^2}
7977 and a vector of @expr{y^2}.
7979 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
7984 2: [2., 2., ..., 2.] 2: [2., 2., ..., 2.]
7985 1: [2., 2., ..., 2.] 1: [1.16, 1.98, ..., 0.81]
7988 v . t . 2. v b 100 @key{RET} @key{RET} V M k r
7995 2: [2., 2., ..., 2.] 1: [0.026, 0.96, ..., 0.036]
7996 1: [0.026, 0.96, ..., 0.036] 2: [0.53, 0.81, ..., 0.094]
7999 1 - 2 V M ^ @key{TAB} V M k r 1 - 2 V M ^
8003 Now we sum the @expr{x^2} and @expr{y^2} values, compare with 1 to
8004 get a vector of 1/0 truth values, then sum the truth values.
8008 1: [0.56, 1.78, ..., 0.13] 1: [1, 0, ..., 1] 1: 84
8016 The ratio @expr{84/100} should approximate the ratio @cpiover{4}.
8020 1: 0.84 1: 3.36 2: 3.36 1: 1.0695
8028 Our estimate, 3.36, is off by about 7%. We could get a better estimate
8029 by taking more points (say, 1000), but it's clear that this method is
8032 (Naturally, since this example uses random numbers your own answer
8033 will be slightly different from the one shown here!)
8035 If you typed @kbd{v .} and @kbd{t .} before, type them again to
8036 return to full-sized display of vectors.
8038 @node List Answer 12, List Answer 13, List Answer 11, Answers to Exercises
8039 @subsection List Tutorial Exercise 12
8042 This problem can be made a lot easier by taking advantage of some
8043 symmetries. First of all, after some thought it's clear that the
8044 @expr{y} axis can be ignored altogether. Just pick a random @expr{x}
8045 component for one end of the match, pick a random direction
8046 @texline @math{\theta},
8047 @infoline @expr{theta},
8048 and see if @expr{x} and
8049 @texline @math{x + \cos \theta}
8050 @infoline @expr{x + cos(theta)}
8051 (which is the @expr{x} coordinate of the other endpoint) cross a line.
8052 The lines are at integer coordinates, so this happens when the two
8053 numbers surround an integer.
8055 Since the two endpoints are equivalent, we may as well choose the leftmost
8056 of the two endpoints as @expr{x}. Then @expr{theta} is an angle pointing
8057 to the right, in the range -90 to 90 degrees. (We could use radians, but
8058 it would feel like cheating to refer to @cpiover{2} radians while trying
8059 to estimate @cpi{}!)
8061 In fact, since the field of lines is infinite we can choose the
8062 coordinates 0 and 1 for the lines on either side of the leftmost
8063 endpoint. The rightmost endpoint will be between 0 and 1 if the
8064 match does not cross a line, or between 1 and 2 if it does. So:
8065 Pick random @expr{x} and
8066 @texline @math{\theta},
8067 @infoline @expr{theta},
8069 @texline @math{x + \cos \theta},
8070 @infoline @expr{x + cos(theta)},
8071 and count how many of the results are greater than one. Simple!
8073 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
8078 1: [0.52, 0.71, ..., 0.72] 2: [0.52, 0.71, ..., 0.72]
8079 . 1: [78.4, 64.5, ..., -42.9]
8082 v . t . 1. v b 100 @key{RET} V M k r 180. v b 100 @key{RET} V M k r 90 -
8087 (The next step may be slow, depending on the speed of your computer.)
8091 2: [0.52, 0.71, ..., 0.72] 1: [0.72, 1.14, ..., 1.45]
8092 1: [0.20, 0.43, ..., 0.73] .
8102 1: [0, 1, ..., 1] 1: 0.64 1: 3.125
8105 1 V M a > V R + 100 / 2 @key{TAB} /
8109 Let's try the third method, too. We'll use random integers up to
8110 one million. The @kbd{k r} command with an integer argument picks
8115 2: [1000000, 1000000, ..., 1000000] 2: [78489, 527587, ..., 814975]
8116 1: [1000000, 1000000, ..., 1000000] 1: [324014, 358783, ..., 955450]
8119 1000000 v b 100 @key{RET} @key{RET} V M k r @key{TAB} V M k r
8126 1: [1, 1, ..., 25] 1: [1, 1, ..., 0] 1: 0.56
8129 V M k g 1 V M a = V R + 100 /
8143 For a proof of this property of the GCD function, see section 4.5.2,
8144 exercise 10, of Knuth's @emph{Art of Computer Programming}, volume II.
8146 If you typed @kbd{v .} and @kbd{t .} before, type them again to
8147 return to full-sized display of vectors.
8149 @node List Answer 13, List Answer 14, List Answer 12, Answers to Exercises
8150 @subsection List Tutorial Exercise 13
8153 First, we put the string on the stack as a vector of ASCII codes.
8157 1: [84, 101, 115, ..., 51]
8160 "Testing, 1, 2, 3 @key{RET}
8165 Note that the @kbd{"} key, like @kbd{$}, initiates algebraic entry so
8166 there was no need to type an apostrophe. Also, Calc didn't mind that
8167 we omitted the closing @kbd{"}. (The same goes for all closing delimiters
8168 like @kbd{)} and @kbd{]} at the end of a formula.
8170 We'll show two different approaches here. In the first, we note that
8171 if the input vector is @expr{[a, b, c, d]}, then the hash code is
8172 @expr{3 (3 (3a + b) + c) + d = 27a + 9b + 3c + d}. In other words,
8173 it's a sum of descending powers of three times the ASCII codes.
8177 2: [84, 101, 115, ..., 51] 2: [84, 101, 115, ..., 51]
8178 1: 16 1: [15, 14, 13, ..., 0]
8181 @key{RET} v l v x 16 @key{RET} -
8188 2: [84, 101, 115, ..., 51] 1: 1960915098 1: 121
8189 1: [14348907, ..., 1] . .
8192 3 @key{TAB} V M ^ * 511 %
8197 Once again, @kbd{*} elegantly summarizes most of the computation.
8198 But there's an even more elegant approach: Reduce the formula
8199 @kbd{3 $$ + $} across the vector. Recall that this represents a
8200 function of two arguments that computes its first argument times three
8201 plus its second argument.
8205 1: [84, 101, 115, ..., 51] 1: 1960915098
8208 "Testing, 1, 2, 3 @key{RET} V R ' 3$$+$ @key{RET}
8213 If you did the decimal arithmetic exercise, this will be familiar.
8214 Basically, we're turning a base-3 vector of digits into an integer,
8215 except that our ``digits'' are much larger than real digits.
8217 Instead of typing @kbd{511 %} again to reduce the result, we can be
8218 cleverer still and notice that rather than computing a huge integer
8219 and taking the modulo at the end, we can take the modulo at each step
8220 without affecting the result. While this means there are more
8221 arithmetic operations, the numbers we operate on remain small so
8222 the operations are faster.
8226 1: [84, 101, 115, ..., 51] 1: 121
8229 "Testing, 1, 2, 3 @key{RET} V R ' (3$$+$)%511 @key{RET}
8233 Why does this work? Think about a two-step computation:
8234 @w{@expr{3 (3a + b) + c}}. Taking a result modulo 511 basically means
8235 subtracting off enough 511's to put the result in the desired range.
8236 So the result when we take the modulo after every step is,
8240 3 (3 a + b - 511 m) + c - 511 n
8246 $$ 3 (3 a + b - 511 m) + c - 511 n $$
8251 for some suitable integers @expr{m} and @expr{n}. Expanding out by
8252 the distributive law yields
8256 9 a + 3 b + c - 511*3 m - 511 n
8262 $$ 9 a + 3 b + c - 511\times3 m - 511 n $$
8267 The @expr{m} term in the latter formula is redundant because any
8268 contribution it makes could just as easily be made by the @expr{n}
8269 term. So we can take it out to get an equivalent formula with
8274 9 a + 3 b + c - 511 n'
8280 $$ 9 a + 3 b + c - 511 n' $$
8285 which is just the formula for taking the modulo only at the end of
8286 the calculation. Therefore the two methods are essentially the same.
8288 Later in the tutorial we will encounter @dfn{modulo forms}, which
8289 basically automate the idea of reducing every intermediate result
8290 modulo some value @var{m}.
8292 @node List Answer 14, Types Answer 1, List Answer 13, Answers to Exercises
8293 @subsection List Tutorial Exercise 14
8295 We want to use @kbd{H V U} to nest a function which adds a random
8296 step to an @expr{(x,y)} coordinate. The function is a bit long, but
8297 otherwise the problem is quite straightforward.
8301 2: [0, 0] 1: [ [ 0, 0 ]
8302 1: 50 [ 0.4288, -0.1695 ]
8303 . [ -0.4787, -0.9027 ]
8306 [0,0] 50 H V U ' <# + [random(2.0)-1, random(2.0)-1]> @key{RET}
8310 Just as the text recommended, we used @samp{< >} nameless function
8311 notation to keep the two @code{random} calls from being evaluated
8312 before nesting even begins.
8314 We now have a vector of @expr{[x, y]} sub-vectors, which by Calc's
8315 rules acts like a matrix. We can transpose this matrix and unpack
8316 to get a pair of vectors, @expr{x} and @expr{y}, suitable for graphing.
8320 2: [ 0, 0.4288, -0.4787, ... ]
8321 1: [ 0, -0.1696, -0.9027, ... ]
8328 Incidentally, because the @expr{x} and @expr{y} are completely
8329 independent in this case, we could have done two separate commands
8330 to create our @expr{x} and @expr{y} vectors of numbers directly.
8332 To make a random walk of unit steps, we note that @code{sincos} of
8333 a random direction exactly gives us an @expr{[x, y]} step of unit
8334 length; in fact, the new nesting function is even briefer, though
8335 we might want to lower the precision a bit for it.
8339 2: [0, 0] 1: [ [ 0, 0 ]
8340 1: 50 [ 0.1318, 0.9912 ]
8341 . [ -0.5965, 0.3061 ]
8344 [0,0] 50 m d p 6 @key{RET} H V U ' <# + sincos(random(360.0))> @key{RET}
8348 Another @kbd{v t v u g f} sequence will graph this new random walk.
8350 An interesting twist on these random walk functions would be to use
8351 complex numbers instead of 2-vectors to represent points on the plane.
8352 In the first example, we'd use something like @samp{random + random*(0,1)},
8353 and in the second we could use polar complex numbers with random phase
8354 angles. (This exercise was first suggested in this form by Randal
8357 @node Types Answer 1, Types Answer 2, List Answer 14, Answers to Exercises
8358 @subsection Types Tutorial Exercise 1
8361 If the number is the square root of @cpi{} times a rational number,
8362 then its square, divided by @cpi{}, should be a rational number.
8366 1: 1.26508260337 1: 0.509433962268 1: 2486645810:4881193627
8374 Technically speaking this is a rational number, but not one that is
8375 likely to have arisen in the original problem. More likely, it just
8376 happens to be the fraction which most closely represents some
8377 irrational number to within 12 digits.
8379 But perhaps our result was not quite exact. Let's reduce the
8380 precision slightly and try again:
8384 1: 0.509433962268 1: 27:53
8387 U p 10 @key{RET} c F
8392 Aha! It's unlikely that an irrational number would equal a fraction
8393 this simple to within ten digits, so our original number was probably
8394 @texline @math{\sqrt{27 \pi / 53}}.
8395 @infoline @expr{sqrt(27 pi / 53)}.
8397 Notice that we didn't need to re-round the number when we reduced the
8398 precision. Remember, arithmetic operations always round their inputs
8399 to the current precision before they begin.
8401 @node Types Answer 2, Types Answer 3, Types Answer 1, Answers to Exercises
8402 @subsection Types Tutorial Exercise 2
8405 @samp{inf / inf = nan}. Perhaps @samp{1} is the ``obvious'' answer.
8406 But if @w{@samp{17 inf = inf}}, then @samp{17 inf / inf = inf / inf = 17}, too.
8408 @samp{exp(inf) = inf}. It's tempting to say that the exponential
8409 of infinity must be ``bigger'' than ``regular'' infinity, but as
8410 far as Calc is concerned all infinities are as just as big.
8411 In other words, as @expr{x} goes to infinity, @expr{e^x} also goes
8412 to infinity, but the fact the @expr{e^x} grows much faster than
8413 @expr{x} is not relevant here.
8415 @samp{exp(-inf) = 0}. Here we have a finite answer even though
8416 the input is infinite.
8418 @samp{sqrt(-inf) = (0, 1) inf}. Remember that @expr{(0, 1)}
8419 represents the imaginary number @expr{i}. Here's a derivation:
8420 @samp{sqrt(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}.
8421 The first part is, by definition, @expr{i}; the second is @code{inf}
8422 because, once again, all infinities are the same size.
8424 @samp{sqrt(uinf) = uinf}. In fact, we do know something about the
8425 direction because @code{sqrt} is defined to return a value in the
8426 right half of the complex plane. But Calc has no notation for this,
8427 so it settles for the conservative answer @code{uinf}.
8429 @samp{abs(uinf) = inf}. No matter which direction @expr{x} points,
8430 @samp{abs(x)} always points along the positive real axis.
8432 @samp{ln(0) = -inf}. Here we have an infinite answer to a finite
8433 input. As in the @expr{1 / 0} case, Calc will only use infinities
8434 here if you have turned on Infinite mode. Otherwise, it will
8435 treat @samp{ln(0)} as an error.
8437 @node Types Answer 3, Types Answer 4, Types Answer 2, Answers to Exercises
8438 @subsection Types Tutorial Exercise 3
8441 We can make @samp{inf - inf} be any real number we like, say,
8442 @expr{a}, just by claiming that we added @expr{a} to the first
8443 infinity but not to the second. This is just as true for complex
8444 values of @expr{a}, so @code{nan} can stand for a complex number.
8445 (And, similarly, @code{uinf} can stand for an infinity that points
8446 in any direction in the complex plane, such as @samp{(0, 1) inf}).
8448 In fact, we can multiply the first @code{inf} by two. Surely
8449 @w{@samp{2 inf - inf = inf}}, but also @samp{2 inf - inf = inf - inf = nan}.
8450 So @code{nan} can even stand for infinity. Obviously it's just
8451 as easy to make it stand for minus infinity as for plus infinity.
8453 The moral of this story is that ``infinity'' is a slippery fish
8454 indeed, and Calc tries to handle it by having a very simple model
8455 for infinities (only the direction counts, not the ``size''); but
8456 Calc is careful to write @code{nan} any time this simple model is
8457 unable to tell what the true answer is.
8459 @node Types Answer 4, Types Answer 5, Types Answer 3, Answers to Exercises
8460 @subsection Types Tutorial Exercise 4
8464 2: 0@@ 47' 26" 1: 0@@ 2' 47.411765"
8468 0@@ 47' 26" @key{RET} 17 /
8473 The average song length is two minutes and 47.4 seconds.
8477 2: 0@@ 2' 47.411765" 1: 0@@ 3' 7.411765" 1: 0@@ 53' 6.000005"
8486 The album would be 53 minutes and 6 seconds long.
8488 @node Types Answer 5, Types Answer 6, Types Answer 4, Answers to Exercises
8489 @subsection Types Tutorial Exercise 5
8492 Let's suppose it's January 14, 1991. The easiest thing to do is
8493 to keep trying 13ths of months until Calc reports a Friday.
8494 We can do this by manually entering dates, or by using @kbd{t I}:
8498 1: <Wed Feb 13, 1991> 1: <Wed Mar 13, 1991> 1: <Sat Apr 13, 1991>
8501 ' <2/13> @key{RET} @key{DEL} ' <3/13> @key{RET} t I
8506 (Calc assumes the current year if you don't say otherwise.)
8508 This is getting tedious---we can keep advancing the date by typing
8509 @kbd{t I} over and over again, but let's automate the job by using
8510 vector mapping. The @kbd{t I} command actually takes a second
8511 ``how-many-months'' argument, which defaults to one. This
8512 argument is exactly what we want to map over:
8516 2: <Sat Apr 13, 1991> 1: [<Mon May 13, 1991>, <Thu Jun 13, 1991>,
8517 1: [1, 2, 3, 4, 5, 6] <Sat Jul 13, 1991>, <Tue Aug 13, 1991>,
8518 . <Fri Sep 13, 1991>, <Sun Oct 13, 1991>]
8521 v x 6 @key{RET} V M t I
8526 Et voil@`a, September 13, 1991 is a Friday.
8533 ' <sep 13> - <jan 14> @key{RET}
8538 And the answer to our original question: 242 days to go.
8540 @node Types Answer 6, Types Answer 7, Types Answer 5, Answers to Exercises
8541 @subsection Types Tutorial Exercise 6
8544 The full rule for leap years is that they occur in every year divisible
8545 by four, except that they don't occur in years divisible by 100, except
8546 that they @emph{do} in years divisible by 400. We could work out the
8547 answer by carefully counting the years divisible by four and the
8548 exceptions, but there is a much simpler way that works even if we
8549 don't know the leap year rule.
8551 Let's assume the present year is 1991. Years have 365 days, except
8552 that leap years (whenever they occur) have 366 days. So let's count
8553 the number of days between now and then, and compare that to the
8554 number of years times 365. The number of extra days we find must be
8555 equal to the number of leap years there were.
8559 1: <Mon Jan 1, 10001> 2: <Mon Jan 1, 10001> 1: 2925593
8560 . 1: <Tue Jan 1, 1991> .
8563 ' <jan 1 10001> @key{RET} ' <jan 1 1991> @key{RET} -
8570 3: 2925593 2: 2925593 2: 2925593 1: 1943
8571 2: 10001 1: 8010 1: 2923650 .
8575 10001 @key{RET} 1991 - 365 * -
8579 @c [fix-ref Date Forms]
8581 There will be 1943 leap years before the year 10001. (Assuming,
8582 of course, that the algorithm for computing leap years remains
8583 unchanged for that long. @xref{Date Forms}, for some interesting
8584 background information in that regard.)
8586 @node Types Answer 7, Types Answer 8, Types Answer 6, Answers to Exercises
8587 @subsection Types Tutorial Exercise 7
8590 The relative errors must be converted to absolute errors so that
8591 @samp{+/-} notation may be used.
8599 20 @key{RET} .05 * 4 @key{RET} .05 *
8603 Now we simply chug through the formula.
8607 1: 19.7392088022 1: 394.78 +/- 19.739 1: 6316.5 +/- 706.21
8610 2 P 2 ^ * 20 p 1 * 4 p .2 @key{RET} 2 ^ *
8614 It turns out the @kbd{v u} command will unpack an error form as
8615 well as a vector. This saves us some retyping of numbers.
8619 3: 6316.5 +/- 706.21 2: 6316.5 +/- 706.21
8624 @key{RET} v u @key{TAB} /
8629 Thus the volume is 6316 cubic centimeters, within about 11 percent.
8631 @node Types Answer 8, Types Answer 9, Types Answer 7, Answers to Exercises
8632 @subsection Types Tutorial Exercise 8
8635 The first answer is pretty simple: @samp{1 / (0 .. 10) = (0.1 .. inf)}.
8636 Since a number in the interval @samp{(0 .. 10)} can get arbitrarily
8637 close to zero, its reciprocal can get arbitrarily large, so the answer
8638 is an interval that effectively means, ``any number greater than 0.1''
8639 but with no upper bound.
8641 The second answer, similarly, is @samp{1 / (-10 .. 0) = (-inf .. -0.1)}.
8643 Calc normally treats division by zero as an error, so that the formula
8644 @w{@samp{1 / 0}} is left unsimplified. Our third problem,
8645 @w{@samp{1 / [0 .. 10]}}, also (potentially) divides by zero because zero
8646 is now a member of the interval. So Calc leaves this one unevaluated, too.
8648 If you turn on Infinite mode by pressing @kbd{m i}, you will
8649 instead get the answer @samp{[0.1 .. inf]}, which includes infinity
8650 as a possible value.
8652 The fourth calculation, @samp{1 / (-10 .. 10)}, has the same problem.
8653 Zero is buried inside the interval, but it's still a possible value.
8654 It's not hard to see that the actual result of @samp{1 / (-10 .. 10)}
8655 will be either greater than @mathit{0.1}, or less than @mathit{-0.1}. Thus
8656 the interval goes from minus infinity to plus infinity, with a ``hole''
8657 in it from @mathit{-0.1} to @mathit{0.1}. Calc doesn't have any way to
8658 represent this, so it just reports @samp{[-inf .. inf]} as the answer.
8659 It may be disappointing to hear ``the answer lies somewhere between
8660 minus infinity and plus infinity, inclusive,'' but that's the best
8661 that interval arithmetic can do in this case.
8663 @node Types Answer 9, Types Answer 10, Types Answer 8, Answers to Exercises
8664 @subsection Types Tutorial Exercise 9
8668 1: [-3 .. 3] 2: [-3 .. 3] 2: [0 .. 9]
8669 . 1: [0 .. 9] 1: [-9 .. 9]
8672 [ 3 n .. 3 ] @key{RET} 2 ^ @key{TAB} @key{RET} *
8677 In the first case the result says, ``if a number is between @mathit{-3} and
8678 3, its square is between 0 and 9.'' The second case says, ``the product
8679 of two numbers each between @mathit{-3} and 3 is between @mathit{-9} and 9.''
8681 An interval form is not a number; it is a symbol that can stand for
8682 many different numbers. Two identical-looking interval forms can stand
8683 for different numbers.
8685 The same issue arises when you try to square an error form.
8687 @node Types Answer 10, Types Answer 11, Types Answer 9, Answers to Exercises
8688 @subsection Types Tutorial Exercise 10
8691 Testing the first number, we might arbitrarily choose 17 for @expr{x}.
8695 1: 17 mod 811749613 2: 17 mod 811749613 1: 533694123 mod 811749613
8699 17 M 811749613 @key{RET} 811749612 ^
8704 Since 533694123 is (considerably) different from 1, the number 811749613
8707 It's awkward to type the number in twice as we did above. There are
8708 various ways to avoid this, and algebraic entry is one. In fact, using
8709 a vector mapping operation we can perform several tests at once. Let's
8710 use this method to test the second number.
8714 2: [17, 42, 100000] 1: [1 mod 15485863, 1 mod ... ]
8718 [17 42 100000] 15485863 @key{RET} V M ' ($$ mod $)^($-1) @key{RET}
8723 The result is three ones (modulo @expr{n}), so it's very probable that
8724 15485863 is prime. (In fact, this number is the millionth prime.)
8726 Note that the functions @samp{($$^($-1)) mod $} or @samp{$$^($-1) % $}
8727 would have been hopelessly inefficient, since they would have calculated
8728 the power using full integer arithmetic.
8730 Calc has a @kbd{k p} command that does primality testing. For small
8731 numbers it does an exact test; for large numbers it uses a variant
8732 of the Fermat test we used here. You can use @kbd{k p} repeatedly
8733 to prove that a large integer is prime with any desired probability.
8735 @node Types Answer 11, Types Answer 12, Types Answer 10, Answers to Exercises
8736 @subsection Types Tutorial Exercise 11
8739 There are several ways to insert a calculated number into an HMS form.
8740 One way to convert a number of seconds to an HMS form is simply to
8741 multiply the number by an HMS form representing one second:
8745 1: 31415926.5359 2: 31415926.5359 1: 8726@@ 38' 46.5359"
8756 2: 8726@@ 38' 46.5359" 1: 6@@ 6' 2.5359" mod 24@@ 0' 0"
8757 1: 15@@ 27' 16" mod 24@@ 0' 0" .
8765 It will be just after six in the morning.
8767 The algebraic @code{hms} function can also be used to build an
8772 1: hms(0, 0, 10000000. pi) 1: 8726@@ 38' 46.5359"
8775 ' hms(0, 0, 1e7 pi) @key{RET} =
8780 The @kbd{=} key is necessary to evaluate the symbol @samp{pi} to
8781 the actual number 3.14159...
8783 @node Types Answer 12, Types Answer 13, Types Answer 11, Answers to Exercises
8784 @subsection Types Tutorial Exercise 12
8787 As we recall, there are 17 songs of about 2 minutes and 47 seconds
8792 2: 0@@ 2' 47" 1: [0@@ 3' 7" .. 0@@ 3' 47"]
8793 1: [0@@ 0' 20" .. 0@@ 1' 0"] .
8796 [ 0@@ 20" .. 0@@ 1' ] +
8803 1: [0@@ 52' 59." .. 1@@ 4' 19."]
8811 No matter how long it is, the album will fit nicely on one CD.
8813 @node Types Answer 13, Types Answer 14, Types Answer 12, Answers to Exercises
8814 @subsection Types Tutorial Exercise 13
8817 Type @kbd{' 1 yr @key{RET} u c s @key{RET}}. The answer is 31557600 seconds.
8819 @node Types Answer 14, Types Answer 15, Types Answer 13, Answers to Exercises
8820 @subsection Types Tutorial Exercise 14
8823 How long will it take for a signal to get from one end of the computer
8828 1: m / c 1: 3.3356 ns
8831 ' 1 m / c @key{RET} u c ns @key{RET}
8836 (Recall, @samp{c} is a ``unit'' corresponding to the speed of light.)
8840 1: 3.3356 ns 1: 0.81356 ns / ns 1: 0.81356
8844 ' 4.1 ns @key{RET} / u s
8849 Thus a signal could take up to 81 percent of a clock cycle just to
8850 go from one place to another inside the computer, assuming the signal
8851 could actually attain the full speed of light. Pretty tight!
8853 @node Types Answer 15, Algebra Answer 1, Types Answer 14, Answers to Exercises
8854 @subsection Types Tutorial Exercise 15
8857 The speed limit is 55 miles per hour on most highways. We want to
8858 find the ratio of Sam's speed to the US speed limit.
8862 1: 55 mph 2: 55 mph 3: 11 hr mph / yd
8866 ' 55 mph @key{RET} ' 5 yd/hr @key{RET} /
8870 The @kbd{u s} command cancels out these units to get a plain
8871 number. Now we take the logarithm base two to find the final
8872 answer, assuming that each successive pill doubles his speed.
8876 1: 19360. 2: 19360. 1: 14.24
8885 Thus Sam can take up to 14 pills without a worry.
8887 @node Algebra Answer 1, Algebra Answer 2, Types Answer 15, Answers to Exercises
8888 @subsection Algebra Tutorial Exercise 1
8891 @c [fix-ref Declarations]
8892 The result @samp{sqrt(x)^2} is simplified back to @expr{x} by the
8893 Calculator, but @samp{sqrt(x^2)} is not. (Consider what happens
8894 if @w{@expr{x = -4}}.) If @expr{x} is real, this formula could be
8895 simplified to @samp{abs(x)}, but for general complex arguments even
8896 that is not safe. (@xref{Declarations}, for a way to tell Calc
8897 that @expr{x} is known to be real.)
8899 @node Algebra Answer 2, Algebra Answer 3, Algebra Answer 1, Answers to Exercises
8900 @subsection Algebra Tutorial Exercise 2
8903 Suppose our roots are @expr{[a, b, c]}. We want a polynomial which
8904 is zero when @expr{x} is any of these values. The trivial polynomial
8905 @expr{x-a} is zero when @expr{x=a}, so the product @expr{(x-a)(x-b)(x-c)}
8906 will do the job. We can use @kbd{a c x} to write this in a more
8911 1: 34 x - 24 x^3 1: [1.19023, -1.19023, 0]
8921 1: [x - 1.19023, x + 1.19023, x] 1: (x - 1.19023) (x + 1.19023) x
8924 V M ' x-$ @key{RET} V R *
8931 1: x^3 - 1.41666 x 1: 34 x - 24 x^3
8934 a c x @key{RET} 24 n * a x
8939 Sure enough, our answer (multiplied by a suitable constant) is the
8940 same as the original polynomial.
8942 @node Algebra Answer 3, Algebra Answer 4, Algebra Answer 2, Answers to Exercises
8943 @subsection Algebra Tutorial Exercise 3
8947 1: x sin(pi x) 1: (sin(pi x) - pi x cos(pi x)) / pi^2
8950 ' x sin(pi x) @key{RET} m r a i x @key{RET}
8958 2: (sin(pi x) - pi x cos(pi x)) / pi^2
8961 ' [y,1] @key{RET} @key{TAB}
8968 1: [(sin(pi y) - pi y cos(pi y)) / pi^2, (sin(pi) - pi cos(pi)) / pi^2]
8978 1: (sin(pi y) - pi y cos(pi y)) / pi^2 + (pi cos(pi) - sin(pi)) / pi^2
8988 1: (sin(3.14159 y) - 3.14159 y cos(3.14159 y)) / 9.8696 - 0.3183
8998 1: [0., -0.95493, 0.63662, -1.5915, 1.2732]
9001 v x 5 @key{RET} @key{TAB} V M $ @key{RET}
9005 @node Algebra Answer 4, Rewrites Answer 1, Algebra Answer 3, Answers to Exercises
9006 @subsection Algebra Tutorial Exercise 4
9009 The hard part is that @kbd{V R +} is no longer sufficient to add up all
9010 the contributions from the slices, since the slices have varying
9011 coefficients. So first we must come up with a vector of these
9012 coefficients. Here's one way:
9016 2: -1 2: 3 1: [4, 2, ..., 4]
9017 1: [1, 2, ..., 9] 1: [-1, 1, ..., -1] .
9020 1 n v x 9 @key{RET} V M ^ 3 @key{TAB} -
9027 1: [4, 2, ..., 4, 1] 1: [1, 4, 2, ..., 4, 1]
9035 Now we compute the function values. Note that for this method we need
9036 eleven values, including both endpoints of the desired interval.
9040 2: [1, 4, 2, ..., 4, 1]
9041 1: [1, 1.1, 1.2, ... , 1.8, 1.9, 2.]
9044 11 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
9051 2: [1, 4, 2, ..., 4, 1]
9052 1: [0., 0.084941, 0.16993, ... ]
9055 ' sin(x) ln(x) @key{RET} m r p 5 @key{RET} V M $ @key{RET}
9060 Once again this calls for @kbd{V M * V R +}; a simple @kbd{*} does the
9065 1: 11.22 1: 1.122 1: 0.374
9073 Wow! That's even better than the result from the Taylor series method.
9075 @node Rewrites Answer 1, Rewrites Answer 2, Algebra Answer 4, Answers to Exercises
9076 @subsection Rewrites Tutorial Exercise 1
9079 We'll use Big mode to make the formulas more readable.
9085 1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: --------
9091 ' (2+sqrt(2)) / (1+sqrt(2)) @key{RET} d B
9096 Multiplying by the conjugate helps because @expr{(a+b) (a-b) = a^2 - b^2}.
9101 1: (2 + V 2 ) (V 2 - 1)
9104 a r a/(b+c) := a*(b-c) / (b^2-c^2) @key{RET}
9112 1: 2 + V 2 - 2 1: V 2
9115 a r a*(b+c) := a*b + a*c a s
9120 (We could have used @kbd{a x} instead of a rewrite rule for the
9123 The multiply-by-conjugate rule turns out to be useful in many
9124 different circumstances, such as when the denominator involves
9125 sines and cosines or the imaginary constant @code{i}.
9127 @node Rewrites Answer 2, Rewrites Answer 3, Rewrites Answer 1, Answers to Exercises
9128 @subsection Rewrites Tutorial Exercise 2
9131 Here is the rule set:
9135 [ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1,
9137 fib(n, x, y) := fib(n-1, y, x+y) ]
9142 The first rule turns a one-argument @code{fib} that people like to write
9143 into a three-argument @code{fib} that makes computation easier. The
9144 second rule converts back from three-argument form once the computation
9145 is done. The third rule does the computation itself. It basically
9146 says that if @expr{x} and @expr{y} are two consecutive Fibonacci numbers,
9147 then @expr{y} and @expr{x+y} are the next (overlapping) pair of Fibonacci
9150 Notice that because the number @expr{n} was ``validated'' by the
9151 conditions on the first rule, there is no need to put conditions on
9152 the other rules because the rule set would never get that far unless
9153 the input were valid. That further speeds computation, since no
9154 extra conditions need to be checked at every step.
9156 Actually, a user with a nasty sense of humor could enter a bad
9157 three-argument @code{fib} call directly, say, @samp{fib(0, 1, 1)},
9158 which would get the rules into an infinite loop. One thing that would
9159 help keep this from happening by accident would be to use something like
9160 @samp{ZzFib} instead of @code{fib} as the name of the three-argument
9163 @node Rewrites Answer 3, Rewrites Answer 4, Rewrites Answer 2, Answers to Exercises
9164 @subsection Rewrites Tutorial Exercise 3
9167 He got an infinite loop. First, Calc did as expected and rewrote
9168 @w{@samp{2 + 3 x}} to @samp{f(2, 3, x)}. Then it looked for ways to
9169 apply the rule again, and found that @samp{f(2, 3, x)} looks like
9170 @samp{a + b x} with @w{@samp{a = 0}} and @samp{b = 1}, so it rewrote to
9171 @samp{f(0, 1, f(2, 3, x))}. It then wrapped another @samp{f(0, 1, ...)}
9172 around that, and so on, ad infinitum. Joe should have used @kbd{M-1 a r}
9173 to make sure the rule applied only once.
9175 (Actually, even the first step didn't work as he expected. What Calc
9176 really gives for @kbd{M-1 a r} in this situation is @samp{f(3 x, 1, 2)},
9177 treating 2 as the ``variable,'' and @samp{3 x} as a constant being added
9178 to it. While this may seem odd, it's just as valid a solution as the
9179 ``obvious'' one. One way to fix this would be to add the condition
9180 @samp{:: variable(x)} to the rule, to make sure the thing that matches
9181 @samp{x} is indeed a variable, or to change @samp{x} to @samp{quote(x)}
9182 on the lefthand side, so that the rule matches the actual variable
9183 @samp{x} rather than letting @samp{x} stand for something else.)
9185 @node Rewrites Answer 4, Rewrites Answer 5, Rewrites Answer 3, Answers to Exercises
9186 @subsection Rewrites Tutorial Exercise 4
9193 Here is a suitable set of rules to solve the first part of the problem:
9197 [ seq(n, c) := seq(n/2, c+1) :: n%2 = 0,
9198 seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]
9202 Given the initial formula @samp{seq(6, 0)}, application of these
9203 rules produces the following sequence of formulas:
9217 whereupon neither of the rules match, and rewriting stops.
9219 We can pretty this up a bit with a couple more rules:
9223 [ seq(n) := seq(n, 0),
9230 Now, given @samp{seq(6)} as the starting configuration, we get 8
9233 The change to return a vector is quite simple:
9237 [ seq(n) := seq(n, []) :: integer(n) :: n > 0,
9239 seq(n, v) := seq(n/2, v | n) :: n%2 = 0,
9240 seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]
9245 Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
9247 Notice that the @expr{n > 1} guard is no longer necessary on the last
9248 rule since the @expr{n = 1} case is now detected by another rule.
9249 But a guard has been added to the initial rule to make sure the
9250 initial value is suitable before the computation begins.
9252 While still a good idea, this guard is not as vitally important as it
9253 was for the @code{fib} function, since calling, say, @samp{seq(x, [])}
9254 will not get into an infinite loop. Calc will not be able to prove
9255 the symbol @samp{x} is either even or odd, so none of the rules will
9256 apply and the rewrites will stop right away.
9258 @node Rewrites Answer 5, Rewrites Answer 6, Rewrites Answer 4, Answers to Exercises
9259 @subsection Rewrites Tutorial Exercise 5
9266 If @expr{x} is the sum @expr{a + b}, then `@tfn{nterms(}@var{x}@tfn{)}' must
9267 be `@tfn{nterms(}@var{a}@tfn{)}' plus `@tfn{nterms(}@var{b}@tfn{)}'. If @expr{x}
9268 is not a sum, then `@tfn{nterms(}@var{x}@tfn{)}' = 1.
9272 [ nterms(a + b) := nterms(a) + nterms(b),
9278 Here we have taken advantage of the fact that earlier rules always
9279 match before later rules; @samp{nterms(x)} will only be tried if we
9280 already know that @samp{x} is not a sum.
9282 @node Rewrites Answer 6, Programming Answer 1, Rewrites Answer 5, Answers to Exercises
9283 @subsection Rewrites Tutorial Exercise 6
9286 Here is a rule set that will do the job:
9290 [ a*(b + c) := a*b + a*c,
9291 opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m
9292 :: constant(a) :: constant(b),
9293 opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m
9294 :: constant(a) :: constant(b),
9295 a O(x^n) := O(x^n) :: constant(a),
9296 x^opt(m) O(x^n) := O(x^(n+m)),
9297 O(x^n) O(x^m) := O(x^(n+m)) ]
9301 If we really want the @kbd{+} and @kbd{*} keys to operate naturally
9302 on power series, we should put these rules in @code{EvalRules}. For
9303 testing purposes, it is better to put them in a different variable,
9304 say, @code{O}, first.
9306 The first rule just expands products of sums so that the rest of the
9307 rules can assume they have an expanded-out polynomial to work with.
9308 Note that this rule does not mention @samp{O} at all, so it will
9309 apply to any product-of-sum it encounters---this rule may surprise
9310 you if you put it into @code{EvalRules}!
9312 In the second rule, the sum of two O's is changed to the smaller O.
9313 The optional constant coefficients are there mostly so that
9314 @samp{O(x^2) - O(x^3)} and @samp{O(x^3) - O(x^2)} are handled
9315 as well as @samp{O(x^2) + O(x^3)}.
9317 The third rule absorbs higher powers of @samp{x} into O's.
9319 The fourth rule says that a constant times a negligible quantity
9320 is still negligible. (This rule will also match @samp{O(x^3) / 4},
9321 with @samp{a = 1/4}.)
9323 The fifth rule rewrites, for example, @samp{x^2 O(x^3)} to @samp{O(x^5)}.
9324 (It is easy to see that if one of these forms is negligible, the other
9325 is, too.) Notice the @samp{x^opt(m)} to pick up terms like
9326 @w{@samp{x O(x^3)}}. Optional powers will match @samp{x} as @samp{x^1}
9327 but not 1 as @samp{x^0}. This turns out to be exactly what we want here.
9329 The sixth rule is the corresponding rule for products of two O's.
9331 Another way to solve this problem would be to create a new ``data type''
9332 that represents truncated power series. We might represent these as
9333 function calls @samp{series(@var{coefs}, @var{x})} where @var{coefs} is
9334 a vector of coefficients for @expr{x^0}, @expr{x^1}, @expr{x^2}, and so
9335 on. Rules would exist for sums and products of such @code{series}
9336 objects, and as an optional convenience could also know how to combine a
9337 @code{series} object with a normal polynomial. (With this, and with a
9338 rule that rewrites @samp{O(x^n)} to the equivalent @code{series} form,
9339 you could still enter power series in exactly the same notation as
9340 before.) Operations on such objects would probably be more efficient,
9341 although the objects would be a bit harder to read.
9343 @c [fix-ref Compositions]
9344 Some other symbolic math programs provide a power series data type
9345 similar to this. Mathematica, for example, has an object that looks
9346 like @samp{PowerSeries[@var{x}, @var{x0}, @var{coefs}, @var{nmin},
9347 @var{nmax}, @var{den}]}, where @var{x0} is the point about which the
9348 power series is taken (we've been assuming this was always zero),
9349 and @var{nmin}, @var{nmax}, and @var{den} allow pseudo-power-series
9350 with fractional or negative powers. Also, the @code{PowerSeries}
9351 objects have a special display format that makes them look like
9352 @samp{2 x^2 + O(x^4)} when they are printed out. (@xref{Compositions},
9353 for a way to do this in Calc, although for something as involved as
9354 this it would probably be better to write the formatting routine
9357 @node Programming Answer 1, Programming Answer 2, Rewrites Answer 6, Answers to Exercises
9358 @subsection Programming Tutorial Exercise 1
9361 Just enter the formula @samp{ninteg(sin(t)/t, t, 0, x)}, type
9362 @kbd{Z F}, and answer the questions. Since this formula contains two
9363 variables, the default argument list will be @samp{(t x)}. We want to
9364 change this to @samp{(x)} since @expr{t} is really a dummy variable
9365 to be used within @code{ninteg}.
9367 The exact keystrokes are @kbd{Z F s Si @key{RET} @key{RET} C-b C-b @key{DEL} @key{DEL} @key{RET} y}.
9368 (The @kbd{C-b C-b @key{DEL} @key{DEL}} are what fix the argument list.)
9370 @node Programming Answer 2, Programming Answer 3, Programming Answer 1, Answers to Exercises
9371 @subsection Programming Tutorial Exercise 2
9374 One way is to move the number to the top of the stack, operate on
9375 it, then move it back: @kbd{C-x ( M-@key{TAB} n M-@key{TAB} M-@key{TAB} C-x )}.
9377 Another way is to negate the top three stack entries, then negate
9378 again the top two stack entries: @kbd{C-x ( M-3 n M-2 n C-x )}.
9380 Finally, it turns out that a negative prefix argument causes a
9381 command like @kbd{n} to operate on the specified stack entry only,
9382 which is just what we want: @kbd{C-x ( M-- 3 n C-x )}.
9384 Just for kicks, let's also do it algebraically:
9385 @w{@kbd{C-x ( ' -$$$, $$, $ @key{RET} C-x )}}.
9387 @node Programming Answer 3, Programming Answer 4, Programming Answer 2, Answers to Exercises
9388 @subsection Programming Tutorial Exercise 3
9391 Each of these functions can be computed using the stack, or using
9392 algebraic entry, whichever way you prefer:
9396 @texline @math{\displaystyle{\sin x \over x}}:
9397 @infoline @expr{sin(x) / x}:
9399 Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}.
9401 Using algebraic entry: @kbd{C-x ( ' sin($)/$ @key{RET} C-x )}.
9404 Computing the logarithm:
9406 Using the stack: @kbd{C-x ( @key{TAB} B C-x )}
9408 Using algebraic entry: @kbd{C-x ( ' log($,$$) @key{RET} C-x )}.
9411 Computing the vector of integers:
9413 Using the stack: @kbd{C-x ( 1 @key{RET} 1 C-u v x C-x )}. (Recall that
9414 @kbd{C-u v x} takes the vector size, starting value, and increment
9417 Alternatively: @kbd{C-x ( ~ v x C-x )}. (The @kbd{~} key pops a
9418 number from the stack and uses it as the prefix argument for the
9421 Using algebraic entry: @kbd{C-x ( ' index($) @key{RET} C-x )}.
9423 @node Programming Answer 4, Programming Answer 5, Programming Answer 3, Answers to Exercises
9424 @subsection Programming Tutorial Exercise 4
9427 Here's one way: @kbd{C-x ( @key{RET} V R + @key{TAB} v l / C-x )}.
9429 @node Programming Answer 5, Programming Answer 6, Programming Answer 4, Answers to Exercises
9430 @subsection Programming Tutorial Exercise 5
9434 2: 1 1: 1.61803398502 2: 1.61803398502
9435 1: 20 . 1: 1.61803398875
9438 1 @key{RET} 20 Z < & 1 + Z > I H P
9443 This answer is quite accurate.
9445 @node Programming Answer 6, Programming Answer 7, Programming Answer 5, Answers to Exercises
9446 @subsection Programming Tutorial Exercise 6
9452 [ [ 0, 1 ] * [a, b] = [b, a + b]
9457 Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @expr{n+1}
9458 and @expr{n+2}. Here's one program that does the job:
9461 C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] @key{RET} v u @key{DEL} C-x )
9465 This program is quite efficient because Calc knows how to raise a
9466 matrix (or other value) to the power @expr{n} in only
9467 @texline @math{\log_2 n}
9468 @infoline @expr{log(n,2)}
9469 steps. For example, this program can compute the 1000th Fibonacci
9470 number (a 209-digit integer!) in about 10 steps; even though the
9471 @kbd{Z < ... Z >} solution had much simpler steps, it would have
9472 required so many steps that it would not have been practical.
9474 @node Programming Answer 7, Programming Answer 8, Programming Answer 6, Answers to Exercises
9475 @subsection Programming Tutorial Exercise 7
9478 The trick here is to compute the harmonic numbers differently, so that
9479 the loop counter itself accumulates the sum of reciprocals. We use
9480 a separate variable to hold the integer counter.
9488 1 t 1 1 @key{RET} 4 Z ( t 2 r 1 1 + s 1 & Z )
9493 The body of the loop goes as follows: First save the harmonic sum
9494 so far in variable 2. Then delete it from the stack; the for loop
9495 itself will take care of remembering it for us. Next, recall the
9496 count from variable 1, add one to it, and feed its reciprocal to
9497 the for loop to use as the step value. The for loop will increase
9498 the ``loop counter'' by that amount and keep going until the
9499 loop counter exceeds 4.
9504 1: 3.99498713092 2: 3.99498713092
9508 r 1 r 2 @key{RET} 31 & +
9512 Thus we find that the 30th harmonic number is 3.99, and the 31st
9513 harmonic number is 4.02.
9515 @node Programming Answer 8, Programming Answer 9, Programming Answer 7, Answers to Exercises
9516 @subsection Programming Tutorial Exercise 8
9519 The first step is to compute the derivative @expr{f'(x)} and thus
9521 @texline @math{\displaystyle{x - {f(x) \over f'(x)}}}.
9522 @infoline @expr{x - f(x)/f'(x)}.
9524 (Because this definition is long, it will be repeated in concise form
9525 below. You can use @w{@kbd{M-# m}} to load it from there. While you are
9526 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9527 keystrokes without executing them. In the following diagrams we'll
9528 pretend Calc actually executed the keystrokes as you typed them,
9529 just for purposes of illustration.)
9533 2: sin(cos(x)) - 0.5 3: 4.5
9534 1: 4.5 2: sin(cos(x)) - 0.5
9535 . 1: -(sin(x) cos(cos(x)))
9538 ' sin(cos(x))-0.5 @key{RET} 4.5 m r C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET}
9546 1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x))
9549 / ' x @key{RET} @key{TAB} - t 1
9553 Now, we enter the loop. We'll use a repeat loop with a 20-repetition
9554 limit just in case the method fails to converge for some reason.
9555 (Normally, the @w{@kbd{Z /}} command will stop the loop before all 20
9556 repetitions are done.)
9560 1: 4.5 3: 4.5 2: 4.5
9561 . 2: x + (sin(cos(x)) ... 1: 5.24196456928
9565 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9569 This is the new guess for @expr{x}. Now we compare it with the
9570 old one to see if we've converged.
9574 3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348
9579 @key{RET} M-@key{TAB} a = Z / Z > Z ' C-x )
9583 The loop converges in just a few steps to this value. To check
9584 the result, we can simply substitute it back into the equation.
9592 @key{RET} ' sin(cos($)) @key{RET}
9596 Let's test the new definition again:
9604 ' x^2-9 @key{RET} 1 X
9608 Once again, here's the full Newton's Method definition:
9612 C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET} / ' x @key{RET} @key{TAB} - t 1
9613 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9614 @key{RET} M-@key{TAB} a = Z /
9621 @c [fix-ref Nesting and Fixed Points]
9622 It turns out that Calc has a built-in command for applying a formula
9623 repeatedly until it converges to a number. @xref{Nesting and Fixed Points},
9624 to see how to use it.
9626 @c [fix-ref Root Finding]
9627 Also, of course, @kbd{a R} is a built-in command that uses Newton's
9628 method (among others) to look for numerical solutions to any equation.
9629 @xref{Root Finding}.
9631 @node Programming Answer 9, Programming Answer 10, Programming Answer 8, Answers to Exercises
9632 @subsection Programming Tutorial Exercise 9
9635 The first step is to adjust @expr{z} to be greater than 5. A simple
9636 ``for'' loop will do the job here. If @expr{z} is less than 5, we
9637 reduce the problem using
9638 @texline @math{\psi(z) = \psi(z+1) - 1/z}.
9639 @infoline @expr{psi(z) = psi(z+1) - 1/z}. We go
9641 @texline @math{\psi(z+1)},
9642 @infoline @expr{psi(z+1)},
9643 and remember to add back a factor of @expr{-1/z} when we're done. This
9644 step is repeated until @expr{z > 5}.
9646 (Because this definition is long, it will be repeated in concise form
9647 below. You can use @w{@kbd{M-# m}} to load it from there. While you are
9648 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9649 keystrokes without executing them. In the following diagrams we'll
9650 pretend Calc actually executed the keystrokes as you typed them,
9651 just for purposes of illustration.)
9658 1.0 @key{RET} C-x ( Z ` s 1 0 t 2
9662 Here, variable 1 holds @expr{z} and variable 2 holds the adjustment
9663 factor. If @expr{z < 5}, we use a loop to increase it.
9665 (By the way, we started with @samp{1.0} instead of the integer 1 because
9666 otherwise the calculation below will try to do exact fractional arithmetic,
9667 and will never converge because fractions compare equal only if they
9668 are exactly equal, not just equal to within the current precision.)
9677 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9681 Now we compute the initial part of the sum:
9682 @texline @math{\ln z - {1 \over 2z}}
9683 @infoline @expr{ln(z) - 1/2z}
9684 minus the adjustment factor.
9688 2: 1.79175946923 2: 1.7084261359 1: -0.57490719743
9689 1: 0.0833333333333 1: 2.28333333333 .
9696 Now we evaluate the series. We'll use another ``for'' loop counting
9697 up the value of @expr{2 n}. (Calc does have a summation command,
9698 @kbd{a +}, but we'll use loops just to get more practice with them.)
9702 3: -0.5749 3: -0.5749 4: -0.5749 2: -0.5749
9703 2: 2 2: 1:6 3: 1:6 1: 2.3148e-3
9708 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9715 3: -0.5749 3: -0.5772 2: -0.5772 1: -0.577215664892
9716 2: -0.5749 2: -0.5772 1: 0 .
9717 1: 2.3148e-3 1: -0.5749 .
9720 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z / 2 Z ) Z ' C-x )
9724 This is the value of
9725 @texline @math{-\gamma},
9726 @infoline @expr{- gamma},
9727 with a slight bit of roundoff error. To get a full 12 digits, let's use
9732 2: -0.577215664892 2: -0.577215664892
9733 1: 1. 1: -0.577215664901532
9735 1. @key{RET} p 16 @key{RET} X
9739 Here's the complete sequence of keystrokes:
9744 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9746 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9747 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z /
9754 @node Programming Answer 10, Programming Answer 11, Programming Answer 9, Answers to Exercises
9755 @subsection Programming Tutorial Exercise 10
9758 Taking the derivative of a term of the form @expr{x^n} will produce
9760 @texline @math{n x^{n-1}}.
9761 @infoline @expr{n x^(n-1)}.
9762 Taking the derivative of a constant
9763 produces zero. From this it is easy to see that the @expr{n}th
9764 derivative of a polynomial, evaluated at @expr{x = 0}, will equal the
9765 coefficient on the @expr{x^n} term times @expr{n!}.
9767 (Because this definition is long, it will be repeated in concise form
9768 below. You can use @w{@kbd{M-# m}} to load it from there. While you are
9769 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9770 keystrokes without executing them. In the following diagrams we'll
9771 pretend Calc actually executed the keystrokes as you typed them,
9772 just for purposes of illustration.)
9776 2: 5 x^4 + (x + 1)^2 3: 5 x^4 + (x + 1)^2
9781 ' 5 x^4 + (x+1)^2 @key{RET} 6 C-x ( Z ` [ ] t 1 0 @key{TAB}
9786 Variable 1 will accumulate the vector of coefficients.
9790 2: 0 3: 0 2: 5 x^4 + ...
9791 1: 5 x^4 + ... 2: 5 x^4 + ... 1: 1
9795 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9800 Note that @kbd{s | 1} appends the top-of-stack value to the vector
9801 in a variable; it is completely analogous to @kbd{s + 1}. We could
9802 have written instead, @kbd{r 1 @key{TAB} | t 1}.
9806 1: 20 x^3 + 2 x + 2 1: 0 1: [1, 2, 1, 0, 5, 0, 0]
9809 a d x @key{RET} 1 Z ) @key{DEL} r 1 Z ' C-x )
9813 To convert back, a simple method is just to map the coefficients
9814 against a table of powers of @expr{x}.
9818 2: [1, 2, 1, 0, 5, 0, 0] 2: [1, 2, 1, 0, 5, 0, 0]
9819 1: 6 1: [0, 1, 2, 3, 4, 5, 6]
9822 6 @key{RET} 1 + 0 @key{RET} 1 C-u v x
9829 2: [1, 2, 1, 0, 5, 0, 0] 2: 1 + 2 x + x^2 + 5 x^4
9830 1: [1, x, x^2, x^3, ... ] .
9833 ' x @key{RET} @key{TAB} V M ^ *
9837 Once again, here are the whole polynomial to/from vector programs:
9841 C-x ( Z ` [ ] t 1 0 @key{TAB}
9842 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9848 C-x ( 1 + 0 @key{RET} 1 C-u v x ' x @key{RET} @key{TAB} V M ^ * C-x )
9852 @node Programming Answer 11, Programming Answer 12, Programming Answer 10, Answers to Exercises
9853 @subsection Programming Tutorial Exercise 11
9856 First we define a dummy program to go on the @kbd{z s} key. The true
9857 @w{@kbd{z s}} key is supposed to take two numbers from the stack and
9858 return one number, so @key{DEL} as a dummy definition will make
9859 sure the stack comes out right.
9867 4 @key{RET} 2 C-x ( @key{DEL} C-x ) Z K s @key{RET} 2
9871 The last step replaces the 2 that was eaten during the creation
9872 of the dummy @kbd{z s} command. Now we move on to the real
9873 definition. The recurrence needs to be rewritten slightly,
9874 to the form @expr{s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)}.
9876 (Because this definition is long, it will be repeated in concise form
9877 below. You can use @kbd{M-# m} to load it from there.)
9887 C-x ( M-2 @key{RET} a = Z [ @key{DEL} @key{DEL} 1 Z :
9894 4: 4 2: 4 2: 3 4: 3 4: 3 3: 3
9895 3: 2 1: 2 1: 2 3: 2 3: 2 2: 2
9896 2: 2 . . 2: 3 2: 3 1: 3
9900 @key{RET} 0 a = Z [ @key{DEL} @key{DEL} 0 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9905 (Note that the value 3 that our dummy @kbd{z s} produces is not correct;
9906 it is merely a placeholder that will do just as well for now.)
9910 3: 3 4: 3 3: 3 2: 3 1: -6
9911 2: 3 3: 3 2: 3 1: 9 .
9916 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9923 1: -6 2: 4 1: 11 2: 11
9927 Z ] Z ] C-x ) Z K s @key{RET} @key{DEL} 4 @key{RET} 2 z s M-@key{RET} k s
9931 Even though the result that we got during the definition was highly
9932 bogus, once the definition is complete the @kbd{z s} command gets
9935 Here's the full program once again:
9939 C-x ( M-2 @key{RET} a =
9940 Z [ @key{DEL} @key{DEL} 1
9942 Z [ @key{DEL} @key{DEL} 0
9943 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9944 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9951 You can read this definition using @kbd{M-# m} (@code{read-kbd-macro})
9952 followed by @kbd{Z K s}, without having to make a dummy definition
9953 first, because @code{read-kbd-macro} doesn't need to execute the
9954 definition as it reads it in. For this reason, @code{M-# m} is often
9955 the easiest way to create recursive programs in Calc.
9957 @node Programming Answer 12, , Programming Answer 11, Answers to Exercises
9958 @subsection Programming Tutorial Exercise 12
9961 This turns out to be a much easier way to solve the problem. Let's
9962 denote Stirling numbers as calls of the function @samp{s}.
9964 First, we store the rewrite rules corresponding to the definition of
9965 Stirling numbers in a convenient variable:
9968 s e StirlingRules @key{RET}
9969 [ s(n,n) := 1 :: n >= 0,
9970 s(n,0) := 0 :: n > 0,
9971 s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ]
9975 Now, it's just a matter of applying the rules:
9979 2: 4 1: s(4, 2) 1: 11
9983 4 @key{RET} 2 C-x ( ' s($$,$) @key{RET} a r StirlingRules @key{RET} C-x )
9987 As in the case of the @code{fib} rules, it would be useful to put these
9988 rules in @code{EvalRules} and to add a @samp{:: remember} condition to
9991 @c This ends the table-of-contents kludge from above:
9993 \global\let\chapternofonts=\oldchapternofonts
9998 @node Introduction, Data Types, Tutorial, Top
9999 @chapter Introduction
10002 This chapter is the beginning of the Calc reference manual.
10003 It covers basic concepts such as the stack, algebraic and
10004 numeric entry, undo, numeric prefix arguments, etc.
10007 @c (Chapter 2, the Tutorial, has been printed in a separate volume.)
10014 * Algebraic Entry::
10015 * Quick Calculator::
10016 * Prefix Arguments::
10019 * Multiple Calculators::
10020 * Troubleshooting Commands::
10023 @node Basic Commands, Help Commands, Introduction, Introduction
10024 @section Basic Commands
10029 @cindex Starting the Calculator
10030 @cindex Running the Calculator
10031 To start the Calculator in its standard interface, type @kbd{M-x calc}.
10032 By default this creates a pair of small windows, @samp{*Calculator*}
10033 and @samp{*Calc Trail*}. The former displays the contents of the
10034 Calculator stack and is manipulated exclusively through Calc commands.
10035 It is possible (though not usually necessary) to create several Calc
10036 mode buffers each of which has an independent stack, undo list, and
10037 mode settings. There is exactly one Calc Trail buffer; it records a
10038 list of the results of all calculations that have been done. The
10039 Calc Trail buffer uses a variant of Calc mode, so Calculator commands
10040 still work when the trail buffer's window is selected. It is possible
10041 to turn the trail window off, but the @samp{*Calc Trail*} buffer itself
10042 still exists and is updated silently. @xref{Trail Commands}.
10050 In most installations, the @kbd{M-# c} key sequence is a more
10051 convenient way to start the Calculator. Also, @kbd{M-# M-#} and
10052 @kbd{M-# #} are synonyms for @kbd{M-# c} unless you last used Calc
10053 in its Keypad mode.
10057 @pindex calc-execute-extended-command
10058 Most Calc commands use one or two keystrokes. Lower- and upper-case
10059 letters are distinct. Commands may also be entered in full @kbd{M-x} form;
10060 for some commands this is the only form. As a convenience, the @kbd{x}
10061 key (@code{calc-execute-extended-command})
10062 is like @kbd{M-x} except that it enters the initial string @samp{calc-}
10063 for you. For example, the following key sequences are equivalent:
10064 @kbd{S}, @kbd{M-x calc-sin @key{RET}}, @kbd{x sin @key{RET}}.
10066 @cindex Extensions module
10067 @cindex @file{calc-ext} module
10068 The Calculator exists in many parts. When you type @kbd{M-# c}, the
10069 Emacs ``auto-load'' mechanism will bring in only the first part, which
10070 contains the basic arithmetic functions. The other parts will be
10071 auto-loaded the first time you use the more advanced commands like trig
10072 functions or matrix operations. This is done to improve the response time
10073 of the Calculator in the common case when all you need to do is a
10074 little arithmetic. If for some reason the Calculator fails to load an
10075 extension module automatically, you can force it to load all the
10076 extensions by using the @kbd{M-# L} (@code{calc-load-everything})
10077 command. @xref{Mode Settings}.
10079 If you type @kbd{M-x calc} or @kbd{M-# c} with any numeric prefix argument,
10080 the Calculator is loaded if necessary, but it is not actually started.
10081 If the argument is positive, the @file{calc-ext} extensions are also
10082 loaded if necessary. User-written Lisp code that wishes to make use
10083 of Calc's arithmetic routines can use @samp{(calc 0)} or @samp{(calc 1)}
10084 to auto-load the Calculator.
10088 If you type @kbd{M-# b}, then next time you use @kbd{M-# c} you
10089 will get a Calculator that uses the full height of the Emacs screen.
10090 When full-screen mode is on, @kbd{M-# c} runs the @code{full-calc}
10091 command instead of @code{calc}. From the Unix shell you can type
10092 @samp{emacs -f full-calc} to start a new Emacs specifically for use
10093 as a calculator. When Calc is started from the Emacs command line
10094 like this, Calc's normal ``quit'' commands actually quit Emacs itself.
10097 @pindex calc-other-window
10098 The @kbd{M-# o} command is like @kbd{M-# c} except that the Calc
10099 window is not actually selected. If you are already in the Calc
10100 window, @kbd{M-# o} switches you out of it. (The regular Emacs
10101 @kbd{C-x o} command would also work for this, but it has a
10102 tendency to drop you into the Calc Trail window instead, which
10103 @kbd{M-# o} takes care not to do.)
10108 For one quick calculation, you can type @kbd{M-# q} (@code{quick-calc})
10109 which prompts you for a formula (like @samp{2+3/4}). The result is
10110 displayed at the bottom of the Emacs screen without ever creating
10111 any special Calculator windows. @xref{Quick Calculator}.
10116 Finally, if you are using the X window system you may want to try
10117 @kbd{M-# k} (@code{calc-keypad}) which runs Calc with a
10118 ``calculator keypad'' picture as well as a stack display. Click on
10119 the keys with the mouse to operate the calculator. @xref{Keypad Mode}.
10123 @cindex Quitting the Calculator
10124 @cindex Exiting the Calculator
10125 The @kbd{q} key (@code{calc-quit}) exits Calc mode and closes the
10126 Calculator's window(s). It does not delete the Calculator buffers.
10127 If you type @kbd{M-x calc} again, the Calculator will reappear with the
10128 contents of the stack intact. Typing @kbd{M-# c} or @kbd{M-# M-#}
10129 again from inside the Calculator buffer is equivalent to executing
10130 @code{calc-quit}; you can think of @kbd{M-# M-#} as toggling the
10131 Calculator on and off.
10134 The @kbd{M-# x} command also turns the Calculator off, no matter which
10135 user interface (standard, Keypad, or Embedded) is currently active.
10136 It also cancels @code{calc-edit} mode if used from there.
10138 @kindex d @key{SPC}
10139 @pindex calc-refresh
10140 @cindex Refreshing a garbled display
10141 @cindex Garbled displays, refreshing
10142 The @kbd{d @key{SPC}} key sequence (@code{calc-refresh}) redraws the contents
10143 of the Calculator buffer from memory. Use this if the contents of the
10144 buffer have been damaged somehow.
10149 The @kbd{o} key (@code{calc-realign}) moves the cursor back to its
10150 ``home'' position at the bottom of the Calculator buffer.
10154 @pindex calc-scroll-left
10155 @pindex calc-scroll-right
10156 @cindex Horizontal scrolling
10158 @cindex Wide text, scrolling
10159 The @kbd{<} and @kbd{>} keys are bound to @code{calc-scroll-left} and
10160 @code{calc-scroll-right}. These are just like the normal horizontal
10161 scrolling commands except that they scroll one half-screen at a time by
10162 default. (Calc formats its output to fit within the bounds of the
10163 window whenever it can.)
10167 @pindex calc-scroll-down
10168 @pindex calc-scroll-up
10169 @cindex Vertical scrolling
10170 The @kbd{@{} and @kbd{@}} keys are bound to @code{calc-scroll-down}
10171 and @code{calc-scroll-up}. They scroll up or down by one-half the
10172 height of the Calc window.
10176 The @kbd{M-# 0} command (@code{calc-reset}; that's @kbd{M-#} followed
10177 by a zero) resets the Calculator to its initial state. This clears
10178 the stack, resets all the modes to their initial values (the values
10179 that were saved with @kbd{m m} (@code{calc-save-modes})), clears the
10180 caches (@pxref{Caches}), and so on. (It does @emph{not} erase the
10181 values of any variables.) With an argument of 0, Calc will be reset to
10182 its default state; namely, the modes will be given their default values.
10183 With a positive prefix argument, @kbd{M-# 0} preserves the contents of
10184 the stack but resets everything else to its initial state; with a
10185 negative prefix argument, @kbd{M-# 0} preserves the contents of the
10186 stack but resets everything else to its default state.
10188 @pindex calc-version
10189 The @kbd{M-x calc-version} command displays the current version number
10190 of Calc and the name of the person who installed it on your system.
10191 (This information is also present in the @samp{*Calc Trail*} buffer,
10192 and in the output of the @kbd{h h} command.)
10194 @node Help Commands, Stack Basics, Basic Commands, Introduction
10195 @section Help Commands
10198 @cindex Help commands
10201 The @kbd{?} key (@code{calc-help}) displays a series of brief help messages.
10202 Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs'
10203 @key{ESC} and @kbd{C-x} prefixes. You can type
10204 @kbd{?} after a prefix to see a list of commands beginning with that
10205 prefix. (If the message includes @samp{[MORE]}, press @kbd{?} again
10206 to see additional commands for that prefix.)
10209 @pindex calc-full-help
10210 The @kbd{h h} (@code{calc-full-help}) command displays all the @kbd{?}
10211 responses at once. When printed, this makes a nice, compact (three pages)
10212 summary of Calc keystrokes.
10214 In general, the @kbd{h} key prefix introduces various commands that
10215 provide help within Calc. Many of the @kbd{h} key functions are
10216 Calc-specific analogues to the @kbd{C-h} functions for Emacs help.
10222 The @kbd{h i} (@code{calc-info}) command runs the Emacs Info system
10223 to read this manual on-line. This is basically the same as typing
10224 @kbd{C-h i} (the regular way to run the Info system), then, if Info
10225 is not already in the Calc manual, selecting the beginning of the
10226 manual. The @kbd{M-# i} command is another way to read the Calc
10227 manual; it is different from @kbd{h i} in that it works any time,
10228 not just inside Calc. The plain @kbd{i} key is also equivalent to
10229 @kbd{h i}, though this key is obsolete and may be replaced with a
10230 different command in a future version of Calc.
10234 @pindex calc-tutorial
10235 The @kbd{h t} (@code{calc-tutorial}) command runs the Info system on
10236 the Tutorial section of the Calc manual. It is like @kbd{h i},
10237 except that it selects the starting node of the tutorial rather
10238 than the beginning of the whole manual. (It actually selects the
10239 node ``Interactive Tutorial'' which tells a few things about
10240 using the Info system before going on to the actual tutorial.)
10241 The @kbd{M-# t} key is equivalent to @kbd{h t} (but it works at
10246 @pindex calc-info-summary
10247 The @kbd{h s} (@code{calc-info-summary}) command runs the Info system
10248 on the Summary node of the Calc manual. @xref{Summary}. The @kbd{M-# s}
10249 key is equivalent to @kbd{h s}.
10252 @pindex calc-describe-key
10253 The @kbd{h k} (@code{calc-describe-key}) command looks up a key
10254 sequence in the Calc manual. For example, @kbd{h k H a S} looks
10255 up the documentation on the @kbd{H a S} (@code{calc-solve-for})
10256 command. This works by looking up the textual description of
10257 the key(s) in the Key Index of the manual, then jumping to the
10258 node indicated by the index.
10260 Most Calc commands do not have traditional Emacs documentation
10261 strings, since the @kbd{h k} command is both more convenient and
10262 more instructive. This means the regular Emacs @kbd{C-h k}
10263 (@code{describe-key}) command will not be useful for Calc keystrokes.
10266 @pindex calc-describe-key-briefly
10267 The @kbd{h c} (@code{calc-describe-key-briefly}) command reads a
10268 key sequence and displays a brief one-line description of it at
10269 the bottom of the screen. It looks for the key sequence in the
10270 Summary node of the Calc manual; if it doesn't find the sequence
10271 there, it acts just like its regular Emacs counterpart @kbd{C-h c}
10272 (@code{describe-key-briefly}). For example, @kbd{h c H a S}
10273 gives the description:
10276 H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes)
10280 which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for}
10281 takes a value @expr{a} from the stack, prompts for a value @expr{v},
10282 then applies the algebraic function @code{fsolve} to these values.
10283 The @samp{?=notes} message means you can now type @kbd{?} to see
10284 additional notes from the summary that apply to this command.
10287 @pindex calc-describe-function
10288 The @kbd{h f} (@code{calc-describe-function}) command looks up an
10289 algebraic function or a command name in the Calc manual. Enter an
10290 algebraic function name to look up that function in the Function
10291 Index or enter a command name beginning with @samp{calc-} to look it
10292 up in the Command Index. This command will also look up operator
10293 symbols that can appear in algebraic formulas, like @samp{%} and
10297 @pindex calc-describe-variable
10298 The @kbd{h v} (@code{calc-describe-variable}) command looks up a
10299 variable in the Calc manual. Enter a variable name like @code{pi} or
10300 @code{PlotRejects}.
10303 @pindex describe-bindings
10304 The @kbd{h b} (@code{calc-describe-bindings}) command is just like
10305 @kbd{C-h b}, except that only local (Calc-related) key bindings are
10309 The @kbd{h n} or @kbd{h C-n} (@code{calc-view-news}) command displays
10310 the ``news'' or change history of Calc. This is kept in the file
10311 @file{README}, which Calc looks for in the same directory as the Calc
10317 The @kbd{h C-c}, @kbd{h C-d}, and @kbd{h C-w} keys display copying,
10318 distribution, and warranty information about Calc. These work by
10319 pulling up the appropriate parts of the ``Copying'' or ``Reporting
10320 Bugs'' sections of the manual.
10322 @node Stack Basics, Numeric Entry, Help Commands, Introduction
10323 @section Stack Basics
10326 @cindex Stack basics
10327 @c [fix-tut RPN Calculations and the Stack]
10328 Calc uses RPN notation. If you are not familiar with RPN, @pxref{RPN
10331 To add the numbers 1 and 2 in Calc you would type the keys:
10332 @kbd{1 @key{RET} 2 +}.
10333 (@key{RET} corresponds to the @key{ENTER} key on most calculators.)
10334 The first three keystrokes ``push'' the numbers 1 and 2 onto the stack. The
10335 @kbd{+} key always ``pops'' the top two numbers from the stack, adds them,
10336 and pushes the result (3) back onto the stack. This number is ready for
10337 further calculations: @kbd{5 -} pushes 5 onto the stack, then pops the
10338 3 and 5, subtracts them, and pushes the result (@mathit{-2}).
10340 Note that the ``top'' of the stack actually appears at the @emph{bottom}
10341 of the buffer. A line containing a single @samp{.} character signifies
10342 the end of the buffer; Calculator commands operate on the number(s)
10343 directly above this line. The @kbd{d t} (@code{calc-truncate-stack})
10344 command allows you to move the @samp{.} marker up and down in the stack;
10345 @pxref{Truncating the Stack}.
10348 @pindex calc-line-numbering
10349 Stack elements are numbered consecutively, with number 1 being the top of
10350 the stack. These line numbers are ordinarily displayed on the lefthand side
10351 of the window. The @kbd{d l} (@code{calc-line-numbering}) command controls
10352 whether these numbers appear. (Line numbers may be turned off since they
10353 slow the Calculator down a bit and also clutter the display.)
10356 @pindex calc-realign
10357 The unshifted letter @kbd{o} (@code{calc-realign}) command repositions
10358 the cursor to its top-of-stack ``home'' position. It also undoes any
10359 horizontal scrolling in the window. If you give it a numeric prefix
10360 argument, it instead moves the cursor to the specified stack element.
10362 The @key{RET} (or equivalent @key{SPC}) key is only required to separate
10363 two consecutive numbers.
10364 (After all, if you typed @kbd{1 2} by themselves the Calculator
10365 would enter the number 12.) If you press @key{RET} or @key{SPC} @emph{not}
10366 right after typing a number, the key duplicates the number on the top of
10367 the stack. @kbd{@key{RET} *} is thus a handy way to square a number.
10369 The @key{DEL} key pops and throws away the top number on the stack.
10370 The @key{TAB} key swaps the top two objects on the stack.
10371 @xref{Stack and Trail}, for descriptions of these and other stack-related
10374 @node Numeric Entry, Algebraic Entry, Stack Basics, Introduction
10375 @section Numeric Entry
10381 @cindex Numeric entry
10382 @cindex Entering numbers
10383 Pressing a digit or other numeric key begins numeric entry using the
10384 minibuffer. The number is pushed on the stack when you press the @key{RET}
10385 or @key{SPC} keys. If you press any other non-numeric key, the number is
10386 pushed onto the stack and the appropriate operation is performed. If
10387 you press a numeric key which is not valid, the key is ignored.
10389 @cindex Minus signs
10390 @cindex Negative numbers, entering
10392 There are three different concepts corresponding to the word ``minus,''
10393 typified by @expr{a-b} (subtraction), @expr{-x}
10394 (change-sign), and @expr{-5} (negative number). Calc uses three
10395 different keys for these operations, respectively:
10396 @kbd{-}, @kbd{n}, and @kbd{_} (the underscore). The @kbd{-} key subtracts
10397 the two numbers on the top of the stack. The @kbd{n} key changes the sign
10398 of the number on the top of the stack or the number currently being entered.
10399 The @kbd{_} key begins entry of a negative number or changes the sign of
10400 the number currently being entered. The following sequences all enter the
10401 number @mathit{-5} onto the stack: @kbd{0 @key{RET} 5 -}, @kbd{5 n @key{RET}},
10402 @kbd{5 @key{RET} n}, @kbd{_ 5 @key{RET}}, @kbd{5 _ @key{RET}}.
10404 Some other keys are active during numeric entry, such as @kbd{#} for
10405 non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms.
10406 These notations are described later in this manual with the corresponding
10407 data types. @xref{Data Types}.
10409 During numeric entry, the only editing key available is @key{DEL}.
10411 @node Algebraic Entry, Quick Calculator, Numeric Entry, Introduction
10412 @section Algebraic Entry
10416 @pindex calc-algebraic-entry
10417 @cindex Algebraic notation
10418 @cindex Formulas, entering
10419 Calculations can also be entered in algebraic form. This is accomplished
10420 by typing the apostrophe key, @kbd{'}, followed by the expression in
10421 standard format: @kbd{@key{'} 2+3*4 @key{RET}} computes
10422 @texline @math{2+(3\times4) = 14}
10423 @infoline @expr{2+(3*4) = 14}
10424 and pushes that on the stack. If you wish you can
10425 ignore the RPN aspect of Calc altogether and simply enter algebraic
10426 expressions in this way. You may want to use @key{DEL} every so often to
10427 clear previous results off the stack.
10429 You can press the apostrophe key during normal numeric entry to switch
10430 the half-entered number into Algebraic entry mode. One reason to do this
10431 would be to use the full Emacs cursor motion and editing keys, which are
10432 available during algebraic entry but not during numeric entry.
10434 In the same vein, during either numeric or algebraic entry you can
10435 press @kbd{`} (backquote) to switch to @code{calc-edit} mode, where
10436 you complete your half-finished entry in a separate buffer.
10437 @xref{Editing Stack Entries}.
10440 @pindex calc-algebraic-mode
10441 @cindex Algebraic Mode
10442 If you prefer algebraic entry, you can use the command @kbd{m a}
10443 (@code{calc-algebraic-mode}) to set Algebraic mode. In this mode,
10444 digits and other keys that would normally start numeric entry instead
10445 start full algebraic entry; as long as your formula begins with a digit
10446 you can omit the apostrophe. Open parentheses and square brackets also
10447 begin algebraic entry. You can still do RPN calculations in this mode,
10448 but you will have to press @key{RET} to terminate every number:
10449 @kbd{2 @key{RET} 3 @key{RET} * 4 @key{RET} +} would accomplish the same
10450 thing as @kbd{2*3+4 @key{RET}}.
10452 @cindex Incomplete Algebraic Mode
10453 If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a}
10454 command, it enables Incomplete Algebraic mode; this is like regular
10455 Algebraic mode except that it applies to the @kbd{(} and @kbd{[} keys
10456 only. Numeric keys still begin a numeric entry in this mode.
10459 @pindex calc-total-algebraic-mode
10460 @cindex Total Algebraic Mode
10461 The @kbd{m t} (@code{calc-total-algebraic-mode}) gives you an even
10462 stronger algebraic-entry mode, in which @emph{all} regular letter and
10463 punctuation keys begin algebraic entry. Use this if you prefer typing
10464 @w{@kbd{sqrt( )}} instead of @kbd{Q}, @w{@kbd{factor( )}} instead of
10465 @kbd{a f}, and so on. To type regular Calc commands when you are in
10466 Total Algebraic mode, hold down the @key{META} key. Thus @kbd{M-q}
10467 is the command to quit Calc, @kbd{M-p} sets the precision, and
10468 @kbd{M-m t} (or @kbd{M-m M-t}, if you prefer) turns Total Algebraic
10469 mode back off again. Meta keys also terminate algebraic entry, so
10470 that @kbd{2+3 M-S} is equivalent to @kbd{2+3 @key{RET} M-S}. The symbol
10471 @samp{Alg*} will appear in the mode line whenever you are in this mode.
10473 Pressing @kbd{'} (the apostrophe) a second time re-enters the previous
10474 algebraic formula. You can then use the normal Emacs editing keys to
10475 modify this formula to your liking before pressing @key{RET}.
10478 @cindex Formulas, referring to stack
10479 Within a formula entered from the keyboard, the symbol @kbd{$}
10480 represents the number on the top of the stack. If an entered formula
10481 contains any @kbd{$} characters, the Calculator replaces the top of
10482 stack with that formula rather than simply pushing the formula onto the
10483 stack. Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{$*2
10484 @key{RET}} replaces it with 6. Note that the @kbd{$} key always
10485 initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{$} is the
10486 first character in the new formula.
10488 Higher stack elements can be accessed from an entered formula with the
10489 symbols @kbd{$$}, @kbd{$$$}, and so on. The number of stack elements
10490 removed (to be replaced by the entered values) equals the number of dollar
10491 signs in the longest such symbol in the formula. For example, @samp{$$+$$$}
10492 adds the second and third stack elements, replacing the top three elements
10493 with the answer. (All information about the top stack element is thus lost
10494 since no single @samp{$} appears in this formula.)
10496 A slightly different way to refer to stack elements is with a dollar
10497 sign followed by a number: @samp{$1}, @samp{$2}, and so on are much
10498 like @samp{$}, @samp{$$}, etc., except that stack entries referred
10499 to numerically are not replaced by the algebraic entry. That is, while
10500 @samp{$+1} replaces 5 on the stack with 6, @samp{$1+1} leaves the 5
10501 on the stack and pushes an additional 6.
10503 If a sequence of formulas are entered separated by commas, each formula
10504 is pushed onto the stack in turn. For example, @samp{1,2,3} pushes
10505 those three numbers onto the stack (leaving the 3 at the top), and
10506 @samp{$+1,$-1} replaces a 5 on the stack with 4 followed by 6. Also,
10507 @samp{$,$$} exchanges the top two elements of the stack, just like the
10510 You can finish an algebraic entry with @kbd{M-=} or @kbd{M-@key{RET}} instead
10511 of @key{RET}. This uses @kbd{=} to evaluate the variables in each
10512 formula that goes onto the stack. (Thus @kbd{' pi @key{RET}} pushes
10513 the variable @samp{pi}, but @kbd{' pi M-@key{RET}} pushes 3.1415.)
10515 If you finish your algebraic entry by pressing @key{LFD} (or @kbd{C-j})
10516 instead of @key{RET}, Calc disables the default simplifications
10517 (as if by @kbd{m O}; @pxref{Simplification Modes}) while the entry
10518 is being pushed on the stack. Thus @kbd{' 1+2 @key{RET}} pushes 3
10519 on the stack, but @kbd{' 1+2 @key{LFD}} pushes the formula @expr{1+2};
10520 you might then press @kbd{=} when it is time to evaluate this formula.
10522 @node Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction
10523 @section ``Quick Calculator'' Mode
10528 @cindex Quick Calculator
10529 There is another way to invoke the Calculator if all you need to do
10530 is make one or two quick calculations. Type @kbd{M-# q} (or
10531 @kbd{M-x quick-calc}), then type any formula as an algebraic entry.
10532 The Calculator will compute the result and display it in the echo
10533 area, without ever actually putting up a Calc window.
10535 You can use the @kbd{$} character in a Quick Calculator formula to
10536 refer to the previous Quick Calculator result. Older results are
10537 not retained; the Quick Calculator has no effect on the full
10538 Calculator's stack or trail. If you compute a result and then
10539 forget what it was, just run @code{M-# q} again and enter
10540 @samp{$} as the formula.
10542 If this is the first time you have used the Calculator in this Emacs
10543 session, the @kbd{M-# q} command will create the @code{*Calculator*}
10544 buffer and perform all the usual initializations; it simply will
10545 refrain from putting that buffer up in a new window. The Quick
10546 Calculator refers to the @code{*Calculator*} buffer for all mode
10547 settings. Thus, for example, to set the precision that the Quick
10548 Calculator uses, simply run the full Calculator momentarily and use
10549 the regular @kbd{p} command.
10551 If you use @code{M-# q} from inside the Calculator buffer, the
10552 effect is the same as pressing the apostrophe key (algebraic entry).
10554 The result of a Quick calculation is placed in the Emacs ``kill ring''
10555 as well as being displayed. A subsequent @kbd{C-y} command will
10556 yank the result into the editing buffer. You can also use this
10557 to yank the result into the next @kbd{M-# q} input line as a more
10558 explicit alternative to @kbd{$} notation, or to yank the result
10559 into the Calculator stack after typing @kbd{M-# c}.
10561 If you finish your formula by typing @key{LFD} (or @kbd{C-j}) instead
10562 of @key{RET}, the result is inserted immediately into the current
10563 buffer rather than going into the kill ring.
10565 Quick Calculator results are actually evaluated as if by the @kbd{=}
10566 key (which replaces variable names by their stored values, if any).
10567 If the formula you enter is an assignment to a variable using the
10568 @samp{:=} operator, say, @samp{foo := 2 + 3} or @samp{foo := foo + 1},
10569 then the result of the evaluation is stored in that Calc variable.
10570 @xref{Store and Recall}.
10572 If the result is an integer and the current display radix is decimal,
10573 the number will also be displayed in hex and octal formats. If the
10574 integer is in the range from 1 to 126, it will also be displayed as
10575 an ASCII character.
10577 For example, the quoted character @samp{"x"} produces the vector
10578 result @samp{[120]} (because 120 is the ASCII code of the lower-case
10579 `x'; @pxref{Strings}). Since this is a vector, not an integer, it
10580 is displayed only according to the current mode settings. But
10581 running Quick Calc again and entering @samp{120} will produce the
10582 result @samp{120 (16#78, 8#170, x)} which shows the number in its
10583 decimal, hexadecimal, octal, and ASCII forms.
10585 Please note that the Quick Calculator is not any faster at loading
10586 or computing the answer than the full Calculator; the name ``quick''
10587 merely refers to the fact that it's much less hassle to use for
10588 small calculations.
10590 @node Prefix Arguments, Undo, Quick Calculator, Introduction
10591 @section Numeric Prefix Arguments
10594 Many Calculator commands use numeric prefix arguments. Some, such as
10595 @kbd{d s} (@code{calc-sci-notation}), set a parameter to the value of
10596 the prefix argument or use a default if you don't use a prefix.
10597 Others (like @kbd{d f} (@code{calc-fix-notation})) require an argument
10598 and prompt for a number if you don't give one as a prefix.
10600 As a rule, stack-manipulation commands accept a numeric prefix argument
10601 which is interpreted as an index into the stack. A positive argument
10602 operates on the top @var{n} stack entries; a negative argument operates
10603 on the @var{n}th stack entry in isolation; and a zero argument operates
10604 on the entire stack.
10606 Most commands that perform computations (such as the arithmetic and
10607 scientific functions) accept a numeric prefix argument that allows the
10608 operation to be applied across many stack elements. For unary operations
10609 (that is, functions of one argument like absolute value or complex
10610 conjugate), a positive prefix argument applies that function to the top
10611 @var{n} stack entries simultaneously, and a negative argument applies it
10612 to the @var{n}th stack entry only. For binary operations (functions of
10613 two arguments like addition, GCD, and vector concatenation), a positive
10614 prefix argument ``reduces'' the function across the top @var{n}
10615 stack elements (for example, @kbd{C-u 5 +} sums the top 5 stack entries;
10616 @pxref{Reducing and Mapping}), and a negative argument maps the next-to-top
10617 @var{n} stack elements with the top stack element as a second argument
10618 (for example, @kbd{7 c-u -5 +} adds 7 to the top 5 stack elements).
10619 This feature is not available for operations which use the numeric prefix
10620 argument for some other purpose.
10622 Numeric prefixes are specified the same way as always in Emacs: Press
10623 a sequence of @key{META}-digits, or press @key{ESC} followed by digits,
10624 or press @kbd{C-u} followed by digits. Some commands treat plain
10625 @kbd{C-u} (without any actual digits) specially.
10628 @pindex calc-num-prefix
10629 You can type @kbd{~} (@code{calc-num-prefix}) to pop an integer from the
10630 top of the stack and enter it as the numeric prefix for the next command.
10631 For example, @kbd{C-u 16 p} sets the precision to 16 digits; an alternate
10632 (silly) way to do this would be @kbd{2 @key{RET} 4 ^ ~ p}, i.e., compute 2
10633 to the fourth power and set the precision to that value.
10635 Conversely, if you have typed a numeric prefix argument the @kbd{~} key
10636 pushes it onto the stack in the form of an integer.
10638 @node Undo, Error Messages, Prefix Arguments, Introduction
10639 @section Undoing Mistakes
10645 @cindex Mistakes, undoing
10646 @cindex Undoing mistakes
10647 @cindex Errors, undoing
10648 The shift-@kbd{U} key (@code{calc-undo}) undoes the most recent operation.
10649 If that operation added or dropped objects from the stack, those objects
10650 are removed or restored. If it was a ``store'' operation, you are
10651 queried whether or not to restore the variable to its original value.
10652 The @kbd{U} key may be pressed any number of times to undo successively
10653 farther back in time; with a numeric prefix argument it undoes a
10654 specified number of operations. The undo history is cleared only by the
10655 @kbd{q} (@code{calc-quit}) command. (Recall that @kbd{M-# c} is
10656 synonymous with @code{calc-quit} while inside the Calculator; this
10657 also clears the undo history.)
10659 Currently the mode-setting commands (like @code{calc-precision}) are not
10660 undoable. You can undo past a point where you changed a mode, but you
10661 will need to reset the mode yourself.
10665 @cindex Redoing after an Undo
10666 The shift-@kbd{D} key (@code{calc-redo}) redoes an operation that was
10667 mistakenly undone. Pressing @kbd{U} with a negative prefix argument is
10668 equivalent to executing @code{calc-redo}. You can redo any number of
10669 times, up to the number of recent consecutive undo commands. Redo
10670 information is cleared whenever you give any command that adds new undo
10671 information, i.e., if you undo, then enter a number on the stack or make
10672 any other change, then it will be too late to redo.
10674 @kindex M-@key{RET}
10675 @pindex calc-last-args
10676 @cindex Last-arguments feature
10677 @cindex Arguments, restoring
10678 The @kbd{M-@key{RET}} key (@code{calc-last-args}) is like undo in that
10679 it restores the arguments of the most recent command onto the stack;
10680 however, it does not remove the result of that command. Given a numeric
10681 prefix argument, this command applies to the @expr{n}th most recent
10682 command which removed items from the stack; it pushes those items back
10685 The @kbd{K} (@code{calc-keep-args}) command provides a related function
10686 to @kbd{M-@key{RET}}. @xref{Stack and Trail}.
10688 It is also possible to recall previous results or inputs using the trail.
10689 @xref{Trail Commands}.
10691 The standard Emacs @kbd{C-_} undo key is recognized as a synonym for @kbd{U}.
10693 @node Error Messages, Multiple Calculators, Undo, Introduction
10694 @section Error Messages
10699 @cindex Errors, messages
10700 @cindex Why did an error occur?
10701 Many situations that would produce an error message in other calculators
10702 simply create unsimplified formulas in the Emacs Calculator. For example,
10703 @kbd{1 @key{RET} 0 /} pushes the formula @expr{1 / 0}; @w{@kbd{0 L}} pushes
10704 the formula @samp{ln(0)}. Floating-point overflow and underflow are also
10705 reasons for this to happen.
10707 When a function call must be left in symbolic form, Calc usually
10708 produces a message explaining why. Messages that are probably
10709 surprising or indicative of user errors are displayed automatically.
10710 Other messages are simply kept in Calc's memory and are displayed only
10711 if you type @kbd{w} (@code{calc-why}). You can also press @kbd{w} if
10712 the same computation results in several messages. (The first message
10713 will end with @samp{[w=more]} in this case.)
10716 @pindex calc-auto-why
10717 The @kbd{d w} (@code{calc-auto-why}) command controls when error messages
10718 are displayed automatically. (Calc effectively presses @kbd{w} for you
10719 after your computation finishes.) By default, this occurs only for
10720 ``important'' messages. The other possible modes are to report
10721 @emph{all} messages automatically, or to report none automatically (so
10722 that you must always press @kbd{w} yourself to see the messages).
10724 @node Multiple Calculators, Troubleshooting Commands, Error Messages, Introduction
10725 @section Multiple Calculators
10728 @pindex another-calc
10729 It is possible to have any number of Calc mode buffers at once.
10730 Usually this is done by executing @kbd{M-x another-calc}, which
10731 is similar to @kbd{M-# c} except that if a @samp{*Calculator*}
10732 buffer already exists, a new, independent one with a name of the
10733 form @samp{*Calculator*<@var{n}>} is created. You can also use the
10734 command @code{calc-mode} to put any buffer into Calculator mode, but
10735 this would ordinarily never be done.
10737 The @kbd{q} (@code{calc-quit}) command does not destroy a Calculator buffer;
10738 it only closes its window. Use @kbd{M-x kill-buffer} to destroy a
10741 Each Calculator buffer keeps its own stack, undo list, and mode settings
10742 such as precision, angular mode, and display formats. In Emacs terms,
10743 variables such as @code{calc-stack} are buffer-local variables. The
10744 global default values of these variables are used only when a new
10745 Calculator buffer is created. The @code{calc-quit} command saves
10746 the stack and mode settings of the buffer being quit as the new defaults.
10748 There is only one trail buffer, @samp{*Calc Trail*}, used by all
10749 Calculator buffers.
10751 @node Troubleshooting Commands, , Multiple Calculators, Introduction
10752 @section Troubleshooting Commands
10755 This section describes commands you can use in case a computation
10756 incorrectly fails or gives the wrong answer.
10758 @xref{Reporting Bugs}, if you find a problem that appears to be due
10759 to a bug or deficiency in Calc.
10762 * Autoloading Problems::
10763 * Recursion Depth::
10768 @node Autoloading Problems, Recursion Depth, Troubleshooting Commands, Troubleshooting Commands
10769 @subsection Autoloading Problems
10772 The Calc program is split into many component files; components are
10773 loaded automatically as you use various commands that require them.
10774 Occasionally Calc may lose track of when a certain component is
10775 necessary; typically this means you will type a command and it won't
10776 work because some function you've never heard of was undefined.
10779 @pindex calc-load-everything
10780 If this happens, the easiest workaround is to type @kbd{M-# L}
10781 (@code{calc-load-everything}) to force all the parts of Calc to be
10782 loaded right away. This will cause Emacs to take up a lot more
10783 memory than it would otherwise, but it's guaranteed to fix the problem.
10785 @node Recursion Depth, Caches, Autoloading Problems, Troubleshooting Commands
10786 @subsection Recursion Depth
10791 @pindex calc-more-recursion-depth
10792 @pindex calc-less-recursion-depth
10793 @cindex Recursion depth
10794 @cindex ``Computation got stuck'' message
10795 @cindex @code{max-lisp-eval-depth}
10796 @cindex @code{max-specpdl-size}
10797 Calc uses recursion in many of its calculations. Emacs Lisp keeps a
10798 variable @code{max-lisp-eval-depth} which limits the amount of recursion
10799 possible in an attempt to recover from program bugs. If a calculation
10800 ever halts incorrectly with the message ``Computation got stuck or
10801 ran too long,'' use the @kbd{M} command (@code{calc-more-recursion-depth})
10802 to increase this limit. (Of course, this will not help if the
10803 calculation really did get stuck due to some problem inside Calc.)
10805 The limit is always increased (multiplied) by a factor of two. There
10806 is also an @kbd{I M} (@code{calc-less-recursion-depth}) command which
10807 decreases this limit by a factor of two, down to a minimum value of 200.
10808 The default value is 1000.
10810 These commands also double or halve @code{max-specpdl-size}, another
10811 internal Lisp recursion limit. The minimum value for this limit is 600.
10813 @node Caches, Debugging Calc, Recursion Depth, Troubleshooting Commands
10818 @cindex Flushing caches
10819 Calc saves certain values after they have been computed once. For
10820 example, the @kbd{P} (@code{calc-pi}) command initially ``knows'' the
10821 constant @cpi{} to about 20 decimal places; if the current precision
10822 is greater than this, it will recompute @cpi{} using a series
10823 approximation. This value will not need to be recomputed ever again
10824 unless you raise the precision still further. Many operations such as
10825 logarithms and sines make use of similarly cached values such as
10827 @texline @math{\ln 2}.
10828 @infoline @expr{ln(2)}.
10829 The visible effect of caching is that
10830 high-precision computations may seem to do extra work the first time.
10831 Other things cached include powers of two (for the binary arithmetic
10832 functions), matrix inverses and determinants, symbolic integrals, and
10833 data points computed by the graphing commands.
10835 @pindex calc-flush-caches
10836 If you suspect a Calculator cache has become corrupt, you can use the
10837 @code{calc-flush-caches} command to reset all caches to the empty state.
10838 (This should only be necessary in the event of bugs in the Calculator.)
10839 The @kbd{M-# 0} (with the zero key) command also resets caches along
10840 with all other aspects of the Calculator's state.
10842 @node Debugging Calc, , Caches, Troubleshooting Commands
10843 @subsection Debugging Calc
10846 A few commands exist to help in the debugging of Calc commands.
10847 @xref{Programming}, to see the various ways that you can write
10848 your own Calc commands.
10851 @pindex calc-timing
10852 The @kbd{Z T} (@code{calc-timing}) command turns on and off a mode
10853 in which the timing of slow commands is reported in the Trail.
10854 Any Calc command that takes two seconds or longer writes a line
10855 to the Trail showing how many seconds it took. This value is
10856 accurate only to within one second.
10858 All steps of executing a command are included; in particular, time
10859 taken to format the result for display in the stack and trail is
10860 counted. Some prompts also count time taken waiting for them to
10861 be answered, while others do not; this depends on the exact
10862 implementation of the command. For best results, if you are timing
10863 a sequence that includes prompts or multiple commands, define a
10864 keyboard macro to run the whole sequence at once. Calc's @kbd{X}
10865 command (@pxref{Keyboard Macros}) will then report the time taken
10866 to execute the whole macro.
10868 Another advantage of the @kbd{X} command is that while it is
10869 executing, the stack and trail are not updated from step to step.
10870 So if you expect the output of your test sequence to leave a result
10871 that may take a long time to format and you don't wish to count
10872 this formatting time, end your sequence with a @key{DEL} keystroke
10873 to clear the result from the stack. When you run the sequence with
10874 @kbd{X}, Calc will never bother to format the large result.
10876 Another thing @kbd{Z T} does is to increase the Emacs variable
10877 @code{gc-cons-threshold} to a much higher value (two million; the
10878 usual default in Calc is 250,000) for the duration of each command.
10879 This generally prevents garbage collection during the timing of
10880 the command, though it may cause your Emacs process to grow
10881 abnormally large. (Garbage collection time is a major unpredictable
10882 factor in the timing of Emacs operations.)
10884 Another command that is useful when debugging your own Lisp
10885 extensions to Calc is @kbd{M-x calc-pass-errors}, which disables
10886 the error handler that changes the ``@code{max-lisp-eval-depth}
10887 exceeded'' message to the much more friendly ``Computation got
10888 stuck or ran too long.'' This handler interferes with the Emacs
10889 Lisp debugger's @code{debug-on-error} mode. Errors are reported
10890 in the handler itself rather than at the true location of the
10891 error. After you have executed @code{calc-pass-errors}, Lisp
10892 errors will be reported correctly but the user-friendly message
10895 @node Data Types, Stack and Trail, Introduction, Top
10896 @chapter Data Types
10899 This chapter discusses the various types of objects that can be placed
10900 on the Calculator stack, how they are displayed, and how they are
10901 entered. (@xref{Data Type Formats}, for information on how these data
10902 types are represented as underlying Lisp objects.)
10904 Integers, fractions, and floats are various ways of describing real
10905 numbers. HMS forms also for many purposes act as real numbers. These
10906 types can be combined to form complex numbers, modulo forms, error forms,
10907 or interval forms. (But these last four types cannot be combined
10908 arbitrarily:@: error forms may not contain modulo forms, for example.)
10909 Finally, all these types of numbers may be combined into vectors,
10910 matrices, or algebraic formulas.
10913 * Integers:: The most basic data type.
10914 * Fractions:: This and above are called @dfn{rationals}.
10915 * Floats:: This and above are called @dfn{reals}.
10916 * Complex Numbers:: This and above are called @dfn{numbers}.
10918 * Vectors and Matrices::
10925 * Incomplete Objects::
10930 @node Integers, Fractions, Data Types, Data Types
10935 The Calculator stores integers to arbitrary precision. Addition,
10936 subtraction, and multiplication of integers always yields an exact
10937 integer result. (If the result of a division or exponentiation of
10938 integers is not an integer, it is expressed in fractional or
10939 floating-point form according to the current Fraction mode.
10940 @xref{Fraction Mode}.)
10942 A decimal integer is represented as an optional sign followed by a
10943 sequence of digits. Grouping (@pxref{Grouping Digits}) can be used to
10944 insert a comma at every third digit for display purposes, but you
10945 must not type commas during the entry of numbers.
10948 A non-decimal integer is represented as an optional sign, a radix
10949 between 2 and 36, a @samp{#} symbol, and one or more digits. For radix 11
10950 and above, the letters A through Z (upper- or lower-case) count as
10951 digits and do not terminate numeric entry mode. @xref{Radix Modes}, for how
10952 to set the default radix for display of integers. Numbers of any radix
10953 may be entered at any time. If you press @kbd{#} at the beginning of a
10954 number, the current display radix is used.
10956 @node Fractions, Floats, Integers, Data Types
10961 A @dfn{fraction} is a ratio of two integers. Fractions are traditionally
10962 written ``2/3'' but Calc uses the notation @samp{2:3}. (The @kbd{/} key
10963 performs RPN division; the following two sequences push the number
10964 @samp{2:3} on the stack: @kbd{2 :@: 3 @key{RET}}, or @kbd{2 @key{RET} 3 /}
10965 assuming Fraction mode has been enabled.)
10966 When the Calculator produces a fractional result it always reduces it to
10967 simplest form, which may in fact be an integer.
10969 Fractions may also be entered in a three-part form, where @samp{2:3:4}
10970 represents two-and-three-quarters. @xref{Fraction Formats}, for fraction
10973 Non-decimal fractions are entered and displayed as
10974 @samp{@var{radix}#@var{num}:@var{denom}} (or in the analogous three-part
10975 form). The numerator and denominator always use the same radix.
10977 @node Floats, Complex Numbers, Fractions, Data Types
10981 @cindex Floating-point numbers
10982 A floating-point number or @dfn{float} is a number stored in scientific
10983 notation. The number of significant digits in the fractional part is
10984 governed by the current floating precision (@pxref{Precision}). The
10985 range of acceptable values is from
10986 @texline @math{10^{-3999999}}
10987 @infoline @expr{10^-3999999}
10989 @texline @math{10^{4000000}}
10990 @infoline @expr{10^4000000}
10991 (exclusive), plus the corresponding negative values and zero.
10993 Calculations that would exceed the allowable range of values (such
10994 as @samp{exp(exp(20))}) are left in symbolic form by Calc. The
10995 messages ``floating-point overflow'' or ``floating-point underflow''
10996 indicate that during the calculation a number would have been produced
10997 that was too large or too close to zero, respectively, to be represented
10998 by Calc. This does not necessarily mean the final result would have
10999 overflowed, just that an overflow occurred while computing the result.
11000 (In fact, it could report an underflow even though the final result
11001 would have overflowed!)
11003 If a rational number and a float are mixed in a calculation, the result
11004 will in general be expressed as a float. Commands that require an integer
11005 value (such as @kbd{k g} [@code{gcd}]) will also accept integer-valued
11006 floats, i.e., floating-point numbers with nothing after the decimal point.
11008 Floats are identified by the presence of a decimal point and/or an
11009 exponent. In general a float consists of an optional sign, digits
11010 including an optional decimal point, and an optional exponent consisting
11011 of an @samp{e}, an optional sign, and up to seven exponent digits.
11012 For example, @samp{23.5e-2} is 23.5 times ten to the minus-second power,
11015 Floating-point numbers are normally displayed in decimal notation with
11016 all significant figures shown. Exceedingly large or small numbers are
11017 displayed in scientific notation. Various other display options are
11018 available. @xref{Float Formats}.
11020 @cindex Accuracy of calculations
11021 Floating-point numbers are stored in decimal, not binary. The result
11022 of each operation is rounded to the nearest value representable in the
11023 number of significant digits specified by the current precision,
11024 rounding away from zero in the case of a tie. Thus (in the default
11025 display mode) what you see is exactly what you get. Some operations such
11026 as square roots and transcendental functions are performed with several
11027 digits of extra precision and then rounded down, in an effort to make the
11028 final result accurate to the full requested precision. However,
11029 accuracy is not rigorously guaranteed. If you suspect the validity of a
11030 result, try doing the same calculation in a higher precision. The
11031 Calculator's arithmetic is not intended to be IEEE-conformant in any
11034 While floats are always @emph{stored} in decimal, they can be entered
11035 and displayed in any radix just like integers and fractions. The
11036 notation @samp{@var{radix}#@var{ddd}.@var{ddd}} is a floating-point
11037 number whose digits are in the specified radix. Note that the @samp{.}
11038 is more aptly referred to as a ``radix point'' than as a decimal
11039 point in this case. The number @samp{8#123.4567} is defined as
11040 @samp{8#1234567 * 8^-4}. If the radix is 14 or less, you can use
11041 @samp{e} notation to write a non-decimal number in scientific notation.
11042 The exponent is written in decimal, and is considered to be a power
11043 of the radix: @samp{8#1234567e-4}. If the radix is 15 or above, the
11044 letter @samp{e} is a digit, so scientific notation must be written
11045 out, e.g., @samp{16#123.4567*16^2}. The first two exercises of the
11046 Modes Tutorial explore some of the properties of non-decimal floats.
11048 @node Complex Numbers, Infinities, Floats, Data Types
11049 @section Complex Numbers
11052 @cindex Complex numbers
11053 There are two supported formats for complex numbers: rectangular and
11054 polar. The default format is rectangular, displayed in the form
11055 @samp{(@var{real},@var{imag})} where @var{real} is the real part and
11056 @var{imag} is the imaginary part, each of which may be any real number.
11057 Rectangular complex numbers can also be displayed in @samp{@var{a}+@var{b}i}
11058 notation; @pxref{Complex Formats}.
11060 Polar complex numbers are displayed in the form
11061 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'
11062 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'
11063 where @var{r} is the nonnegative magnitude and
11064 @texline @math{\theta}
11065 @infoline @var{theta}
11066 is the argument or phase angle. The range of
11067 @texline @math{\theta}
11068 @infoline @var{theta}
11069 depends on the current angular mode (@pxref{Angular Modes}); it is
11070 generally between @mathit{-180} and @mathit{+180} degrees or the equivalent range
11073 Complex numbers are entered in stages using incomplete objects.
11074 @xref{Incomplete Objects}.
11076 Operations on rectangular complex numbers yield rectangular complex
11077 results, and similarly for polar complex numbers. Where the two types
11078 are mixed, or where new complex numbers arise (as for the square root of
11079 a negative real), the current @dfn{Polar mode} is used to determine the
11080 type. @xref{Polar Mode}.
11082 A complex result in which the imaginary part is zero (or the phase angle
11083 is 0 or 180 degrees or @cpi{} radians) is automatically converted to a real
11086 @node Infinities, Vectors and Matrices, Complex Numbers, Data Types
11087 @section Infinities
11091 @cindex @code{inf} variable
11092 @cindex @code{uinf} variable
11093 @cindex @code{nan} variable
11097 The word @code{inf} represents the mathematical concept of @dfn{infinity}.
11098 Calc actually has three slightly different infinity-like values:
11099 @code{inf}, @code{uinf}, and @code{nan}. These are just regular
11100 variable names (@pxref{Variables}); you should avoid using these
11101 names for your own variables because Calc gives them special
11102 treatment. Infinities, like all variable names, are normally
11103 entered using algebraic entry.
11105 Mathematically speaking, it is not rigorously correct to treat
11106 ``infinity'' as if it were a number, but mathematicians often do
11107 so informally. When they say that @samp{1 / inf = 0}, what they
11108 really mean is that @expr{1 / x}, as @expr{x} becomes larger and
11109 larger, becomes arbitrarily close to zero. So you can imagine
11110 that if @expr{x} got ``all the way to infinity,'' then @expr{1 / x}
11111 would go all the way to zero. Similarly, when they say that
11112 @samp{exp(inf) = inf}, they mean that
11113 @texline @math{e^x}
11114 @infoline @expr{exp(x)}
11115 grows without bound as @expr{x} grows. The symbol @samp{-inf} likewise
11116 stands for an infinitely negative real value; for example, we say that
11117 @samp{exp(-inf) = 0}. You can have an infinity pointing in any
11118 direction on the complex plane: @samp{sqrt(-inf) = i inf}.
11120 The same concept of limits can be used to define @expr{1 / 0}. We
11121 really want the value that @expr{1 / x} approaches as @expr{x}
11122 approaches zero. But if all we have is @expr{1 / 0}, we can't
11123 tell which direction @expr{x} was coming from. If @expr{x} was
11124 positive and decreasing toward zero, then we should say that
11125 @samp{1 / 0 = inf}. But if @expr{x} was negative and increasing
11126 toward zero, the answer is @samp{1 / 0 = -inf}. In fact, @expr{x}
11127 could be an imaginary number, giving the answer @samp{i inf} or
11128 @samp{-i inf}. Calc uses the special symbol @samp{uinf} to mean
11129 @dfn{undirected infinity}, i.e., a value which is infinitely
11130 large but with an unknown sign (or direction on the complex plane).
11132 Calc actually has three modes that say how infinities are handled.
11133 Normally, infinities never arise from calculations that didn't
11134 already have them. Thus, @expr{1 / 0} is treated simply as an
11135 error and left unevaluated. The @kbd{m i} (@code{calc-infinite-mode})
11136 command (@pxref{Infinite Mode}) enables a mode in which
11137 @expr{1 / 0} evaluates to @code{uinf} instead. There is also
11138 an alternative type of infinite mode which says to treat zeros
11139 as if they were positive, so that @samp{1 / 0 = inf}. While this
11140 is less mathematically correct, it may be the answer you want in
11143 Since all infinities are ``as large'' as all others, Calc simplifies,
11144 e.g., @samp{5 inf} to @samp{inf}. Another example is
11145 @samp{5 - inf = -inf}, where the @samp{-inf} is so large that
11146 adding a finite number like five to it does not affect it.
11147 Note that @samp{a - inf} also results in @samp{-inf}; Calc assumes
11148 that variables like @code{a} always stand for finite quantities.
11149 Just to show that infinities really are all the same size,
11150 note that @samp{sqrt(inf) = inf^2 = exp(inf) = inf} in Calc's
11153 It's not so easy to define certain formulas like @samp{0 * inf} and
11154 @samp{inf / inf}. Depending on where these zeros and infinities
11155 came from, the answer could be literally anything. The latter
11156 formula could be the limit of @expr{x / x} (giving a result of one),
11157 or @expr{2 x / x} (giving two), or @expr{x^2 / x} (giving @code{inf}),
11158 or @expr{x / x^2} (giving zero). Calc uses the symbol @code{nan}
11159 to represent such an @dfn{indeterminate} value. (The name ``nan''
11160 comes from analogy with the ``NAN'' concept of IEEE standard
11161 arithmetic; it stands for ``Not A Number.'' This is somewhat of a
11162 misnomer, since @code{nan} @emph{does} stand for some number or
11163 infinity, it's just that @emph{which} number it stands for
11164 cannot be determined.) In Calc's notation, @samp{0 * inf = nan}
11165 and @samp{inf / inf = nan}. A few other common indeterminate
11166 expressions are @samp{inf - inf} and @samp{inf ^ 0}. Also,
11167 @samp{0 / 0 = nan} if you have turned on Infinite mode
11168 (as described above).
11170 Infinities are especially useful as parts of @dfn{intervals}.
11171 @xref{Interval Forms}.
11173 @node Vectors and Matrices, Strings, Infinities, Data Types
11174 @section Vectors and Matrices
11178 @cindex Plain vectors
11180 The @dfn{vector} data type is flexible and general. A vector is simply a
11181 list of zero or more data objects. When these objects are numbers, the
11182 whole is a vector in the mathematical sense. When these objects are
11183 themselves vectors of equal (nonzero) length, the whole is a @dfn{matrix}.
11184 A vector which is not a matrix is referred to here as a @dfn{plain vector}.
11186 A vector is displayed as a list of values separated by commas and enclosed
11187 in square brackets: @samp{[1, 2, 3]}. Thus the following is a 2 row by
11188 3 column matrix: @samp{[[1, 2, 3], [4, 5, 6]]}. Vectors, like complex
11189 numbers, are entered as incomplete objects. @xref{Incomplete Objects}.
11190 During algebraic entry, vectors are entered all at once in the usual
11191 brackets-and-commas form. Matrices may be entered algebraically as nested
11192 vectors, or using the shortcut notation @w{@samp{[1, 2, 3; 4, 5, 6]}},
11193 with rows separated by semicolons. The commas may usually be omitted
11194 when entering vectors: @samp{[1 2 3]}. Curly braces may be used in
11195 place of brackets: @samp{@{1, 2, 3@}}, but the commas are required in
11198 Traditional vector and matrix arithmetic is also supported;
11199 @pxref{Basic Arithmetic} and @pxref{Matrix Functions}.
11200 Many other operations are applied to vectors element-wise. For example,
11201 the complex conjugate of a vector is a vector of the complex conjugates
11208 Algebraic functions for building vectors include @samp{vec(a, b, c)}
11209 to build @samp{[a, b, c]}, @samp{cvec(a, n, m)} to build an
11210 @texline @math{n\times m}
11211 @infoline @var{n}x@var{m}
11212 matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers
11213 from 1 to @samp{n}.
11215 @node Strings, HMS Forms, Vectors and Matrices, Data Types
11221 @cindex Character strings
11222 Character strings are not a special data type in the Calculator.
11223 Rather, a string is represented simply as a vector all of whose
11224 elements are integers in the range 0 to 255 (ASCII codes). You can
11225 enter a string at any time by pressing the @kbd{"} key. Quotation
11226 marks and backslashes are written @samp{\"} and @samp{\\}, respectively,
11227 inside strings. Other notations introduced by backslashes are:
11243 Finally, a backslash followed by three octal digits produces any
11244 character from its ASCII code.
11247 @pindex calc-display-strings
11248 Strings are normally displayed in vector-of-integers form. The
11249 @w{@kbd{d "}} (@code{calc-display-strings}) command toggles a mode in
11250 which any vectors of small integers are displayed as quoted strings
11253 The backslash notations shown above are also used for displaying
11254 strings. Characters 128 and above are not translated by Calc; unless
11255 you have an Emacs modified for 8-bit fonts, these will show up in
11256 backslash-octal-digits notation. For characters below 32, and
11257 for character 127, Calc uses the backslash-letter combination if
11258 there is one, or otherwise uses a @samp{\^} sequence.
11260 The only Calc feature that uses strings is @dfn{compositions};
11261 @pxref{Compositions}. Strings also provide a convenient
11262 way to do conversions between ASCII characters and integers.
11268 There is a @code{string} function which provides a different display
11269 format for strings. Basically, @samp{string(@var{s})}, where @var{s}
11270 is a vector of integers in the proper range, is displayed as the
11271 corresponding string of characters with no surrounding quotation
11272 marks or other modifications. Thus @samp{string("ABC")} (or
11273 @samp{string([65 66 67])}) will look like @samp{ABC} on the stack.
11274 This happens regardless of whether @w{@kbd{d "}} has been used. The
11275 only way to turn it off is to use @kbd{d U} (unformatted language
11276 mode) which will display @samp{string("ABC")} instead.
11278 Control characters are displayed somewhat differently by @code{string}.
11279 Characters below 32, and character 127, are shown using @samp{^} notation
11280 (same as shown above, but without the backslash). The quote and
11281 backslash characters are left alone, as are characters 128 and above.
11287 The @code{bstring} function is just like @code{string} except that
11288 the resulting string is breakable across multiple lines if it doesn't
11289 fit all on one line. Potential break points occur at every space
11290 character in the string.
11292 @node HMS Forms, Date Forms, Strings, Data Types
11296 @cindex Hours-minutes-seconds forms
11297 @cindex Degrees-minutes-seconds forms
11298 @dfn{HMS} stands for Hours-Minutes-Seconds; when used as an angular
11299 argument, the interpretation is Degrees-Minutes-Seconds. All functions
11300 that operate on angles accept HMS forms. These are interpreted as
11301 degrees regardless of the current angular mode. It is also possible to
11302 use HMS as the angular mode so that calculated angles are expressed in
11303 degrees, minutes, and seconds.
11309 @kindex ' (HMS forms)
11313 @kindex " (HMS forms)
11317 @kindex h (HMS forms)
11321 @kindex o (HMS forms)
11325 @kindex m (HMS forms)
11329 @kindex s (HMS forms)
11330 The default format for HMS values is
11331 @samp{@var{hours}@@ @var{mins}' @var{secs}"}. During entry, the letters
11332 @samp{h} (for ``hours'') or
11333 @samp{o} (approximating the ``degrees'' symbol) are accepted as well as
11334 @samp{@@}, @samp{m} is accepted in place of @samp{'}, and @samp{s} is
11335 accepted in place of @samp{"}.
11336 The @var{hours} value is an integer (or integer-valued float).
11337 The @var{mins} value is an integer or integer-valued float between 0 and 59.
11338 The @var{secs} value is a real number between 0 (inclusive) and 60
11339 (exclusive). A positive HMS form is interpreted as @var{hours} +
11340 @var{mins}/60 + @var{secs}/3600. A negative HMS form is interpreted
11341 as @mathit{- @var{hours}} @mathit{-} @var{mins}/60 @mathit{-} @var{secs}/3600.
11342 Display format for HMS forms is quite flexible. @xref{HMS Formats}.
11344 HMS forms can be added and subtracted. When they are added to numbers,
11345 the numbers are interpreted according to the current angular mode. HMS
11346 forms can also be multiplied and divided by real numbers. Dividing
11347 two HMS forms produces a real-valued ratio of the two angles.
11350 @cindex Time of day
11351 Just for kicks, @kbd{M-x calc-time} pushes the current time of day on
11352 the stack as an HMS form.
11354 @node Date Forms, Modulo Forms, HMS Forms, Data Types
11355 @section Date Forms
11359 A @dfn{date form} represents a date and possibly an associated time.
11360 Simple date arithmetic is supported: Adding a number to a date
11361 produces a new date shifted by that many days; adding an HMS form to
11362 a date shifts it by that many hours. Subtracting two date forms
11363 computes the number of days between them (represented as a simple
11364 number). Many other operations, such as multiplying two date forms,
11365 are nonsensical and are not allowed by Calc.
11367 Date forms are entered and displayed enclosed in @samp{< >} brackets.
11368 The default format is, e.g., @samp{<Wed Jan 9, 1991>} for dates,
11369 or @samp{<3:32:20pm Wed Jan 9, 1991>} for dates with times.
11370 Input is flexible; date forms can be entered in any of the usual
11371 notations for dates and times. @xref{Date Formats}.
11373 Date forms are stored internally as numbers, specifically the number
11374 of days since midnight on the morning of January 1 of the year 1 AD.
11375 If the internal number is an integer, the form represents a date only;
11376 if the internal number is a fraction or float, the form represents
11377 a date and time. For example, @samp{<6:00am Wed Jan 9, 1991>}
11378 is represented by the number 726842.25. The standard precision of
11379 12 decimal digits is enough to ensure that a (reasonable) date and
11380 time can be stored without roundoff error.
11382 If the current precision is greater than 12, date forms will keep
11383 additional digits in the seconds position. For example, if the
11384 precision is 15, the seconds will keep three digits after the
11385 decimal point. Decreasing the precision below 12 may cause the
11386 time part of a date form to become inaccurate. This can also happen
11387 if astronomically high years are used, though this will not be an
11388 issue in everyday (or even everymillennium) use. Note that date
11389 forms without times are stored as exact integers, so roundoff is
11390 never an issue for them.
11392 You can use the @kbd{v p} (@code{calc-pack}) and @kbd{v u}
11393 (@code{calc-unpack}) commands to get at the numerical representation
11394 of a date form. @xref{Packing and Unpacking}.
11396 Date forms can go arbitrarily far into the future or past. Negative
11397 year numbers represent years BC. Calc uses a combination of the
11398 Gregorian and Julian calendars, following the history of Great
11399 Britain and the British colonies. This is the same calendar that
11400 is used by the @code{cal} program in most Unix implementations.
11402 @cindex Julian calendar
11403 @cindex Gregorian calendar
11404 Some historical background: The Julian calendar was created by
11405 Julius Caesar in the year 46 BC as an attempt to fix the gradual
11406 drift caused by the lack of leap years in the calendar used
11407 until that time. The Julian calendar introduced an extra day in
11408 all years divisible by four. After some initial confusion, the
11409 calendar was adopted around the year we call 8 AD. Some centuries
11410 later it became apparent that the Julian year of 365.25 days was
11411 itself not quite right. In 1582 Pope Gregory XIII introduced the
11412 Gregorian calendar, which added the new rule that years divisible
11413 by 100, but not by 400, were not to be considered leap years
11414 despite being divisible by four. Many countries delayed adoption
11415 of the Gregorian calendar because of religious differences;
11416 in Britain it was put off until the year 1752, by which time
11417 the Julian calendar had fallen eleven days behind the true
11418 seasons. So the switch to the Gregorian calendar in early
11419 September 1752 introduced a discontinuity: The day after
11420 Sep 2, 1752 is Sep 14, 1752. Calc follows this convention.
11421 To take another example, Russia waited until 1918 before
11422 adopting the new calendar, and thus needed to remove thirteen
11423 days (between Feb 1, 1918 and Feb 14, 1918). This means that
11424 Calc's reckoning will be inconsistent with Russian history between
11425 1752 and 1918, and similarly for various other countries.
11427 Today's timekeepers introduce an occasional ``leap second'' as
11428 well, but Calc does not take these minor effects into account.
11429 (If it did, it would have to report a non-integer number of days
11430 between, say, @samp{<12:00am Mon Jan 1, 1900>} and
11431 @samp{<12:00am Sat Jan 1, 2000>}.)
11433 Calc uses the Julian calendar for all dates before the year 1752,
11434 including dates BC when the Julian calendar technically had not
11435 yet been invented. Thus the claim that day number @mathit{-10000} is
11436 called ``August 16, 28 BC'' should be taken with a grain of salt.
11438 Please note that there is no ``year 0''; the day before
11439 @samp{<Sat Jan 1, +1>} is @samp{<Fri Dec 31, -1>}. These are
11440 days 0 and @mathit{-1} respectively in Calc's internal numbering scheme.
11442 @cindex Julian day counting
11443 Another day counting system in common use is, confusingly, also
11444 called ``Julian.'' It was invented in 1583 by Joseph Justus
11445 Scaliger, who named it in honor of his father Julius Caesar
11446 Scaliger. For obscure reasons he chose to start his day
11447 numbering on Jan 1, 4713 BC at noon, which in Calc's scheme
11448 is @mathit{-1721423.5} (recall that Calc starts at midnight instead
11449 of noon). Thus to convert a Calc date code obtained by
11450 unpacking a date form into a Julian day number, simply add
11451 1721423.5. The Julian code for @samp{6:00am Jan 9, 1991}
11452 is 2448265.75. The built-in @kbd{t J} command performs
11453 this conversion for you.
11455 @cindex Unix time format
11456 The Unix operating system measures time as an integer number of
11457 seconds since midnight, Jan 1, 1970. To convert a Calc date
11458 value into a Unix time stamp, first subtract 719164 (the code
11459 for @samp{<Jan 1, 1970>}), then multiply by 86400 (the number of
11460 seconds in a day) and press @kbd{R} to round to the nearest
11461 integer. If you have a date form, you can simply subtract the
11462 day @samp{<Jan 1, 1970>} instead of unpacking and subtracting
11463 719164. Likewise, divide by 86400 and add @samp{<Jan 1, 1970>}
11464 to convert from Unix time to a Calc date form. (Note that
11465 Unix normally maintains the time in the GMT time zone; you may
11466 need to subtract five hours to get New York time, or eight hours
11467 for California time. The same is usually true of Julian day
11468 counts.) The built-in @kbd{t U} command performs these
11471 @node Modulo Forms, Error Forms, Date Forms, Data Types
11472 @section Modulo Forms
11475 @cindex Modulo forms
11476 A @dfn{modulo form} is a real number which is taken modulo (i.e., within
11477 an integer multiple of) some value @var{M}. Arithmetic modulo @var{M}
11478 often arises in number theory. Modulo forms are written
11479 `@var{a} @tfn{mod} @var{M}',
11480 where @var{a} and @var{M} are real numbers or HMS forms, and
11481 @texline @math{0 \le a < M}.
11482 @infoline @expr{0 <= a < @var{M}}.
11483 In many applications @expr{a} and @expr{M} will be
11484 integers but this is not required.
11489 @kindex M (modulo forms)
11493 @tindex mod (operator)
11494 To create a modulo form during numeric entry, press the shift-@kbd{M}
11495 key to enter the word @samp{mod}. As a special convenience, pressing
11496 shift-@kbd{M} a second time automatically enters the value of @expr{M}
11497 that was most recently used before. During algebraic entry, either
11498 type @samp{mod} by hand or press @kbd{M-m} (that's @kbd{@key{META}-m}).
11499 Once again, pressing this a second time enters the current modulo.
11501 Modulo forms are not to be confused with the modulo operator @samp{%}.
11502 The expression @samp{27 % 10} means to compute 27 modulo 10 to produce
11503 the result 7. Further computations treat this 7 as just a regular integer.
11504 The expression @samp{27 mod 10} produces the result @samp{7 mod 10};
11505 further computations with this value are again reduced modulo 10 so that
11506 the result always lies in the desired range.
11508 When two modulo forms with identical @expr{M}'s are added or multiplied,
11509 the Calculator simply adds or multiplies the values, then reduces modulo
11510 @expr{M}. If one argument is a modulo form and the other a plain number,
11511 the plain number is treated like a compatible modulo form. It is also
11512 possible to raise modulo forms to powers; the result is the value raised
11513 to the power, then reduced modulo @expr{M}. (When all values involved
11514 are integers, this calculation is done much more efficiently than
11515 actually computing the power and then reducing.)
11517 @cindex Modulo division
11518 Two modulo forms `@var{a} @tfn{mod} @var{M}' and `@var{b} @tfn{mod} @var{M}'
11519 can be divided if @expr{a}, @expr{b}, and @expr{M} are all
11520 integers. The result is the modulo form which, when multiplied by
11521 `@var{b} @tfn{mod} @var{M}', produces `@var{a} @tfn{mod} @var{M}'. If
11522 there is no solution to this equation (which can happen only when
11523 @expr{M} is non-prime), or if any of the arguments are non-integers, the
11524 division is left in symbolic form. Other operations, such as square
11525 roots, are not yet supported for modulo forms. (Note that, although
11526 @w{`@tfn{(}@var{a} @tfn{mod} @var{M}@tfn{)^.5}'} will compute a ``modulo square root''
11527 in the sense of reducing
11528 @texline @math{\sqrt a}
11529 @infoline @expr{sqrt(a)}
11530 modulo @expr{M}, this is not a useful definition from the
11531 number-theoretical point of view.)
11533 It is possible to mix HMS forms and modulo forms. For example, an
11534 HMS form modulo 24 could be used to manipulate clock times; an HMS
11535 form modulo 360 would be suitable for angles. Making the modulo @expr{M}
11536 also be an HMS form eliminates troubles that would arise if the angular
11537 mode were inadvertently set to Radians, in which case
11538 @w{@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees modulo
11541 Modulo forms cannot have variables or formulas for components. If you
11542 enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus
11543 to each of the coefficients: @samp{(1 mod 5) x + (2 mod 5)}.
11545 You can use @kbd{v p} and @kbd{%} to modify modulo forms.
11546 @xref{Packing and Unpacking}. @xref{Basic Arithmetic}.
11552 The algebraic function @samp{makemod(a, m)} builds the modulo form
11553 @w{@samp{a mod m}}.
11555 @node Error Forms, Interval Forms, Modulo Forms, Data Types
11556 @section Error Forms
11559 @cindex Error forms
11560 @cindex Standard deviations
11561 An @dfn{error form} is a number with an associated standard
11562 deviation, as in @samp{2.3 +/- 0.12}. The notation
11563 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11564 @infoline `@var{x} @tfn{+/-} sigma'
11565 stands for an uncertain value which follows
11566 a normal or Gaussian distribution of mean @expr{x} and standard
11567 deviation or ``error''
11568 @texline @math{\sigma}.
11569 @infoline @expr{sigma}.
11570 Both the mean and the error can be either numbers or
11571 formulas. Generally these are real numbers but the mean may also be
11572 complex. If the error is negative or complex, it is changed to its
11573 absolute value. An error form with zero error is converted to a
11574 regular number by the Calculator.
11576 All arithmetic and transcendental functions accept error forms as input.
11577 Operations on the mean-value part work just like operations on regular
11578 numbers. The error part for any function @expr{f(x)} (such as
11579 @texline @math{\sin x}
11580 @infoline @expr{sin(x)})
11581 is defined by the error of @expr{x} times the derivative of @expr{f}
11582 evaluated at the mean value of @expr{x}. For a two-argument function
11583 @expr{f(x,y)} (such as addition) the error is the square root of the sum
11584 of the squares of the errors due to @expr{x} and @expr{y}.
11587 f(x \hbox{\code{ +/- }} \sigma)
11588 &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr
11589 f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y)
11590 &= f(x,y) \hbox{\code{ +/- }}
11591 \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x}
11593 +\left(\sigma_y \left| {\partial f(x,y) \over \partial y}
11594 \right| \right)^2 } \cr
11598 definition assumes the errors in @expr{x} and @expr{y} are uncorrelated.
11599 A side effect of this definition is that @samp{(2 +/- 1) * (2 +/- 1)}
11600 is not the same as @samp{(2 +/- 1)^2}; the former represents the product
11601 of two independent values which happen to have the same probability
11602 distributions, and the latter is the product of one random value with itself.
11603 The former will produce an answer with less error, since on the average
11604 the two independent errors can be expected to cancel out.
11606 Consult a good text on error analysis for a discussion of the proper use
11607 of standard deviations. Actual errors often are neither Gaussian-distributed
11608 nor uncorrelated, and the above formulas are valid only when errors
11609 are small. As an example, the error arising from
11610 @texline `@tfn{sin(}@var{x} @tfn{+/-} @math{\sigma}@tfn{)}'
11611 @infoline `@tfn{sin(}@var{x} @tfn{+/-} @var{sigma}@tfn{)}'
11613 @texline `@math{\sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11614 @infoline `@var{sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11615 When @expr{x} is close to zero,
11616 @texline @math{\cos x}
11617 @infoline @expr{cos(x)}
11618 is close to one so the error in the sine is close to
11619 @texline @math{\sigma};
11620 @infoline @expr{sigma};
11621 this makes sense, since
11622 @texline @math{\sin x}
11623 @infoline @expr{sin(x)}
11624 is approximately @expr{x} near zero, so a given error in @expr{x} will
11625 produce about the same error in the sine. Likewise, near 90 degrees
11626 @texline @math{\cos x}
11627 @infoline @expr{cos(x)}
11628 is nearly zero and so the computed error is
11629 small: The sine curve is nearly flat in that region, so an error in @expr{x}
11630 has relatively little effect on the value of
11631 @texline @math{\sin x}.
11632 @infoline @expr{sin(x)}.
11633 However, consider @samp{sin(90 +/- 1000)}. The cosine of 90 is zero, so
11634 Calc will report zero error! We get an obviously wrong result because
11635 we have violated the small-error approximation underlying the error
11636 analysis. If the error in @expr{x} had been small, the error in
11637 @texline @math{\sin x}
11638 @infoline @expr{sin(x)}
11639 would indeed have been negligible.
11644 @kindex p (error forms)
11646 To enter an error form during regular numeric entry, use the @kbd{p}
11647 (``plus-or-minus'') key to type the @samp{+/-} symbol. (If you try actually
11648 typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's
11649 @kbd{+} command!) Within an algebraic formula, you can press @kbd{M-p} to
11650 type the @samp{+/-} symbol, or type it out by hand.
11652 Error forms and complex numbers can be mixed; the formulas shown above
11653 are used for complex numbers, too; note that if the error part evaluates
11654 to a complex number its absolute value (or the square root of the sum of
11655 the squares of the absolute values of the two error contributions) is
11656 used. Mathematically, this corresponds to a radially symmetric Gaussian
11657 distribution of numbers on the complex plane. However, note that Calc
11658 considers an error form with real components to represent a real number,
11659 not a complex distribution around a real mean.
11661 Error forms may also be composed of HMS forms. For best results, both
11662 the mean and the error should be HMS forms if either one is.
11668 The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}.
11670 @node Interval Forms, Incomplete Objects, Error Forms, Data Types
11671 @section Interval Forms
11674 @cindex Interval forms
11675 An @dfn{interval} is a subset of consecutive real numbers. For example,
11676 the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4,
11677 inclusive. If you multiply it by the interval @samp{[0.5 ..@: 2]} you
11678 obtain @samp{[1 ..@: 8]}. This calculation represents the fact that if
11679 you multiply some number in the range @samp{[2 ..@: 4]} by some other
11680 number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range
11681 from 1 to 8. Interval arithmetic is used to get a worst-case estimate
11682 of the possible range of values a computation will produce, given the
11683 set of possible values of the input.
11686 Calc supports several varieties of intervals, including @dfn{closed}
11687 intervals of the type shown above, @dfn{open} intervals such as
11688 @samp{(2 ..@: 4)}, which represents the range of numbers from 2 to 4
11689 @emph{exclusive}, and @dfn{semi-open} intervals in which one end
11690 uses a round parenthesis and the other a square bracket. In mathematical
11692 @samp{[2 ..@: 4]} means @expr{2 <= x <= 4}, whereas
11693 @samp{[2 ..@: 4)} represents @expr{2 <= x < 4},
11694 @samp{(2 ..@: 4]} represents @expr{2 < x <= 4}, and
11695 @samp{(2 ..@: 4)} represents @expr{2 < x < 4}.
11698 Calc supports several varieties of intervals, including \dfn{closed}
11699 intervals of the type shown above, \dfn{open} intervals such as
11700 \samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4
11701 \emph{exclusive}, and \dfn{semi-open} intervals in which one end
11702 uses a round parenthesis and the other a square bracket. In mathematical
11705 [2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 \le x \le 4 \cr
11706 [2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 \le x < 4 \cr
11707 (2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 < x \le 4 \cr
11708 (2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 < x < 4 \cr
11712 The lower and upper limits of an interval must be either real numbers
11713 (or HMS or date forms), or symbolic expressions which are assumed to be
11714 real-valued, or @samp{-inf} and @samp{inf}. In general the lower limit
11715 must be less than the upper limit. A closed interval containing only
11716 one value, @samp{[3 ..@: 3]}, is converted to a plain number (3)
11717 automatically. An interval containing no values at all (such as
11718 @samp{[3 ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not
11719 guaranteed to behave well when used in arithmetic. Note that the
11720 interval @samp{[3 .. inf)} represents all real numbers greater than
11721 or equal to 3, and @samp{(-inf .. inf)} represents all real numbers.
11722 In fact, @samp{[-inf .. inf]} represents all real numbers including
11723 the real infinities.
11725 Intervals are entered in the notation shown here, either as algebraic
11726 formulas, or using incomplete forms. (@xref{Incomplete Objects}.)
11727 In algebraic formulas, multiple periods in a row are collected from
11728 left to right, so that @samp{1...1e2} is interpreted as @samp{1.0 ..@: 1e2}
11729 rather than @samp{1 ..@: 0.1e2}. Add spaces or zeros if you want to
11730 get the other interpretation. If you omit the lower or upper limit,
11731 a default of @samp{-inf} or @samp{inf} (respectively) is furnished.
11733 Infinite mode also affects operations on intervals
11734 (@pxref{Infinities}). Calc will always introduce an open infinity,
11735 as in @samp{1 / (0 .. 2] = [0.5 .. inf)}. But closed infinities,
11736 @w{@samp{1 / [0 .. 2] = [0.5 .. inf]}}, arise only in Infinite mode;
11737 otherwise they are left unevaluated. Note that the ``direction'' of
11738 a zero is not an issue in this case since the zero is always assumed
11739 to be continuous with the rest of the interval. For intervals that
11740 contain zero inside them Calc is forced to give the result,
11741 @samp{1 / (-2 .. 2) = [-inf .. inf]}.
11743 While it may seem that intervals and error forms are similar, they are
11744 based on entirely different concepts of inexact quantities. An error
11746 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11747 @infoline `@var{x} @tfn{+/-} @var{sigma}'
11748 means a variable is random, and its value could
11749 be anything but is ``probably'' within one
11750 @texline @math{\sigma}
11751 @infoline @var{sigma}
11752 of the mean value @expr{x}. An interval
11753 `@tfn{[}@var{a} @tfn{..@:} @var{b}@tfn{]}' means a
11754 variable's value is unknown, but guaranteed to lie in the specified
11755 range. Error forms are statistical or ``average case'' approximations;
11756 interval arithmetic tends to produce ``worst case'' bounds on an
11759 Intervals may not contain complex numbers, but they may contain
11760 HMS forms or date forms.
11762 @xref{Set Operations}, for commands that interpret interval forms
11763 as subsets of the set of real numbers.
11769 The algebraic function @samp{intv(n, a, b)} builds an interval form
11770 from @samp{a} to @samp{b}; @samp{n} is an integer code which must
11771 be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or
11774 Please note that in fully rigorous interval arithmetic, care would be
11775 taken to make sure that the computation of the lower bound rounds toward
11776 minus infinity, while upper bound computations round toward plus
11777 infinity. Calc's arithmetic always uses a round-to-nearest mode,
11778 which means that roundoff errors could creep into an interval
11779 calculation to produce intervals slightly smaller than they ought to
11780 be. For example, entering @samp{[1..2]} and pressing @kbd{Q 2 ^}
11781 should yield the interval @samp{[1..2]} again, but in fact it yields the
11782 (slightly too small) interval @samp{[1..1.9999999]} due to roundoff
11785 @node Incomplete Objects, Variables, Interval Forms, Data Types
11786 @section Incomplete Objects
11806 @cindex Incomplete vectors
11807 @cindex Incomplete complex numbers
11808 @cindex Incomplete interval forms
11809 When @kbd{(} or @kbd{[} is typed to begin entering a complex number or
11810 vector, respectively, the effect is to push an @dfn{incomplete} complex
11811 number or vector onto the stack. The @kbd{,} key adds the value(s) at
11812 the top of the stack onto the current incomplete object. The @kbd{)}
11813 and @kbd{]} keys ``close'' the incomplete object after adding any values
11814 on the top of the stack in front of the incomplete object.
11816 As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]}
11817 pushes the vector @samp{[2, 6, 9]} onto the stack. Likewise, @kbd{( 1 , 2 Q )}
11818 pushes the complex number @samp{(1, 1.414)} (approximately).
11820 If several values lie on the stack in front of the incomplete object,
11821 all are collected and appended to the object. Thus the @kbd{,} key
11822 is redundant: @kbd{[ 2 @key{RET} 3 @key{RET} 2 * 9 ]}. Some people
11823 prefer the equivalent @key{SPC} key to @key{RET}.
11825 As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or
11826 @kbd{,} adds a zero or duplicates the preceding value in the list being
11827 formed. Typing @key{DEL} during incomplete entry removes the last item
11831 The @kbd{;} key is used in the same way as @kbd{,} to create polar complex
11832 numbers: @kbd{( 1 ; 2 )}. When entering a vector, @kbd{;} is useful for
11833 creating a matrix. In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is
11834 equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}.
11838 Incomplete entry is also used to enter intervals. For example,
11839 @kbd{[ 2 ..@: 4 )} enters a semi-open interval. Note that when you type
11840 the first period, it will be interpreted as a decimal point, but when
11841 you type a second period immediately afterward, it is re-interpreted as
11842 part of the interval symbol. Typing @kbd{..} corresponds to executing
11843 the @code{calc-dots} command.
11845 If you find incomplete entry distracting, you may wish to enter vectors
11846 and complex numbers as algebraic formulas by pressing the apostrophe key.
11848 @node Variables, Formulas, Incomplete Objects, Data Types
11852 @cindex Variables, in formulas
11853 A @dfn{variable} is somewhere between a storage register on a conventional
11854 calculator, and a variable in a programming language. (In fact, a Calc
11855 variable is really just an Emacs Lisp variable that contains a Calc number
11856 or formula.) A variable's name is normally composed of letters and digits.
11857 Calc also allows apostrophes and @code{#} signs in variable names.
11858 (The Calc variable @code{foo} corresponds to the Emacs Lisp variable
11859 @code{var-foo}, but unless you access the variable from within Emacs
11860 Lisp, you don't need to worry about it. Variable names in algebraic
11861 formulas implicitly have @samp{var-} prefixed to their names. The
11862 @samp{#} character in variable names used in algebraic formulas
11863 corresponds to a dash @samp{-} in the Lisp variable name. If the name
11864 contains any dashes, the prefix @samp{var-} is @emph{not} automatically
11865 added. Thus the two formulas @samp{foo + 1} and @samp{var#foo + 1} both
11866 refer to the same variable.)
11868 In a command that takes a variable name, you can either type the full
11869 name of a variable, or type a single digit to use one of the special
11870 convenience variables @code{q0} through @code{q9}. For example,
11871 @kbd{3 s s 2} stores the number 3 in variable @code{q2}, and
11872 @w{@kbd{3 s s foo @key{RET}}} stores that number in variable
11875 To push a variable itself (as opposed to the variable's value) on the
11876 stack, enter its name as an algebraic expression using the apostrophe
11880 @pindex calc-evaluate
11881 @cindex Evaluation of variables in a formula
11882 @cindex Variables, evaluation
11883 @cindex Formulas, evaluation
11884 The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by
11885 replacing all variables in the formula which have been given values by a
11886 @code{calc-store} or @code{calc-let} command by their stored values.
11887 Other variables are left alone. Thus a variable that has not been
11888 stored acts like an abstract variable in algebra; a variable that has
11889 been stored acts more like a register in a traditional calculator.
11890 With a positive numeric prefix argument, @kbd{=} evaluates the top
11891 @var{n} stack entries; with a negative argument, @kbd{=} evaluates
11892 the @var{n}th stack entry.
11894 @cindex @code{e} variable
11895 @cindex @code{pi} variable
11896 @cindex @code{i} variable
11897 @cindex @code{phi} variable
11898 @cindex @code{gamma} variable
11904 A few variables are called @dfn{special constants}. Their names are
11905 @samp{e}, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}.
11906 (@xref{Scientific Functions}.) When they are evaluated with @kbd{=},
11907 their values are calculated if necessary according to the current precision
11908 or complex polar mode. If you wish to use these symbols for other purposes,
11909 simply undefine or redefine them using @code{calc-store}.
11911 The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for
11912 infinite or indeterminate values. It's best not to use them as
11913 regular variables, since Calc uses special algebraic rules when
11914 it manipulates them. Calc displays a warning message if you store
11915 a value into any of these special variables.
11917 @xref{Store and Recall}, for a discussion of commands dealing with variables.
11919 @node Formulas, , Variables, Data Types
11924 @cindex Expressions
11925 @cindex Operators in formulas
11926 @cindex Precedence of operators
11927 When you press the apostrophe key you may enter any expression or formula
11928 in algebraic form. (Calc uses the terms ``expression'' and ``formula''
11929 interchangeably.) An expression is built up of numbers, variable names,
11930 and function calls, combined with various arithmetic operators.
11932 be used to indicate grouping. Spaces are ignored within formulas, except
11933 that spaces are not permitted within variable names or numbers.
11934 Arithmetic operators, in order from highest to lowest precedence, and
11935 with their equivalent function names, are:
11937 @samp{_} [@code{subscr}] (subscripts);
11939 postfix @samp{%} [@code{percent}] (as in @samp{25% = 0.25});
11941 prefix @samp{+} and @samp{-} [@code{neg}] (as in @samp{-x})
11942 and prefix @samp{!} [@code{lnot}] (logical ``not,'' as in @samp{!x});
11944 @samp{+/-} [@code{sdev}] (the standard deviation symbol) and
11945 @samp{mod} [@code{makemod}] (the symbol for modulo forms);
11947 postfix @samp{!} [@code{fact}] (factorial, as in @samp{n!})
11948 and postfix @samp{!!} [@code{dfact}] (double factorial);
11950 @samp{^} [@code{pow}] (raised-to-the-power-of);
11952 @samp{*} [@code{mul}];
11954 @samp{/} [@code{div}], @samp{%} [@code{mod}] (modulo), and
11955 @samp{\} [@code{idiv}] (integer division);
11957 infix @samp{+} [@code{add}] and @samp{-} [@code{sub}] (as in @samp{x-y});
11959 @samp{|} [@code{vconcat}] (vector concatenation);
11961 relations @samp{=} [@code{eq}], @samp{!=} [@code{neq}], @samp{<} [@code{lt}],
11962 @samp{>} [@code{gt}], @samp{<=} [@code{leq}], and @samp{>=} [@code{geq}];
11964 @samp{&&} [@code{land}] (logical ``and'');
11966 @samp{||} [@code{lor}] (logical ``or'');
11968 the C-style ``if'' operator @samp{a?b:c} [@code{if}];
11970 @samp{!!!} [@code{pnot}] (rewrite pattern ``not'');
11972 @samp{&&&} [@code{pand}] (rewrite pattern ``and'');
11974 @samp{|||} [@code{por}] (rewrite pattern ``or'');
11976 @samp{:=} [@code{assign}] (for assignments and rewrite rules);
11978 @samp{::} [@code{condition}] (rewrite pattern condition);
11980 @samp{=>} [@code{evalto}].
11982 Note that, unlike in usual computer notation, multiplication binds more
11983 strongly than division: @samp{a*b/c*d} is equivalent to
11984 @texline @math{a b \over c d}.
11985 @infoline @expr{(a*b)/(c*d)}.
11987 @cindex Multiplication, implicit
11988 @cindex Implicit multiplication
11989 The multiplication sign @samp{*} may be omitted in many cases. In particular,
11990 if the righthand side is a number, variable name, or parenthesized
11991 expression, the @samp{*} may be omitted. Implicit multiplication has the
11992 same precedence as the explicit @samp{*} operator. The one exception to
11993 the rule is that a variable name followed by a parenthesized expression,
11995 is interpreted as a function call, not an implicit @samp{*}. In many
11996 cases you must use a space if you omit the @samp{*}: @samp{2a} is the
11997 same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab}
11998 is a variable called @code{ab}, @emph{not} the product of @samp{a} and
11999 @samp{b}! Also note that @samp{f (x)} is still a function call.
12001 @cindex Implicit comma in vectors
12002 The rules are slightly different for vectors written with square brackets.
12003 In vectors, the space character is interpreted (like the comma) as a
12004 separator of elements of the vector. Thus @w{@samp{[ 2a b+c d ]}} is
12005 equivalent to @samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent
12006 to @samp{2*a*b + c*d}.
12007 Note that spaces around the brackets, and around explicit commas, are
12008 ignored. To force spaces to be interpreted as multiplication you can
12009 enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is
12010 interpreted as @samp{[a*b, 2*c*d]}. An implicit comma is also inserted
12011 between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.
12013 Vectors that contain commas (not embedded within nested parentheses or
12014 brackets) do not treat spaces specially: @samp{[a b, 2 c d]} is a vector
12015 of two elements. Also, if it would be an error to treat spaces as
12016 separators, but not otherwise, then Calc will ignore spaces:
12017 @w{@samp{[a - b]}} is a vector of one element, but @w{@samp{[a -b]}} is
12018 a vector of two elements. Finally, vectors entered with curly braces
12019 instead of square brackets do not give spaces any special treatment.
12020 When Calc displays a vector that does not contain any commas, it will
12021 insert parentheses if necessary to make the meaning clear:
12022 @w{@samp{[(a b)]}}.
12024 The expression @samp{5%-2} is ambiguous; is this five-percent minus two,
12025 or five modulo minus-two? Calc always interprets the leftmost symbol as
12026 an infix operator preferentially (modulo, in this case), so you would
12027 need to write @samp{(5%)-2} to get the former interpretation.
12029 @cindex Function call notation
12030 A function call is, e.g., @samp{sin(1+x)}. (The Calc algebraic function
12031 @code{foo} corresponds to the Emacs Lisp function @code{calcFunc-foo},
12032 but unless you access the function from within Emacs Lisp, you don't
12033 need to worry about it.) Most mathematical Calculator commands like
12034 @code{calc-sin} have function equivalents like @code{sin}.
12035 If no Lisp function is defined for a function called by a formula, the
12036 call is left as it is during algebraic manipulation: @samp{f(x+y)} is
12037 left alone. Beware that many innocent-looking short names like @code{in}
12038 and @code{re} have predefined meanings which could surprise you; however,
12039 single letters or single letters followed by digits are always safe to
12040 use for your own function names. @xref{Function Index}.
12042 In the documentation for particular commands, the notation @kbd{H S}
12043 (@code{calc-sinh}) [@code{sinh}] means that the key sequence @kbd{H S}, the
12044 command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all
12045 represent the same operation.
12047 Commands that interpret (``parse'') text as algebraic formulas include
12048 algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse
12049 the contents of the editing buffer when you finish, the @kbd{M-# g}
12050 and @w{@kbd{M-# r}} commands, the @kbd{C-y} command, the X window system
12051 ``paste'' mouse operation, and Embedded mode. All of these operations
12052 use the same rules for parsing formulas; in particular, language modes
12053 (@pxref{Language Modes}) affect them all in the same way.
12055 When you read a large amount of text into the Calculator (say a vector
12056 which represents a big set of rewrite rules; @pxref{Rewrite Rules}),
12057 you may wish to include comments in the text. Calc's formula parser
12058 ignores the symbol @samp{%%} and anything following it on a line:
12061 [ a + b, %% the sum of "a" and "b"
12063 %% last line is coming up:
12068 This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}.
12070 @xref{Syntax Tables}, for a way to create your own operators and other
12071 input notations. @xref{Compositions}, for a way to create new display
12074 @xref{Algebra}, for commands for manipulating formulas symbolically.
12076 @node Stack and Trail, Mode Settings, Data Types, Top
12077 @chapter Stack and Trail Commands
12080 This chapter describes the Calc commands for manipulating objects on the
12081 stack and in the trail buffer. (These commands operate on objects of any
12082 type, such as numbers, vectors, formulas, and incomplete objects.)
12085 * Stack Manipulation::
12086 * Editing Stack Entries::
12091 @node Stack Manipulation, Editing Stack Entries, Stack and Trail, Stack and Trail
12092 @section Stack Manipulation Commands
12098 @cindex Duplicating stack entries
12099 To duplicate the top object on the stack, press @key{RET} or @key{SPC}
12100 (two equivalent keys for the @code{calc-enter} command).
12101 Given a positive numeric prefix argument, these commands duplicate
12102 several elements at the top of the stack.
12103 Given a negative argument,
12104 these commands duplicate the specified element of the stack.
12105 Given an argument of zero, they duplicate the entire stack.
12106 For example, with @samp{10 20 30} on the stack,
12107 @key{RET} creates @samp{10 20 30 30},
12108 @kbd{C-u 2 @key{RET}} creates @samp{10 20 30 20 30},
12109 @kbd{C-u - 2 @key{RET}} creates @samp{10 20 30 20}, and
12110 @kbd{C-u 0 @key{RET}} creates @samp{10 20 30 10 20 30}.
12114 The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you
12115 have it, else on @kbd{C-j}) is like @code{calc-enter}
12116 except that the sign of the numeric prefix argument is interpreted
12117 oppositely. Also, with no prefix argument the default argument is 2.
12118 Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}}
12119 are both equivalent to @kbd{C-u - 2 @key{RET}}, producing
12120 @samp{10 20 30 20}.
12125 @cindex Removing stack entries
12126 @cindex Deleting stack entries
12127 To remove the top element from the stack, press @key{DEL} (@code{calc-pop}).
12128 The @kbd{C-d} key is a synonym for @key{DEL}.
12129 (If the top element is an incomplete object with at least one element, the
12130 last element is removed from it.) Given a positive numeric prefix argument,
12131 several elements are removed. Given a negative argument, the specified
12132 element of the stack is deleted. Given an argument of zero, the entire
12134 For example, with @samp{10 20 30} on the stack,
12135 @key{DEL} leaves @samp{10 20},
12136 @kbd{C-u 2 @key{DEL}} leaves @samp{10},
12137 @kbd{C-u - 2 @key{DEL}} leaves @samp{10 30}, and
12138 @kbd{C-u 0 @key{DEL}} leaves an empty stack.
12140 @kindex M-@key{DEL}
12141 @pindex calc-pop-above
12142 The @kbd{M-@key{DEL}} (@code{calc-pop-above}) command is to @key{DEL} what
12143 @key{LFD} is to @key{RET}: It interprets the sign of the numeric
12144 prefix argument in the opposite way, and the default argument is 2.
12145 Thus @kbd{M-@key{DEL}} by itself removes the second-from-top stack element,
12146 leaving the first, third, fourth, and so on; @kbd{M-3 M-@key{DEL}} deletes
12147 the third stack element.
12150 @pindex calc-roll-down
12151 To exchange the top two elements of the stack, press @key{TAB}
12152 (@code{calc-roll-down}). Given a positive numeric prefix argument, the
12153 specified number of elements at the top of the stack are rotated downward.
12154 Given a negative argument, the entire stack is rotated downward the specified
12155 number of times. Given an argument of zero, the entire stack is reversed
12157 For example, with @samp{10 20 30 40 50} on the stack,
12158 @key{TAB} creates @samp{10 20 30 50 40},
12159 @kbd{C-u 3 @key{TAB}} creates @samp{10 20 50 30 40},
12160 @kbd{C-u - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and
12161 @kbd{C-u 0 @key{TAB}} creates @samp{50 40 30 20 10}.
12163 @kindex M-@key{TAB}
12164 @pindex calc-roll-up
12165 The command @kbd{M-@key{TAB}} (@code{calc-roll-up}) is analogous to @key{TAB}
12166 except that it rotates upward instead of downward. Also, the default
12167 with no prefix argument is to rotate the top 3 elements.
12168 For example, with @samp{10 20 30 40 50} on the stack,
12169 @kbd{M-@key{TAB}} creates @samp{10 20 40 50 30},
12170 @kbd{C-u 4 M-@key{TAB}} creates @samp{10 30 40 50 20},
12171 @kbd{C-u - 2 M-@key{TAB}} creates @samp{30 40 50 10 20}, and
12172 @kbd{C-u 0 M-@key{TAB}} creates @samp{50 40 30 20 10}.
12174 A good way to view the operation of @key{TAB} and @kbd{M-@key{TAB}} is in
12175 terms of moving a particular element to a new position in the stack.
12176 With a positive argument @var{n}, @key{TAB} moves the top stack
12177 element down to level @var{n}, making room for it by pulling all the
12178 intervening stack elements toward the top. @kbd{M-@key{TAB}} moves the
12179 element at level @var{n} up to the top. (Compare with @key{LFD},
12180 which copies instead of moving the element in level @var{n}.)
12182 With a negative argument @mathit{-@var{n}}, @key{TAB} rotates the stack
12183 to move the object in level @var{n} to the deepest place in the
12184 stack, and the object in level @mathit{@var{n}+1} to the top. @kbd{M-@key{TAB}}
12185 rotates the deepest stack element to be in level @mathit{n}, also
12186 putting the top stack element in level @mathit{@var{n}+1}.
12188 @xref{Selecting Subformulas}, for a way to apply these commands to
12189 any portion of a vector or formula on the stack.
12191 @node Editing Stack Entries, Trail Commands, Stack Manipulation, Stack and Trail
12192 @section Editing Stack Entries
12197 @pindex calc-edit-finish
12198 @cindex Editing the stack with Emacs
12199 The backquote, @kbd{`} (@code{calc-edit}) command creates a temporary
12200 buffer (@samp{*Calc Edit*}) for editing the top-of-stack value using
12201 regular Emacs commands. With a numeric prefix argument, it edits the
12202 specified number of stack entries at once. (An argument of zero edits
12203 the entire stack; a negative argument edits one specific stack entry.)
12205 When you are done editing, press @kbd{C-c C-c} to finish and return
12206 to Calc. The @key{RET} and @key{LFD} keys also work to finish most
12207 sorts of editing, though in some cases Calc leaves @key{RET} with its
12208 usual meaning (``insert a newline'') if it's a situation where you
12209 might want to insert new lines into the editing buffer.
12211 When you finish editing, the Calculator parses the lines of text in
12212 the @samp{*Calc Edit*} buffer as numbers or formulas, replaces the
12213 original stack elements in the original buffer with these new values,
12214 then kills the @samp{*Calc Edit*} buffer. The original Calculator buffer
12215 continues to exist during editing, but for best results you should be
12216 careful not to change it until you have finished the edit. You can
12217 also cancel the edit by killing the buffer with @kbd{C-x k}.
12219 The formula is normally reevaluated as it is put onto the stack.
12220 For example, editing @samp{a + 2} to @samp{3 + 2} and pressing
12221 @kbd{C-c C-c} will push 5 on the stack. If you use @key{LFD} to
12222 finish, Calc will put the result on the stack without evaluating it.
12224 If you give a prefix argument to @kbd{C-c C-c},
12225 Calc will not kill the @samp{*Calc Edit*} buffer. You can switch
12226 back to that buffer and continue editing if you wish. However, you
12227 should understand that if you initiated the edit with @kbd{`}, the
12228 @kbd{C-c C-c} operation will be programmed to replace the top of the
12229 stack with the new edited value, and it will do this even if you have
12230 rearranged the stack in the meanwhile. This is not so much of a problem
12231 with other editing commands, though, such as @kbd{s e}
12232 (@code{calc-edit-variable}; @pxref{Operations on Variables}).
12234 If the @code{calc-edit} command involves more than one stack entry,
12235 each line of the @samp{*Calc Edit*} buffer is interpreted as a
12236 separate formula. Otherwise, the entire buffer is interpreted as
12237 one formula, with line breaks ignored. (You can use @kbd{C-o} or
12238 @kbd{C-q C-j} to insert a newline in the buffer without pressing @key{RET}.)
12240 The @kbd{`} key also works during numeric or algebraic entry. The
12241 text entered so far is moved to the @code{*Calc Edit*} buffer for
12242 more extensive editing than is convenient in the minibuffer.
12244 @node Trail Commands, Keep Arguments, Editing Stack Entries, Stack and Trail
12245 @section Trail Commands
12248 @cindex Trail buffer
12249 The commands for manipulating the Calc Trail buffer are two-key sequences
12250 beginning with the @kbd{t} prefix.
12253 @pindex calc-trail-display
12254 The @kbd{t d} (@code{calc-trail-display}) command turns display of the
12255 trail on and off. Normally the trail display is toggled on if it was off,
12256 off if it was on. With a numeric prefix of zero, this command always
12257 turns the trail off; with a prefix of one, it always turns the trail on.
12258 The other trail-manipulation commands described here automatically turn
12259 the trail on. Note that when the trail is off values are still recorded
12260 there; they are simply not displayed. To set Emacs to turn the trail
12261 off by default, type @kbd{t d} and then save the mode settings with
12262 @kbd{m m} (@code{calc-save-modes}).
12265 @pindex calc-trail-in
12267 @pindex calc-trail-out
12268 The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o}
12269 (@code{calc-trail-out}) commands switch the cursor into and out of the
12270 Calc Trail window. In practice they are rarely used, since the commands
12271 shown below are a more convenient way to move around in the
12272 trail, and they work ``by remote control'' when the cursor is still
12273 in the Calculator window.
12275 @cindex Trail pointer
12276 There is a @dfn{trail pointer} which selects some entry of the trail at
12277 any given time. The trail pointer looks like a @samp{>} symbol right
12278 before the selected number. The following commands operate on the
12279 trail pointer in various ways.
12282 @pindex calc-trail-yank
12283 @cindex Retrieving previous results
12284 The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in
12285 the trail and pushes it onto the Calculator stack. It allows you to
12286 re-use any previously computed value without retyping. With a numeric
12287 prefix argument @var{n}, it yanks the value @var{n} lines above the current
12291 @pindex calc-trail-scroll-left
12293 @pindex calc-trail-scroll-right
12294 The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >}
12295 (@code{calc-trail-scroll-right}) commands horizontally scroll the trail
12296 window left or right by one half of its width.
12299 @pindex calc-trail-next
12301 @pindex calc-trail-previous
12303 @pindex calc-trail-forward
12305 @pindex calc-trail-backward
12306 The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p}
12307 (@code{calc-trail-previous)} commands move the trail pointer down or up
12308 one line. The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b}
12309 (@code{calc-trail-backward}) commands move the trail pointer down or up
12310 one screenful at a time. All of these commands accept numeric prefix
12311 arguments to move several lines or screenfuls at a time.
12314 @pindex calc-trail-first
12316 @pindex calc-trail-last
12318 @pindex calc-trail-here
12319 The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]}
12320 (@code{calc-trail-last}) commands move the trail pointer to the first or
12321 last line of the trail. The @kbd{t h} (@code{calc-trail-here}) command
12322 moves the trail pointer to the cursor position; unlike the other trail
12323 commands, @kbd{t h} works only when Calc Trail is the selected window.
12326 @pindex calc-trail-isearch-forward
12328 @pindex calc-trail-isearch-backward
12330 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
12331 (@code{calc-trail-isearch-backward}) commands perform an incremental
12332 search forward or backward through the trail. You can press @key{RET}
12333 to terminate the search; the trail pointer moves to the current line.
12334 If you cancel the search with @kbd{C-g}, the trail pointer stays where
12335 it was when the search began.
12338 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
12339 (@code{calc-trail-isearch-backward}) com\-mands perform an incremental
12340 search forward or backward through the trail. You can press @key{RET}
12341 to terminate the search; the trail pointer moves to the current line.
12342 If you cancel the search with @kbd{C-g}, the trail pointer stays where
12343 it was when the search began.
12347 @pindex calc-trail-marker
12348 The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a
12349 line of text of your own choosing into the trail. The text is inserted
12350 after the line containing the trail pointer; this usually means it is
12351 added to the end of the trail. Trail markers are useful mainly as the
12352 targets for later incremental searches in the trail.
12355 @pindex calc-trail-kill
12356 The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line
12357 from the trail. The line is saved in the Emacs kill ring suitable for
12358 yanking into another buffer, but it is not easy to yank the text back
12359 into the trail buffer. With a numeric prefix argument, this command
12360 kills the @var{n} lines below or above the selected one.
12362 The @kbd{t .} (@code{calc-full-trail-vectors}) command is described
12363 elsewhere; @pxref{Vector and Matrix Formats}.
12365 @node Keep Arguments, , Trail Commands, Stack and Trail
12366 @section Keep Arguments
12370 @pindex calc-keep-args
12371 The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for
12372 the following command. It prevents that command from removing its
12373 arguments from the stack. For example, after @kbd{2 @key{RET} 3 +},
12374 the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +},
12375 the stack contains the arguments and the result: @samp{2 3 5}.
12377 With the exception of keyboard macros, this works for all commands that
12378 take arguments off the stack. (To avoid potentially unpleasant behavior,
12379 a @kbd{K} prefix before a keyboard macro will be ignored. A @kbd{K}
12380 prefix called @emph{within} the keyboard macro will still take effect.)
12381 As another example, @kbd{K a s} simplifies a formula, pushing the
12382 simplified version of the formula onto the stack after the original
12383 formula (rather than replacing the original formula). Note that you
12384 could get the same effect by typing @kbd{@key{RET} a s}, copying the
12385 formula and then simplifying the copy. One difference is that for a very
12386 large formula the time taken to format the intermediate copy in
12387 @kbd{@key{RET} a s} could be noticeable; @kbd{K a s} would avoid this
12390 Even stack manipulation commands are affected. @key{TAB} works by
12391 popping two values and pushing them back in the opposite order,
12392 so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}.
12394 A few Calc commands provide other ways of doing the same thing.
12395 For example, @kbd{' sin($)} replaces the number on the stack with
12396 its sine using algebraic entry; to push the sine and keep the
12397 original argument you could use either @kbd{' sin($1)} or
12398 @kbd{K ' sin($)}. @xref{Algebraic Entry}. Also, the @kbd{s s}
12399 command is effectively the same as @kbd{K s t}. @xref{Storing Variables}.
12401 If you execute a command and then decide you really wanted to keep
12402 the argument, you can press @kbd{M-@key{RET}} (@code{calc-last-args}).
12403 This command pushes the last arguments that were popped by any command
12404 onto the stack. Note that the order of things on the stack will be
12405 different than with @kbd{K}: @kbd{2 @key{RET} 3 + M-@key{RET}} leaves
12406 @samp{5 2 3} on the stack instead of @samp{2 3 5}. @xref{Undo}.
12408 @node Mode Settings, Arithmetic, Stack and Trail, Top
12409 @chapter Mode Settings
12412 This chapter describes commands that set modes in the Calculator.
12413 They do not affect the contents of the stack, although they may change
12414 the @emph{appearance} or @emph{interpretation} of the stack's contents.
12417 * General Mode Commands::
12419 * Inverse and Hyperbolic::
12420 * Calculation Modes::
12421 * Simplification Modes::
12429 @node General Mode Commands, Precision, Mode Settings, Mode Settings
12430 @section General Mode Commands
12434 @pindex calc-save-modes
12435 @cindex Continuous memory
12436 @cindex Saving mode settings
12437 @cindex Permanent mode settings
12438 @cindex Calc init file, mode settings
12439 You can save all of the current mode settings in your Calc init file
12440 (the file given by the variable @code{calc-settings-file}, typically
12441 @file{~/.calc.el}) with the @kbd{m m} (@code{calc-save-modes}) command.
12442 This will cause Emacs to reestablish these modes each time it starts up.
12443 The modes saved in the file include everything controlled by the @kbd{m}
12444 and @kbd{d} prefix keys, the current precision and binary word size,
12445 whether or not the trail is displayed, the current height of the Calc
12446 window, and more. The current interface (used when you type @kbd{M-#
12447 M-#}) is also saved. If there were already saved mode settings in the
12448 file, they are replaced. Otherwise, the new mode information is
12449 appended to the end of the file.
12452 @pindex calc-mode-record-mode
12453 The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to
12454 record all the mode settings (as if by pressing @kbd{m m}) every
12455 time a mode setting changes. If the modes are saved this way, then this
12456 ``automatic mode recording'' mode is also saved.
12457 Type @kbd{m R} again to disable this method of recording the mode
12458 settings. To turn it off permanently, the @kbd{m m} command will also be
12459 necessary. (If Embedded mode is enabled, other options for recording
12460 the modes are available; @pxref{Mode Settings in Embedded Mode}.)
12463 @pindex calc-settings-file-name
12464 The @kbd{m F} (@code{calc-settings-file-name}) command allows you to
12465 choose a different file than the current value of @code{calc-settings-file}
12466 for @kbd{m m}, @kbd{Z P}, and similar commands to save permanent information.
12467 You are prompted for a file name. All Calc modes are then reset to
12468 their default values, then settings from the file you named are loaded
12469 if this file exists, and this file becomes the one that Calc will
12470 use in the future for commands like @kbd{m m}. The default settings
12471 file name is @file{~/.calc.el}. You can see the current file name by
12472 giving a blank response to the @kbd{m F} prompt. See also the
12473 discussion of the @code{calc-settings-file} variable; @pxref{Customizable Variables}.
12475 If the file name you give is your user init file (typically
12476 @file{~/.emacs}), @kbd{m F} will not automatically load the new file. This
12477 is because your user init file may contain other things you don't want
12478 to reread. You can give
12479 a numeric prefix argument of 1 to @kbd{m F} to force it to read the
12480 file no matter what. Conversely, an argument of @mathit{-1} tells
12481 @kbd{m F} @emph{not} to read the new file. An argument of 2 or @mathit{-2}
12482 tells @kbd{m F} not to reset the modes to their defaults beforehand,
12483 which is useful if you intend your new file to have a variant of the
12484 modes present in the file you were using before.
12487 @pindex calc-always-load-extensions
12488 The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode
12489 in which the first use of Calc loads the entire program, including all
12490 extensions modules. Otherwise, the extensions modules will not be loaded
12491 until the various advanced Calc features are used. Since this mode only
12492 has effect when Calc is first loaded, @kbd{m x} is usually followed by
12493 @kbd{m m} to make the mode-setting permanent. To load all of Calc just
12494 once, rather than always in the future, you can press @kbd{M-# L}.
12497 @pindex calc-shift-prefix
12498 The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which
12499 all of Calc's letter prefix keys may be typed shifted as well as unshifted.
12500 If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often
12501 you might find it easier to turn this mode on so that you can type
12502 @kbd{A S} instead. When this mode is enabled, the commands that used to
12503 be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can
12504 now be invoked by pressing the shifted letter twice: @kbd{A A}. Note
12505 that the @kbd{v} prefix key always works both shifted and unshifted, and
12506 the @kbd{z} and @kbd{Z} prefix keys are always distinct. Also, the @kbd{h}
12507 prefix is not affected by this mode. Press @kbd{m S} again to disable
12508 shifted-prefix mode.
12510 @node Precision, Inverse and Hyperbolic, General Mode Commands, Mode Settings
12515 @pindex calc-precision
12516 @cindex Precision of calculations
12517 The @kbd{p} (@code{calc-precision}) command controls the precision to
12518 which floating-point calculations are carried. The precision must be
12519 at least 3 digits and may be arbitrarily high, within the limits of
12520 memory and time. This affects only floats: Integer and rational
12521 calculations are always carried out with as many digits as necessary.
12523 The @kbd{p} key prompts for the current precision. If you wish you
12524 can instead give the precision as a numeric prefix argument.
12526 Many internal calculations are carried to one or two digits higher
12527 precision than normal. Results are rounded down afterward to the
12528 current precision. Unless a special display mode has been selected,
12529 floats are always displayed with their full stored precision, i.e.,
12530 what you see is what you get. Reducing the current precision does not
12531 round values already on the stack, but those values will be rounded
12532 down before being used in any calculation. The @kbd{c 0} through
12533 @kbd{c 9} commands (@pxref{Conversions}) can be used to round an
12534 existing value to a new precision.
12536 @cindex Accuracy of calculations
12537 It is important to distinguish the concepts of @dfn{precision} and
12538 @dfn{accuracy}. In the normal usage of these words, the number
12539 123.4567 has a precision of 7 digits but an accuracy of 4 digits.
12540 The precision is the total number of digits not counting leading
12541 or trailing zeros (regardless of the position of the decimal point).
12542 The accuracy is simply the number of digits after the decimal point
12543 (again not counting trailing zeros). In Calc you control the precision,
12544 not the accuracy of computations. If you were to set the accuracy
12545 instead, then calculations like @samp{exp(100)} would generate many
12546 more digits than you would typically need, while @samp{exp(-100)} would
12547 probably round to zero! In Calc, both these computations give you
12548 exactly 12 (or the requested number of) significant digits.
12550 The only Calc features that deal with accuracy instead of precision
12551 are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}),
12552 and the rounding functions like @code{floor} and @code{round}
12553 (@pxref{Integer Truncation}). Also, @kbd{c 0} through @kbd{c 9}
12554 deal with both precision and accuracy depending on the magnitudes
12555 of the numbers involved.
12557 If you need to work with a particular fixed accuracy (say, dollars and
12558 cents with two digits after the decimal point), one solution is to work
12559 with integers and an ``implied'' decimal point. For example, $8.99
12560 divided by 6 would be entered @kbd{899 @key{RET} 6 /}, yielding 149.833
12561 (actually $1.49833 with our implied decimal point); pressing @kbd{R}
12562 would round this to 150 cents, i.e., $1.50.
12564 @xref{Floats}, for still more on floating-point precision and related
12567 @node Inverse and Hyperbolic, Calculation Modes, Precision, Mode Settings
12568 @section Inverse and Hyperbolic Flags
12572 @pindex calc-inverse
12573 There is no single-key equivalent to the @code{calc-arcsin} function.
12574 Instead, you must first press @kbd{I} (@code{calc-inverse}) to set
12575 the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}).
12576 The @kbd{I} key actually toggles the Inverse Flag. When this flag
12577 is set, the word @samp{Inv} appears in the mode line.
12580 @pindex calc-hyperbolic
12581 Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the
12582 Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}.
12583 If both of these flags are set at once, the effect will be
12584 @code{calc-arcsinh}. (The Hyperbolic flag is also used by some
12585 non-trigonometric commands; for example @kbd{H L} computes a base-10,
12586 instead of base-@mathit{e}, logarithm.)
12588 Command names like @code{calc-arcsin} are provided for completeness, and
12589 may be executed with @kbd{x} or @kbd{M-x}. Their effect is simply to
12590 toggle the Inverse and/or Hyperbolic flags and then execute the
12591 corresponding base command (@code{calc-sin} in this case).
12593 The Inverse and Hyperbolic flags apply only to the next Calculator
12594 command, after which they are automatically cleared. (They are also
12595 cleared if the next keystroke is not a Calc command.) Digits you
12596 type after @kbd{I} or @kbd{H} (or @kbd{K}) are treated as prefix
12597 arguments for the next command, not as numeric entries. The same
12598 is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means to
12599 subtract and keep arguments).
12601 The third Calc prefix flag, @kbd{K} (keep-arguments), is discussed
12602 elsewhere. @xref{Keep Arguments}.
12604 @node Calculation Modes, Simplification Modes, Inverse and Hyperbolic, Mode Settings
12605 @section Calculation Modes
12608 The commands in this section are two-key sequences beginning with
12609 the @kbd{m} prefix. (That's the letter @kbd{m}, not the @key{META} key.)
12610 The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere
12611 (@pxref{Algebraic Entry}).
12620 * Automatic Recomputation::
12621 * Working Message::
12624 @node Angular Modes, Polar Mode, Calculation Modes, Calculation Modes
12625 @subsection Angular Modes
12628 @cindex Angular mode
12629 The Calculator supports three notations for angles: radians, degrees,
12630 and degrees-minutes-seconds. When a number is presented to a function
12631 like @code{sin} that requires an angle, the current angular mode is
12632 used to interpret the number as either radians or degrees. If an HMS
12633 form is presented to @code{sin}, it is always interpreted as
12634 degrees-minutes-seconds.
12636 Functions that compute angles produce a number in radians, a number in
12637 degrees, or an HMS form depending on the current angular mode. If the
12638 result is a complex number and the current mode is HMS, the number is
12639 instead expressed in degrees. (Complex-number calculations would
12640 normally be done in Radians mode, though. Complex numbers are converted
12641 to degrees by calculating the complex result in radians and then
12642 multiplying by 180 over @cpi{}.)
12645 @pindex calc-radians-mode
12647 @pindex calc-degrees-mode
12649 @pindex calc-hms-mode
12650 The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}),
12651 and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode.
12652 The current angular mode is displayed on the Emacs mode line.
12653 The default angular mode is Degrees.
12655 @node Polar Mode, Fraction Mode, Angular Modes, Calculation Modes
12656 @subsection Polar Mode
12660 The Calculator normally ``prefers'' rectangular complex numbers in the
12661 sense that rectangular form is used when the proper form can not be
12662 decided from the input. This might happen by multiplying a rectangular
12663 number by a polar one, by taking the square root of a negative real
12664 number, or by entering @kbd{( 2 @key{SPC} 3 )}.
12667 @pindex calc-polar-mode
12668 The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number
12669 preference between rectangular and polar forms. In Polar mode, all
12670 of the above example situations would produce polar complex numbers.
12672 @node Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes
12673 @subsection Fraction Mode
12676 @cindex Fraction mode
12677 @cindex Division of integers
12678 Division of two integers normally yields a floating-point number if the
12679 result cannot be expressed as an integer. In some cases you would
12680 rather get an exact fractional answer. One way to accomplish this is
12681 to use the @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command, which
12682 divides the two integers on the top of the stack to produce a fraction:
12683 @kbd{6 @key{RET} 4 :} produces @expr{3:2} even though
12684 @kbd{6 @key{RET} 4 /} produces @expr{1.5}.
12687 @pindex calc-frac-mode
12688 To set the Calculator to produce fractional results for normal integer
12689 divisions, use the @kbd{m f} (@code{calc-frac-mode}) command.
12690 For example, @expr{8/4} produces @expr{2} in either mode,
12691 but @expr{6/4} produces @expr{3:2} in Fraction mode, @expr{1.5} in
12694 At any time you can use @kbd{c f} (@code{calc-float}) to convert a
12695 fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a
12696 float to a fraction. @xref{Conversions}.
12698 @node Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes
12699 @subsection Infinite Mode
12702 @cindex Infinite mode
12703 The Calculator normally treats results like @expr{1 / 0} as errors;
12704 formulas like this are left in unsimplified form. But Calc can be
12705 put into a mode where such calculations instead produce ``infinite''
12709 @pindex calc-infinite-mode
12710 The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode
12711 on and off. When the mode is off, infinities do not arise except
12712 in calculations that already had infinities as inputs. (One exception
12713 is that infinite open intervals like @samp{[0 .. inf)} can be
12714 generated; however, intervals closed at infinity (@samp{[0 .. inf]})
12715 will not be generated when Infinite mode is off.)
12717 With Infinite mode turned on, @samp{1 / 0} will generate @code{uinf},
12718 an undirected infinity. @xref{Infinities}, for a discussion of the
12719 difference between @code{inf} and @code{uinf}. Also, @expr{0 / 0}
12720 evaluates to @code{nan}, the ``indeterminate'' symbol. Various other
12721 functions can also return infinities in this mode; for example,
12722 @samp{ln(0) = -inf}, and @samp{gamma(-7) = uinf}. Once again,
12723 note that @samp{exp(inf) = inf} regardless of Infinite mode because
12724 this calculation has infinity as an input.
12726 @cindex Positive Infinite mode
12727 The @kbd{m i} command with a numeric prefix argument of zero,
12728 i.e., @kbd{C-u 0 m i}, turns on a Positive Infinite mode in
12729 which zero is treated as positive instead of being directionless.
12730 Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode.
12731 Note that zero never actually has a sign in Calc; there are no
12732 separate representations for @mathit{+0} and @mathit{-0}. Positive
12733 Infinite mode merely changes the interpretation given to the
12734 single symbol, @samp{0}. One consequence of this is that, while
12735 you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0}
12736 is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}.
12738 @node Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes
12739 @subsection Symbolic Mode
12742 @cindex Symbolic mode
12743 @cindex Inexact results
12744 Calculations are normally performed numerically wherever possible.
12745 For example, the @code{calc-sqrt} command, or @code{sqrt} function in an
12746 algebraic expression, produces a numeric answer if the argument is a
12747 number or a symbolic expression if the argument is an expression:
12748 @kbd{2 Q} pushes 1.4142 but @kbd{@key{'} x+1 @key{RET} Q} pushes @samp{sqrt(x+1)}.
12751 @pindex calc-symbolic-mode
12752 In @dfn{Symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode})
12753 command, functions which would produce inexact, irrational results are
12754 left in symbolic form. Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes
12758 @pindex calc-eval-num
12759 The shift-@kbd{N} (@code{calc-eval-num}) command evaluates numerically
12760 the expression at the top of the stack, by temporarily disabling
12761 @code{calc-symbolic-mode} and executing @kbd{=} (@code{calc-evaluate}).
12762 Given a numeric prefix argument, it also
12763 sets the floating-point precision to the specified value for the duration
12766 To evaluate a formula numerically without expanding the variables it
12767 contains, you can use the key sequence @kbd{m s a v m s} (this uses
12768 @code{calc-alg-evaluate}, which resimplifies but doesn't evaluate
12771 @node Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation Modes
12772 @subsection Matrix and Scalar Modes
12775 @cindex Matrix mode
12776 @cindex Scalar mode
12777 Calc sometimes makes assumptions during algebraic manipulation that
12778 are awkward or incorrect when vectors and matrices are involved.
12779 Calc has two modes, @dfn{Matrix mode} and @dfn{Scalar mode}, which
12780 modify its behavior around vectors in useful ways.
12783 @pindex calc-matrix-mode
12784 Press @kbd{m v} (@code{calc-matrix-mode}) once to enter Matrix mode.
12785 In this mode, all objects are assumed to be matrices unless provably
12786 otherwise. One major effect is that Calc will no longer consider
12787 multiplication to be commutative. (Recall that in matrix arithmetic,
12788 @samp{A*B} is not the same as @samp{B*A}.) This assumption affects
12789 rewrite rules and algebraic simplification. Another effect of this
12790 mode is that calculations that would normally produce constants like
12791 0 and 1 (e.g., @expr{a - a} and @expr{a / a}, respectively) will now
12792 produce function calls that represent ``generic'' zero or identity
12793 matrices: @samp{idn(0)}, @samp{idn(1)}. The @code{idn} function
12794 @samp{idn(@var{a},@var{n})} returns @var{a} times an @var{n}x@var{n}
12795 identity matrix; if @var{n} is omitted, it doesn't know what
12796 dimension to use and so the @code{idn} call remains in symbolic
12797 form. However, if this generic identity matrix is later combined
12798 with a matrix whose size is known, it will be converted into
12799 a true identity matrix of the appropriate size. On the other hand,
12800 if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc
12801 will assume it really was a scalar after all and produce, e.g., 3.
12803 Press @kbd{m v} a second time to get Scalar mode. Here, objects are
12804 assumed @emph{not} to be vectors or matrices unless provably so.
12805 For example, normally adding a variable to a vector, as in
12806 @samp{[x, y, z] + a}, will leave the sum in symbolic form because
12807 as far as Calc knows, @samp{a} could represent either a number or
12808 another 3-vector. In Scalar mode, @samp{a} is assumed to be a
12809 non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}.
12811 Press @kbd{m v} a third time to return to the normal mode of operation.
12813 If you press @kbd{m v} with a numeric prefix argument @var{n}, you
12814 get a special ``dimensioned'' Matrix mode in which matrices of
12815 unknown size are assumed to be @var{n}x@var{n} square matrices.
12816 Then, the function call @samp{idn(1)} will expand into an actual
12817 matrix rather than representing a ``generic'' matrix.
12819 @cindex Declaring scalar variables
12820 Of course these modes are approximations to the true state of
12821 affairs, which is probably that some quantities will be matrices
12822 and others will be scalars. One solution is to ``declare''
12823 certain variables or functions to be scalar-valued.
12824 @xref{Declarations}, to see how to make declarations in Calc.
12826 There is nothing stopping you from declaring a variable to be
12827 scalar and then storing a matrix in it; however, if you do, the
12828 results you get from Calc may not be valid. Suppose you let Calc
12829 get the result @samp{[x+a, y+a, z+a]} shown above, and then stored
12830 @samp{[1, 2, 3]} in @samp{a}. The result would not be the same as
12831 for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken
12832 your earlier promise to Calc that @samp{a} would be scalar.
12834 Another way to mix scalars and matrices is to use selections
12835 (@pxref{Selecting Subformulas}). Use Matrix mode when operating on
12836 your formula normally; then, to apply Scalar mode to a certain part
12837 of the formula without affecting the rest just select that part,
12838 change into Scalar mode and press @kbd{=} to resimplify the part
12839 under this mode, then change back to Matrix mode before deselecting.
12841 @node Automatic Recomputation, Working Message, Matrix Mode, Calculation Modes
12842 @subsection Automatic Recomputation
12845 The @dfn{evaluates-to} operator, @samp{=>}, has the special
12846 property that any @samp{=>} formulas on the stack are recomputed
12847 whenever variable values or mode settings that might affect them
12848 are changed. @xref{Evaluates-To Operator}.
12851 @pindex calc-auto-recompute
12852 The @kbd{m C} (@code{calc-auto-recompute}) command turns this
12853 automatic recomputation on and off. If you turn it off, Calc will
12854 not update @samp{=>} operators on the stack (nor those in the
12855 attached Embedded mode buffer, if there is one). They will not
12856 be updated unless you explicitly do so by pressing @kbd{=} or until
12857 you press @kbd{m C} to turn recomputation back on. (While automatic
12858 recomputation is off, you can think of @kbd{m C m C} as a command
12859 to update all @samp{=>} operators while leaving recomputation off.)
12861 To update @samp{=>} operators in an Embedded buffer while
12862 automatic recomputation is off, use @w{@kbd{M-# u}}.
12863 @xref{Embedded Mode}.
12865 @node Working Message, , Automatic Recomputation, Calculation Modes
12866 @subsection Working Messages
12869 @cindex Performance
12870 @cindex Working messages
12871 Since the Calculator is written entirely in Emacs Lisp, which is not
12872 designed for heavy numerical work, many operations are quite slow.
12873 The Calculator normally displays the message @samp{Working...} in the
12874 echo area during any command that may be slow. In addition, iterative
12875 operations such as square roots and trigonometric functions display the
12876 intermediate result at each step. Both of these types of messages can
12877 be disabled if you find them distracting.
12880 @pindex calc-working
12881 Type @kbd{m w} (@code{calc-working}) with a numeric prefix of 0 to
12882 disable all ``working'' messages. Use a numeric prefix of 1 to enable
12883 only the plain @samp{Working...} message. Use a numeric prefix of 2 to
12884 see intermediate results as well. With no numeric prefix this displays
12887 While it may seem that the ``working'' messages will slow Calc down
12888 considerably, experiments have shown that their impact is actually
12889 quite small. But if your terminal is slow you may find that it helps
12890 to turn the messages off.
12892 @node Simplification Modes, Declarations, Calculation Modes, Mode Settings
12893 @section Simplification Modes
12896 The current @dfn{simplification mode} controls how numbers and formulas
12897 are ``normalized'' when being taken from or pushed onto the stack.
12898 Some normalizations are unavoidable, such as rounding floating-point
12899 results to the current precision, and reducing fractions to simplest
12900 form. Others, such as simplifying a formula like @expr{a+a} (or @expr{2+3}),
12901 are done by default but can be turned off when necessary.
12903 When you press a key like @kbd{+} when @expr{2} and @expr{3} are on the
12904 stack, Calc pops these numbers, normalizes them, creates the formula
12905 @expr{2+3}, normalizes it, and pushes the result. Of course the standard
12906 rules for normalizing @expr{2+3} will produce the result @expr{5}.
12908 Simplification mode commands consist of the lower-case @kbd{m} prefix key
12909 followed by a shifted letter.
12912 @pindex calc-no-simplify-mode
12913 The @kbd{m O} (@code{calc-no-simplify-mode}) command turns off all optional
12914 simplifications. These would leave a formula like @expr{2+3} alone. In
12915 fact, nothing except simple numbers are ever affected by normalization
12919 @pindex calc-num-simplify-mode
12920 The @kbd{m N} (@code{calc-num-simplify-mode}) command turns off simplification
12921 of any formulas except those for which all arguments are constants. For
12922 example, @expr{1+2} is simplified to @expr{3}, and @expr{a+(2-2)} is
12923 simplified to @expr{a+0} but no further, since one argument of the sum
12924 is not a constant. Unfortunately, @expr{(a+2)-2} is @emph{not} simplified
12925 because the top-level @samp{-} operator's arguments are not both
12926 constant numbers (one of them is the formula @expr{a+2}).
12927 A constant is a number or other numeric object (such as a constant
12928 error form or modulo form), or a vector all of whose
12929 elements are constant.
12932 @pindex calc-default-simplify-mode
12933 The @kbd{m D} (@code{calc-default-simplify-mode}) command restores the
12934 default simplifications for all formulas. This includes many easy and
12935 fast algebraic simplifications such as @expr{a+0} to @expr{a}, and
12936 @expr{a + 2 a} to @expr{3 a}, as well as evaluating functions like
12937 @expr{@tfn{deriv}(x^2, x)} to @expr{2 x}.
12940 @pindex calc-bin-simplify-mode
12941 The @kbd{m B} (@code{calc-bin-simplify-mode}) mode applies the default
12942 simplifications to a result and then, if the result is an integer,
12943 uses the @kbd{b c} (@code{calc-clip}) command to clip the integer according
12944 to the current binary word size. @xref{Binary Functions}. Real numbers
12945 are rounded to the nearest integer and then clipped; other kinds of
12946 results (after the default simplifications) are left alone.
12949 @pindex calc-alg-simplify-mode
12950 The @kbd{m A} (@code{calc-alg-simplify-mode}) mode does algebraic
12951 simplification; it applies all the default simplifications, and also
12952 the more powerful (and slower) simplifications made by @kbd{a s}
12953 (@code{calc-simplify}). @xref{Algebraic Simplifications}.
12956 @pindex calc-ext-simplify-mode
12957 The @kbd{m E} (@code{calc-ext-simplify-mode}) mode does ``extended''
12958 algebraic simplification, as by the @kbd{a e} (@code{calc-simplify-extended})
12959 command. @xref{Unsafe Simplifications}.
12962 @pindex calc-units-simplify-mode
12963 The @kbd{m U} (@code{calc-units-simplify-mode}) mode does units
12964 simplification; it applies the command @kbd{u s}
12965 (@code{calc-simplify-units}), which in turn
12966 is a superset of @kbd{a s}. In this mode, variable names which
12967 are identifiable as unit names (like @samp{mm} for ``millimeters'')
12968 are simplified with their unit definitions in mind.
12970 A common technique is to set the simplification mode down to the lowest
12971 amount of simplification you will allow to be applied automatically, then
12972 use manual commands like @kbd{a s} and @kbd{c c} (@code{calc-clean}) to
12973 perform higher types of simplifications on demand. @xref{Algebraic
12974 Definitions}, for another sample use of No-Simplification mode.
12976 @node Declarations, Display Modes, Simplification Modes, Mode Settings
12977 @section Declarations
12980 A @dfn{declaration} is a statement you make that promises you will
12981 use a certain variable or function in a restricted way. This may
12982 give Calc the freedom to do things that it couldn't do if it had to
12983 take the fully general situation into account.
12986 * Declaration Basics::
12987 * Kinds of Declarations::
12988 * Functions for Declarations::
12991 @node Declaration Basics, Kinds of Declarations, Declarations, Declarations
12992 @subsection Declaration Basics
12996 @pindex calc-declare-variable
12997 The @kbd{s d} (@code{calc-declare-variable}) command is the easiest
12998 way to make a declaration for a variable. This command prompts for
12999 the variable name, then prompts for the declaration. The default
13000 at the declaration prompt is the previous declaration, if any.
13001 You can edit this declaration, or press @kbd{C-k} to erase it and
13002 type a new declaration. (Or, erase it and press @key{RET} to clear
13003 the declaration, effectively ``undeclaring'' the variable.)
13005 A declaration is in general a vector of @dfn{type symbols} and
13006 @dfn{range} values. If there is only one type symbol or range value,
13007 you can write it directly rather than enclosing it in a vector.
13008 For example, @kbd{s d foo @key{RET} real @key{RET}} declares @code{foo} to
13009 be a real number, and @kbd{s d bar @key{RET} [int, const, [1..6]] @key{RET}}
13010 declares @code{bar} to be a constant integer between 1 and 6.
13011 (Actually, you can omit the outermost brackets and Calc will
13012 provide them for you: @kbd{s d bar @key{RET} int, const, [1..6] @key{RET}}.)
13014 @cindex @code{Decls} variable
13016 Declarations in Calc are kept in a special variable called @code{Decls}.
13017 This variable encodes the set of all outstanding declarations in
13018 the form of a matrix. Each row has two elements: A variable or
13019 vector of variables declared by that row, and the declaration
13020 specifier as described above. You can use the @kbd{s D} command to
13021 edit this variable if you wish to see all the declarations at once.
13022 @xref{Operations on Variables}, for a description of this command
13023 and the @kbd{s p} command that allows you to save your declarations
13024 permanently if you wish.
13026 Items being declared can also be function calls. The arguments in
13027 the call are ignored; the effect is to say that this function returns
13028 values of the declared type for any valid arguments. The @kbd{s d}
13029 command declares only variables, so if you wish to make a function
13030 declaration you will have to edit the @code{Decls} matrix yourself.
13032 For example, the declaration matrix
13038 [ f(1,2,3), [0 .. inf) ] ]
13043 declares that @code{foo} represents a real number, @code{j}, @code{k}
13044 and @code{n} represent integers, and the function @code{f} always
13045 returns a real number in the interval shown.
13048 If there is a declaration for the variable @code{All}, then that
13049 declaration applies to all variables that are not otherwise declared.
13050 It does not apply to function names. For example, using the row
13051 @samp{[All, real]} says that all your variables are real unless they
13052 are explicitly declared without @code{real} in some other row.
13053 The @kbd{s d} command declares @code{All} if you give a blank
13054 response to the variable-name prompt.
13056 @node Kinds of Declarations, Functions for Declarations, Declaration Basics, Declarations
13057 @subsection Kinds of Declarations
13060 The type-specifier part of a declaration (that is, the second prompt
13061 in the @kbd{s d} command) can be a type symbol, an interval, or a
13062 vector consisting of zero or more type symbols followed by zero or
13063 more intervals or numbers that represent the set of possible values
13068 [ [ a, [1, 2, 3, 4, 5] ]
13070 [ c, [int, 1 .. 5] ] ]
13074 Here @code{a} is declared to contain one of the five integers shown;
13075 @code{b} is any number in the interval from 1 to 5 (any real number
13076 since we haven't specified), and @code{c} is any integer in that
13077 interval. Thus the declarations for @code{a} and @code{c} are
13078 nearly equivalent (see below).
13080 The type-specifier can be the empty vector @samp{[]} to say that
13081 nothing is known about a given variable's value. This is the same
13082 as not declaring the variable at all except that it overrides any
13083 @code{All} declaration which would otherwise apply.
13085 The initial value of @code{Decls} is the empty vector @samp{[]}.
13086 If @code{Decls} has no stored value or if the value stored in it
13087 is not valid, it is ignored and there are no declarations as far
13088 as Calc is concerned. (The @kbd{s d} command will replace such a
13089 malformed value with a fresh empty matrix, @samp{[]}, before recording
13090 the new declaration.) Unrecognized type symbols are ignored.
13092 The following type symbols describe what sorts of numbers will be
13093 stored in a variable:
13099 Numerical integers. (Integers or integer-valued floats.)
13101 Fractions. (Rational numbers which are not integers.)
13103 Rational numbers. (Either integers or fractions.)
13105 Floating-point numbers.
13107 Real numbers. (Integers, fractions, or floats. Actually,
13108 intervals and error forms with real components also count as
13111 Positive real numbers. (Strictly greater than zero.)
13113 Nonnegative real numbers. (Greater than or equal to zero.)
13115 Numbers. (Real or complex.)
13118 Calc uses this information to determine when certain simplifications
13119 of formulas are safe. For example, @samp{(x^y)^z} cannot be
13120 simplified to @samp{x^(y z)} in general; for example,
13121 @samp{((-3)^2)^1:2} is 3, but @samp{(-3)^(2*1:2) = (-3)^1} is @mathit{-3}.
13122 However, this simplification @emph{is} safe if @code{z} is known
13123 to be an integer, or if @code{x} is known to be a nonnegative
13124 real number. If you have given declarations that allow Calc to
13125 deduce either of these facts, Calc will perform this simplification
13128 Calc can apply a certain amount of logic when using declarations.
13129 For example, @samp{(x^y)^(2n+1)} will be simplified if @code{n}
13130 has been declared @code{int}; Calc knows that an integer times an
13131 integer, plus an integer, must always be an integer. (In fact,
13132 Calc would simplify @samp{(-x)^(2n+1)} to @samp{-(x^(2n+1))} since
13133 it is able to determine that @samp{2n+1} must be an odd integer.)
13135 Similarly, @samp{(abs(x)^y)^z} will be simplified to @samp{abs(x)^(y z)}
13136 because Calc knows that the @code{abs} function always returns a
13137 nonnegative real. If you had a @code{myabs} function that also had
13138 this property, you could get Calc to recognize it by adding the row
13139 @samp{[myabs(), nonneg]} to the @code{Decls} matrix.
13141 One instance of this simplification is @samp{sqrt(x^2)} (since the
13142 @code{sqrt} function is effectively a one-half power). Normally
13143 Calc leaves this formula alone. After the command
13144 @kbd{s d x @key{RET} real @key{RET}}, however, it can simplify the formula to
13145 @samp{abs(x)}. And after @kbd{s d x @key{RET} nonneg @key{RET}}, Calc can
13146 simplify this formula all the way to @samp{x}.
13148 If there are any intervals or real numbers in the type specifier,
13149 they comprise the set of possible values that the variable or
13150 function being declared can have. In particular, the type symbol
13151 @code{real} is effectively the same as the range @samp{[-inf .. inf]}
13152 (note that infinity is included in the range of possible values);
13153 @code{pos} is the same as @samp{(0 .. inf]}, and @code{nonneg} is
13154 the same as @samp{[0 .. inf]}. Saying @samp{[real, [-5 .. 5]]} is
13155 redundant because the fact that the variable is real can be
13156 deduced just from the interval, but @samp{[int, [-5 .. 5]]} and
13157 @samp{[rat, [-5 .. 5]]} are useful combinations.
13159 Note that the vector of intervals or numbers is in the same format
13160 used by Calc's set-manipulation commands. @xref{Set Operations}.
13162 The type specifier @samp{[1, 2, 3]} is equivalent to
13163 @samp{[numint, 1, 2, 3]}, @emph{not} to @samp{[int, 1, 2, 3]}.
13164 In other words, the range of possible values means only that
13165 the variable's value must be numerically equal to a number in
13166 that range, but not that it must be equal in type as well.
13167 Calc's set operations act the same way; @samp{in(2, [1., 2., 3.])}
13168 and @samp{in(1.5, [1:2, 3:2, 5:2])} both report ``true.''
13170 If you use a conflicting combination of type specifiers, the
13171 results are unpredictable. An example is @samp{[pos, [0 .. 5]]},
13172 where the interval does not lie in the range described by the
13175 ``Real'' declarations mostly affect simplifications involving powers
13176 like the one described above. Another case where they are used
13177 is in the @kbd{a P} command which returns a list of all roots of a
13178 polynomial; if the variable has been declared real, only the real
13179 roots (if any) will be included in the list.
13181 ``Integer'' declarations are used for simplifications which are valid
13182 only when certain values are integers (such as @samp{(x^y)^z}
13185 Another command that makes use of declarations is @kbd{a s}, when
13186 simplifying equations and inequalities. It will cancel @code{x}
13187 from both sides of @samp{a x = b x} only if it is sure @code{x}
13188 is non-zero, say, because it has a @code{pos} declaration.
13189 To declare specifically that @code{x} is real and non-zero,
13190 use @samp{[[-inf .. 0), (0 .. inf]]}. (There is no way in the
13191 current notation to say that @code{x} is nonzero but not necessarily
13192 real.) The @kbd{a e} command does ``unsafe'' simplifications,
13193 including cancelling @samp{x} from the equation when @samp{x} is
13194 not known to be nonzero.
13196 Another set of type symbols distinguish between scalars and vectors.
13200 The value is not a vector.
13202 The value is a vector.
13204 The value is a matrix (a rectangular vector of vectors).
13207 These type symbols can be combined with the other type symbols
13208 described above; @samp{[int, matrix]} describes an object which
13209 is a matrix of integers.
13211 Scalar/vector declarations are used to determine whether certain
13212 algebraic operations are safe. For example, @samp{[a, b, c] + x}
13213 is normally not simplified to @samp{[a + x, b + x, c + x]}, but
13214 it will be if @code{x} has been declared @code{scalar}. On the
13215 other hand, multiplication is usually assumed to be commutative,
13216 but the terms in @samp{x y} will never be exchanged if both @code{x}
13217 and @code{y} are known to be vectors or matrices. (Calc currently
13218 never distinguishes between @code{vector} and @code{matrix}
13221 @xref{Matrix Mode}, for a discussion of Matrix mode and
13222 Scalar mode, which are similar to declaring @samp{[All, matrix]}
13223 or @samp{[All, scalar]} but much more convenient.
13225 One more type symbol that is recognized is used with the @kbd{H a d}
13226 command for taking total derivatives of a formula. @xref{Calculus}.
13230 The value is a constant with respect to other variables.
13233 Calc does not check the declarations for a variable when you store
13234 a value in it. However, storing @mathit{-3.5} in a variable that has
13235 been declared @code{pos}, @code{int}, or @code{matrix} may have
13236 unexpected effects; Calc may evaluate @samp{sqrt(x^2)} to @expr{3.5}
13237 if it substitutes the value first, or to @expr{-3.5} if @code{x}
13238 was declared @code{pos} and the formula @samp{sqrt(x^2)} is
13239 simplified to @samp{x} before the value is substituted. Before
13240 using a variable for a new purpose, it is best to use @kbd{s d}
13241 or @kbd{s D} to check to make sure you don't still have an old
13242 declaration for the variable that will conflict with its new meaning.
13244 @node Functions for Declarations, , Kinds of Declarations, Declarations
13245 @subsection Functions for Declarations
13248 Calc has a set of functions for accessing the current declarations
13249 in a convenient manner. These functions return 1 if the argument
13250 can be shown to have the specified property, or 0 if the argument
13251 can be shown @emph{not} to have that property; otherwise they are
13252 left unevaluated. These functions are suitable for use with rewrite
13253 rules (@pxref{Conditional Rewrite Rules}) or programming constructs
13254 (@pxref{Conditionals in Macros}). They can be entered only using
13255 algebraic notation. @xref{Logical Operations}, for functions
13256 that perform other tests not related to declarations.
13258 For example, @samp{dint(17)} returns 1 because 17 is an integer, as
13259 do @samp{dint(n)} and @samp{dint(2 n - 3)} if @code{n} has been declared
13260 @code{int}, but @samp{dint(2.5)} and @samp{dint(n + 0.5)} return 0.
13261 Calc consults knowledge of its own built-in functions as well as your
13262 own declarations: @samp{dint(floor(x))} returns 1.
13276 The @code{dint} function checks if its argument is an integer.
13277 The @code{dnatnum} function checks if its argument is a natural
13278 number, i.e., a nonnegative integer. The @code{dnumint} function
13279 checks if its argument is numerically an integer, i.e., either an
13280 integer or an integer-valued float. Note that these and the other
13281 data type functions also accept vectors or matrices composed of
13282 suitable elements, and that real infinities @samp{inf} and @samp{-inf}
13283 are considered to be integers for the purposes of these functions.
13289 The @code{drat} function checks if its argument is rational, i.e.,
13290 an integer or fraction. Infinities count as rational, but intervals
13291 and error forms do not.
13297 The @code{dreal} function checks if its argument is real. This
13298 includes integers, fractions, floats, real error forms, and intervals.
13304 The @code{dimag} function checks if its argument is imaginary,
13305 i.e., is mathematically equal to a real number times @expr{i}.
13319 The @code{dpos} function checks for positive (but nonzero) reals.
13320 The @code{dneg} function checks for negative reals. The @code{dnonneg}
13321 function checks for nonnegative reals, i.e., reals greater than or
13322 equal to zero. Note that the @kbd{a s} command can simplify an
13323 expression like @expr{x > 0} to 1 or 0 using @code{dpos}, and that
13324 @kbd{a s} is effectively applied to all conditions in rewrite rules,
13325 so the actual functions @code{dpos}, @code{dneg}, and @code{dnonneg}
13326 are rarely necessary.
13332 The @code{dnonzero} function checks that its argument is nonzero.
13333 This includes all nonzero real or complex numbers, all intervals that
13334 do not include zero, all nonzero modulo forms, vectors all of whose
13335 elements are nonzero, and variables or formulas whose values can be
13336 deduced to be nonzero. It does not include error forms, since they
13337 represent values which could be anything including zero. (This is
13338 also the set of objects considered ``true'' in conditional contexts.)
13348 The @code{deven} function returns 1 if its argument is known to be
13349 an even integer (or integer-valued float); it returns 0 if its argument
13350 is known not to be even (because it is known to be odd or a non-integer).
13351 The @kbd{a s} command uses this to simplify a test of the form
13352 @samp{x % 2 = 0}. There is also an analogous @code{dodd} function.
13358 The @code{drange} function returns a set (an interval or a vector
13359 of intervals and/or numbers; @pxref{Set Operations}) that describes
13360 the set of possible values of its argument. If the argument is
13361 a variable or a function with a declaration, the range is copied
13362 from the declaration. Otherwise, the possible signs of the
13363 expression are determined using a method similar to @code{dpos},
13364 etc., and a suitable set like @samp{[0 .. inf]} is returned. If
13365 the expression is not provably real, the @code{drange} function
13366 remains unevaluated.
13372 The @code{dscalar} function returns 1 if its argument is provably
13373 scalar, or 0 if its argument is provably non-scalar. It is left
13374 unevaluated if this cannot be determined. (If Matrix mode or Scalar
13375 mode is in effect, this function returns 1 or 0, respectively,
13376 if it has no other information.) When Calc interprets a condition
13377 (say, in a rewrite rule) it considers an unevaluated formula to be
13378 ``false.'' Thus, @samp{dscalar(a)} is ``true'' only if @code{a} is
13379 provably scalar, and @samp{!dscalar(a)} is ``true'' only if @code{a}
13380 is provably non-scalar; both are ``false'' if there is insufficient
13381 information to tell.
13383 @node Display Modes, Language Modes, Declarations, Mode Settings
13384 @section Display Modes
13387 The commands in this section are two-key sequences beginning with the
13388 @kbd{d} prefix. The @kbd{d l} (@code{calc-line-numbering}) and @kbd{d b}
13389 (@code{calc-line-breaking}) commands are described elsewhere;
13390 @pxref{Stack Basics} and @pxref{Normal Language Modes}, respectively.
13391 Display formats for vectors and matrices are also covered elsewhere;
13392 @pxref{Vector and Matrix Formats}.
13394 One thing all display modes have in common is their treatment of the
13395 @kbd{H} prefix. This prefix causes any mode command that would normally
13396 refresh the stack to leave the stack display alone. The word ``Dirty''
13397 will appear in the mode line when Calc thinks the stack display may not
13398 reflect the latest mode settings.
13400 @kindex d @key{RET}
13401 @pindex calc-refresh-top
13402 The @kbd{d @key{RET}} (@code{calc-refresh-top}) command reformats the
13403 top stack entry according to all the current modes. Positive prefix
13404 arguments reformat the top @var{n} entries; negative prefix arguments
13405 reformat the specified entry, and a prefix of zero is equivalent to
13406 @kbd{d @key{SPC}} (@code{calc-refresh}), which reformats the entire stack.
13407 For example, @kbd{H d s M-2 d @key{RET}} changes to scientific notation
13408 but reformats only the top two stack entries in the new mode.
13410 The @kbd{I} prefix has another effect on the display modes. The mode
13411 is set only temporarily; the top stack entry is reformatted according
13412 to that mode, then the original mode setting is restored. In other
13413 words, @kbd{I d s} is equivalent to @kbd{H d s d @key{RET} H d (@var{old mode})}.
13417 * Grouping Digits::
13419 * Complex Formats::
13420 * Fraction Formats::
13423 * Truncating the Stack::
13428 @node Radix Modes, Grouping Digits, Display Modes, Display Modes
13429 @subsection Radix Modes
13432 @cindex Radix display
13433 @cindex Non-decimal numbers
13434 @cindex Decimal and non-decimal numbers
13435 Calc normally displays numbers in decimal (@dfn{base-10} or @dfn{radix-10})
13436 notation. Calc can actually display in any radix from two (binary) to 36.
13437 When the radix is above 10, the letters @code{A} to @code{Z} are used as
13438 digits. When entering such a number, letter keys are interpreted as
13439 potential digits rather than terminating numeric entry mode.
13445 @cindex Hexadecimal integers
13446 @cindex Octal integers
13447 The key sequences @kbd{d 2}, @kbd{d 8}, @kbd{d 6}, and @kbd{d 0} select
13448 binary, octal, hexadecimal, and decimal as the current display radix,
13449 respectively. Numbers can always be entered in any radix, though the
13450 current radix is used as a default if you press @kbd{#} without any initial
13451 digits. A number entered without a @kbd{#} is @emph{always} interpreted
13456 To set the radix generally, use @kbd{d r} (@code{calc-radix}) and enter
13457 an integer from 2 to 36. You can specify the radix as a numeric prefix
13458 argument; otherwise you will be prompted for it.
13461 @pindex calc-leading-zeros
13462 @cindex Leading zeros
13463 Integers normally are displayed with however many digits are necessary to
13464 represent the integer and no more. The @kbd{d z} (@code{calc-leading-zeros})
13465 command causes integers to be padded out with leading zeros according to the
13466 current binary word size. (@xref{Binary Functions}, for a discussion of
13467 word size.) If the absolute value of the word size is @expr{w}, all integers
13468 are displayed with at least enough digits to represent
13469 @texline @math{2^w-1}
13470 @infoline @expr{(2^w)-1}
13471 in the current radix. (Larger integers will still be displayed in their
13474 @node Grouping Digits, Float Formats, Radix Modes, Display Modes
13475 @subsection Grouping Digits
13479 @pindex calc-group-digits
13480 @cindex Grouping digits
13481 @cindex Digit grouping
13482 Long numbers can be hard to read if they have too many digits. For
13483 example, the factorial of 30 is 33 digits long! Press @kbd{d g}
13484 (@code{calc-group-digits}) to enable @dfn{Grouping} mode, in which digits
13485 are displayed in clumps of 3 or 4 (depending on the current radix)
13486 separated by commas.
13488 The @kbd{d g} command toggles grouping on and off.
13489 With a numeric prefix of 0, this command displays the current state of
13490 the grouping flag; with an argument of minus one it disables grouping;
13491 with a positive argument @expr{N} it enables grouping on every @expr{N}
13492 digits. For floating-point numbers, grouping normally occurs only
13493 before the decimal point. A negative prefix argument @expr{-N} enables
13494 grouping every @expr{N} digits both before and after the decimal point.
13497 @pindex calc-group-char
13498 The @kbd{d ,} (@code{calc-group-char}) command allows you to choose any
13499 character as the grouping separator. The default is the comma character.
13500 If you find it difficult to read vectors of large integers grouped with
13501 commas, you may wish to use spaces or some other character instead.
13502 This command takes the next character you type, whatever it is, and
13503 uses it as the digit separator. As a special case, @kbd{d , \} selects
13504 @samp{\,} (@TeX{}'s thin-space symbol) as the digit separator.
13506 Please note that grouped numbers will not generally be parsed correctly
13507 if re-read in textual form, say by the use of @kbd{M-# y} and @kbd{M-# g}.
13508 (@xref{Kill and Yank}, for details on these commands.) One exception is
13509 the @samp{\,} separator, which doesn't interfere with parsing because it
13510 is ignored by @TeX{} language mode.
13512 @node Float Formats, Complex Formats, Grouping Digits, Display Modes
13513 @subsection Float Formats
13516 Floating-point quantities are normally displayed in standard decimal
13517 form, with scientific notation used if the exponent is especially high
13518 or low. All significant digits are normally displayed. The commands
13519 in this section allow you to choose among several alternative display
13520 formats for floats.
13523 @pindex calc-normal-notation
13524 The @kbd{d n} (@code{calc-normal-notation}) command selects the normal
13525 display format. All significant figures in a number are displayed.
13526 With a positive numeric prefix, numbers are rounded if necessary to
13527 that number of significant digits. With a negative numerix prefix,
13528 the specified number of significant digits less than the current
13529 precision is used. (Thus @kbd{C-u -2 d n} displays 10 digits if the
13530 current precision is 12.)
13533 @pindex calc-fix-notation
13534 The @kbd{d f} (@code{calc-fix-notation}) command selects fixed-point
13535 notation. The numeric argument is the number of digits after the
13536 decimal point, zero or more. This format will relax into scientific
13537 notation if a nonzero number would otherwise have been rounded all the
13538 way to zero. Specifying a negative number of digits is the same as
13539 for a positive number, except that small nonzero numbers will be rounded
13540 to zero rather than switching to scientific notation.
13543 @pindex calc-sci-notation
13544 @cindex Scientific notation, display of
13545 The @kbd{d s} (@code{calc-sci-notation}) command selects scientific
13546 notation. A positive argument sets the number of significant figures
13547 displayed, of which one will be before and the rest after the decimal
13548 point. A negative argument works the same as for @kbd{d n} format.
13549 The default is to display all significant digits.
13552 @pindex calc-eng-notation
13553 @cindex Engineering notation, display of
13554 The @kbd{d e} (@code{calc-eng-notation}) command selects engineering
13555 notation. This is similar to scientific notation except that the
13556 exponent is rounded down to a multiple of three, with from one to three
13557 digits before the decimal point. An optional numeric prefix sets the
13558 number of significant digits to display, as for @kbd{d s}.
13560 It is important to distinguish between the current @emph{precision} and
13561 the current @emph{display format}. After the commands @kbd{C-u 10 p}
13562 and @kbd{C-u 6 d n} the Calculator computes all results to ten
13563 significant figures but displays only six. (In fact, intermediate
13564 calculations are often carried to one or two more significant figures,
13565 but values placed on the stack will be rounded down to ten figures.)
13566 Numbers are never actually rounded to the display precision for storage,
13567 except by commands like @kbd{C-k} and @kbd{M-# y} which operate on the
13568 actual displayed text in the Calculator buffer.
13571 @pindex calc-point-char
13572 The @kbd{d .} (@code{calc-point-char}) command selects the character used
13573 as a decimal point. Normally this is a period; users in some countries
13574 may wish to change this to a comma. Note that this is only a display
13575 style; on entry, periods must always be used to denote floating-point
13576 numbers, and commas to separate elements in a list.
13578 @node Complex Formats, Fraction Formats, Float Formats, Display Modes
13579 @subsection Complex Formats
13583 @pindex calc-complex-notation
13584 There are three supported notations for complex numbers in rectangular
13585 form. The default is as a pair of real numbers enclosed in parentheses
13586 and separated by a comma: @samp{(a,b)}. The @kbd{d c}
13587 (@code{calc-complex-notation}) command selects this style.
13590 @pindex calc-i-notation
13592 @pindex calc-j-notation
13593 The other notations are @kbd{d i} (@code{calc-i-notation}), in which
13594 numbers are displayed in @samp{a+bi} form, and @kbd{d j}
13595 (@code{calc-j-notation}) which displays the form @samp{a+bj} preferred
13596 in some disciplines.
13598 @cindex @code{i} variable
13600 Complex numbers are normally entered in @samp{(a,b)} format.
13601 If you enter @samp{2+3i} as an algebraic formula, it will be stored as
13602 the formula @samp{2 + 3 * i}. However, if you use @kbd{=} to evaluate
13603 this formula and you have not changed the variable @samp{i}, the @samp{i}
13604 will be interpreted as @samp{(0,1)} and the formula will be simplified
13605 to @samp{(2,3)}. Other commands (like @code{calc-sin}) will @emph{not}
13606 interpret the formula @samp{2 + 3 * i} as a complex number.
13607 @xref{Variables}, under ``special constants.''
13609 @node Fraction Formats, HMS Formats, Complex Formats, Display Modes
13610 @subsection Fraction Formats
13614 @pindex calc-over-notation
13615 Display of fractional numbers is controlled by the @kbd{d o}
13616 (@code{calc-over-notation}) command. By default, a number like
13617 eight thirds is displayed in the form @samp{8:3}. The @kbd{d o} command
13618 prompts for a one- or two-character format. If you give one character,
13619 that character is used as the fraction separator. Common separators are
13620 @samp{:} and @samp{/}. (During input of numbers, the @kbd{:} key must be
13621 used regardless of the display format; in particular, the @kbd{/} is used
13622 for RPN-style division, @emph{not} for entering fractions.)
13624 If you give two characters, fractions use ``integer-plus-fractional-part''
13625 notation. For example, the format @samp{+/} would display eight thirds
13626 as @samp{2+2/3}. If two colons are present in a number being entered,
13627 the number is interpreted in this form (so that the entries @kbd{2:2:3}
13628 and @kbd{8:3} are equivalent).
13630 It is also possible to follow the one- or two-character format with
13631 a number. For example: @samp{:10} or @samp{+/3}. In this case,
13632 Calc adjusts all fractions that are displayed to have the specified
13633 denominator, if possible. Otherwise it adjusts the denominator to
13634 be a multiple of the specified value. For example, in @samp{:6} mode
13635 the fraction @expr{1:6} will be unaffected, but @expr{2:3} will be
13636 displayed as @expr{4:6}, @expr{1:2} will be displayed as @expr{3:6},
13637 and @expr{1:8} will be displayed as @expr{3:24}. Integers are also
13638 affected by this mode: 3 is displayed as @expr{18:6}. Note that the
13639 format @samp{:1} writes fractions the same as @samp{:}, but it writes
13640 integers as @expr{n:1}.
13642 The fraction format does not affect the way fractions or integers are
13643 stored, only the way they appear on the screen. The fraction format
13644 never affects floats.
13646 @node HMS Formats, Date Formats, Fraction Formats, Display Modes
13647 @subsection HMS Formats
13651 @pindex calc-hms-notation
13652 The @kbd{d h} (@code{calc-hms-notation}) command controls the display of
13653 HMS (hours-minutes-seconds) forms. It prompts for a string which
13654 consists basically of an ``hours'' marker, optional punctuation, a
13655 ``minutes'' marker, more optional punctuation, and a ``seconds'' marker.
13656 Punctuation is zero or more spaces, commas, or semicolons. The hours
13657 marker is one or more non-punctuation characters. The minutes and
13658 seconds markers must be single non-punctuation characters.
13660 The default HMS format is @samp{@@ ' "}, producing HMS values of the form
13661 @samp{23@@ 30' 15.75"}. The format @samp{deg, ms} would display this same
13662 value as @samp{23deg, 30m15.75s}. During numeric entry, the @kbd{h} or @kbd{o}
13663 keys are recognized as synonyms for @kbd{@@} regardless of display format.
13664 The @kbd{m} and @kbd{s} keys are recognized as synonyms for @kbd{'} and
13665 @kbd{"}, respectively, but only if an @kbd{@@} (or @kbd{h} or @kbd{o}) has
13666 already been typed; otherwise, they have their usual meanings
13667 (@kbd{m-} prefix and @kbd{s-} prefix). Thus, @kbd{5 "}, @kbd{0 @@ 5 "}, and
13668 @kbd{0 h 5 s} are some of the ways to enter the quantity ``five seconds.''
13669 The @kbd{'} key is recognized as ``minutes'' only if @kbd{@@} (or @kbd{h} or
13670 @kbd{o}) has already been pressed; otherwise it means to switch to algebraic
13673 @node Date Formats, Truncating the Stack, HMS Formats, Display Modes
13674 @subsection Date Formats
13678 @pindex calc-date-notation
13679 The @kbd{d d} (@code{calc-date-notation}) command controls the display
13680 of date forms (@pxref{Date Forms}). It prompts for a string which
13681 contains letters that represent the various parts of a date and time.
13682 To show which parts should be omitted when the form represents a pure
13683 date with no time, parts of the string can be enclosed in @samp{< >}
13684 marks. If you don't include @samp{< >} markers in the format, Calc
13685 guesses at which parts, if any, should be omitted when formatting
13688 The default format is: @samp{<H:mm:SSpp >Www Mmm D, YYYY}.
13689 An example string in this format is @samp{3:32pm Wed Jan 9, 1991}.
13690 If you enter a blank format string, this default format is
13693 Calc uses @samp{< >} notation for nameless functions as well as for
13694 dates. @xref{Specifying Operators}. To avoid confusion with nameless
13695 functions, your date formats should avoid using the @samp{#} character.
13698 * Date Formatting Codes::
13699 * Free-Form Dates::
13700 * Standard Date Formats::
13703 @node Date Formatting Codes, Free-Form Dates, Date Formats, Date Formats
13704 @subsubsection Date Formatting Codes
13707 When displaying a date, the current date format is used. All
13708 characters except for letters and @samp{<} and @samp{>} are
13709 copied literally when dates are formatted. The portion between
13710 @samp{< >} markers is omitted for pure dates, or included for
13711 date/time forms. Letters are interpreted according to the table
13714 When dates are read in during algebraic entry, Calc first tries to
13715 match the input string to the current format either with or without
13716 the time part. The punctuation characters (including spaces) must
13717 match exactly; letter fields must correspond to suitable text in
13718 the input. If this doesn't work, Calc checks if the input is a
13719 simple number; if so, the number is interpreted as a number of days
13720 since Jan 1, 1 AD. Otherwise, Calc tries a much more relaxed and
13721 flexible algorithm which is described in the next section.
13723 Weekday names are ignored during reading.
13725 Two-digit year numbers are interpreted as lying in the range
13726 from 1941 to 2039. Years outside that range are always
13727 entered and displayed in full. Year numbers with a leading
13728 @samp{+} sign are always interpreted exactly, allowing the
13729 entry and display of the years 1 through 99 AD.
13731 Here is a complete list of the formatting codes for dates:
13735 Year: ``91'' for 1991, ``7'' for 2007, ``+23'' for 23 AD.
13737 Year: ``91'' for 1991, ``07'' for 2007, ``+23'' for 23 AD.
13739 Year: ``91'' for 1991, `` 7'' for 2007, ``+23'' for 23 AD.
13741 Year: ``1991'' for 1991, ``23'' for 23 AD.
13743 Year: ``1991'' for 1991, ``+23'' for 23 AD.
13745 Year: ``ad'' or blank.
13747 Year: ``AD'' or blank.
13749 Year: ``ad '' or blank. (Note trailing space.)
13751 Year: ``AD '' or blank.
13753 Year: ``a.d.'' or blank.
13755 Year: ``A.D.'' or blank.
13757 Year: ``bc'' or blank.
13759 Year: ``BC'' or blank.
13761 Year: `` bc'' or blank. (Note leading space.)
13763 Year: `` BC'' or blank.
13765 Year: ``b.c.'' or blank.
13767 Year: ``B.C.'' or blank.
13769 Month: ``8'' for August.
13771 Month: ``08'' for August.
13773 Month: `` 8'' for August.
13775 Month: ``AUG'' for August.
13777 Month: ``Aug'' for August.
13779 Month: ``aug'' for August.
13781 Month: ``AUGUST'' for August.
13783 Month: ``August'' for August.
13785 Day: ``7'' for 7th day of month.
13787 Day: ``07'' for 7th day of month.
13789 Day: `` 7'' for 7th day of month.
13791 Weekday: ``0'' for Sunday, ``6'' for Saturday.
13793 Weekday: ``SUN'' for Sunday.
13795 Weekday: ``Sun'' for Sunday.
13797 Weekday: ``sun'' for Sunday.
13799 Weekday: ``SUNDAY'' for Sunday.
13801 Weekday: ``Sunday'' for Sunday.
13803 Day of year: ``34'' for Feb. 3.
13805 Day of year: ``034'' for Feb. 3.
13807 Day of year: `` 34'' for Feb. 3.
13809 Hour: ``5'' for 5 AM; ``17'' for 5 PM.
13811 Hour: ``05'' for 5 AM; ``17'' for 5 PM.
13813 Hour: `` 5'' for 5 AM; ``17'' for 5 PM.
13815 Hour: ``5'' for 5 AM and 5 PM.
13817 Hour: ``05'' for 5 AM and 5 PM.
13819 Hour: `` 5'' for 5 AM and 5 PM.
13821 AM/PM: ``a'' or ``p''.
13823 AM/PM: ``A'' or ``P''.
13825 AM/PM: ``am'' or ``pm''.
13827 AM/PM: ``AM'' or ``PM''.
13829 AM/PM: ``a.m.'' or ``p.m.''.
13831 AM/PM: ``A.M.'' or ``P.M.''.
13833 Minutes: ``7'' for 7.
13835 Minutes: ``07'' for 7.
13837 Minutes: `` 7'' for 7.
13839 Seconds: ``7'' for 7; ``7.23'' for 7.23.
13841 Seconds: ``07'' for 7; ``07.23'' for 7.23.
13843 Seconds: `` 7'' for 7; `` 7.23'' for 7.23.
13845 Optional seconds: ``07'' for 7; blank for 0.
13847 Optional seconds: `` 7'' for 7; blank for 0.
13849 Numeric date/time: ``726842.25'' for 6:00am Wed Jan 9, 1991.
13851 Numeric date: ``726842'' for any time on Wed Jan 9, 1991.
13853 Julian date/time: ``2448265.75'' for 6:00am Wed Jan 9, 1991.
13855 Julian date: ``2448266'' for any time on Wed Jan 9, 1991.
13857 Unix time: ``663400800'' for 6:00am Wed Jan 9, 1991.
13859 Brackets suppression. An ``X'' at the front of the format
13860 causes the surrounding @w{@samp{< >}} delimiters to be omitted
13861 when formatting dates. Note that the brackets are still
13862 required for algebraic entry.
13865 If ``SS'' or ``BS'' (optional seconds) is preceded by a colon, the
13866 colon is also omitted if the seconds part is zero.
13868 If ``bb,'' ``bbb'' or ``bbbb'' or their upper-case equivalents
13869 appear in the format, then negative year numbers are displayed
13870 without a minus sign. Note that ``aa'' and ``bb'' are mutually
13871 exclusive. Some typical usages would be @samp{YYYY AABB};
13872 @samp{AAAYYYYBBB}; @samp{YYYYBBB}.
13874 The formats ``YY,'' ``YYYY,'' ``MM,'' ``DD,'' ``ddd,'' ``hh,'' ``HH,''
13875 ``mm,'' ``ss,'' and ``SS'' actually match any number of digits during
13876 reading unless several of these codes are strung together with no
13877 punctuation in between, in which case the input must have exactly as
13878 many digits as there are letters in the format.
13880 The ``j,'' ``J,'' and ``U'' formats do not make any time zone
13881 adjustment. They effectively use @samp{julian(x,0)} and
13882 @samp{unixtime(x,0)} to make the conversion; @pxref{Date Arithmetic}.
13884 @node Free-Form Dates, Standard Date Formats, Date Formatting Codes, Date Formats
13885 @subsubsection Free-Form Dates
13888 When reading a date form during algebraic entry, Calc falls back
13889 on the algorithm described here if the input does not exactly
13890 match the current date format. This algorithm generally
13891 ``does the right thing'' and you don't have to worry about it,
13892 but it is described here in full detail for the curious.
13894 Calc does not distinguish between upper- and lower-case letters
13895 while interpreting dates.
13897 First, the time portion, if present, is located somewhere in the
13898 text and then removed. The remaining text is then interpreted as
13901 A time is of the form @samp{hh:mm:ss}, possibly with the seconds
13902 part omitted and possibly with an AM/PM indicator added to indicate
13903 12-hour time. If the AM/PM is present, the minutes may also be
13904 omitted. The AM/PM part may be any of the words @samp{am},
13905 @samp{pm}, @samp{noon}, or @samp{midnight}; each of these may be
13906 abbreviated to one letter, and the alternate forms @samp{a.m.},
13907 @samp{p.m.}, and @samp{mid} are also understood. Obviously
13908 @samp{noon} and @samp{midnight} are allowed only on 12:00:00.
13909 The words @samp{noon}, @samp{mid}, and @samp{midnight} are also
13910 recognized with no number attached.
13912 If there is no AM/PM indicator, the time is interpreted in 24-hour
13915 To read the date portion, all words and numbers are isolated
13916 from the string; other characters are ignored. All words must
13917 be either month names or day-of-week names (the latter of which
13918 are ignored). Names can be written in full or as three-letter
13921 Large numbers, or numbers with @samp{+} or @samp{-} signs,
13922 are interpreted as years. If one of the other numbers is
13923 greater than 12, then that must be the day and the remaining
13924 number in the input is therefore the month. Otherwise, Calc
13925 assumes the month, day and year are in the same order that they
13926 appear in the current date format. If the year is omitted, the
13927 current year is taken from the system clock.
13929 If there are too many or too few numbers, or any unrecognizable
13930 words, then the input is rejected.
13932 If there are any large numbers (of five digits or more) other than
13933 the year, they are ignored on the assumption that they are something
13934 like Julian dates that were included along with the traditional
13935 date components when the date was formatted.
13937 One of the words @samp{ad}, @samp{a.d.}, @samp{bc}, or @samp{b.c.}
13938 may optionally be used; the latter two are equivalent to a
13939 minus sign on the year value.
13941 If you always enter a four-digit year, and use a name instead
13942 of a number for the month, there is no danger of ambiguity.
13944 @node Standard Date Formats, , Free-Form Dates, Date Formats
13945 @subsubsection Standard Date Formats
13948 There are actually ten standard date formats, numbered 0 through 9.
13949 Entering a blank line at the @kbd{d d} command's prompt gives
13950 you format number 1, Calc's usual format. You can enter any digit
13951 to select the other formats.
13953 To create your own standard date formats, give a numeric prefix
13954 argument from 0 to 9 to the @w{@kbd{d d}} command. The format you
13955 enter will be recorded as the new standard format of that
13956 number, as well as becoming the new current date format.
13957 You can save your formats permanently with the @w{@kbd{m m}}
13958 command (@pxref{Mode Settings}).
13962 @samp{N} (Numerical format)
13964 @samp{<H:mm:SSpp >Www Mmm D, YYYY} (American format)
13966 @samp{D Mmm YYYY<, h:mm:SS>} (European format)
13968 @samp{Www Mmm BD< hh:mm:ss> YYYY} (Unix written date format)
13970 @samp{M/D/Y< H:mm:SSpp>} (American slashed format)
13972 @samp{D.M.Y< h:mm:SS>} (European dotted format)
13974 @samp{M-D-Y< H:mm:SSpp>} (American dashed format)
13976 @samp{D-M-Y< h:mm:SS>} (European dashed format)
13978 @samp{j<, h:mm:ss>} (Julian day plus time)
13980 @samp{YYddd< hh:mm:ss>} (Year-day format)
13983 @node Truncating the Stack, Justification, Date Formats, Display Modes
13984 @subsection Truncating the Stack
13988 @pindex calc-truncate-stack
13989 @cindex Truncating the stack
13990 @cindex Narrowing the stack
13991 The @kbd{d t} (@code{calc-truncate-stack}) command moves the @samp{.}@:
13992 line that marks the top-of-stack up or down in the Calculator buffer.
13993 The number right above that line is considered to the be at the top of
13994 the stack. Any numbers below that line are ``hidden'' from all stack
13995 operations (although still visible to the user). This is similar to the
13996 Emacs ``narrowing'' feature, except that the values below the @samp{.}
13997 are @emph{visible}, just temporarily frozen. This feature allows you to
13998 keep several independent calculations running at once in different parts
13999 of the stack, or to apply a certain command to an element buried deep in
14002 Pressing @kbd{d t} by itself moves the @samp{.} to the line the cursor
14003 is on. Thus, this line and all those below it become hidden. To un-hide
14004 these lines, move down to the end of the buffer and press @w{@kbd{d t}}.
14005 With a positive numeric prefix argument @expr{n}, @kbd{d t} hides the
14006 bottom @expr{n} values in the buffer. With a negative argument, it hides
14007 all but the top @expr{n} values. With an argument of zero, it hides zero
14008 values, i.e., moves the @samp{.} all the way down to the bottom.
14011 @pindex calc-truncate-up
14013 @pindex calc-truncate-down
14014 The @kbd{d [} (@code{calc-truncate-up}) and @kbd{d ]}
14015 (@code{calc-truncate-down}) commands move the @samp{.} up or down one
14016 line at a time (or several lines with a prefix argument).
14018 @node Justification, Labels, Truncating the Stack, Display Modes
14019 @subsection Justification
14023 @pindex calc-left-justify
14025 @pindex calc-center-justify
14027 @pindex calc-right-justify
14028 Values on the stack are normally left-justified in the window. You can
14029 control this arrangement by typing @kbd{d <} (@code{calc-left-justify}),
14030 @kbd{d >} (@code{calc-right-justify}), or @kbd{d =}
14031 (@code{calc-center-justify}). For example, in Right-Justification mode,
14032 stack entries are displayed flush-right against the right edge of the
14035 If you change the width of the Calculator window you may have to type
14036 @kbd{d @key{SPC}} (@code{calc-refresh}) to re-align right-justified or centered
14039 Right-justification is especially useful together with fixed-point
14040 notation (see @code{d f}; @code{calc-fix-notation}). With these modes
14041 together, the decimal points on numbers will always line up.
14043 With a numeric prefix argument, the justification commands give you
14044 a little extra control over the display. The argument specifies the
14045 horizontal ``origin'' of a display line. It is also possible to
14046 specify a maximum line width using the @kbd{d b} command (@pxref{Normal
14047 Language Modes}). For reference, the precise rules for formatting and
14048 breaking lines are given below. Notice that the interaction between
14049 origin and line width is slightly different in each justification
14052 In Left-Justified mode, the line is indented by a number of spaces
14053 given by the origin (default zero). If the result is longer than the
14054 maximum line width, if given, or too wide to fit in the Calc window
14055 otherwise, then it is broken into lines which will fit; each broken
14056 line is indented to the origin.
14058 In Right-Justified mode, lines are shifted right so that the rightmost
14059 character is just before the origin, or just before the current
14060 window width if no origin was specified. If the line is too long
14061 for this, then it is broken; the current line width is used, if
14062 specified, or else the origin is used as a width if that is
14063 specified, or else the line is broken to fit in the window.
14065 In Centering mode, the origin is the column number of the center of
14066 each stack entry. If a line width is specified, lines will not be
14067 allowed to go past that width; Calc will either indent less or
14068 break the lines if necessary. If no origin is specified, half the
14069 line width or Calc window width is used.
14071 Note that, in each case, if line numbering is enabled the display
14072 is indented an additional four spaces to make room for the line
14073 number. The width of the line number is taken into account when
14074 positioning according to the current Calc window width, but not
14075 when positioning by explicit origins and widths. In the latter
14076 case, the display is formatted as specified, and then uniformly
14077 shifted over four spaces to fit the line numbers.
14079 @node Labels, , Justification, Display Modes
14084 @pindex calc-left-label
14085 The @kbd{d @{} (@code{calc-left-label}) command prompts for a string,
14086 then displays that string to the left of every stack entry. If the
14087 entries are left-justified (@pxref{Justification}), then they will
14088 appear immediately after the label (unless you specified an origin
14089 greater than the length of the label). If the entries are centered
14090 or right-justified, the label appears on the far left and does not
14091 affect the horizontal position of the stack entry.
14093 Give a blank string (with @kbd{d @{ @key{RET}}) to turn the label off.
14096 @pindex calc-right-label
14097 The @kbd{d @}} (@code{calc-right-label}) command similarly adds a
14098 label on the righthand side. It does not affect positioning of
14099 the stack entries unless they are right-justified. Also, if both
14100 a line width and an origin are given in Right-Justified mode, the
14101 stack entry is justified to the origin and the righthand label is
14102 justified to the line width.
14104 One application of labels would be to add equation numbers to
14105 formulas you are manipulating in Calc and then copying into a
14106 document (possibly using Embedded mode). The equations would
14107 typically be centered, and the equation numbers would be on the
14108 left or right as you prefer.
14110 @node Language Modes, Modes Variable, Display Modes, Mode Settings
14111 @section Language Modes
14114 The commands in this section change Calc to use a different notation for
14115 entry and display of formulas, corresponding to the conventions of some
14116 other common language such as Pascal or La@TeX{}. Objects displayed on the
14117 stack or yanked from the Calculator to an editing buffer will be formatted
14118 in the current language; objects entered in algebraic entry or yanked from
14119 another buffer will be interpreted according to the current language.
14121 The current language has no effect on things written to or read from the
14122 trail buffer, nor does it affect numeric entry. Only algebraic entry is
14123 affected. You can make even algebraic entry ignore the current language
14124 and use the standard notation by giving a numeric prefix, e.g., @kbd{C-u '}.
14126 For example, suppose the formula @samp{2*a[1] + atan(a[2])} occurs in a C
14127 program; elsewhere in the program you need the derivatives of this formula
14128 with respect to @samp{a[1]} and @samp{a[2]}. First, type @kbd{d C}
14129 to switch to C notation. Now use @code{C-u M-# g} to grab the formula
14130 into the Calculator, @kbd{a d a[1] @key{RET}} to differentiate with respect
14131 to the first variable, and @kbd{M-# y} to yank the formula for the derivative
14132 back into your C program. Press @kbd{U} to undo the differentiation and
14133 repeat with @kbd{a d a[2] @key{RET}} for the other derivative.
14135 Without being switched into C mode first, Calc would have misinterpreted
14136 the brackets in @samp{a[1]} and @samp{a[2]}, would not have known that
14137 @code{atan} was equivalent to Calc's built-in @code{arctan} function,
14138 and would have written the formula back with notations (like implicit
14139 multiplication) which would not have been valid for a C program.
14141 As another example, suppose you are maintaining a C program and a La@TeX{}
14142 document, each of which needs a copy of the same formula. You can grab the
14143 formula from the program in C mode, switch to La@TeX{} mode, and yank the
14144 formula into the document in La@TeX{} math-mode format.
14146 Language modes are selected by typing the letter @kbd{d} followed by a
14147 shifted letter key.
14150 * Normal Language Modes::
14151 * C FORTRAN Pascal::
14152 * TeX and LaTeX Language Modes::
14153 * Eqn Language Mode::
14154 * Mathematica Language Mode::
14155 * Maple Language Mode::
14160 @node Normal Language Modes, C FORTRAN Pascal, Language Modes, Language Modes
14161 @subsection Normal Language Modes
14165 @pindex calc-normal-language
14166 The @kbd{d N} (@code{calc-normal-language}) command selects the usual
14167 notation for Calc formulas, as described in the rest of this manual.
14168 Matrices are displayed in a multi-line tabular format, but all other
14169 objects are written in linear form, as they would be typed from the
14173 @pindex calc-flat-language
14174 @cindex Matrix display
14175 The @kbd{d O} (@code{calc-flat-language}) command selects a language
14176 identical with the normal one, except that matrices are written in
14177 one-line form along with everything else. In some applications this
14178 form may be more suitable for yanking data into other buffers.
14181 @pindex calc-line-breaking
14182 @cindex Line breaking
14183 @cindex Breaking up long lines
14184 Even in one-line mode, long formulas or vectors will still be split
14185 across multiple lines if they exceed the width of the Calculator window.
14186 The @kbd{d b} (@code{calc-line-breaking}) command turns this line-breaking
14187 feature on and off. (It works independently of the current language.)
14188 If you give a numeric prefix argument of five or greater to the @kbd{d b}
14189 command, that argument will specify the line width used when breaking
14193 @pindex calc-big-language
14194 The @kbd{d B} (@code{calc-big-language}) command selects a language
14195 which uses textual approximations to various mathematical notations,
14196 such as powers, quotients, and square roots:
14206 in place of @samp{sqrt((a+1)/b + c^2)}.
14208 Subscripts like @samp{a_i} are displayed as actual subscripts in Big
14209 mode. Double subscripts, @samp{a_i_j} (@samp{subscr(subscr(a, i), j)})
14210 are displayed as @samp{a} with subscripts separated by commas:
14211 @samp{i, j}. They must still be entered in the usual underscore
14214 One slight ambiguity of Big notation is that
14223 can represent either the negative rational number @expr{-3:4}, or the
14224 actual expression @samp{-(3/4)}; but the latter formula would normally
14225 never be displayed because it would immediately be evaluated to
14226 @expr{-3:4} or @expr{-0.75}, so this ambiguity is not a problem in
14229 Non-decimal numbers are displayed with subscripts. Thus there is no
14230 way to tell the difference between @samp{16#C2} and @samp{C2_16},
14231 though generally you will know which interpretation is correct.
14232 Logarithms @samp{log(x,b)} and @samp{log10(x)} also use subscripts
14235 In Big mode, stack entries often take up several lines. To aid
14236 readability, stack entries are separated by a blank line in this mode.
14237 You may find it useful to expand the Calc window's height using
14238 @kbd{C-x ^} (@code{enlarge-window}) or to make the Calc window the only
14239 one on the screen with @kbd{C-x 1} (@code{delete-other-windows}).
14241 Long lines are currently not rearranged to fit the window width in
14242 Big mode, so you may need to use the @kbd{<} and @kbd{>} keys
14243 to scroll across a wide formula. For really big formulas, you may
14244 even need to use @kbd{@{} and @kbd{@}} to scroll up and down.
14247 @pindex calc-unformatted-language
14248 The @kbd{d U} (@code{calc-unformatted-language}) command altogether disables
14249 the use of operator notation in formulas. In this mode, the formula
14250 shown above would be displayed:
14253 sqrt(add(div(add(a, 1), b), pow(c, 2)))
14256 These four modes differ only in display format, not in the format
14257 expected for algebraic entry. The standard Calc operators work in
14258 all four modes, and unformatted notation works in any language mode
14259 (except that Mathematica mode expects square brackets instead of
14262 @node C FORTRAN Pascal, TeX and LaTeX Language Modes, Normal Language Modes, Language Modes
14263 @subsection C, FORTRAN, and Pascal Modes
14267 @pindex calc-c-language
14269 The @kbd{d C} (@code{calc-c-language}) command selects the conventions
14270 of the C language for display and entry of formulas. This differs from
14271 the normal language mode in a variety of (mostly minor) ways. In
14272 particular, C language operators and operator precedences are used in
14273 place of Calc's usual ones. For example, @samp{a^b} means @samp{xor(a,b)}
14274 in C mode; a value raised to a power is written as a function call,
14277 In C mode, vectors and matrices use curly braces instead of brackets.
14278 Octal and hexadecimal values are written with leading @samp{0} or @samp{0x}
14279 rather than using the @samp{#} symbol. Array subscripting is
14280 translated into @code{subscr} calls, so that @samp{a[i]} in C
14281 mode is the same as @samp{a_i} in Normal mode. Assignments
14282 turn into the @code{assign} function, which Calc normally displays
14283 using the @samp{:=} symbol.
14285 The variables @code{pi} and @code{e} would be displayed @samp{pi}
14286 and @samp{e} in Normal mode, but in C mode they are displayed as
14287 @samp{M_PI} and @samp{M_E}, corresponding to the names of constants
14288 typically provided in the @file{<math.h>} header. Functions whose
14289 names are different in C are translated automatically for entry and
14290 display purposes. For example, entering @samp{asin(x)} will push the
14291 formula @samp{arcsin(x)} onto the stack; this formula will be displayed
14292 as @samp{asin(x)} as long as C mode is in effect.
14295 @pindex calc-pascal-language
14296 @cindex Pascal language
14297 The @kbd{d P} (@code{calc-pascal-language}) command selects Pascal
14298 conventions. Like C mode, Pascal mode interprets array brackets and uses
14299 a different table of operators. Hexadecimal numbers are entered and
14300 displayed with a preceding dollar sign. (Thus the regular meaning of
14301 @kbd{$2} during algebraic entry does not work in Pascal mode, though
14302 @kbd{$} (and @kbd{$$}, etc.) not followed by digits works the same as
14303 always.) No special provisions are made for other non-decimal numbers,
14304 vectors, and so on, since there is no universally accepted standard way
14305 of handling these in Pascal.
14308 @pindex calc-fortran-language
14309 @cindex FORTRAN language
14310 The @kbd{d F} (@code{calc-fortran-language}) command selects FORTRAN
14311 conventions. Various function names are transformed into FORTRAN
14312 equivalents. Vectors are written as @samp{/1, 2, 3/}, and may be
14313 entered this way or using square brackets. Since FORTRAN uses round
14314 parentheses for both function calls and array subscripts, Calc displays
14315 both in the same way; @samp{a(i)} is interpreted as a function call
14316 upon reading, and subscripts must be entered as @samp{subscr(a, i)}.
14317 Also, if the variable @code{a} has been declared to have type
14318 @code{vector} or @code{matrix} then @samp{a(i)} will be parsed as a
14319 subscript. (@xref{Declarations}.) Usually it doesn't matter, though;
14320 if you enter the subscript expression @samp{a(i)} and Calc interprets
14321 it as a function call, you'll never know the difference unless you
14322 switch to another language mode or replace @code{a} with an actual
14323 vector (or unless @code{a} happens to be the name of a built-in
14326 Underscores are allowed in variable and function names in all of these
14327 language modes. The underscore here is equivalent to the @samp{#} in
14328 Normal mode, or to hyphens in the underlying Emacs Lisp variable names.
14330 FORTRAN and Pascal modes normally do not adjust the case of letters in
14331 formulas. Most built-in Calc names use lower-case letters. If you use a
14332 positive numeric prefix argument with @kbd{d P} or @kbd{d F}, these
14333 modes will use upper-case letters exclusively for display, and will
14334 convert to lower-case on input. With a negative prefix, these modes
14335 convert to lower-case for display and input.
14337 @node TeX and LaTeX Language Modes, Eqn Language Mode, C FORTRAN Pascal, Language Modes
14338 @subsection @TeX{} and La@TeX{} Language Modes
14342 @pindex calc-tex-language
14343 @cindex TeX language
14345 @pindex calc-latex-language
14346 @cindex LaTeX language
14347 The @kbd{d T} (@code{calc-tex-language}) command selects the conventions
14348 of ``math mode'' in Donald Knuth's @TeX{} typesetting language,
14349 and the @kbd{d L} (@code{calc-latex-language}) command selects the
14350 conventions of ``math mode'' in La@TeX{}, a typesetting language that
14351 uses @TeX{} as its formatting engine. Calc's La@TeX{} language mode can
14352 read any formula that the @TeX{} language mode can, although La@TeX{}
14353 mode may display it differently.
14355 Formulas are entered and displayed in the appropriate notation;
14356 @texline @math{\sin(a/b)}
14357 @infoline @expr{sin(a/b)}
14358 will appear as @samp{\sin\left( a \over b \right)} in @TeX{} mode and
14359 @samp{\sin\left(\frac@{a@}@{b@}\right)} in La@TeX{} mode.
14360 Math formulas are often enclosed by @samp{$ $} signs in @TeX{} and
14361 La@TeX{}; these should be omitted when interfacing with Calc. To Calc,
14362 the @samp{$} sign has the same meaning it always does in algebraic
14363 formulas (a reference to an existing entry on the stack).
14365 Complex numbers are displayed as in @samp{3 + 4i}. Fractions and
14366 quotients are written using @code{\over} in @TeX{} mode (as in
14367 @code{@{a \over b@}}) and @code{\frac} in La@TeX{} mode (as in
14368 @code{\frac@{a@}@{b@}}); binomial coefficients are written with
14369 @code{\choose} in @TeX{} mode (as in @code{@{a \choose b@}}) and
14370 @code{\binom} in La@TeX{} mode (as in @code{\binom@{a@}@{b@}}).
14371 Interval forms are written with @code{\ldots}, and error forms are
14372 written with @code{\pm}. Absolute values are written as in
14373 @samp{|x + 1|}, and the floor and ceiling functions are written with
14374 @code{\lfloor}, @code{\rfloor}, etc. The words @code{\left} and
14375 @code{\right} are ignored when reading formulas in @TeX{} and La@TeX{}
14376 modes. Both @code{inf} and @code{uinf} are written as @code{\infty};
14377 when read, @code{\infty} always translates to @code{inf}.
14379 Function calls are written the usual way, with the function name followed
14380 by the arguments in parentheses. However, functions for which @TeX{}
14381 and La@TeX{} have special names (like @code{\sin}) will use curly braces
14382 instead of parentheses for very simple arguments. During input, curly
14383 braces and parentheses work equally well for grouping, but when the
14384 document is formatted the curly braces will be invisible. Thus the
14386 @texline @math{\sin{2 x}}
14387 @infoline @expr{sin 2x}
14389 @texline @math{\sin(2 + x)}.
14390 @infoline @expr{sin(2 + x)}.
14392 Function and variable names not treated specially by @TeX{} and La@TeX{}
14393 are simply written out as-is, which will cause them to come out in
14394 italic letters in the printed document. If you invoke @kbd{d T} or
14395 @kbd{d L} with a positive numeric prefix argument, names of more than
14396 one character will instead be enclosed in a protective commands that
14397 will prevent them from being typeset in the math italics; they will be
14398 written @samp{\hbox@{@var{name}@}} in @TeX{} mode and
14399 @samp{\text@{@var{name}@}} in La@TeX{} mode. The
14400 @samp{\hbox@{ @}} and @samp{\text@{ @}} notations are ignored during
14401 reading. If you use a negative prefix argument, such function names are
14402 written @samp{\@var{name}}, and function names that begin with @code{\} during
14403 reading have the @code{\} removed. (Note that in this mode, long
14404 variable names are still written with @code{\hbox} or @code{\text}.
14405 However, you can always make an actual variable name like @code{\bar} in
14408 During reading, text of the form @samp{\matrix@{ ...@: @}} is replaced
14409 by @samp{[ ...@: ]}. The same also applies to @code{\pmatrix} and
14410 @code{\bmatrix}. In La@TeX{} mode this also applies to
14411 @samp{\begin@{matrix@} ... \end@{matrix@}},
14412 @samp{\begin@{bmatrix@} ... \end@{bmatrix@}},
14413 @samp{\begin@{pmatrix@} ... \end@{pmatrix@}}, as well as
14414 @samp{\begin@{smallmatrix@} ... \end@{smallmatrix@}}.
14415 The symbol @samp{&} is interpreted as a comma,
14416 and the symbols @samp{\cr} and @samp{\\} are interpreted as semicolons.
14417 During output, matrices are displayed in @samp{\matrix@{ a & b \\ c & d@}}
14418 format in @TeX{} mode and in
14419 @samp{\begin@{pmatrix@} a & b \\ c & d \end@{pmatrix@}} format in
14420 La@TeX{} mode; you may need to edit this afterwards to change to your
14421 preferred matrix form. If you invoke @kbd{d T} or @kbd{d L} with an
14422 argument of 2 or -2, then matrices will be displayed in two-dimensional
14433 This may be convenient for isolated matrices, but could lead to
14434 expressions being displayed like
14437 \begin@{pmatrix@} \times x
14444 While this wouldn't bother Calc, it is incorrect La@TeX{}.
14445 (Similarly for @TeX{}.)
14447 Accents like @code{\tilde} and @code{\bar} translate into function
14448 calls internally (@samp{tilde(x)}, @samp{bar(x)}). The @code{\underline}
14449 sequence is treated as an accent. The @code{\vec} accent corresponds
14450 to the function name @code{Vec}, because @code{vec} is the name of
14451 a built-in Calc function. The following table shows the accents
14452 in Calc, @TeX{}, La@TeX{} and @dfn{eqn} (described in the next section):
14456 @let@calcindexershow=@calcindexernoshow @c Suppress marginal notes
14457 @let@calcindexersh=@calcindexernoshow
14565 acute \acute \acute
14569 breve \breve \breve
14571 check \check \check
14577 dotdot \ddot \ddot dotdot
14580 grave \grave \grave
14585 tilde \tilde \tilde tilde
14587 under \underline \underline under
14592 The @samp{=>} (evaluates-to) operator appears as a @code{\to} symbol:
14593 @samp{@{@var{a} \to @var{b}@}}. @TeX{} defines @code{\to} as an
14594 alias for @code{\rightarrow}. However, if the @samp{=>} is the
14595 top-level expression being formatted, a slightly different notation
14596 is used: @samp{\evalto @var{a} \to @var{b}}. The @code{\evalto}
14597 word is ignored by Calc's input routines, and is undefined in @TeX{}.
14598 You will typically want to include one of the following definitions
14599 at the top of a @TeX{} file that uses @code{\evalto}:
14603 \def\evalto#1\to@{@}
14606 The first definition formats evaluates-to operators in the usual
14607 way. The second causes only the @var{b} part to appear in the
14608 printed document; the @var{a} part and the arrow are hidden.
14609 Another definition you may wish to use is @samp{\let\to=\Rightarrow}
14610 which causes @code{\to} to appear more like Calc's @samp{=>} symbol.
14611 @xref{Evaluates-To Operator}, for a discussion of @code{evalto}.
14613 The complete set of @TeX{} control sequences that are ignored during
14617 \hbox \mbox \text \left \right
14618 \, \> \: \; \! \quad \qquad \hfil \hfill
14619 \displaystyle \textstyle \dsize \tsize
14620 \scriptstyle \scriptscriptstyle \ssize \ssize
14621 \rm \bf \it \sl \roman \bold \italic \slanted
14622 \cal \mit \Cal \Bbb \frak \goth
14626 Note that, because these symbols are ignored, reading a @TeX{} or
14627 La@TeX{} formula into Calc and writing it back out may lose spacing and
14630 Also, the ``discretionary multiplication sign'' @samp{\*} is read
14631 the same as @samp{*}.
14634 The @TeX{} version of this manual includes some printed examples at the
14635 end of this section.
14638 Here are some examples of how various Calc formulas are formatted in @TeX{}:
14643 \sin\left( {a^2 \over b_i} \right)
14647 $$ \sin\left( a^2 \over b_i \right) $$
14653 [(3, 4), 3:4, 3 +/- 4, [3 .. inf)]
14654 [3 + 4i, @{3 \over 4@}, 3 \pm 4, [3 \ldots \infty)]
14659 $$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
14665 [abs(a), abs(a / b), floor(a), ceil(a / b)]
14666 [|a|, \left| a \over b \right|,
14667 \lfloor a \rfloor, \left\lceil a \over b \right\rceil]
14671 $$ [|a|, \left| a \over b \right|,
14672 \lfloor a \rfloor, \left\lceil a \over b \right\rceil] $$
14678 [sin(a), sin(2 a), sin(2 + a), sin(a / b)]
14679 [\sin@{a@}, \sin@{2 a@}, \sin(2 + a),
14680 \sin\left( @{a \over b@} \right)]
14685 $$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
14689 First with plain @kbd{d T}, then with @kbd{C-u d T}, then finally with
14690 @kbd{C-u - d T} (using the example definition
14691 @samp{\def\foo#1@{\tilde F(#1)@}}:
14695 [f(a), foo(bar), sin(pi)]
14696 [f(a), foo(bar), \sin{\pi}]
14697 [f(a), \hbox@{foo@}(\hbox@{bar@}), \sin@{\pi@}]
14698 [f(a), \foo@{\hbox@{bar@}@}, \sin@{\pi@}]
14702 $$ [f(a), foo(bar), \sin{\pi}] $$
14703 $$ [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}] $$
14704 $$ [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$
14708 First with @samp{\def\evalto@{@}}, then with @samp{\def\evalto#1\to@{@}}:
14713 \evalto 2 + 3 \to 5
14723 First with standard @code{\to}, then with @samp{\let\to\Rightarrow}:
14727 [2 + 3 => 5, a / 2 => (b + c) / 2]
14728 [@{2 + 3 \to 5@}, @{@{a \over 2@} \to @{b + c \over 2@}@}]
14733 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
14734 {\let\to\Rightarrow
14735 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
14739 Matrices normally, then changing @code{\matrix} to @code{\pmatrix}:
14743 [ [ a / b, 0 ], [ 0, 2^(x + 1) ] ]
14744 \matrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14745 \pmatrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14750 $$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14751 $$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14756 @node Eqn Language Mode, Mathematica Language Mode, TeX and LaTeX Language Modes, Language Modes
14757 @subsection Eqn Language Mode
14761 @pindex calc-eqn-language
14762 @dfn{Eqn} is another popular formatter for math formulas. It is
14763 designed for use with the TROFF text formatter, and comes standard
14764 with many versions of Unix. The @kbd{d E} (@code{calc-eqn-language})
14765 command selects @dfn{eqn} notation.
14767 The @dfn{eqn} language's main idiosyncrasy is that whitespace plays
14768 a significant part in the parsing of the language. For example,
14769 @samp{sqrt x+1 + y} treats @samp{x+1} as the argument of the
14770 @code{sqrt} operator. @dfn{Eqn} also understands more conventional
14771 grouping using curly braces: @samp{sqrt@{x+1@} + y}. Braces are
14772 required only when the argument contains spaces.
14774 In Calc's @dfn{eqn} mode, however, curly braces are required to
14775 delimit arguments of operators like @code{sqrt}. The first of the
14776 above examples would treat only the @samp{x} as the argument of
14777 @code{sqrt}, and in fact @samp{sin x+1} would be interpreted as
14778 @samp{sin * x + 1}, because @code{sin} is not a special operator
14779 in the @dfn{eqn} language. If you always surround the argument
14780 with curly braces, Calc will never misunderstand.
14782 Calc also understands parentheses as grouping characters. Another
14783 peculiarity of @dfn{eqn}'s syntax makes it advisable to separate
14784 words with spaces from any surrounding characters that aren't curly
14785 braces, so Calc writes @samp{sin ( x + y )} in @dfn{eqn} mode.
14786 (The spaces around @code{sin} are important to make @dfn{eqn}
14787 recognize that @code{sin} should be typeset in a roman font, and
14788 the spaces around @code{x} and @code{y} are a good idea just in
14789 case the @dfn{eqn} document has defined special meanings for these
14792 Powers and subscripts are written with the @code{sub} and @code{sup}
14793 operators, respectively. Note that the caret symbol @samp{^} is
14794 treated the same as a space in @dfn{eqn} mode, as is the @samp{~}
14795 symbol (these are used to introduce spaces of various widths into
14796 the typeset output of @dfn{eqn}).
14798 As in La@TeX{} mode, Calc's formatter omits parentheses around the
14799 arguments of functions like @code{ln} and @code{sin} if they are
14800 ``simple-looking''; in this case Calc surrounds the argument with
14801 braces, separated by a @samp{~} from the function name: @samp{sin~@{x@}}.
14803 Font change codes (like @samp{roman @var{x}}) and positioning codes
14804 (like @samp{~} and @samp{down @var{n} @var{x}}) are ignored by the
14805 @dfn{eqn} reader. Also ignored are the words @code{left}, @code{right},
14806 @code{mark}, and @code{lineup}. Quotation marks in @dfn{eqn} mode input
14807 are treated the same as curly braces: @samp{sqrt "1+x"} is equivalent to
14808 @samp{sqrt @{1+x@}}; this is only an approximation to the true meaning
14809 of quotes in @dfn{eqn}, but it is good enough for most uses.
14811 Accent codes (@samp{@var{x} dot}) are handled by treating them as
14812 function calls (@samp{dot(@var{x})}) internally.
14813 @xref{TeX and LaTeX Language Modes}, for a table of these accent
14814 functions. The @code{prime} accent is treated specially if it occurs on
14815 a variable or function name: @samp{f prime prime @w{( x prime )}} is
14816 stored internally as @samp{f'@w{'}(x')}. For example, taking the
14817 derivative of @samp{f(2 x)} with @kbd{a d x} will produce @samp{2 f'(2
14818 x)}, which @dfn{eqn} mode will display as @samp{2 f prime ( 2 x )}.
14820 Assignments are written with the @samp{<-} (left-arrow) symbol,
14821 and @code{evalto} operators are written with @samp{->} or
14822 @samp{evalto ... ->} (@pxref{TeX and LaTeX Language Modes}, for a discussion
14823 of this). The regular Calc symbols @samp{:=} and @samp{=>} are also
14824 recognized for these operators during reading.
14826 Vectors in @dfn{eqn} mode use regular Calc square brackets, but
14827 matrices are formatted as @samp{matrix @{ ccol @{ a above b @} ... @}}.
14828 The words @code{lcol} and @code{rcol} are recognized as synonyms
14829 for @code{ccol} during input, and are generated instead of @code{ccol}
14830 if the matrix justification mode so specifies.
14832 @node Mathematica Language Mode, Maple Language Mode, Eqn Language Mode, Language Modes
14833 @subsection Mathematica Language Mode
14837 @pindex calc-mathematica-language
14838 @cindex Mathematica language
14839 The @kbd{d M} (@code{calc-mathematica-language}) command selects the
14840 conventions of Mathematica. Notable differences in Mathematica mode
14841 are that the names of built-in functions are capitalized, and function
14842 calls use square brackets instead of parentheses. Thus the Calc
14843 formula @samp{sin(2 x)} is entered and displayed @w{@samp{Sin[2 x]}} in
14846 Vectors and matrices use curly braces in Mathematica. Complex numbers
14847 are written @samp{3 + 4 I}. The standard special constants in Calc are
14848 written @code{Pi}, @code{E}, @code{I}, @code{GoldenRatio}, @code{EulerGamma},
14849 @code{Infinity}, @code{ComplexInfinity}, and @code{Indeterminate} in
14851 Non-decimal numbers are written, e.g., @samp{16^^7fff}. Floating-point
14852 numbers in scientific notation are written @samp{1.23*10.^3}.
14853 Subscripts use double square brackets: @samp{a[[i]]}.
14855 @node Maple Language Mode, Compositions, Mathematica Language Mode, Language Modes
14856 @subsection Maple Language Mode
14860 @pindex calc-maple-language
14861 @cindex Maple language
14862 The @kbd{d W} (@code{calc-maple-language}) command selects the
14863 conventions of Maple.
14865 Maple's language is much like C. Underscores are allowed in symbol
14866 names; square brackets are used for subscripts; explicit @samp{*}s for
14867 multiplications are required. Use either @samp{^} or @samp{**} to
14870 Maple uses square brackets for lists and curly braces for sets. Calc
14871 interprets both notations as vectors, and displays vectors with square
14872 brackets. This means Maple sets will be converted to lists when they
14873 pass through Calc. As a special case, matrices are written as calls
14874 to the function @code{matrix}, given a list of lists as the argument,
14875 and can be read in this form or with all-capitals @code{MATRIX}.
14877 The Maple interval notation @samp{2 .. 3} has no surrounding brackets;
14878 Calc reads @samp{2 .. 3} as the closed interval @samp{[2 .. 3]}, and
14879 writes any kind of interval as @samp{2 .. 3}. This means you cannot
14880 see the difference between an open and a closed interval while in
14881 Maple display mode.
14883 Maple writes complex numbers as @samp{3 + 4*I}. Its special constants
14884 are @code{Pi}, @code{E}, @code{I}, and @code{infinity} (all three of
14885 @code{inf}, @code{uinf}, and @code{nan} display as @code{infinity}).
14886 Floating-point numbers are written @samp{1.23*10.^3}.
14888 Among things not currently handled by Calc's Maple mode are the
14889 various quote symbols, procedures and functional operators, and
14890 inert (@samp{&}) operators.
14892 @node Compositions, Syntax Tables, Maple Language Mode, Language Modes
14893 @subsection Compositions
14896 @cindex Compositions
14897 There are several @dfn{composition functions} which allow you to get
14898 displays in a variety of formats similar to those in Big language
14899 mode. Most of these functions do not evaluate to anything; they are
14900 placeholders which are left in symbolic form by Calc's evaluator but
14901 are recognized by Calc's display formatting routines.
14903 Two of these, @code{string} and @code{bstring}, are described elsewhere.
14904 @xref{Strings}. For example, @samp{string("ABC")} is displayed as
14905 @samp{ABC}. When viewed on the stack it will be indistinguishable from
14906 the variable @code{ABC}, but internally it will be stored as
14907 @samp{string([65, 66, 67])} and can still be manipulated this way; for
14908 example, the selection and vector commands @kbd{j 1 v v j u} would
14909 select the vector portion of this object and reverse the elements, then
14910 deselect to reveal a string whose characters had been reversed.
14912 The composition functions do the same thing in all language modes
14913 (although their components will of course be formatted in the current
14914 language mode). The one exception is Unformatted mode (@kbd{d U}),
14915 which does not give the composition functions any special treatment.
14916 The functions are discussed here because of their relationship to
14917 the language modes.
14920 * Composition Basics::
14921 * Horizontal Compositions::
14922 * Vertical Compositions::
14923 * Other Compositions::
14924 * Information about Compositions::
14925 * User-Defined Compositions::
14928 @node Composition Basics, Horizontal Compositions, Compositions, Compositions
14929 @subsubsection Composition Basics
14932 Compositions are generally formed by stacking formulas together
14933 horizontally or vertically in various ways. Those formulas are
14934 themselves compositions. @TeX{} users will find this analogous
14935 to @TeX{}'s ``boxes.'' Each multi-line composition has a
14936 @dfn{baseline}; horizontal compositions use the baselines to
14937 decide how formulas should be positioned relative to one another.
14938 For example, in the Big mode formula
14950 the second term of the sum is four lines tall and has line three as
14951 its baseline. Thus when the term is combined with 17, line three
14952 is placed on the same level as the baseline of 17.
14958 Another important composition concept is @dfn{precedence}. This is
14959 an integer that represents the binding strength of various operators.
14960 For example, @samp{*} has higher precedence (195) than @samp{+} (180),
14961 which means that @samp{(a * b) + c} will be formatted without the
14962 parentheses, but @samp{a * (b + c)} will keep the parentheses.
14964 The operator table used by normal and Big language modes has the
14965 following precedences:
14968 _ 1200 @r{(subscripts)}
14969 % 1100 @r{(as in n}%@r{)}
14970 - 1000 @r{(as in }-@r{n)}
14971 ! 1000 @r{(as in }!@r{n)}
14974 !! 210 @r{(as in n}!!@r{)}
14975 ! 210 @r{(as in n}!@r{)}
14977 * 195 @r{(or implicit multiplication)}
14979 + - 180 @r{(as in a}+@r{b)}
14981 < = 160 @r{(and other relations)}
14993 The general rule is that if an operator with precedence @expr{n}
14994 occurs as an argument to an operator with precedence @expr{m}, then
14995 the argument is enclosed in parentheses if @expr{n < m}. Top-level
14996 expressions and expressions which are function arguments, vector
14997 components, etc., are formatted with precedence zero (so that they
14998 normally never get additional parentheses).
15000 For binary left-associative operators like @samp{+}, the righthand
15001 argument is actually formatted with one-higher precedence than shown
15002 in the table. This makes sure @samp{(a + b) + c} omits the parentheses,
15003 but the unnatural form @samp{a + (b + c)} keeps its parentheses.
15004 Right-associative operators like @samp{^} format the lefthand argument
15005 with one-higher precedence.
15011 The @code{cprec} function formats an expression with an arbitrary
15012 precedence. For example, @samp{cprec(abc, 185)} will combine into
15013 sums and products as follows: @samp{7 + abc}, @samp{7 (abc)} (because
15014 this @code{cprec} form has higher precedence than addition, but lower
15015 precedence than multiplication).
15021 A final composition issue is @dfn{line breaking}. Calc uses two
15022 different strategies for ``flat'' and ``non-flat'' compositions.
15023 A non-flat composition is anything that appears on multiple lines
15024 (not counting line breaking). Examples would be matrices and Big
15025 mode powers and quotients. Non-flat compositions are displayed
15026 exactly as specified. If they come out wider than the current
15027 window, you must use horizontal scrolling (@kbd{<} and @kbd{>}) to
15030 Flat compositions, on the other hand, will be broken across several
15031 lines if they are too wide to fit the window. Certain points in a
15032 composition are noted internally as @dfn{break points}. Calc's
15033 general strategy is to fill each line as much as possible, then to
15034 move down to the next line starting at the first break point that
15035 didn't fit. However, the line breaker understands the hierarchical
15036 structure of formulas. It will not break an ``inner'' formula if
15037 it can use an earlier break point from an ``outer'' formula instead.
15038 For example, a vector of sums might be formatted as:
15042 [ a + b + c, d + e + f,
15043 g + h + i, j + k + l, m ]
15048 If the @samp{m} can fit, then so, it seems, could the @samp{g}.
15049 But Calc prefers to break at the comma since the comma is part
15050 of a ``more outer'' formula. Calc would break at a plus sign
15051 only if it had to, say, if the very first sum in the vector had
15052 itself been too large to fit.
15054 Of the composition functions described below, only @code{choriz}
15055 generates break points. The @code{bstring} function (@pxref{Strings})
15056 also generates breakable items: A break point is added after every
15057 space (or group of spaces) except for spaces at the very beginning or
15060 Composition functions themselves count as levels in the formula
15061 hierarchy, so a @code{choriz} that is a component of a larger
15062 @code{choriz} will be less likely to be broken. As a special case,
15063 if a @code{bstring} occurs as a component of a @code{choriz} or
15064 @code{choriz}-like object (such as a vector or a list of arguments
15065 in a function call), then the break points in that @code{bstring}
15066 will be on the same level as the break points of the surrounding
15069 @node Horizontal Compositions, Vertical Compositions, Composition Basics, Compositions
15070 @subsubsection Horizontal Compositions
15077 The @code{choriz} function takes a vector of objects and composes
15078 them horizontally. For example, @samp{choriz([17, a b/c, d])} formats
15079 as @w{@samp{17a b / cd}} in Normal language mode, or as
15090 in Big language mode. This is actually one case of the general
15091 function @samp{choriz(@var{vec}, @var{sep}, @var{prec})}, where
15092 either or both of @var{sep} and @var{prec} may be omitted.
15093 @var{Prec} gives the @dfn{precedence} to use when formatting
15094 each of the components of @var{vec}. The default precedence is
15095 the precedence from the surrounding environment.
15097 @var{Sep} is a string (i.e., a vector of character codes as might
15098 be entered with @code{" "} notation) which should separate components
15099 of the composition. Also, if @var{sep} is given, the line breaker
15100 will allow lines to be broken after each occurrence of @var{sep}.
15101 If @var{sep} is omitted, the composition will not be breakable
15102 (unless any of its component compositions are breakable).
15104 For example, @samp{2 choriz([a, b c, d = e], " + ", 180)} is
15105 formatted as @samp{2 a + b c + (d = e)}. To get the @code{choriz}
15106 to have precedence 180 ``outwards'' as well as ``inwards,''
15107 enclose it in a @code{cprec} form: @samp{2 cprec(choriz(...), 180)}
15108 formats as @samp{2 (a + b c + (d = e))}.
15110 The baseline of a horizontal composition is the same as the
15111 baselines of the component compositions, which are all aligned.
15113 @node Vertical Compositions, Other Compositions, Horizontal Compositions, Compositions
15114 @subsubsection Vertical Compositions
15121 The @code{cvert} function makes a vertical composition. Each
15122 component of the vector is centered in a column. The baseline of
15123 the result is by default the top line of the resulting composition.
15124 For example, @samp{f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))}
15125 formats in Big mode as
15140 There are several special composition functions that work only as
15141 components of a vertical composition. The @code{cbase} function
15142 controls the baseline of the vertical composition; the baseline
15143 will be the same as the baseline of whatever component is enclosed
15144 in @code{cbase}. Thus @samp{f(cvert([a, cbase(bb), ccc]),
15145 cvert([a^2 + 1, cbase(b^2)]))} displays as
15165 There are also @code{ctbase} and @code{cbbase} functions which
15166 make the baseline of the vertical composition equal to the top
15167 or bottom line (rather than the baseline) of that component.
15168 Thus @samp{cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) +
15169 cvert([cbbase(a / b)])} gives
15181 There should be only one @code{cbase}, @code{ctbase}, or @code{cbbase}
15182 function in a given vertical composition. These functions can also
15183 be written with no arguments: @samp{ctbase()} is a zero-height object
15184 which means the baseline is the top line of the following item, and
15185 @samp{cbbase()} means the baseline is the bottom line of the preceding
15192 The @code{crule} function builds a ``rule,'' or horizontal line,
15193 across a vertical composition. By itself @samp{crule()} uses @samp{-}
15194 characters to build the rule. You can specify any other character,
15195 e.g., @samp{crule("=")}. The argument must be a character code or
15196 vector of exactly one character code. It is repeated to match the
15197 width of the widest item in the stack. For example, a quotient
15198 with a thick line is @samp{cvert([a + 1, cbase(crule("=")), b^2])}:
15217 Finally, the functions @code{clvert} and @code{crvert} act exactly
15218 like @code{cvert} except that the items are left- or right-justified
15219 in the stack. Thus @samp{clvert([a, bb, ccc]) + crvert([a, bb, ccc])}
15230 Like @code{choriz}, the vertical compositions accept a second argument
15231 which gives the precedence to use when formatting the components.
15232 Vertical compositions do not support separator strings.
15234 @node Other Compositions, Information about Compositions, Vertical Compositions, Compositions
15235 @subsubsection Other Compositions
15242 The @code{csup} function builds a superscripted expression. For
15243 example, @samp{csup(a, b)} looks the same as @samp{a^b} does in Big
15244 language mode. This is essentially a horizontal composition of
15245 @samp{a} and @samp{b}, where @samp{b} is shifted up so that its
15246 bottom line is one above the baseline.
15252 Likewise, the @code{csub} function builds a subscripted expression.
15253 This shifts @samp{b} down so that its top line is one below the
15254 bottom line of @samp{a} (note that this is not quite analogous to
15255 @code{csup}). Other arrangements can be obtained by using
15256 @code{choriz} and @code{cvert} directly.
15262 The @code{cflat} function formats its argument in ``flat'' mode,
15263 as obtained by @samp{d O}, if the current language mode is normal
15264 or Big. It has no effect in other language modes. For example,
15265 @samp{a^(b/c)} is formatted by Big mode like @samp{csup(a, cflat(b/c))}
15266 to improve its readability.
15272 The @code{cspace} function creates horizontal space. For example,
15273 @samp{cspace(4)} is effectively the same as @samp{string(" ")}.
15274 A second string (i.e., vector of characters) argument is repeated
15275 instead of the space character. For example, @samp{cspace(4, "ab")}
15276 looks like @samp{abababab}. If the second argument is not a string,
15277 it is formatted in the normal way and then several copies of that
15278 are composed together: @samp{cspace(4, a^2)} yields
15288 If the number argument is zero, this is a zero-width object.
15294 The @code{cvspace} function creates vertical space, or a vertical
15295 stack of copies of a certain string or formatted object. The
15296 baseline is the center line of the resulting stack. A numerical
15297 argument of zero will produce an object which contributes zero
15298 height if used in a vertical composition.
15308 There are also @code{ctspace} and @code{cbspace} functions which
15309 create vertical space with the baseline the same as the baseline
15310 of the top or bottom copy, respectively, of the second argument.
15311 Thus @samp{cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)}
15328 @node Information about Compositions, User-Defined Compositions, Other Compositions, Compositions
15329 @subsubsection Information about Compositions
15332 The functions in this section are actual functions; they compose their
15333 arguments according to the current language and other display modes,
15334 then return a certain measurement of the composition as an integer.
15340 The @code{cwidth} function measures the width, in characters, of a
15341 composition. For example, @samp{cwidth(a + b)} is 5, and
15342 @samp{cwidth(a / b)} is 5 in Normal mode, 1 in Big mode, and 11 in
15343 @TeX{} mode (for @samp{@{a \over b@}}). The argument may involve
15344 the composition functions described in this section.
15350 The @code{cheight} function measures the height of a composition.
15351 This is the total number of lines in the argument's printed form.
15361 The functions @code{cascent} and @code{cdescent} measure the amount
15362 of the height that is above (and including) the baseline, or below
15363 the baseline, respectively. Thus @samp{cascent(@var{x}) + cdescent(@var{x})}
15364 always equals @samp{cheight(@var{x})}. For a one-line formula like
15365 @samp{a + b}, @code{cascent} returns 1 and @code{cdescent} returns 0.
15366 For @samp{a / b} in Big mode, @code{cascent} returns 2 and @code{cdescent}
15367 returns 1. The only formula for which @code{cascent} will return zero
15368 is @samp{cvspace(0)} or equivalents.
15370 @node User-Defined Compositions, , Information about Compositions, Compositions
15371 @subsubsection User-Defined Compositions
15375 @pindex calc-user-define-composition
15376 The @kbd{Z C} (@code{calc-user-define-composition}) command lets you
15377 define the display format for any algebraic function. You provide a
15378 formula containing a certain number of argument variables on the stack.
15379 Any time Calc formats a call to the specified function in the current
15380 language mode and with that number of arguments, Calc effectively
15381 replaces the function call with that formula with the arguments
15384 Calc builds the default argument list by sorting all the variable names
15385 that appear in the formula into alphabetical order. You can edit this
15386 argument list before pressing @key{RET} if you wish. Any variables in
15387 the formula that do not appear in the argument list will be displayed
15388 literally; any arguments that do not appear in the formula will not
15389 affect the display at all.
15391 You can define formats for built-in functions, for functions you have
15392 defined with @kbd{Z F} (@pxref{Algebraic Definitions}), or for functions
15393 which have no definitions but are being used as purely syntactic objects.
15394 You can define different formats for each language mode, and for each
15395 number of arguments, using a succession of @kbd{Z C} commands. When
15396 Calc formats a function call, it first searches for a format defined
15397 for the current language mode (and number of arguments); if there is
15398 none, it uses the format defined for the Normal language mode. If
15399 neither format exists, Calc uses its built-in standard format for that
15400 function (usually just @samp{@var{func}(@var{args})}).
15402 If you execute @kbd{Z C} with the number 0 on the stack instead of a
15403 formula, any defined formats for the function in the current language
15404 mode will be removed. The function will revert to its standard format.
15406 For example, the default format for the binomial coefficient function
15407 @samp{choose(n, m)} in the Big language mode is
15418 You might prefer the notation,
15428 To define this notation, first make sure you are in Big mode,
15429 then put the formula
15432 choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])
15436 on the stack and type @kbd{Z C}. Answer the first prompt with
15437 @code{choose}. The second prompt will be the default argument list
15438 of @samp{(C m n)}. Edit this list to be @samp{(n m)} and press
15439 @key{RET}. Now, try it out: For example, turn simplification
15440 off with @kbd{m O} and enter @samp{choose(a,b) + choose(7,3)}
15441 as an algebraic entry.
15450 As another example, let's define the usual notation for Stirling
15451 numbers of the first kind, @samp{stir1(n, m)}. This is just like
15452 the regular format for binomial coefficients but with square brackets
15453 instead of parentheses.
15456 choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])
15459 Now type @kbd{Z C stir1 @key{RET}}, edit the argument list to
15460 @samp{(n m)}, and type @key{RET}.
15462 The formula provided to @kbd{Z C} usually will involve composition
15463 functions, but it doesn't have to. Putting the formula @samp{a + b + c}
15464 onto the stack and typing @kbd{Z C foo @key{RET} @key{RET}} would define
15465 the function @samp{foo(x,y,z)} to display like @samp{x + y + z}.
15466 This ``sum'' will act exactly like a real sum for all formatting
15467 purposes (it will be parenthesized the same, and so on). However
15468 it will be computationally unrelated to a sum. For example, the
15469 formula @samp{2 * foo(1, 2, 3)} will display as @samp{2 (1 + 2 + 3)}.
15470 Operator precedences have caused the ``sum'' to be written in
15471 parentheses, but the arguments have not actually been summed.
15472 (Generally a display format like this would be undesirable, since
15473 it can easily be confused with a real sum.)
15475 The special function @code{eval} can be used inside a @kbd{Z C}
15476 composition formula to cause all or part of the formula to be
15477 evaluated at display time. For example, if the formula is
15478 @samp{a + eval(b + c)}, then @samp{foo(1, 2, 3)} will be displayed
15479 as @samp{1 + 5}. Evaluation will use the default simplifications,
15480 regardless of the current simplification mode. There are also
15481 @code{evalsimp} and @code{evalextsimp} which simplify as if by
15482 @kbd{a s} and @kbd{a e} (respectively). Note that these ``functions''
15483 operate only in the context of composition formulas (and also in
15484 rewrite rules, where they serve a similar purpose; @pxref{Rewrite
15485 Rules}). On the stack, a call to @code{eval} will be left in
15488 It is not a good idea to use @code{eval} except as a last resort.
15489 It can cause the display of formulas to be extremely slow. For
15490 example, while @samp{eval(a + b)} might seem quite fast and simple,
15491 there are several situations where it could be slow. For example,
15492 @samp{a} and/or @samp{b} could be polar complex numbers, in which
15493 case doing the sum requires trigonometry. Or, @samp{a} could be
15494 the factorial @samp{fact(100)} which is unevaluated because you
15495 have typed @kbd{m O}; @code{eval} will evaluate it anyway to
15496 produce a large, unwieldy integer.
15498 You can save your display formats permanently using the @kbd{Z P}
15499 command (@pxref{Creating User Keys}).
15501 @node Syntax Tables, , Compositions, Language Modes
15502 @subsection Syntax Tables
15505 @cindex Syntax tables
15506 @cindex Parsing formulas, customized
15507 Syntax tables do for input what compositions do for output: They
15508 allow you to teach custom notations to Calc's formula parser.
15509 Calc keeps a separate syntax table for each language mode.
15511 (Note that the Calc ``syntax tables'' discussed here are completely
15512 unrelated to the syntax tables described in the Emacs manual.)
15515 @pindex calc-edit-user-syntax
15516 The @kbd{Z S} (@code{calc-edit-user-syntax}) command edits the
15517 syntax table for the current language mode. If you want your
15518 syntax to work in any language, define it in the Normal language
15519 mode. Type @kbd{C-c C-c} to finish editing the syntax table, or
15520 @kbd{C-x k} to cancel the edit. The @kbd{m m} command saves all
15521 the syntax tables along with the other mode settings;
15522 @pxref{General Mode Commands}.
15525 * Syntax Table Basics::
15526 * Precedence in Syntax Tables::
15527 * Advanced Syntax Patterns::
15528 * Conditional Syntax Rules::
15531 @node Syntax Table Basics, Precedence in Syntax Tables, Syntax Tables, Syntax Tables
15532 @subsubsection Syntax Table Basics
15535 @dfn{Parsing} is the process of converting a raw string of characters,
15536 such as you would type in during algebraic entry, into a Calc formula.
15537 Calc's parser works in two stages. First, the input is broken down
15538 into @dfn{tokens}, such as words, numbers, and punctuation symbols
15539 like @samp{+}, @samp{:=}, and @samp{+/-}. Space between tokens is
15540 ignored (except when it serves to separate adjacent words). Next,
15541 the parser matches this string of tokens against various built-in
15542 syntactic patterns, such as ``an expression followed by @samp{+}
15543 followed by another expression'' or ``a name followed by @samp{(},
15544 zero or more expressions separated by commas, and @samp{)}.''
15546 A @dfn{syntax table} is a list of user-defined @dfn{syntax rules},
15547 which allow you to specify new patterns to define your own
15548 favorite input notations. Calc's parser always checks the syntax
15549 table for the current language mode, then the table for the Normal
15550 language mode, before it uses its built-in rules to parse an
15551 algebraic formula you have entered. Each syntax rule should go on
15552 its own line; it consists of a @dfn{pattern}, a @samp{:=} symbol,
15553 and a Calc formula with an optional @dfn{condition}. (Syntax rules
15554 resemble algebraic rewrite rules, but the notation for patterns is
15555 completely different.)
15557 A syntax pattern is a list of tokens, separated by spaces.
15558 Except for a few special symbols, tokens in syntax patterns are
15559 matched literally, from left to right. For example, the rule,
15566 would cause Calc to parse the formula @samp{4+foo()*5} as if it
15567 were @samp{4+(2+3)*5}. Notice that the parentheses were written
15568 as two separate tokens in the rule. As a result, the rule works
15569 for both @samp{foo()} and @w{@samp{foo ( )}}. If we had written
15570 the rule as @samp{foo () := 2+3}, then Calc would treat @samp{()}
15571 as a single, indivisible token, so that @w{@samp{foo( )}} would
15572 not be recognized by the rule. (It would be parsed as a regular
15573 zero-argument function call instead.) In fact, this rule would
15574 also make trouble for the rest of Calc's parser: An unrelated
15575 formula like @samp{bar()} would now be tokenized into @samp{bar ()}
15576 instead of @samp{bar ( )}, so that the standard parser for function
15577 calls would no longer recognize it!
15579 While it is possible to make a token with a mixture of letters
15580 and punctuation symbols, this is not recommended. It is better to
15581 break it into several tokens, as we did with @samp{foo()} above.
15583 The symbol @samp{#} in a syntax pattern matches any Calc expression.
15584 On the righthand side, the things that matched the @samp{#}s can
15585 be referred to as @samp{#1}, @samp{#2}, and so on (where @samp{#1}
15586 matches the leftmost @samp{#} in the pattern). For example, these
15587 rules match a user-defined function, prefix operator, infix operator,
15588 and postfix operator, respectively:
15591 foo ( # ) := myfunc(#1)
15592 foo # := myprefix(#1)
15593 # foo # := myinfix(#1,#2)
15594 # foo := mypostfix(#1)
15597 Thus @samp{foo(3)} will parse as @samp{myfunc(3)}, and @samp{2+3 foo}
15598 will parse as @samp{mypostfix(2+3)}.
15600 It is important to write the first two rules in the order shown,
15601 because Calc tries rules in order from first to last. If the
15602 pattern @samp{foo #} came first, it would match anything that could
15603 match the @samp{foo ( # )} rule, since an expression in parentheses
15604 is itself a valid expression. Thus the @w{@samp{foo ( # )}} rule would
15605 never get to match anything. Likewise, the last two rules must be
15606 written in the order shown or else @samp{3 foo 4} will be parsed as
15607 @samp{mypostfix(3) * 4}. (Of course, the best way to avoid these
15608 ambiguities is not to use the same symbol in more than one way at
15609 the same time! In case you're not convinced, try the following
15610 exercise: How will the above rules parse the input @samp{foo(3,4)},
15611 if at all? Work it out for yourself, then try it in Calc and see.)
15613 Calc is quite flexible about what sorts of patterns are allowed.
15614 The only rule is that every pattern must begin with a literal
15615 token (like @samp{foo} in the first two patterns above), or with
15616 a @samp{#} followed by a literal token (as in the last two
15617 patterns). After that, any mixture is allowed, although putting
15618 two @samp{#}s in a row will not be very useful since two
15619 expressions with nothing between them will be parsed as one
15620 expression that uses implicit multiplication.
15622 As a more practical example, Maple uses the notation
15623 @samp{sum(a(i), i=1..10)} for sums, which Calc's Maple mode doesn't
15624 recognize at present. To handle this syntax, we simply add the
15628 sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)
15632 to the Maple mode syntax table. As another example, C mode can't
15633 read assignment operators like @samp{++} and @samp{*=}. We can
15634 define these operators quite easily:
15637 # *= # := muleq(#1,#2)
15638 # ++ := postinc(#1)
15643 To complete the job, we would use corresponding composition functions
15644 and @kbd{Z C} to cause these functions to display in their respective
15645 Maple and C notations. (Note that the C example ignores issues of
15646 operator precedence, which are discussed in the next section.)
15648 You can enclose any token in quotes to prevent its usual
15649 interpretation in syntax patterns:
15652 # ":=" # := becomes(#1,#2)
15655 Quotes also allow you to include spaces in a token, although once
15656 again it is generally better to use two tokens than one token with
15657 an embedded space. To include an actual quotation mark in a quoted
15658 token, precede it with a backslash. (This also works to include
15659 backslashes in tokens.)
15662 # "bad token" # "/\"\\" # := silly(#1,#2,#3)
15666 This will parse @samp{3 bad token 4 /"\ 5} to @samp{silly(3,4,5)}.
15668 The token @kbd{#} has a predefined meaning in Calc's formula parser;
15669 it is not valid to use @samp{"#"} in a syntax rule. However, longer
15670 tokens that include the @samp{#} character are allowed. Also, while
15671 @samp{"$"} and @samp{"\""} are allowed as tokens, their presence in
15672 the syntax table will prevent those characters from working in their
15673 usual ways (referring to stack entries and quoting strings,
15676 Finally, the notation @samp{%%} anywhere in a syntax table causes
15677 the rest of the line to be ignored as a comment.
15679 @node Precedence in Syntax Tables, Advanced Syntax Patterns, Syntax Table Basics, Syntax Tables
15680 @subsubsection Precedence
15683 Different operators are generally assigned different @dfn{precedences}.
15684 By default, an operator defined by a rule like
15687 # foo # := foo(#1,#2)
15691 will have an extremely low precedence, so that @samp{2*3+4 foo 5 == 6}
15692 will be parsed as @samp{(2*3+4) foo (5 == 6)}. To change the
15693 precedence of an operator, use the notation @samp{#/@var{p}} in
15694 place of @samp{#}, where @var{p} is an integer precedence level.
15695 For example, 185 lies between the precedences for @samp{+} and
15696 @samp{*}, so if we change this rule to
15699 #/185 foo #/186 := foo(#1,#2)
15703 then @samp{2+3 foo 4*5} will be parsed as @samp{2+(3 foo (4*5))}.
15704 Also, because we've given the righthand expression slightly higher
15705 precedence, our new operator will be left-associative:
15706 @samp{1 foo 2 foo 3} will be parsed as @samp{(1 foo 2) foo 3}.
15707 By raising the precedence of the lefthand expression instead, we
15708 can create a right-associative operator.
15710 @xref{Composition Basics}, for a table of precedences of the
15711 standard Calc operators. For the precedences of operators in other
15712 language modes, look in the Calc source file @file{calc-lang.el}.
15714 @node Advanced Syntax Patterns, Conditional Syntax Rules, Precedence in Syntax Tables, Syntax Tables
15715 @subsubsection Advanced Syntax Patterns
15718 To match a function with a variable number of arguments, you could
15722 foo ( # ) := myfunc(#1)
15723 foo ( # , # ) := myfunc(#1,#2)
15724 foo ( # , # , # ) := myfunc(#1,#2,#3)
15728 but this isn't very elegant. To match variable numbers of items,
15729 Calc uses some notations inspired regular expressions and the
15730 ``extended BNF'' style used by some language designers.
15733 foo ( @{ # @}*, ) := apply(myfunc,#1)
15736 The token @samp{@{} introduces a repeated or optional portion.
15737 One of the three tokens @samp{@}*}, @samp{@}+}, or @samp{@}?}
15738 ends the portion. These will match zero or more, one or more,
15739 or zero or one copies of the enclosed pattern, respectively.
15740 In addition, @samp{@}*} and @samp{@}+} can be followed by a
15741 separator token (with no space in between, as shown above).
15742 Thus @samp{@{ # @}*,} matches nothing, or one expression, or
15743 several expressions separated by commas.
15745 A complete @samp{@{ ... @}} item matches as a vector of the
15746 items that matched inside it. For example, the above rule will
15747 match @samp{foo(1,2,3)} to get @samp{apply(myfunc,[1,2,3])}.
15748 The Calc @code{apply} function takes a function name and a vector
15749 of arguments and builds a call to the function with those
15750 arguments, so the net result is the formula @samp{myfunc(1,2,3)}.
15752 If the body of a @samp{@{ ... @}} contains several @samp{#}s
15753 (or nested @samp{@{ ... @}} constructs), then the items will be
15754 strung together into the resulting vector. If the body
15755 does not contain anything but literal tokens, the result will
15756 always be an empty vector.
15759 foo ( @{ # , # @}+, ) := bar(#1)
15760 foo ( @{ @{ # @}*, @}*; ) := matrix(#1)
15764 will parse @samp{foo(1, 2, 3, 4)} as @samp{bar([1, 2, 3, 4])}, and
15765 @samp{foo(1, 2; 3, 4)} as @samp{matrix([[1, 2], [3, 4]])}. Also, after
15766 some thought it's easy to see how this pair of rules will parse
15767 @samp{foo(1, 2, 3)} as @samp{matrix([[1, 2, 3]])}, since the first
15768 rule will only match an even number of arguments. The rule
15771 foo ( # @{ , # , # @}? ) := bar(#1,#2)
15775 will parse @samp{foo(2,3,4)} as @samp{bar(2,[3,4])}, and
15776 @samp{foo(2)} as @samp{bar(2,[])}.
15778 The notation @samp{@{ ... @}?.} (note the trailing period) works
15779 just the same as regular @samp{@{ ... @}?}, except that it does not
15780 count as an argument; the following two rules are equivalent:
15783 foo ( # , @{ also @}? # ) := bar(#1,#3)
15784 foo ( # , @{ also @}?. # ) := bar(#1,#2)
15788 Note that in the first case the optional text counts as @samp{#2},
15789 which will always be an empty vector, but in the second case no
15790 empty vector is produced.
15792 Another variant is @samp{@{ ... @}?$}, which means the body is
15793 optional only at the end of the input formula. All built-in syntax
15794 rules in Calc use this for closing delimiters, so that during
15795 algebraic entry you can type @kbd{[sqrt(2), sqrt(3 @key{RET}}, omitting
15796 the closing parenthesis and bracket. Calc does this automatically
15797 for trailing @samp{)}, @samp{]}, and @samp{>} tokens in syntax
15798 rules, but you can use @samp{@{ ... @}?$} explicitly to get
15799 this effect with any token (such as @samp{"@}"} or @samp{end}).
15800 Like @samp{@{ ... @}?.}, this notation does not count as an
15801 argument. Conversely, you can use quotes, as in @samp{")"}, to
15802 prevent a closing-delimiter token from being automatically treated
15805 Calc's parser does not have full backtracking, which means some
15806 patterns will not work as you might expect:
15809 foo ( @{ # , @}? # , # ) := bar(#1,#2,#3)
15813 Here we are trying to make the first argument optional, so that
15814 @samp{foo(2,3)} parses as @samp{bar([],2,3)}. Unfortunately, Calc
15815 first tries to match @samp{2,} against the optional part of the
15816 pattern, finds a match, and so goes ahead to match the rest of the
15817 pattern. Later on it will fail to match the second comma, but it
15818 doesn't know how to go back and try the other alternative at that
15819 point. One way to get around this would be to use two rules:
15822 foo ( # , # , # ) := bar([#1],#2,#3)
15823 foo ( # , # ) := bar([],#1,#2)
15826 More precisely, when Calc wants to match an optional or repeated
15827 part of a pattern, it scans forward attempting to match that part.
15828 If it reaches the end of the optional part without failing, it
15829 ``finalizes'' its choice and proceeds. If it fails, though, it
15830 backs up and tries the other alternative. Thus Calc has ``partial''
15831 backtracking. A fully backtracking parser would go on to make sure
15832 the rest of the pattern matched before finalizing the choice.
15834 @node Conditional Syntax Rules, , Advanced Syntax Patterns, Syntax Tables
15835 @subsubsection Conditional Syntax Rules
15838 It is possible to attach a @dfn{condition} to a syntax rule. For
15842 foo ( # ) := ifoo(#1) :: integer(#1)
15843 foo ( # ) := gfoo(#1)
15847 will parse @samp{foo(3)} as @samp{ifoo(3)}, but will parse
15848 @samp{foo(3.5)} and @samp{foo(x)} as calls to @code{gfoo}. Any
15849 number of conditions may be attached; all must be true for the
15850 rule to succeed. A condition is ``true'' if it evaluates to a
15851 nonzero number. @xref{Logical Operations}, for a list of Calc
15852 functions like @code{integer} that perform logical tests.
15854 The exact sequence of events is as follows: When Calc tries a
15855 rule, it first matches the pattern as usual. It then substitutes
15856 @samp{#1}, @samp{#2}, etc., in the conditions, if any. Next, the
15857 conditions are simplified and evaluated in order from left to right,
15858 as if by the @w{@kbd{a s}} algebra command (@pxref{Simplifying Formulas}).
15859 Each result is true if it is a nonzero number, or an expression
15860 that can be proven to be nonzero (@pxref{Declarations}). If the
15861 results of all conditions are true, the expression (such as
15862 @samp{ifoo(#1)}) has its @samp{#}s substituted, and that is the
15863 result of the parse. If the result of any condition is false, Calc
15864 goes on to try the next rule in the syntax table.
15866 Syntax rules also support @code{let} conditions, which operate in
15867 exactly the same way as they do in algebraic rewrite rules.
15868 @xref{Other Features of Rewrite Rules}, for details. A @code{let}
15869 condition is always true, but as a side effect it defines a
15870 variable which can be used in later conditions, and also in the
15871 expression after the @samp{:=} sign:
15874 foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)
15878 The @code{dnumint} function tests if a value is numerically an
15879 integer, i.e., either a true integer or an integer-valued float.
15880 This rule will parse @code{foo} with a half-integer argument,
15881 like @samp{foo(3.5)}, to a call like @samp{hifoo(4.)}.
15883 The lefthand side of a syntax rule @code{let} must be a simple
15884 variable, not the arbitrary pattern that is allowed in rewrite
15887 The @code{matches} function is also treated specially in syntax
15888 rule conditions (again, in the same way as in rewrite rules).
15889 @xref{Matching Commands}. If the matching pattern contains
15890 meta-variables, then those meta-variables may be used in later
15891 conditions and in the result expression. The arguments to
15892 @code{matches} are not evaluated in this situation.
15895 sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])
15899 This is another way to implement the Maple mode @code{sum} notation.
15900 In this approach, we allow @samp{#2} to equal the whole expression
15901 @samp{i=1..10}. Then, we use @code{matches} to break it apart into
15902 its components. If the expression turns out not to match the pattern,
15903 the syntax rule will fail. Note that @kbd{Z S} always uses Calc's
15904 Normal language mode for editing expressions in syntax rules, so we
15905 must use regular Calc notation for the interval @samp{[b..c]} that
15906 will correspond to the Maple mode interval @samp{1..10}.
15908 @node Modes Variable, Calc Mode Line, Language Modes, Mode Settings
15909 @section The @code{Modes} Variable
15913 @pindex calc-get-modes
15914 The @kbd{m g} (@code{calc-get-modes}) command pushes onto the stack
15915 a vector of numbers that describes the various mode settings that
15916 are in effect. With a numeric prefix argument, it pushes only the
15917 @var{n}th mode, i.e., the @var{n}th element of this vector. Keyboard
15918 macros can use the @kbd{m g} command to modify their behavior based
15919 on the current mode settings.
15921 @cindex @code{Modes} variable
15923 The modes vector is also available in the special variable
15924 @code{Modes}. In other words, @kbd{m g} is like @kbd{s r Modes @key{RET}}.
15925 It will not work to store into this variable; in fact, if you do,
15926 @code{Modes} will cease to track the current modes. (The @kbd{m g}
15927 command will continue to work, however.)
15929 In general, each number in this vector is suitable as a numeric
15930 prefix argument to the associated mode-setting command. (Recall
15931 that the @kbd{~} key takes a number from the stack and gives it as
15932 a numeric prefix to the next command.)
15934 The elements of the modes vector are as follows:
15938 Current precision. Default is 12; associated command is @kbd{p}.
15941 Binary word size. Default is 32; associated command is @kbd{b w}.
15944 Stack size (not counting the value about to be pushed by @kbd{m g}).
15945 This is zero if @kbd{m g} is executed with an empty stack.
15948 Number radix. Default is 10; command is @kbd{d r}.
15951 Floating-point format. This is the number of digits, plus the
15952 constant 0 for normal notation, 10000 for scientific notation,
15953 20000 for engineering notation, or 30000 for fixed-point notation.
15954 These codes are acceptable as prefix arguments to the @kbd{d n}
15955 command, but note that this may lose information: For example,
15956 @kbd{d s} and @kbd{C-u 12 d s} have similar (but not quite
15957 identical) effects if the current precision is 12, but they both
15958 produce a code of 10012, which will be treated by @kbd{d n} as
15959 @kbd{C-u 12 d s}. If the precision then changes, the float format
15960 will still be frozen at 12 significant figures.
15963 Angular mode. Default is 1 (degrees). Other values are 2 (radians)
15964 and 3 (HMS). The @kbd{m d} command accepts these prefixes.
15967 Symbolic mode. Value is 0 or 1; default is 0. Command is @kbd{m s}.
15970 Fraction mode. Value is 0 or 1; default is 0. Command is @kbd{m f}.
15973 Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0.
15974 Command is @kbd{m p}.
15977 Matrix/Scalar mode. Default value is @mathit{-1}. Value is 0 for Scalar
15978 mode, @mathit{-2} for Matrix mode, or @var{N} for
15979 @texline @math{N\times N}
15980 @infoline @var{N}x@var{N}
15981 Matrix mode. Command is @kbd{m v}.
15984 Simplification mode. Default is 1. Value is @mathit{-1} for off (@kbd{m O}),
15985 0 for @kbd{m N}, 2 for @kbd{m B}, 3 for @kbd{m A}, 4 for @kbd{m E},
15986 or 5 for @w{@kbd{m U}}. The @kbd{m D} command accepts these prefixes.
15989 Infinite mode. Default is @mathit{-1} (off). Value is 1 if the mode is on,
15990 or 0 if the mode is on with positive zeros. Command is @kbd{m i}.
15993 For example, the sequence @kbd{M-1 m g @key{RET} 2 + ~ p} increases the
15994 precision by two, leaving a copy of the old precision on the stack.
15995 Later, @kbd{~ p} will restore the original precision using that
15996 stack value. (This sequence might be especially useful inside a
15999 As another example, @kbd{M-3 m g 1 - ~ @key{DEL}} deletes all but the
16000 oldest (bottommost) stack entry.
16002 Yet another example: The HP-48 ``round'' command rounds a number
16003 to the current displayed precision. You could roughly emulate this
16004 in Calc with the sequence @kbd{M-5 m g 10000 % ~ c c}. (This
16005 would not work for fixed-point mode, but it wouldn't be hard to
16006 do a full emulation with the help of the @kbd{Z [} and @kbd{Z ]}
16007 programming commands. @xref{Conditionals in Macros}.)
16009 @node Calc Mode Line, , Modes Variable, Mode Settings
16010 @section The Calc Mode Line
16013 @cindex Mode line indicators
16014 This section is a summary of all symbols that can appear on the
16015 Calc mode line, the highlighted bar that appears under the Calc
16016 stack window (or under an editing window in Embedded mode).
16018 The basic mode line format is:
16021 --%%-Calc: 12 Deg @var{other modes} (Calculator)
16024 The @samp{%%} is the Emacs symbol for ``read-only''; it shows that
16025 regular Emacs commands are not allowed to edit the stack buffer
16026 as if it were text.
16028 The word @samp{Calc:} changes to @samp{CalcEmbed:} if Embedded mode
16029 is enabled. The words after this describe the various Calc modes
16030 that are in effect.
16032 The first mode is always the current precision, an integer.
16033 The second mode is always the angular mode, either @code{Deg},
16034 @code{Rad}, or @code{Hms}.
16036 Here is a complete list of the remaining symbols that can appear
16041 Algebraic mode (@kbd{m a}; @pxref{Algebraic Entry}).
16044 Incomplete algebraic mode (@kbd{C-u m a}).
16047 Total algebraic mode (@kbd{m t}).
16050 Symbolic mode (@kbd{m s}; @pxref{Symbolic Mode}).
16053 Matrix mode (@kbd{m v}; @pxref{Matrix Mode}).
16055 @item Matrix@var{n}
16056 Dimensioned Matrix mode (@kbd{C-u @var{n} m v}).
16059 Scalar mode (@kbd{m v}; @pxref{Matrix Mode}).
16062 Polar complex mode (@kbd{m p}; @pxref{Polar Mode}).
16065 Fraction mode (@kbd{m f}; @pxref{Fraction Mode}).
16068 Infinite mode (@kbd{m i}; @pxref{Infinite Mode}).
16071 Positive Infinite mode (@kbd{C-u 0 m i}).
16074 Default simplifications off (@kbd{m O}; @pxref{Simplification Modes}).
16077 Default simplifications for numeric arguments only (@kbd{m N}).
16079 @item BinSimp@var{w}
16080 Binary-integer simplification mode; word size @var{w} (@kbd{m B}, @kbd{b w}).
16083 Algebraic simplification mode (@kbd{m A}).
16086 Extended algebraic simplification mode (@kbd{m E}).
16089 Units simplification mode (@kbd{m U}).
16092 Current radix is 2 (@kbd{d 2}; @pxref{Radix Modes}).
16095 Current radix is 8 (@kbd{d 8}).
16098 Current radix is 16 (@kbd{d 6}).
16101 Current radix is @var{n} (@kbd{d r}).
16104 Leading zeros (@kbd{d z}; @pxref{Radix Modes}).
16107 Big language mode (@kbd{d B}; @pxref{Normal Language Modes}).
16110 One-line normal language mode (@kbd{d O}).
16113 Unformatted language mode (@kbd{d U}).
16116 C language mode (@kbd{d C}; @pxref{C FORTRAN Pascal}).
16119 Pascal language mode (@kbd{d P}).
16122 FORTRAN language mode (@kbd{d F}).
16125 @TeX{} language mode (@kbd{d T}; @pxref{TeX and LaTeX Language Modes}).
16128 La@TeX{} language mode (@kbd{d L}; @pxref{TeX and LaTeX Language Modes}).
16131 @dfn{Eqn} language mode (@kbd{d E}; @pxref{Eqn Language Mode}).
16134 Mathematica language mode (@kbd{d M}; @pxref{Mathematica Language Mode}).
16137 Maple language mode (@kbd{d W}; @pxref{Maple Language Mode}).
16140 Normal float mode with @var{n} digits (@kbd{d n}; @pxref{Float Formats}).
16143 Fixed point mode with @var{n} digits after the point (@kbd{d f}).
16146 Scientific notation mode (@kbd{d s}).
16149 Scientific notation with @var{n} digits (@kbd{d s}).
16152 Engineering notation mode (@kbd{d e}).
16155 Engineering notation with @var{n} digits (@kbd{d e}).
16158 Left-justified display indented by @var{n} (@kbd{d <}; @pxref{Justification}).
16161 Right-justified display (@kbd{d >}).
16164 Right-justified display with width @var{n} (@kbd{d >}).
16167 Centered display (@kbd{d =}).
16169 @item Center@var{n}
16170 Centered display with center column @var{n} (@kbd{d =}).
16173 Line breaking with width @var{n} (@kbd{d b}; @pxref{Normal Language Modes}).
16176 No line breaking (@kbd{d b}).
16179 Selections show deep structure (@kbd{j b}; @pxref{Making Selections}).
16182 Record modes in @file{~/.calc.el} (@kbd{m R}; @pxref{General Mode Commands}).
16185 Record modes in Embedded buffer (@kbd{m R}).
16188 Record modes as editing-only in Embedded buffer (@kbd{m R}).
16191 Record modes as permanent-only in Embedded buffer (@kbd{m R}).
16194 Record modes as global in Embedded buffer (@kbd{m R}).
16197 Automatic recomputation turned off (@kbd{m C}; @pxref{Automatic
16201 GNUPLOT process is alive in background (@pxref{Graphics}).
16204 Top-of-stack has a selection (Embedded only; @pxref{Making Selections}).
16207 The stack display may not be up-to-date (@pxref{Display Modes}).
16210 ``Inverse'' prefix was pressed (@kbd{I}; @pxref{Inverse and Hyperbolic}).
16213 ``Hyperbolic'' prefix was pressed (@kbd{H}).
16216 ``Keep-arguments'' prefix was pressed (@kbd{K}).
16219 Stack is truncated (@kbd{d t}; @pxref{Truncating the Stack}).
16222 In addition, the symbols @code{Active} and @code{~Active} can appear
16223 as minor modes on an Embedded buffer's mode line. @xref{Embedded Mode}.
16225 @node Arithmetic, Scientific Functions, Mode Settings, Top
16226 @chapter Arithmetic Functions
16229 This chapter describes the Calc commands for doing simple calculations
16230 on numbers, such as addition, absolute value, and square roots. These
16231 commands work by removing the top one or two values from the stack,
16232 performing the desired operation, and pushing the result back onto the
16233 stack. If the operation cannot be performed, the result pushed is a
16234 formula instead of a number, such as @samp{2/0} (because division by zero
16235 is invalid) or @samp{sqrt(x)} (because the argument @samp{x} is a formula).
16237 Most of the commands described here can be invoked by a single keystroke.
16238 Some of the more obscure ones are two-letter sequences beginning with
16239 the @kbd{f} (``functions'') prefix key.
16241 @xref{Prefix Arguments}, for a discussion of the effect of numeric
16242 prefix arguments on commands in this chapter which do not otherwise
16243 interpret a prefix argument.
16246 * Basic Arithmetic::
16247 * Integer Truncation::
16248 * Complex Number Functions::
16250 * Date Arithmetic::
16251 * Financial Functions::
16252 * Binary Functions::
16255 @node Basic Arithmetic, Integer Truncation, Arithmetic, Arithmetic
16256 @section Basic Arithmetic
16265 The @kbd{+} (@code{calc-plus}) command adds two numbers. The numbers may
16266 be any of the standard Calc data types. The resulting sum is pushed back
16269 If both arguments of @kbd{+} are vectors or matrices (of matching dimensions),
16270 the result is a vector or matrix sum. If one argument is a vector and the
16271 other a scalar (i.e., a non-vector), the scalar is added to each of the
16272 elements of the vector to form a new vector. If the scalar is not a
16273 number, the operation is left in symbolic form: Suppose you added @samp{x}
16274 to the vector @samp{[1,2]}. You may want the result @samp{[1+x,2+x]}, or
16275 you may plan to substitute a 2-vector for @samp{x} in the future. Since
16276 the Calculator can't tell which interpretation you want, it makes the
16277 safest assumption. @xref{Reducing and Mapping}, for a way to add @samp{x}
16278 to every element of a vector.
16280 If either argument of @kbd{+} is a complex number, the result will in general
16281 be complex. If one argument is in rectangular form and the other polar,
16282 the current Polar mode determines the form of the result. If Symbolic
16283 mode is enabled, the sum may be left as a formula if the necessary
16284 conversions for polar addition are non-trivial.
16286 If both arguments of @kbd{+} are HMS forms, the forms are added according to
16287 the usual conventions of hours-minutes-seconds notation. If one argument
16288 is an HMS form and the other is a number, that number is converted from
16289 degrees or radians (depending on the current Angular mode) to HMS format
16290 and then the two HMS forms are added.
16292 If one argument of @kbd{+} is a date form, the other can be either a
16293 real number, which advances the date by a certain number of days, or
16294 an HMS form, which advances the date by a certain amount of time.
16295 Subtracting two date forms yields the number of days between them.
16296 Adding two date forms is meaningless, but Calc interprets it as the
16297 subtraction of one date form and the negative of the other. (The
16298 negative of a date form can be understood by remembering that dates
16299 are stored as the number of days before or after Jan 1, 1 AD.)
16301 If both arguments of @kbd{+} are error forms, the result is an error form
16302 with an appropriately computed standard deviation. If one argument is an
16303 error form and the other is a number, the number is taken to have zero error.
16304 Error forms may have symbolic formulas as their mean and/or error parts;
16305 adding these will produce a symbolic error form result. However, adding an
16306 error form to a plain symbolic formula (as in @samp{(a +/- b) + c}) will not
16307 work, for the same reasons just mentioned for vectors. Instead you must
16308 write @samp{(a +/- b) + (c +/- 0)}.
16310 If both arguments of @kbd{+} are modulo forms with equal values of @expr{M},
16311 or if one argument is a modulo form and the other a plain number, the
16312 result is a modulo form which represents the sum, modulo @expr{M}, of
16315 If both arguments of @kbd{+} are intervals, the result is an interval
16316 which describes all possible sums of the possible input values. If
16317 one argument is a plain number, it is treated as the interval
16318 @w{@samp{[x ..@: x]}}.
16320 If one argument of @kbd{+} is an infinity and the other is not, the
16321 result is that same infinity. If both arguments are infinite and in
16322 the same direction, the result is the same infinity, but if they are
16323 infinite in different directions the result is @code{nan}.
16331 The @kbd{-} (@code{calc-minus}) command subtracts two values. The top
16332 number on the stack is subtracted from the one behind it, so that the
16333 computation @kbd{5 @key{RET} 2 -} produces 3, not @mathit{-3}. All options
16334 available for @kbd{+} are available for @kbd{-} as well.
16342 The @kbd{*} (@code{calc-times}) command multiplies two numbers. If one
16343 argument is a vector and the other a scalar, the scalar is multiplied by
16344 the elements of the vector to produce a new vector. If both arguments
16345 are vectors, the interpretation depends on the dimensions of the
16346 vectors: If both arguments are matrices, a matrix multiplication is
16347 done. If one argument is a matrix and the other a plain vector, the
16348 vector is interpreted as a row vector or column vector, whichever is
16349 dimensionally correct. If both arguments are plain vectors, the result
16350 is a single scalar number which is the dot product of the two vectors.
16352 If one argument of @kbd{*} is an HMS form and the other a number, the
16353 HMS form is multiplied by that amount. It is an error to multiply two
16354 HMS forms together, or to attempt any multiplication involving date
16355 forms. Error forms, modulo forms, and intervals can be multiplied;
16356 see the comments for addition of those forms. When two error forms
16357 or intervals are multiplied they are considered to be statistically
16358 independent; thus, @samp{[-2 ..@: 3] * [-2 ..@: 3]} is @samp{[-6 ..@: 9]},
16359 whereas @w{@samp{[-2 ..@: 3] ^ 2}} is @samp{[0 ..@: 9]}.
16362 @pindex calc-divide
16367 The @kbd{/} (@code{calc-divide}) command divides two numbers. When
16368 dividing a scalar @expr{B} by a square matrix @expr{A}, the computation
16369 performed is @expr{B} times the inverse of @expr{A}. This also occurs
16370 if @expr{B} is itself a vector or matrix, in which case the effect is
16371 to solve the set of linear equations represented by @expr{B}. If @expr{B}
16372 is a matrix with the same number of rows as @expr{A}, or a plain vector
16373 (which is interpreted here as a column vector), then the equation
16374 @expr{A X = B} is solved for the vector or matrix @expr{X}. Otherwise,
16375 if @expr{B} is a non-square matrix with the same number of @emph{columns}
16376 as @expr{A}, the equation @expr{X A = B} is solved. If you wish a vector
16377 @expr{B} to be interpreted as a row vector to be solved as @expr{X A = B},
16378 make it into a one-row matrix with @kbd{C-u 1 v p} first. To force a
16379 left-handed solution with a square matrix @expr{B}, transpose @expr{A} and
16380 @expr{B} before dividing, then transpose the result.
16382 HMS forms can be divided by real numbers or by other HMS forms. Error
16383 forms can be divided in any combination of ways. Modulo forms where both
16384 values and the modulo are integers can be divided to get an integer modulo
16385 form result. Intervals can be divided; dividing by an interval that
16386 encompasses zero or has zero as a limit will result in an infinite
16395 The @kbd{^} (@code{calc-power}) command raises a number to a power. If
16396 the power is an integer, an exact result is computed using repeated
16397 multiplications. For non-integer powers, Calc uses Newton's method or
16398 logarithms and exponentials. Square matrices can be raised to integer
16399 powers. If either argument is an error (or interval or modulo) form,
16400 the result is also an error (or interval or modulo) form.
16404 If you press the @kbd{I} (inverse) key first, the @kbd{I ^} command
16405 computes an Nth root: @kbd{125 @key{RET} 3 I ^} computes the number 5.
16406 (This is entirely equivalent to @kbd{125 @key{RET} 1:3 ^}.)
16415 The @kbd{\} (@code{calc-idiv}) command divides two numbers on the stack
16416 to produce an integer result. It is equivalent to dividing with
16417 @key{/}, then rounding down with @kbd{F} (@code{calc-floor}), only a bit
16418 more convenient and efficient. Also, since it is an all-integer
16419 operation when the arguments are integers, it avoids problems that
16420 @kbd{/ F} would have with floating-point roundoff.
16428 The @kbd{%} (@code{calc-mod}) command performs a ``modulo'' (or ``remainder'')
16429 operation. Mathematically, @samp{a%b = a - (a\b)*b}, and is defined
16430 for all real numbers @expr{a} and @expr{b} (except @expr{b=0}). For
16431 positive @expr{b}, the result will always be between 0 (inclusive) and
16432 @expr{b} (exclusive). Modulo does not work for HMS forms and error forms.
16433 If @expr{a} is a modulo form, its modulo is changed to @expr{b}, which
16434 must be positive real number.
16439 The @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command
16440 divides the two integers on the top of the stack to produce a fractional
16441 result. This is a convenient shorthand for enabling Fraction mode (with
16442 @kbd{m f}) temporarily and using @samp{/}. Note that during numeric entry
16443 the @kbd{:} key is interpreted as a fraction separator, so to divide 8 by 6
16444 you would have to type @kbd{8 @key{RET} 6 @key{RET} :}. (Of course, in
16445 this case, it would be much easier simply to enter the fraction directly
16446 as @kbd{8:6 @key{RET}}!)
16449 @pindex calc-change-sign
16450 The @kbd{n} (@code{calc-change-sign}) command negates the number on the top
16451 of the stack. It works on numbers, vectors and matrices, HMS forms, date
16452 forms, error forms, intervals, and modulo forms.
16457 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the absolute
16458 value of a number. The result of @code{abs} is always a nonnegative
16459 real number: With a complex argument, it computes the complex magnitude.
16460 With a vector or matrix argument, it computes the Frobenius norm, i.e.,
16461 the square root of the sum of the squares of the absolute values of the
16462 elements. The absolute value of an error form is defined by replacing
16463 the mean part with its absolute value and leaving the error part the same.
16464 The absolute value of a modulo form is undefined. The absolute value of
16465 an interval is defined in the obvious way.
16468 @pindex calc-abssqr
16470 The @kbd{f A} (@code{calc-abssqr}) [@code{abssqr}] command computes the
16471 absolute value squared of a number, vector or matrix, or error form.
16476 The @kbd{f s} (@code{calc-sign}) [@code{sign}] command returns 1 if its
16477 argument is positive, @mathit{-1} if its argument is negative, or 0 if its
16478 argument is zero. In algebraic form, you can also write @samp{sign(a,x)}
16479 which evaluates to @samp{x * sign(a)}, i.e., either @samp{x}, @samp{-x}, or
16480 zero depending on the sign of @samp{a}.
16486 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
16487 reciprocal of a number, i.e., @expr{1 / x}. Operating on a square
16488 matrix, it computes the inverse of that matrix.
16493 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] command computes the square
16494 root of a number. For a negative real argument, the result will be a
16495 complex number whose form is determined by the current Polar mode.
16500 The @kbd{f h} (@code{calc-hypot}) [@code{hypot}] command computes the square
16501 root of the sum of the squares of two numbers. That is, @samp{hypot(a,b)}
16502 is the length of the hypotenuse of a right triangle with sides @expr{a}
16503 and @expr{b}. If the arguments are complex numbers, their squared
16504 magnitudes are used.
16509 The @kbd{f Q} (@code{calc-isqrt}) [@code{isqrt}] command computes the
16510 integer square root of an integer. This is the true square root of the
16511 number, rounded down to an integer. For example, @samp{isqrt(10)}
16512 produces 3. Note that, like @kbd{\} [@code{idiv}], this uses exact
16513 integer arithmetic throughout to avoid roundoff problems. If the input
16514 is a floating-point number or other non-integer value, this is exactly
16515 the same as @samp{floor(sqrt(x))}.
16523 The @kbd{f n} (@code{calc-min}) [@code{min}] and @kbd{f x} (@code{calc-max})
16524 [@code{max}] commands take the minimum or maximum of two real numbers,
16525 respectively. These commands also work on HMS forms, date forms,
16526 intervals, and infinities. (In algebraic expressions, these functions
16527 take any number of arguments and return the maximum or minimum among
16528 all the arguments.)
16532 @pindex calc-mant-part
16534 @pindex calc-xpon-part
16536 The @kbd{f M} (@code{calc-mant-part}) [@code{mant}] function extracts
16537 the ``mantissa'' part @expr{m} of its floating-point argument; @kbd{f X}
16538 (@code{calc-xpon-part}) [@code{xpon}] extracts the ``exponent'' part
16539 @expr{e}. The original number is equal to
16540 @texline @math{m \times 10^e},
16541 @infoline @expr{m * 10^e},
16542 where @expr{m} is in the interval @samp{[1.0 ..@: 10.0)} except that
16543 @expr{m=e=0} if the original number is zero. For integers
16544 and fractions, @code{mant} returns the number unchanged and @code{xpon}
16545 returns zero. The @kbd{v u} (@code{calc-unpack}) command can also be
16546 used to ``unpack'' a floating-point number; this produces an integer
16547 mantissa and exponent, with the constraint that the mantissa is not
16548 a multiple of ten (again except for the @expr{m=e=0} case).
16551 @pindex calc-scale-float
16553 The @kbd{f S} (@code{calc-scale-float}) [@code{scf}] function scales a number
16554 by a given power of ten. Thus, @samp{scf(mant(x), xpon(x)) = x} for any
16555 real @samp{x}. The second argument must be an integer, but the first
16556 may actually be any numeric value. For example, @samp{scf(5,-2) = 0.05}
16557 or @samp{1:20} depending on the current Fraction mode.
16561 @pindex calc-decrement
16562 @pindex calc-increment
16565 The @kbd{f [} (@code{calc-decrement}) [@code{decr}] and @kbd{f ]}
16566 (@code{calc-increment}) [@code{incr}] functions decrease or increase
16567 a number by one unit. For integers, the effect is obvious. For
16568 floating-point numbers, the change is by one unit in the last place.
16569 For example, incrementing @samp{12.3456} when the current precision
16570 is 6 digits yields @samp{12.3457}. If the current precision had been
16571 8 digits, the result would have been @samp{12.345601}. Incrementing
16572 @samp{0.0} produces
16573 @texline @math{10^{-p}},
16574 @infoline @expr{10^-p},
16575 where @expr{p} is the current
16576 precision. These operations are defined only on integers and floats.
16577 With numeric prefix arguments, they change the number by @expr{n} units.
16579 Note that incrementing followed by decrementing, or vice-versa, will
16580 almost but not quite always cancel out. Suppose the precision is
16581 6 digits and the number @samp{9.99999} is on the stack. Incrementing
16582 will produce @samp{10.0000}; decrementing will produce @samp{9.9999}.
16583 One digit has been dropped. This is an unavoidable consequence of the
16584 way floating-point numbers work.
16586 Incrementing a date/time form adjusts it by a certain number of seconds.
16587 Incrementing a pure date form adjusts it by a certain number of days.
16589 @node Integer Truncation, Complex Number Functions, Basic Arithmetic, Arithmetic
16590 @section Integer Truncation
16593 There are four commands for truncating a real number to an integer,
16594 differing mainly in their treatment of negative numbers. All of these
16595 commands have the property that if the argument is an integer, the result
16596 is the same integer. An integer-valued floating-point argument is converted
16599 If you press @kbd{H} (@code{calc-hyperbolic}) first, the result will be
16600 expressed as an integer-valued floating-point number.
16602 @cindex Integer part of a number
16611 The @kbd{F} (@code{calc-floor}) [@code{floor} or @code{ffloor}] command
16612 truncates a real number to the next lower integer, i.e., toward minus
16613 infinity. Thus @kbd{3.6 F} produces 3, but @kbd{_3.6 F} produces
16617 @pindex calc-ceiling
16624 The @kbd{I F} (@code{calc-ceiling}) [@code{ceil} or @code{fceil}]
16625 command truncates toward positive infinity. Thus @kbd{3.6 I F} produces
16626 4, and @kbd{_3.6 I F} produces @mathit{-3}.
16636 The @kbd{R} (@code{calc-round}) [@code{round} or @code{fround}] command
16637 rounds to the nearest integer. When the fractional part is .5 exactly,
16638 this command rounds away from zero. (All other rounding in the
16639 Calculator uses this convention as well.) Thus @kbd{3.5 R} produces 4
16640 but @kbd{3.4 R} produces 3; @kbd{_3.5 R} produces @mathit{-4}.
16650 The @kbd{I R} (@code{calc-trunc}) [@code{trunc} or @code{ftrunc}]
16651 command truncates toward zero. In other words, it ``chops off''
16652 everything after the decimal point. Thus @kbd{3.6 I R} produces 3 and
16653 @kbd{_3.6 I R} produces @mathit{-3}.
16655 These functions may not be applied meaningfully to error forms, but they
16656 do work for intervals. As a convenience, applying @code{floor} to a
16657 modulo form floors the value part of the form. Applied to a vector,
16658 these functions operate on all elements of the vector one by one.
16659 Applied to a date form, they operate on the internal numerical
16660 representation of dates, converting a date/time form into a pure date.
16678 There are two more rounding functions which can only be entered in
16679 algebraic notation. The @code{roundu} function is like @code{round}
16680 except that it rounds up, toward plus infinity, when the fractional
16681 part is .5. This distinction matters only for negative arguments.
16682 Also, @code{rounde} rounds to an even number in the case of a tie,
16683 rounding up or down as necessary. For example, @samp{rounde(3.5)} and
16684 @samp{rounde(4.5)} both return 4, but @samp{rounde(5.5)} returns 6.
16685 The advantage of round-to-even is that the net error due to rounding
16686 after a long calculation tends to cancel out to zero. An important
16687 subtle point here is that the number being fed to @code{rounde} will
16688 already have been rounded to the current precision before @code{rounde}
16689 begins. For example, @samp{rounde(2.500001)} with a current precision
16690 of 6 will incorrectly, or at least surprisingly, yield 2 because the
16691 argument will first have been rounded down to @expr{2.5} (which
16692 @code{rounde} sees as an exact tie between 2 and 3).
16694 Each of these functions, when written in algebraic formulas, allows
16695 a second argument which specifies the number of digits after the
16696 decimal point to keep. For example, @samp{round(123.4567, 2)} will
16697 produce the answer 123.46, and @samp{round(123.4567, -1)} will
16698 produce 120 (i.e., the cutoff is one digit to the @emph{left} of
16699 the decimal point). A second argument of zero is equivalent to
16700 no second argument at all.
16702 @cindex Fractional part of a number
16703 To compute the fractional part of a number (i.e., the amount which, when
16704 added to `@tfn{floor(}@var{n}@tfn{)}', will produce @var{n}) just take @var{n}
16705 modulo 1 using the @code{%} command.
16707 Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm),
16708 and @kbd{f Q} (integer square root) commands, which are analogous to
16709 @kbd{/}, @kbd{B}, and @kbd{Q}, respectively, except that they take integer
16710 arguments and return the result rounded down to an integer.
16712 @node Complex Number Functions, Conversions, Integer Truncation, Arithmetic
16713 @section Complex Number Functions
16719 The @kbd{J} (@code{calc-conj}) [@code{conj}] command computes the
16720 complex conjugate of a number. For complex number @expr{a+bi}, the
16721 complex conjugate is @expr{a-bi}. If the argument is a real number,
16722 this command leaves it the same. If the argument is a vector or matrix,
16723 this command replaces each element by its complex conjugate.
16726 @pindex calc-argument
16728 The @kbd{G} (@code{calc-argument}) [@code{arg}] command computes the
16729 ``argument'' or polar angle of a complex number. For a number in polar
16730 notation, this is simply the second component of the pair
16731 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'.
16732 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'.
16733 The result is expressed according to the current angular mode and will
16734 be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees
16735 (inclusive), or the equivalent range in radians.
16737 @pindex calc-imaginary
16738 The @code{calc-imaginary} command multiplies the number on the
16739 top of the stack by the imaginary number @expr{i = (0,1)}. This
16740 command is not normally bound to a key in Calc, but it is available
16741 on the @key{IMAG} button in Keypad mode.
16746 The @kbd{f r} (@code{calc-re}) [@code{re}] command replaces a complex number
16747 by its real part. This command has no effect on real numbers. (As an
16748 added convenience, @code{re} applied to a modulo form extracts
16754 The @kbd{f i} (@code{calc-im}) [@code{im}] command replaces a complex number
16755 by its imaginary part; real numbers are converted to zero. With a vector
16756 or matrix argument, these functions operate element-wise.
16761 @kindex v p (complex)
16763 The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on
16764 the stack into a composite object such as a complex number. With
16765 a prefix argument of @mathit{-1}, it produces a rectangular complex number;
16766 with an argument of @mathit{-2}, it produces a polar complex number.
16767 (Also, @pxref{Building Vectors}.)
16772 @kindex v u (complex)
16773 @pindex calc-unpack
16774 The @kbd{v u} (@code{calc-unpack}) command takes the complex number
16775 (or other composite object) on the top of the stack and unpacks it
16776 into its separate components.
16778 @node Conversions, Date Arithmetic, Complex Number Functions, Arithmetic
16779 @section Conversions
16782 The commands described in this section convert numbers from one form
16783 to another; they are two-key sequences beginning with the letter @kbd{c}.
16788 The @kbd{c f} (@code{calc-float}) [@code{pfloat}] command converts the
16789 number on the top of the stack to floating-point form. For example,
16790 @expr{23} is converted to @expr{23.0}, @expr{3:2} is converted to
16791 @expr{1.5}, and @expr{2.3} is left the same. If the value is a composite
16792 object such as a complex number or vector, each of the components is
16793 converted to floating-point. If the value is a formula, all numbers
16794 in the formula are converted to floating-point. Note that depending
16795 on the current floating-point precision, conversion to floating-point
16796 format may lose information.
16798 As a special exception, integers which appear as powers or subscripts
16799 are not floated by @kbd{c f}. If you really want to float a power,
16800 you can use a @kbd{j s} command to select the power followed by @kbd{c f}.
16801 Because @kbd{c f} cannot examine the formula outside of the selection,
16802 it does not notice that the thing being floated is a power.
16803 @xref{Selecting Subformulas}.
16805 The normal @kbd{c f} command is ``pervasive'' in the sense that it
16806 applies to all numbers throughout the formula. The @code{pfloat}
16807 algebraic function never stays around in a formula; @samp{pfloat(a + 1)}
16808 changes to @samp{a + 1.0} as soon as it is evaluated.
16812 With the Hyperbolic flag, @kbd{H c f} [@code{float}] operates
16813 only on the number or vector of numbers at the top level of its
16814 argument. Thus, @samp{float(1)} is 1.0, but @samp{float(a + 1)}
16815 is left unevaluated because its argument is not a number.
16817 You should use @kbd{H c f} if you wish to guarantee that the final
16818 value, once all the variables have been assigned, is a float; you
16819 would use @kbd{c f} if you wish to do the conversion on the numbers
16820 that appear right now.
16823 @pindex calc-fraction
16825 The @kbd{c F} (@code{calc-fraction}) [@code{pfrac}] command converts a
16826 floating-point number into a fractional approximation. By default, it
16827 produces a fraction whose decimal representation is the same as the
16828 input number, to within the current precision. You can also give a
16829 numeric prefix argument to specify a tolerance, either directly, or,
16830 if the prefix argument is zero, by using the number on top of the stack
16831 as the tolerance. If the tolerance is a positive integer, the fraction
16832 is correct to within that many significant figures. If the tolerance is
16833 a non-positive integer, it specifies how many digits fewer than the current
16834 precision to use. If the tolerance is a floating-point number, the
16835 fraction is correct to within that absolute amount.
16839 The @code{pfrac} function is pervasive, like @code{pfloat}.
16840 There is also a non-pervasive version, @kbd{H c F} [@code{frac}],
16841 which is analogous to @kbd{H c f} discussed above.
16844 @pindex calc-to-degrees
16846 The @kbd{c d} (@code{calc-to-degrees}) [@code{deg}] command converts a
16847 number into degrees form. The value on the top of the stack may be an
16848 HMS form (interpreted as degrees-minutes-seconds), or a real number which
16849 will be interpreted in radians regardless of the current angular mode.
16852 @pindex calc-to-radians
16854 The @kbd{c r} (@code{calc-to-radians}) [@code{rad}] command converts an
16855 HMS form or angle in degrees into an angle in radians.
16858 @pindex calc-to-hms
16860 The @kbd{c h} (@code{calc-to-hms}) [@code{hms}] command converts a real
16861 number, interpreted according to the current angular mode, to an HMS
16862 form describing the same angle. In algebraic notation, the @code{hms}
16863 function also accepts three arguments: @samp{hms(@var{h}, @var{m}, @var{s})}.
16864 (The three-argument version is independent of the current angular mode.)
16866 @pindex calc-from-hms
16867 The @code{calc-from-hms} command converts the HMS form on the top of the
16868 stack into a real number according to the current angular mode.
16875 The @kbd{c p} (@code{calc-polar}) command converts the complex number on
16876 the top of the stack from polar to rectangular form, or from rectangular
16877 to polar form, whichever is appropriate. Real numbers are left the same.
16878 This command is equivalent to the @code{rect} or @code{polar}
16879 functions in algebraic formulas, depending on the direction of
16880 conversion. (It uses @code{polar}, except that if the argument is
16881 already a polar complex number, it uses @code{rect} instead. The
16882 @kbd{I c p} command always uses @code{rect}.)
16887 The @kbd{c c} (@code{calc-clean}) [@code{pclean}] command ``cleans'' the
16888 number on the top of the stack. Floating point numbers are re-rounded
16889 according to the current precision. Polar numbers whose angular
16890 components have strayed from the @mathit{-180} to @mathit{+180} degree range
16891 are normalized. (Note that results will be undesirable if the current
16892 angular mode is different from the one under which the number was
16893 produced!) Integers and fractions are generally unaffected by this
16894 operation. Vectors and formulas are cleaned by cleaning each component
16895 number (i.e., pervasively).
16897 If the simplification mode is set below the default level, it is raised
16898 to the default level for the purposes of this command. Thus, @kbd{c c}
16899 applies the default simplifications even if their automatic application
16900 is disabled. @xref{Simplification Modes}.
16902 @cindex Roundoff errors, correcting
16903 A numeric prefix argument to @kbd{c c} sets the floating-point precision
16904 to that value for the duration of the command. A positive prefix (of at
16905 least 3) sets the precision to the specified value; a negative or zero
16906 prefix decreases the precision by the specified amount.
16909 @pindex calc-clean-num
16910 The keystroke sequences @kbd{c 0} through @kbd{c 9} are equivalent
16911 to @kbd{c c} with the corresponding negative prefix argument. If roundoff
16912 errors have changed 2.0 into 1.999999, typing @kbd{c 1} to clip off one
16913 decimal place often conveniently does the trick.
16915 The @kbd{c c} command with a numeric prefix argument, and the @kbd{c 0}
16916 through @kbd{c 9} commands, also ``clip'' very small floating-point
16917 numbers to zero. If the exponent is less than or equal to the negative
16918 of the specified precision, the number is changed to 0.0. For example,
16919 if the current precision is 12, then @kbd{c 2} changes the vector
16920 @samp{[1e-8, 1e-9, 1e-10, 1e-11]} to @samp{[1e-8, 1e-9, 0, 0]}.
16921 Numbers this small generally arise from roundoff noise.
16923 If the numbers you are using really are legitimately this small,
16924 you should avoid using the @kbd{c 0} through @kbd{c 9} commands.
16925 (The plain @kbd{c c} command rounds to the current precision but
16926 does not clip small numbers.)
16928 One more property of @kbd{c 0} through @kbd{c 9}, and of @kbd{c c} with
16929 a prefix argument, is that integer-valued floats are converted to
16930 plain integers, so that @kbd{c 1} on @samp{[1., 1.5, 2., 2.5, 3.]}
16931 produces @samp{[1, 1.5, 2, 2.5, 3]}. This is not done for huge
16932 numbers (@samp{1e100} is technically an integer-valued float, but
16933 you wouldn't want it automatically converted to a 100-digit integer).
16938 With the Hyperbolic flag, @kbd{H c c} and @kbd{H c 0} through @kbd{H c 9}
16939 operate non-pervasively [@code{clean}].
16941 @node Date Arithmetic, Financial Functions, Conversions, Arithmetic
16942 @section Date Arithmetic
16945 @cindex Date arithmetic, additional functions
16946 The commands described in this section perform various conversions
16947 and calculations involving date forms (@pxref{Date Forms}). They
16948 use the @kbd{t} (for time/date) prefix key followed by shifted
16951 The simplest date arithmetic is done using the regular @kbd{+} and @kbd{-}
16952 commands. In particular, adding a number to a date form advances the
16953 date form by a certain number of days; adding an HMS form to a date
16954 form advances the date by a certain amount of time; and subtracting two
16955 date forms produces a difference measured in days. The commands
16956 described here provide additional, more specialized operations on dates.
16958 Many of these commands accept a numeric prefix argument; if you give
16959 plain @kbd{C-u} as the prefix, these commands will instead take the
16960 additional argument from the top of the stack.
16963 * Date Conversions::
16969 @node Date Conversions, Date Functions, Date Arithmetic, Date Arithmetic
16970 @subsection Date Conversions
16976 The @kbd{t D} (@code{calc-date}) [@code{date}] command converts a
16977 date form into a number, measured in days since Jan 1, 1 AD. The
16978 result will be an integer if @var{date} is a pure date form, or a
16979 fraction or float if @var{date} is a date/time form. Or, if its
16980 argument is a number, it converts this number into a date form.
16982 With a numeric prefix argument, @kbd{t D} takes that many objects
16983 (up to six) from the top of the stack and interprets them in one
16984 of the following ways:
16986 The @samp{date(@var{year}, @var{month}, @var{day})} function
16987 builds a pure date form out of the specified year, month, and
16988 day, which must all be integers. @var{Year} is a year number,
16989 such as 1991 (@emph{not} the same as 91!). @var{Month} must be
16990 an integer in the range 1 to 12; @var{day} must be in the range
16991 1 to 31. If the specified month has fewer than 31 days and
16992 @var{day} is too large, the equivalent day in the following
16993 month will be used.
16995 The @samp{date(@var{month}, @var{day})} function builds a
16996 pure date form using the current year, as determined by the
16999 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hms})}
17000 function builds a date/time form using an @var{hms} form.
17002 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hour},
17003 @var{minute}, @var{second})} function builds a date/time form.
17004 @var{hour} should be an integer in the range 0 to 23;
17005 @var{minute} should be an integer in the range 0 to 59;
17006 @var{second} should be any real number in the range @samp{[0 .. 60)}.
17007 The last two arguments default to zero if omitted.
17010 @pindex calc-julian
17012 @cindex Julian day counts, conversions
17013 The @kbd{t J} (@code{calc-julian}) [@code{julian}] command converts
17014 a date form into a Julian day count, which is the number of days
17015 since noon on Jan 1, 4713 BC. A pure date is converted to an integer
17016 Julian count representing noon of that day. A date/time form is
17017 converted to an exact floating-point Julian count, adjusted to
17018 interpret the date form in the current time zone but the Julian
17019 day count in Greenwich Mean Time. A numeric prefix argument allows
17020 you to specify the time zone; @pxref{Time Zones}. Use a prefix of
17021 zero to suppress the time zone adjustment. Note that pure date forms
17022 are never time-zone adjusted.
17024 This command can also do the opposite conversion, from a Julian day
17025 count (either an integer day, or a floating-point day and time in
17026 the GMT zone), into a pure date form or a date/time form in the
17027 current or specified time zone.
17030 @pindex calc-unix-time
17032 @cindex Unix time format, conversions
17033 The @kbd{t U} (@code{calc-unix-time}) [@code{unixtime}] command
17034 converts a date form into a Unix time value, which is the number of
17035 seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result
17036 will be an integer if the current precision is 12 or less; for higher
17037 precisions, the result may be a float with (@var{precision}@minus{}12)
17038 digits after the decimal. Just as for @kbd{t J}, the numeric time
17039 is interpreted in the GMT time zone and the date form is interpreted
17040 in the current or specified zone. Some systems use Unix-like
17041 numbering but with the local time zone; give a prefix of zero to
17042 suppress the adjustment if so.
17045 @pindex calc-convert-time-zones
17047 @cindex Time Zones, converting between
17048 The @kbd{t C} (@code{calc-convert-time-zones}) [@code{tzconv}]
17049 command converts a date form from one time zone to another. You
17050 are prompted for each time zone name in turn; you can answer with
17051 any suitable Calc time zone expression (@pxref{Time Zones}).
17052 If you answer either prompt with a blank line, the local time
17053 zone is used for that prompt. You can also answer the first
17054 prompt with @kbd{$} to take the two time zone names from the
17055 stack (and the date to be converted from the third stack level).
17057 @node Date Functions, Business Days, Date Conversions, Date Arithmetic
17058 @subsection Date Functions
17064 The @kbd{t N} (@code{calc-now}) [@code{now}] command pushes the
17065 current date and time on the stack as a date form. The time is
17066 reported in terms of the specified time zone; with no numeric prefix
17067 argument, @kbd{t N} reports for the current time zone.
17070 @pindex calc-date-part
17071 The @kbd{t P} (@code{calc-date-part}) command extracts one part
17072 of a date form. The prefix argument specifies the part; with no
17073 argument, this command prompts for a part code from 1 to 9.
17074 The various part codes are described in the following paragraphs.
17077 The @kbd{M-1 t P} [@code{year}] function extracts the year number
17078 from a date form as an integer, e.g., 1991. This and the
17079 following functions will also accept a real number for an
17080 argument, which is interpreted as a standard Calc day number.
17081 Note that this function will never return zero, since the year
17082 1 BC immediately precedes the year 1 AD.
17085 The @kbd{M-2 t P} [@code{month}] function extracts the month number
17086 from a date form as an integer in the range 1 to 12.
17089 The @kbd{M-3 t P} [@code{day}] function extracts the day number
17090 from a date form as an integer in the range 1 to 31.
17093 The @kbd{M-4 t P} [@code{hour}] function extracts the hour from
17094 a date form as an integer in the range 0 (midnight) to 23. Note
17095 that 24-hour time is always used. This returns zero for a pure
17096 date form. This function (and the following two) also accept
17097 HMS forms as input.
17100 The @kbd{M-5 t P} [@code{minute}] function extracts the minute
17101 from a date form as an integer in the range 0 to 59.
17104 The @kbd{M-6 t P} [@code{second}] function extracts the second
17105 from a date form. If the current precision is 12 or less,
17106 the result is an integer in the range 0 to 59. For higher
17107 precisions, the result may instead be a floating-point number.
17110 The @kbd{M-7 t P} [@code{weekday}] function extracts the weekday
17111 number from a date form as an integer in the range 0 (Sunday)
17115 The @kbd{M-8 t P} [@code{yearday}] function extracts the day-of-year
17116 number from a date form as an integer in the range 1 (January 1)
17117 to 366 (December 31 of a leap year).
17120 The @kbd{M-9 t P} [@code{time}] function extracts the time portion
17121 of a date form as an HMS form. This returns @samp{0@@ 0' 0"}
17122 for a pure date form.
17125 @pindex calc-new-month
17127 The @kbd{t M} (@code{calc-new-month}) [@code{newmonth}] command
17128 computes a new date form that represents the first day of the month
17129 specified by the input date. The result is always a pure date
17130 form; only the year and month numbers of the input are retained.
17131 With a numeric prefix argument @var{n} in the range from 1 to 31,
17132 @kbd{t M} computes the @var{n}th day of the month. (If @var{n}
17133 is greater than the actual number of days in the month, or if
17134 @var{n} is zero, the last day of the month is used.)
17137 @pindex calc-new-year
17139 The @kbd{t Y} (@code{calc-new-year}) [@code{newyear}] command
17140 computes a new pure date form that represents the first day of
17141 the year specified by the input. The month, day, and time
17142 of the input date form are lost. With a numeric prefix argument
17143 @var{n} in the range from 1 to 366, @kbd{t Y} computes the
17144 @var{n}th day of the year (366 is treated as 365 in non-leap
17145 years). A prefix argument of 0 computes the last day of the
17146 year (December 31). A negative prefix argument from @mathit{-1} to
17147 @mathit{-12} computes the first day of the @var{n}th month of the year.
17150 @pindex calc-new-week
17152 The @kbd{t W} (@code{calc-new-week}) [@code{newweek}] command
17153 computes a new pure date form that represents the Sunday on or before
17154 the input date. With a numeric prefix argument, it can be made to
17155 use any day of the week as the starting day; the argument must be in
17156 the range from 0 (Sunday) to 6 (Saturday). This function always
17157 subtracts between 0 and 6 days from the input date.
17159 Here's an example use of @code{newweek}: Find the date of the next
17160 Wednesday after a given date. Using @kbd{M-3 t W} or @samp{newweek(d, 3)}
17161 will give you the @emph{preceding} Wednesday, so @samp{newweek(d+7, 3)}
17162 will give you the following Wednesday. A further look at the definition
17163 of @code{newweek} shows that if the input date is itself a Wednesday,
17164 this formula will return the Wednesday one week in the future. An
17165 exercise for the reader is to modify this formula to yield the same day
17166 if the input is already a Wednesday. Another interesting exercise is
17167 to preserve the time-of-day portion of the input (@code{newweek} resets
17168 the time to midnight; hint:@: how can @code{newweek} be defined in terms
17169 of the @code{weekday} function?).
17175 The @samp{pwday(@var{date})} function (not on any key) computes the
17176 day-of-month number of the Sunday on or before @var{date}. With
17177 two arguments, @samp{pwday(@var{date}, @var{day})} computes the day
17178 number of the Sunday on or before day number @var{day} of the month
17179 specified by @var{date}. The @var{day} must be in the range from
17180 7 to 31; if the day number is greater than the actual number of days
17181 in the month, the true number of days is used instead. Thus
17182 @samp{pwday(@var{date}, 7)} finds the first Sunday of the month, and
17183 @samp{pwday(@var{date}, 31)} finds the last Sunday of the month.
17184 With a third @var{weekday} argument, @code{pwday} can be made to look
17185 for any day of the week instead of Sunday.
17188 @pindex calc-inc-month
17190 The @kbd{t I} (@code{calc-inc-month}) [@code{incmonth}] command
17191 increases a date form by one month, or by an arbitrary number of
17192 months specified by a numeric prefix argument. The time portion,
17193 if any, of the date form stays the same. The day also stays the
17194 same, except that if the new month has fewer days the day
17195 number may be reduced to lie in the valid range. For example,
17196 @samp{incmonth(<Jan 31, 1991>)} produces @samp{<Feb 28, 1991>}.
17197 Because of this, @kbd{t I t I} and @kbd{M-2 t I} do not always give
17198 the same results (@samp{<Mar 28, 1991>} versus @samp{<Mar 31, 1991>}
17205 The @samp{incyear(@var{date}, @var{step})} function increases
17206 a date form by the specified number of years, which may be
17207 any positive or negative integer. Note that @samp{incyear(d, n)}
17208 is equivalent to @w{@samp{incmonth(d, 12*n)}}, but these do not have
17209 simple equivalents in terms of day arithmetic because
17210 months and years have varying lengths. If the @var{step}
17211 argument is omitted, 1 year is assumed. There is no keyboard
17212 command for this function; use @kbd{C-u 12 t I} instead.
17214 There is no @code{newday} function at all because @kbd{F} [@code{floor}]
17215 serves this purpose. Similarly, instead of @code{incday} and
17216 @code{incweek} simply use @expr{d + n} or @expr{d + 7 n}.
17218 @xref{Basic Arithmetic}, for the @kbd{f ]} [@code{incr}] command
17219 which can adjust a date/time form by a certain number of seconds.
17221 @node Business Days, Time Zones, Date Functions, Date Arithmetic
17222 @subsection Business Days
17225 Often time is measured in ``business days'' or ``working days,''
17226 where weekends and holidays are skipped. Calc's normal date
17227 arithmetic functions use calendar days, so that subtracting two
17228 consecutive Mondays will yield a difference of 7 days. By contrast,
17229 subtracting two consecutive Mondays would yield 5 business days
17230 (assuming two-day weekends and the absence of holidays).
17236 @pindex calc-business-days-plus
17237 @pindex calc-business-days-minus
17238 The @kbd{t +} (@code{calc-business-days-plus}) [@code{badd}]
17239 and @kbd{t -} (@code{calc-business-days-minus}) [@code{bsub}]
17240 commands perform arithmetic using business days. For @kbd{t +},
17241 one argument must be a date form and the other must be a real
17242 number (positive or negative). If the number is not an integer,
17243 then a certain amount of time is added as well as a number of
17244 days; for example, adding 0.5 business days to a time in Friday
17245 evening will produce a time in Monday morning. It is also
17246 possible to add an HMS form; adding @samp{12@@ 0' 0"} also adds
17247 half a business day. For @kbd{t -}, the arguments are either a
17248 date form and a number or HMS form, or two date forms, in which
17249 case the result is the number of business days between the two
17252 @cindex @code{Holidays} variable
17254 By default, Calc considers any day that is not a Saturday or
17255 Sunday to be a business day. You can define any number of
17256 additional holidays by editing the variable @code{Holidays}.
17257 (There is an @w{@kbd{s H}} convenience command for editing this
17258 variable.) Initially, @code{Holidays} contains the vector
17259 @samp{[sat, sun]}. Entries in the @code{Holidays} vector may
17260 be any of the following kinds of objects:
17264 Date forms (pure dates, not date/time forms). These specify
17265 particular days which are to be treated as holidays.
17268 Intervals of date forms. These specify a range of days, all of
17269 which are holidays (e.g., Christmas week). @xref{Interval Forms}.
17272 Nested vectors of date forms. Each date form in the vector is
17273 considered to be a holiday.
17276 Any Calc formula which evaluates to one of the above three things.
17277 If the formula involves the variable @expr{y}, it stands for a
17278 yearly repeating holiday; @expr{y} will take on various year
17279 numbers like 1992. For example, @samp{date(y, 12, 25)} specifies
17280 Christmas day, and @samp{newweek(date(y, 11, 7), 4) + 21} specifies
17281 Thanksgiving (which is held on the fourth Thursday of November).
17282 If the formula involves the variable @expr{m}, that variable
17283 takes on month numbers from 1 to 12: @samp{date(y, m, 15)} is
17284 a holiday that takes place on the 15th of every month.
17287 A weekday name, such as @code{sat} or @code{sun}. This is really
17288 a variable whose name is a three-letter, lower-case day name.
17291 An interval of year numbers (integers). This specifies the span of
17292 years over which this holiday list is to be considered valid. Any
17293 business-day arithmetic that goes outside this range will result
17294 in an error message. Use this if you are including an explicit
17295 list of holidays, rather than a formula to generate them, and you
17296 want to make sure you don't accidentally go beyond the last point
17297 where the holidays you entered are complete. If there is no
17298 limiting interval in the @code{Holidays} vector, the default
17299 @samp{[1 .. 2737]} is used. (This is the absolute range of years
17300 for which Calc's business-day algorithms will operate.)
17303 An interval of HMS forms. This specifies the span of hours that
17304 are to be considered one business day. For example, if this
17305 range is @samp{[9@@ 0' 0" .. 17@@ 0' 0"]} (i.e., 9am to 5pm), then
17306 the business day is only eight hours long, so that @kbd{1.5 t +}
17307 on @samp{<4:00pm Fri Dec 13, 1991>} will add one business day and
17308 four business hours to produce @samp{<12:00pm Tue Dec 17, 1991>}.
17309 Likewise, @kbd{t -} will now express differences in time as
17310 fractions of an eight-hour day. Times before 9am will be treated
17311 as 9am by business date arithmetic, and times at or after 5pm will
17312 be treated as 4:59:59pm. If there is no HMS interval in @code{Holidays},
17313 the full 24-hour day @samp{[0@ 0' 0" .. 24@ 0' 0"]} is assumed.
17314 (Regardless of the type of bounds you specify, the interval is
17315 treated as inclusive on the low end and exclusive on the high end,
17316 so that the work day goes from 9am up to, but not including, 5pm.)
17319 If the @code{Holidays} vector is empty, then @kbd{t +} and
17320 @kbd{t -} will act just like @kbd{+} and @kbd{-} because there will
17321 then be no difference between business days and calendar days.
17323 Calc expands the intervals and formulas you give into a complete
17324 list of holidays for internal use. This is done mainly to make
17325 sure it can detect multiple holidays. (For example,
17326 @samp{<Jan 1, 1989>} is both New Year's Day and a Sunday, but
17327 Calc's algorithms take care to count it only once when figuring
17328 the number of holidays between two dates.)
17330 Since the complete list of holidays for all the years from 1 to
17331 2737 would be huge, Calc actually computes only the part of the
17332 list between the smallest and largest years that have been involved
17333 in business-day calculations so far. Normally, you won't have to
17334 worry about this. Keep in mind, however, that if you do one
17335 calculation for 1992, and another for 1792, even if both involve
17336 only a small range of years, Calc will still work out all the
17337 holidays that fall in that 200-year span.
17339 If you add a (positive) number of days to a date form that falls on a
17340 weekend or holiday, the date form is treated as if it were the most
17341 recent business day. (Thus adding one business day to a Friday,
17342 Saturday, or Sunday will all yield the following Monday.) If you
17343 subtract a number of days from a weekend or holiday, the date is
17344 effectively on the following business day. (So subtracting one business
17345 day from Saturday, Sunday, or Monday yields the preceding Friday.) The
17346 difference between two dates one or both of which fall on holidays
17347 equals the number of actual business days between them. These
17348 conventions are consistent in the sense that, if you add @var{n}
17349 business days to any date, the difference between the result and the
17350 original date will come out to @var{n} business days. (It can't be
17351 completely consistent though; a subtraction followed by an addition
17352 might come out a bit differently, since @kbd{t +} is incapable of
17353 producing a date that falls on a weekend or holiday.)
17359 There is a @code{holiday} function, not on any keys, that takes
17360 any date form and returns 1 if that date falls on a weekend or
17361 holiday, as defined in @code{Holidays}, or 0 if the date is a
17364 @node Time Zones, , Business Days, Date Arithmetic
17365 @subsection Time Zones
17369 @cindex Daylight savings time
17370 Time zones and daylight savings time are a complicated business.
17371 The conversions to and from Julian and Unix-style dates automatically
17372 compute the correct time zone and daylight savings adjustment to use,
17373 provided they can figure out this information. This section describes
17374 Calc's time zone adjustment algorithm in detail, in case you want to
17375 do conversions in different time zones or in case Calc's algorithms
17376 can't determine the right correction to use.
17378 Adjustments for time zones and daylight savings time are done by
17379 @kbd{t U}, @kbd{t J}, @kbd{t N}, and @kbd{t C}, but not by any other
17380 commands. In particular, @samp{<may 1 1991> - <apr 1 1991>} evaluates
17381 to exactly 30 days even though there is a daylight-savings
17382 transition in between. This is also true for Julian pure dates:
17383 @samp{julian(<may 1 1991>) - julian(<apr 1 1991>)}. But Julian
17384 and Unix date/times will adjust for daylight savings time:
17385 @samp{julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)}
17386 evaluates to @samp{29.95834} (that's 29 days and 23 hours)
17387 because one hour was lost when daylight savings commenced on
17390 In brief, the idiom @samp{julian(@var{date1}) - julian(@var{date2})}
17391 computes the actual number of 24-hour periods between two dates, whereas
17392 @samp{@var{date1} - @var{date2}} computes the number of calendar
17393 days between two dates without taking daylight savings into account.
17395 @pindex calc-time-zone
17400 The @code{calc-time-zone} [@code{tzone}] command converts the time
17401 zone specified by its numeric prefix argument into a number of
17402 seconds difference from Greenwich mean time (GMT). If the argument
17403 is a number, the result is simply that value multiplied by 3600.
17404 Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If
17405 Daylight Savings time is in effect, one hour should be subtracted from
17406 the normal difference.
17408 If you give a prefix of plain @kbd{C-u}, @code{calc-time-zone} (like other
17409 date arithmetic commands that include a time zone argument) takes the
17410 zone argument from the top of the stack. (In the case of @kbd{t J}
17411 and @kbd{t U}, the normal argument is then taken from the second-to-top
17412 stack position.) This allows you to give a non-integer time zone
17413 adjustment. The time-zone argument can also be an HMS form, or
17414 it can be a variable which is a time zone name in upper- or lower-case.
17415 For example @samp{tzone(PST) = tzone(8)} and @samp{tzone(pdt) = tzone(7)}
17416 (for Pacific standard and daylight savings times, respectively).
17418 North American and European time zone names are defined as follows;
17419 note that for each time zone there is one name for standard time,
17420 another for daylight savings time, and a third for ``generalized'' time
17421 in which the daylight savings adjustment is computed from context.
17425 YST PST MST CST EST AST NST GMT WET MET MEZ
17426 9 8 7 6 5 4 3.5 0 -1 -2 -2
17428 YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ
17429 8 7 6 5 4 3 2.5 -1 -2 -3 -3
17431 YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ
17432 9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3
17436 @vindex math-tzone-names
17437 To define time zone names that do not appear in the above table,
17438 you must modify the Lisp variable @code{math-tzone-names}. This
17439 is a list of lists describing the different time zone names; its
17440 structure is best explained by an example. The three entries for
17441 Pacific Time look like this:
17445 ( ( "PST" 8 0 ) ; Name as an upper-case string, then standard
17446 ( "PDT" 8 -1 ) ; adjustment, then daylight savings adjustment.
17447 ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone.
17451 @cindex @code{TimeZone} variable
17453 With no arguments, @code{calc-time-zone} or @samp{tzone()} obtains an
17454 argument from the Calc variable @code{TimeZone} if a value has been
17455 stored for that variable. If not, Calc runs the Unix @samp{date}
17456 command and looks for one of the above time zone names in the output;
17457 if this does not succeed, @samp{tzone()} leaves itself unevaluated.
17458 The time zone name in the @samp{date} output may be followed by a signed
17459 adjustment, e.g., @samp{GMT+5} or @samp{GMT+0500} which specifies a
17460 number of hours and minutes to be added to the base time zone.
17461 Calc stores the time zone it finds into @code{TimeZone} to speed
17462 later calls to @samp{tzone()}.
17464 The special time zone name @code{local} is equivalent to no argument,
17465 i.e., it uses the local time zone as obtained from the @code{date}
17468 If the time zone name found is one of the standard or daylight
17469 savings zone names from the above table, and Calc's internal
17470 daylight savings algorithm says that time and zone are consistent
17471 (e.g., @code{PDT} accompanies a date that Calc's algorithm would also
17472 consider to be daylight savings, or @code{PST} accompanies a date
17473 that Calc would consider to be standard time), then Calc substitutes
17474 the corresponding generalized time zone (like @code{PGT}).
17476 If your system does not have a suitable @samp{date} command, you
17477 may wish to put a @samp{(setq var-TimeZone ...)} in your Emacs
17478 initialization file to set the time zone. (Since you are interacting
17479 with the variable @code{TimeZone} directly from Emacs Lisp, the
17480 @code{var-} prefix needs to be present.) The easiest way to do
17481 this is to edit the @code{TimeZone} variable using Calc's @kbd{s T}
17482 command, then use the @kbd{s p} (@code{calc-permanent-variable})
17483 command to save the value of @code{TimeZone} permanently.
17485 The @kbd{t J} and @code{t U} commands with no numeric prefix
17486 arguments do the same thing as @samp{tzone()}. If the current
17487 time zone is a generalized time zone, e.g., @code{EGT}, Calc
17488 examines the date being converted to tell whether to use standard
17489 or daylight savings time. But if the current time zone is explicit,
17490 e.g., @code{EST} or @code{EDT}, then that adjustment is used exactly
17491 and Calc's daylight savings algorithm is not consulted.
17493 Some places don't follow the usual rules for daylight savings time.
17494 The state of Arizona, for example, does not observe daylight savings
17495 time. If you run Calc during the winter season in Arizona, the
17496 Unix @code{date} command will report @code{MST} time zone, which
17497 Calc will change to @code{MGT}. If you then convert a time that
17498 lies in the summer months, Calc will apply an incorrect daylight
17499 savings time adjustment. To avoid this, set your @code{TimeZone}
17500 variable explicitly to @code{MST} to force the use of standard,
17501 non-daylight-savings time.
17503 @vindex math-daylight-savings-hook
17504 @findex math-std-daylight-savings
17505 By default Calc always considers daylight savings time to begin at
17506 2 a.m.@: on the first Sunday of April, and to end at 2 a.m.@: on the
17507 last Sunday of October. This is the rule that has been in effect
17508 in North America since 1987. If you are in a country that uses
17509 different rules for computing daylight savings time, you have two
17510 choices: Write your own daylight savings hook, or control time
17511 zones explicitly by setting the @code{TimeZone} variable and/or
17512 always giving a time-zone argument for the conversion functions.
17514 The Lisp variable @code{math-daylight-savings-hook} holds the
17515 name of a function that is used to compute the daylight savings
17516 adjustment for a given date. The default is
17517 @code{math-std-daylight-savings}, which computes an adjustment
17518 (either 0 or @mathit{-1}) using the North American rules given above.
17520 The daylight savings hook function is called with four arguments:
17521 The date, as a floating-point number in standard Calc format;
17522 a six-element list of the date decomposed into year, month, day,
17523 hour, minute, and second, respectively; a string which contains
17524 the generalized time zone name in upper-case, e.g., @code{"WEGT"};
17525 and a special adjustment to be applied to the hour value when
17526 converting into a generalized time zone (see below).
17528 @findex math-prev-weekday-in-month
17529 The Lisp function @code{math-prev-weekday-in-month} is useful for
17530 daylight savings computations. This is an internal version of
17531 the user-level @code{pwday} function described in the previous
17532 section. It takes four arguments: The floating-point date value,
17533 the corresponding six-element date list, the day-of-month number,
17534 and the weekday number (0-6).
17536 The default daylight savings hook ignores the time zone name, but a
17537 more sophisticated hook could use different algorithms for different
17538 time zones. It would also be possible to use different algorithms
17539 depending on the year number, but the default hook always uses the
17540 algorithm for 1987 and later. Here is a listing of the default
17541 daylight savings hook:
17544 (defun math-std-daylight-savings (date dt zone bump)
17545 (cond ((< (nth 1 dt) 4) 0)
17547 (let ((sunday (math-prev-weekday-in-month date dt 7 0)))
17548 (cond ((< (nth 2 dt) sunday) 0)
17549 ((= (nth 2 dt) sunday)
17550 (if (>= (nth 3 dt) (+ 3 bump)) -1 0))
17552 ((< (nth 1 dt) 10) -1)
17554 (let ((sunday (math-prev-weekday-in-month date dt 31 0)))
17555 (cond ((< (nth 2 dt) sunday) -1)
17556 ((= (nth 2 dt) sunday)
17557 (if (>= (nth 3 dt) (+ 2 bump)) 0 -1))
17564 The @code{bump} parameter is equal to zero when Calc is converting
17565 from a date form in a generalized time zone into a GMT date value.
17566 It is @mathit{-1} when Calc is converting in the other direction. The
17567 adjustments shown above ensure that the conversion behaves correctly
17568 and reasonably around the 2 a.m.@: transition in each direction.
17570 There is a ``missing'' hour between 2 a.m.@: and 3 a.m.@: at the
17571 beginning of daylight savings time; converting a date/time form that
17572 falls in this hour results in a time value for the following hour,
17573 from 3 a.m.@: to 4 a.m. At the end of daylight savings time, the
17574 hour from 1 a.m.@: to 2 a.m.@: repeats itself; converting a date/time
17575 form that falls in in this hour results in a time value for the first
17576 manifestation of that time (@emph{not} the one that occurs one hour later).
17578 If @code{math-daylight-savings-hook} is @code{nil}, then the
17579 daylight savings adjustment is always taken to be zero.
17581 In algebraic formulas, @samp{tzone(@var{zone}, @var{date})}
17582 computes the time zone adjustment for a given zone name at a
17583 given date. The @var{date} is ignored unless @var{zone} is a
17584 generalized time zone. If @var{date} is a date form, the
17585 daylight savings computation is applied to it as it appears.
17586 If @var{date} is a numeric date value, it is adjusted for the
17587 daylight-savings version of @var{zone} before being given to
17588 the daylight savings hook. This odd-sounding rule ensures
17589 that the daylight-savings computation is always done in
17590 local time, not in the GMT time that a numeric @var{date}
17591 is typically represented in.
17597 The @samp{dsadj(@var{date}, @var{zone})} function computes the
17598 daylight savings adjustment that is appropriate for @var{date} in
17599 time zone @var{zone}. If @var{zone} is explicitly in or not in
17600 daylight savings time (e.g., @code{PDT} or @code{PST}) the
17601 @var{date} is ignored. If @var{zone} is a generalized time zone,
17602 the algorithms described above are used. If @var{zone} is omitted,
17603 the computation is done for the current time zone.
17605 @xref{Reporting Bugs}, for the address of Calc's author, if you
17606 should wish to contribute your improved versions of
17607 @code{math-tzone-names} and @code{math-daylight-savings-hook}
17608 to the Calc distribution.
17610 @node Financial Functions, Binary Functions, Date Arithmetic, Arithmetic
17611 @section Financial Functions
17614 Calc's financial or business functions use the @kbd{b} prefix
17615 key followed by a shifted letter. (The @kbd{b} prefix followed by
17616 a lower-case letter is used for operations on binary numbers.)
17618 Note that the rate and the number of intervals given to these
17619 functions must be on the same time scale, e.g., both months or
17620 both years. Mixing an annual interest rate with a time expressed
17621 in months will give you very wrong answers!
17623 It is wise to compute these functions to a higher precision than
17624 you really need, just to make sure your answer is correct to the
17625 last penny; also, you may wish to check the definitions at the end
17626 of this section to make sure the functions have the meaning you expect.
17632 * Related Financial Functions::
17633 * Depreciation Functions::
17634 * Definitions of Financial Functions::
17637 @node Percentages, Future Value, Financial Functions, Financial Functions
17638 @subsection Percentages
17641 @pindex calc-percent
17644 The @kbd{M-%} (@code{calc-percent}) command takes a percentage value,
17645 say 5.4, and converts it to an equivalent actual number. For example,
17646 @kbd{5.4 M-%} enters 0.054 on the stack. (That's the @key{META} or
17647 @key{ESC} key combined with @kbd{%}.)
17649 Actually, @kbd{M-%} creates a formula of the form @samp{5.4%}.
17650 You can enter @samp{5.4%} yourself during algebraic entry. The
17651 @samp{%} operator simply means, ``the preceding value divided by
17652 100.'' The @samp{%} operator has very high precedence, so that
17653 @samp{1+8%} is interpreted as @samp{1+(8%)}, not as @samp{(1+8)%}.
17654 (The @samp{%} operator is just a postfix notation for the
17655 @code{percent} function, just like @samp{20!} is the notation for
17656 @samp{fact(20)}, or twenty-factorial.)
17658 The formula @samp{5.4%} would normally evaluate immediately to
17659 0.054, but the @kbd{M-%} command suppresses evaluation as it puts
17660 the formula onto the stack. However, the next Calc command that
17661 uses the formula @samp{5.4%} will evaluate it as its first step.
17662 The net effect is that you get to look at @samp{5.4%} on the stack,
17663 but Calc commands see it as @samp{0.054}, which is what they expect.
17665 In particular, @samp{5.4%} and @samp{0.054} are suitable values
17666 for the @var{rate} arguments of the various financial functions,
17667 but the number @samp{5.4} is probably @emph{not} suitable---it
17668 represents a rate of 540 percent!
17670 The key sequence @kbd{M-% *} effectively means ``percent-of.''
17671 For example, @kbd{68 @key{RET} 25 M-% *} computes 17, which is 25% of
17672 68 (and also 68% of 25, which comes out to the same thing).
17675 @pindex calc-convert-percent
17676 The @kbd{c %} (@code{calc-convert-percent}) command converts the
17677 value on the top of the stack from numeric to percentage form.
17678 For example, if 0.08 is on the stack, @kbd{c %} converts it to
17679 @samp{8%}. The quantity is the same, it's just represented
17680 differently. (Contrast this with @kbd{M-%}, which would convert
17681 this number to @samp{0.08%}.) The @kbd{=} key is a convenient way
17682 to convert a formula like @samp{8%} back to numeric form, 0.08.
17684 To compute what percentage one quantity is of another quantity,
17685 use @kbd{/ c %}. For example, @w{@kbd{17 @key{RET} 68 / c %}} displays
17689 @pindex calc-percent-change
17691 The @kbd{b %} (@code{calc-percent-change}) [@code{relch}] command
17692 calculates the percentage change from one number to another.
17693 For example, @kbd{40 @key{RET} 50 b %} produces the answer @samp{25%},
17694 since 50 is 25% larger than 40. A negative result represents a
17695 decrease: @kbd{50 @key{RET} 40 b %} produces @samp{-20%}, since 40 is
17696 20% smaller than 50. (The answers are different in magnitude
17697 because, in the first case, we're increasing by 25% of 40, but
17698 in the second case, we're decreasing by 20% of 50.) The effect
17699 of @kbd{40 @key{RET} 50 b %} is to compute @expr{(50-40)/40}, converting
17700 the answer to percentage form as if by @kbd{c %}.
17702 @node Future Value, Present Value, Percentages, Financial Functions
17703 @subsection Future Value
17707 @pindex calc-fin-fv
17709 The @kbd{b F} (@code{calc-fin-fv}) [@code{fv}] command computes
17710 the future value of an investment. It takes three arguments
17711 from the stack: @samp{fv(@var{rate}, @var{n}, @var{payment})}.
17712 If you give payments of @var{payment} every year for @var{n}
17713 years, and the money you have paid earns interest at @var{rate} per
17714 year, then this function tells you what your investment would be
17715 worth at the end of the period. (The actual interval doesn't
17716 have to be years, as long as @var{n} and @var{rate} are expressed
17717 in terms of the same intervals.) This function assumes payments
17718 occur at the @emph{end} of each interval.
17722 The @kbd{I b F} [@code{fvb}] command does the same computation,
17723 but assuming your payments are at the beginning of each interval.
17724 Suppose you plan to deposit $1000 per year in a savings account
17725 earning 5.4% interest, starting right now. How much will be
17726 in the account after five years? @code{fvb(5.4%, 5, 1000) = 5870.73}.
17727 Thus you will have earned $870 worth of interest over the years.
17728 Using the stack, this calculation would have been
17729 @kbd{5.4 M-% 5 @key{RET} 1000 I b F}. Note that the rate is expressed
17730 as a number between 0 and 1, @emph{not} as a percentage.
17734 The @kbd{H b F} [@code{fvl}] command computes the future value
17735 of an initial lump sum investment. Suppose you could deposit
17736 those five thousand dollars in the bank right now; how much would
17737 they be worth in five years? @code{fvl(5.4%, 5, 5000) = 6503.89}.
17739 The algebraic functions @code{fv} and @code{fvb} accept an optional
17740 fourth argument, which is used as an initial lump sum in the sense
17741 of @code{fvl}. In other words, @code{fv(@var{rate}, @var{n},
17742 @var{payment}, @var{initial}) = fv(@var{rate}, @var{n}, @var{payment})
17743 + fvl(@var{rate}, @var{n}, @var{initial})}.
17745 To illustrate the relationships between these functions, we could
17746 do the @code{fvb} calculation ``by hand'' using @code{fvl}. The
17747 final balance will be the sum of the contributions of our five
17748 deposits at various times. The first deposit earns interest for
17749 five years: @code{fvl(5.4%, 5, 1000) = 1300.78}. The second
17750 deposit only earns interest for four years: @code{fvl(5.4%, 4, 1000) =
17751 1234.13}. And so on down to the last deposit, which earns one
17752 year's interest: @code{fvl(5.4%, 1, 1000) = 1054.00}. The sum of
17753 these five values is, sure enough, $5870.73, just as was computed
17754 by @code{fvb} directly.
17756 What does @code{fv(5.4%, 5, 1000) = 5569.96} mean? The payments
17757 are now at the ends of the periods. The end of one year is the same
17758 as the beginning of the next, so what this really means is that we've
17759 lost the payment at year zero (which contributed $1300.78), but we're
17760 now counting the payment at year five (which, since it didn't have
17761 a chance to earn interest, counts as $1000). Indeed, @expr{5569.96 =
17762 5870.73 - 1300.78 + 1000} (give or take a bit of roundoff error).
17764 @node Present Value, Related Financial Functions, Future Value, Financial Functions
17765 @subsection Present Value
17769 @pindex calc-fin-pv
17771 The @kbd{b P} (@code{calc-fin-pv}) [@code{pv}] command computes
17772 the present value of an investment. Like @code{fv}, it takes
17773 three arguments: @code{pv(@var{rate}, @var{n}, @var{payment})}.
17774 It computes the present value of a series of regular payments.
17775 Suppose you have the chance to make an investment that will
17776 pay $2000 per year over the next four years; as you receive
17777 these payments you can put them in the bank at 9% interest.
17778 You want to know whether it is better to make the investment, or
17779 to keep the money in the bank where it earns 9% interest right
17780 from the start. The calculation @code{pv(9%, 4, 2000)} gives the
17781 result 6479.44. If your initial investment must be less than this,
17782 say, $6000, then the investment is worthwhile. But if you had to
17783 put up $7000, then it would be better just to leave it in the bank.
17785 Here is the interpretation of the result of @code{pv}: You are
17786 trying to compare the return from the investment you are
17787 considering, which is @code{fv(9%, 4, 2000) = 9146.26}, with
17788 the return from leaving the money in the bank, which is
17789 @code{fvl(9%, 4, @var{x})} where @var{x} is the amount of money
17790 you would have to put up in advance. The @code{pv} function
17791 finds the break-even point, @expr{x = 6479.44}, at which
17792 @code{fvl(9%, 4, 6479.44)} is also equal to 9146.26. This is
17793 the largest amount you should be willing to invest.
17797 The @kbd{I b P} [@code{pvb}] command solves the same problem,
17798 but with payments occurring at the beginning of each interval.
17799 It has the same relationship to @code{fvb} as @code{pv} has
17800 to @code{fv}. For example @code{pvb(9%, 4, 2000) = 7062.59},
17801 a larger number than @code{pv} produced because we get to start
17802 earning interest on the return from our investment sooner.
17806 The @kbd{H b P} [@code{pvl}] command computes the present value of
17807 an investment that will pay off in one lump sum at the end of the
17808 period. For example, if we get our $8000 all at the end of the
17809 four years, @code{pvl(9%, 4, 8000) = 5667.40}. This is much
17810 less than @code{pv} reported, because we don't earn any interest
17811 on the return from this investment. Note that @code{pvl} and
17812 @code{fvl} are simple inverses: @code{fvl(9%, 4, 5667.40) = 8000}.
17814 You can give an optional fourth lump-sum argument to @code{pv}
17815 and @code{pvb}; this is handled in exactly the same way as the
17816 fourth argument for @code{fv} and @code{fvb}.
17819 @pindex calc-fin-npv
17821 The @kbd{b N} (@code{calc-fin-npv}) [@code{npv}] command computes
17822 the net present value of a series of irregular investments.
17823 The first argument is the interest rate. The second argument is
17824 a vector which represents the expected return from the investment
17825 at the end of each interval. For example, if the rate represents
17826 a yearly interest rate, then the vector elements are the return
17827 from the first year, second year, and so on.
17829 Thus, @code{npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44}.
17830 Obviously this function is more interesting when the payments are
17833 The @code{npv} function can actually have two or more arguments.
17834 Multiple arguments are interpreted in the same way as for the
17835 vector statistical functions like @code{vsum}.
17836 @xref{Single-Variable Statistics}. Basically, if there are several
17837 payment arguments, each either a vector or a plain number, all these
17838 values are collected left-to-right into the complete list of payments.
17839 A numeric prefix argument on the @kbd{b N} command says how many
17840 payment values or vectors to take from the stack.
17844 The @kbd{I b N} [@code{npvb}] command computes the net present
17845 value where payments occur at the beginning of each interval
17846 rather than at the end.
17848 @node Related Financial Functions, Depreciation Functions, Present Value, Financial Functions
17849 @subsection Related Financial Functions
17852 The functions in this section are basically inverses of the
17853 present value functions with respect to the various arguments.
17856 @pindex calc-fin-pmt
17858 The @kbd{b M} (@code{calc-fin-pmt}) [@code{pmt}] command computes
17859 the amount of periodic payment necessary to amortize a loan.
17860 Thus @code{pmt(@var{rate}, @var{n}, @var{amount})} equals the
17861 value of @var{payment} such that @code{pv(@var{rate}, @var{n},
17862 @var{payment}) = @var{amount}}.
17866 The @kbd{I b M} [@code{pmtb}] command does the same computation
17867 but using @code{pvb} instead of @code{pv}. Like @code{pv} and
17868 @code{pvb}, these functions can also take a fourth argument which
17869 represents an initial lump-sum investment.
17872 The @kbd{H b M} key just invokes the @code{fvl} function, which is
17873 the inverse of @code{pvl}. There is no explicit @code{pmtl} function.
17876 @pindex calc-fin-nper
17878 The @kbd{b #} (@code{calc-fin-nper}) [@code{nper}] command computes
17879 the number of regular payments necessary to amortize a loan.
17880 Thus @code{nper(@var{rate}, @var{payment}, @var{amount})} equals
17881 the value of @var{n} such that @code{pv(@var{rate}, @var{n},
17882 @var{payment}) = @var{amount}}. If @var{payment} is too small
17883 ever to amortize a loan for @var{amount} at interest rate @var{rate},
17884 the @code{nper} function is left in symbolic form.
17888 The @kbd{I b #} [@code{nperb}] command does the same computation
17889 but using @code{pvb} instead of @code{pv}. You can give a fourth
17890 lump-sum argument to these functions, but the computation will be
17891 rather slow in the four-argument case.
17895 The @kbd{H b #} [@code{nperl}] command does the same computation
17896 using @code{pvl}. By exchanging @var{payment} and @var{amount} you
17897 can also get the solution for @code{fvl}. For example,
17898 @code{nperl(8%, 2000, 1000) = 9.006}, so if you place $1000 in a
17899 bank account earning 8%, it will take nine years to grow to $2000.
17902 @pindex calc-fin-rate
17904 The @kbd{b T} (@code{calc-fin-rate}) [@code{rate}] command computes
17905 the rate of return on an investment. This is also an inverse of @code{pv}:
17906 @code{rate(@var{n}, @var{payment}, @var{amount})} computes the value of
17907 @var{rate} such that @code{pv(@var{rate}, @var{n}, @var{payment}) =
17908 @var{amount}}. The result is expressed as a formula like @samp{6.3%}.
17914 The @kbd{I b T} [@code{rateb}] and @kbd{H b T} [@code{ratel}]
17915 commands solve the analogous equations with @code{pvb} or @code{pvl}
17916 in place of @code{pv}. Also, @code{rate} and @code{rateb} can
17917 accept an optional fourth argument just like @code{pv} and @code{pvb}.
17918 To redo the above example from a different perspective,
17919 @code{ratel(9, 2000, 1000) = 8.00597%}, which says you will need an
17920 interest rate of 8% in order to double your account in nine years.
17923 @pindex calc-fin-irr
17925 The @kbd{b I} (@code{calc-fin-irr}) [@code{irr}] command is the
17926 analogous function to @code{rate} but for net present value.
17927 Its argument is a vector of payments. Thus @code{irr(@var{payments})}
17928 computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0};
17929 this rate is known as the @dfn{internal rate of return}.
17933 The @kbd{I b I} [@code{irrb}] command computes the internal rate of
17934 return assuming payments occur at the beginning of each period.
17936 @node Depreciation Functions, Definitions of Financial Functions, Related Financial Functions, Financial Functions
17937 @subsection Depreciation Functions
17940 The functions in this section calculate @dfn{depreciation}, which is
17941 the amount of value that a possession loses over time. These functions
17942 are characterized by three parameters: @var{cost}, the original cost
17943 of the asset; @var{salvage}, the value the asset will have at the end
17944 of its expected ``useful life''; and @var{life}, the number of years
17945 (or other periods) of the expected useful life.
17947 There are several methods for calculating depreciation that differ in
17948 the way they spread the depreciation over the lifetime of the asset.
17951 @pindex calc-fin-sln
17953 The @kbd{b S} (@code{calc-fin-sln}) [@code{sln}] command computes the
17954 ``straight-line'' depreciation. In this method, the asset depreciates
17955 by the same amount every year (or period). For example,
17956 @samp{sln(12000, 2000, 5)} returns 2000. The asset costs $12000
17957 initially and will be worth $2000 after five years; it loses $2000
17961 @pindex calc-fin-syd
17963 The @kbd{b Y} (@code{calc-fin-syd}) [@code{syd}] command computes the
17964 accelerated ``sum-of-years'-digits'' depreciation. Here the depreciation
17965 is higher during the early years of the asset's life. Since the
17966 depreciation is different each year, @kbd{b Y} takes a fourth @var{period}
17967 parameter which specifies which year is requested, from 1 to @var{life}.
17968 If @var{period} is outside this range, the @code{syd} function will
17972 @pindex calc-fin-ddb
17974 The @kbd{b D} (@code{calc-fin-ddb}) [@code{ddb}] command computes an
17975 accelerated depreciation using the double-declining balance method.
17976 It also takes a fourth @var{period} parameter.
17978 For symmetry, the @code{sln} function will accept a @var{period}
17979 parameter as well, although it will ignore its value except that the
17980 return value will as usual be zero if @var{period} is out of range.
17982 For example, pushing the vector @expr{[1,2,3,4,5]} (perhaps with @kbd{v x 5})
17983 and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
17984 ddb(12000,2000,5,$)] @key{RET}} produces a matrix that allows us to compare
17985 the three depreciation methods:
17989 [ [ 2000, 3333, 4800 ]
17990 [ 2000, 2667, 2880 ]
17991 [ 2000, 2000, 1728 ]
17992 [ 2000, 1333, 592 ]
17998 (Values have been rounded to nearest integers in this figure.)
17999 We see that @code{sln} depreciates by the same amount each year,
18000 @kbd{syd} depreciates more at the beginning and less at the end,
18001 and @kbd{ddb} weights the depreciation even more toward the beginning.
18003 Summing columns with @kbd{V R : +} yields @expr{[10000, 10000, 10000]};
18004 the total depreciation in any method is (by definition) the
18005 difference between the cost and the salvage value.
18007 @node Definitions of Financial Functions, , Depreciation Functions, Financial Functions
18008 @subsection Definitions
18011 For your reference, here are the actual formulas used to compute
18012 Calc's financial functions.
18014 Calc will not evaluate a financial function unless the @var{rate} or
18015 @var{n} argument is known. However, @var{payment} or @var{amount} can
18016 be a variable. Calc expands these functions according to the
18017 formulas below for symbolic arguments only when you use the @kbd{a "}
18018 (@code{calc-expand-formula}) command, or when taking derivatives or
18019 integrals or solving equations involving the functions.
18022 These formulas are shown using the conventions of Big display
18023 mode (@kbd{d B}); for example, the formula for @code{fv} written
18024 linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}.
18029 fv(rate, n, pmt) = pmt * ---------------
18033 ((1 + rate) - 1) (1 + rate)
18034 fvb(rate, n, pmt) = pmt * ----------------------------
18038 fvl(rate, n, pmt) = pmt * (1 + rate)
18042 pv(rate, n, pmt) = pmt * ----------------
18046 (1 - (1 + rate) ) (1 + rate)
18047 pvb(rate, n, pmt) = pmt * -----------------------------
18051 pvl(rate, n, pmt) = pmt * (1 + rate)
18054 npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate)
18057 npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate)
18060 (amt - x * (1 + rate) ) * rate
18061 pmt(rate, n, amt, x) = -------------------------------
18066 (amt - x * (1 + rate) ) * rate
18067 pmtb(rate, n, amt, x) = -------------------------------
18069 (1 - (1 + rate) ) (1 + rate)
18072 nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate)
18076 nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
18080 nperl(rate, pmt, amt) = - log(---, 1 + rate)
18085 ratel(n, pmt, amt) = ------ - 1
18090 sln(cost, salv, life) = -----------
18093 (cost - salv) * (life - per + 1)
18094 syd(cost, salv, life, per) = --------------------------------
18095 life * (life + 1) / 2
18098 ddb(cost, salv, life, per) = --------, book = cost - depreciation so far
18104 $$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
18105 $$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
18106 $$ \code{fvl}(r, n, p) = p (1 + r)^n $$
18107 $$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$
18108 $$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$
18109 $$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$
18110 $$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$
18111 $$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$
18112 $$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$
18113 $$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over
18114 (1 - (1 + r)^{-n}) (1 + r) } $$
18115 $$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$
18116 $$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$
18117 $$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$
18118 $$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$
18119 $$ \code{sln}(c, s, l) = { c - s \over l } $$
18120 $$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$
18121 $$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$
18125 In @code{pmt} and @code{pmtb}, @expr{x=0} if omitted.
18127 These functions accept any numeric objects, including error forms,
18128 intervals, and even (though not very usefully) complex numbers. The
18129 above formulas specify exactly the behavior of these functions with
18130 all sorts of inputs.
18132 Note that if the first argument to the @code{log} in @code{nper} is
18133 negative, @code{nper} leaves itself in symbolic form rather than
18134 returning a (financially meaningless) complex number.
18136 @samp{rate(num, pmt, amt)} solves the equation
18137 @samp{pv(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R}
18138 (@code{calc-find-root}), with the interval @samp{[.01% .. 100%]}
18139 for an initial guess. The @code{rateb} function is the same except
18140 that it uses @code{pvb}. Note that @code{ratel} can be solved
18141 directly; its formula is shown in the above list.
18143 Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0}
18146 If you give a fourth argument to @code{nper} or @code{nperb}, Calc
18147 will also use @kbd{H a R} to solve the equation using an initial
18148 guess interval of @samp{[0 .. 100]}.
18150 A fourth argument to @code{fv} simply sums the two components
18151 calculated from the above formulas for @code{fv} and @code{fvl}.
18152 The same is true of @code{fvb}, @code{pv}, and @code{pvb}.
18154 The @kbd{ddb} function is computed iteratively; the ``book'' value
18155 starts out equal to @var{cost}, and decreases according to the above
18156 formula for the specified number of periods. If the book value
18157 would decrease below @var{salvage}, it only decreases to @var{salvage}
18158 and the depreciation is zero for all subsequent periods. The @code{ddb}
18159 function returns the amount the book value decreased in the specified
18162 @node Binary Functions, , Financial Functions, Arithmetic
18163 @section Binary Number Functions
18166 The commands in this chapter all use two-letter sequences beginning with
18167 the @kbd{b} prefix.
18169 @cindex Binary numbers
18170 The ``binary'' operations actually work regardless of the currently
18171 displayed radix, although their results make the most sense in a radix
18172 like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @w{@kbd{d 6}}
18173 commands, respectively). You may also wish to enable display of leading
18174 zeros with @kbd{d z}. @xref{Radix Modes}.
18176 @cindex Word size for binary operations
18177 The Calculator maintains a current @dfn{word size} @expr{w}, an
18178 arbitrary positive or negative integer. For a positive word size, all
18179 of the binary operations described here operate modulo @expr{2^w}. In
18180 particular, negative arguments are converted to positive integers modulo
18181 @expr{2^w} by all binary functions.
18183 If the word size is negative, binary operations produce 2's complement
18185 @texline @math{-2^{-w-1}}
18186 @infoline @expr{-(2^(-w-1))}
18188 @texline @math{2^{-w-1}-1}
18189 @infoline @expr{2^(-w-1)-1}
18190 inclusive. Either mode accepts inputs in any range; the sign of
18191 @expr{w} affects only the results produced.
18196 The @kbd{b c} (@code{calc-clip})
18197 [@code{clip}] command can be used to clip a number by reducing it modulo
18198 @expr{2^w}. The commands described in this chapter automatically clip
18199 their results to the current word size. Note that other operations like
18200 addition do not use the current word size, since integer addition
18201 generally is not ``binary.'' (However, @pxref{Simplification Modes},
18202 @code{calc-bin-simplify-mode}.) For example, with a word size of 8
18203 bits @kbd{b c} converts a number to the range 0 to 255; with a word
18204 size of @mathit{-8} @kbd{b c} converts to the range @mathit{-128} to 127.
18207 @pindex calc-word-size
18208 The default word size is 32 bits. All operations except the shifts and
18209 rotates allow you to specify a different word size for that one
18210 operation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the
18211 top of stack to the range 0 to 255 regardless of the current word size.
18212 To set the word size permanently, use @kbd{b w} (@code{calc-word-size}).
18213 This command displays a prompt with the current word size; press @key{RET}
18214 immediately to keep this word size, or type a new word size at the prompt.
18216 When the binary operations are written in symbolic form, they take an
18217 optional second (or third) word-size parameter. When a formula like
18218 @samp{and(a,b)} is finally evaluated, the word size current at that time
18219 will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of
18220 @mathit{-8} will always be used. A symbolic binary function will be left
18221 in symbolic form unless the all of its argument(s) are integers or
18222 integer-valued floats.
18224 If either or both arguments are modulo forms for which @expr{M} is a
18225 power of two, that power of two is taken as the word size unless a
18226 numeric prefix argument overrides it. The current word size is never
18227 consulted when modulo-power-of-two forms are involved.
18232 The @kbd{b a} (@code{calc-and}) [@code{and}] command computes the bitwise
18233 AND of the two numbers on the top of the stack. In other words, for each
18234 of the @expr{w} binary digits of the two numbers (pairwise), the corresponding
18235 bit of the result is 1 if and only if both input bits are 1:
18236 @samp{and(2#1100, 2#1010) = 2#1000}.
18241 The @kbd{b o} (@code{calc-or}) [@code{or}] command computes the bitwise
18242 inclusive OR of two numbers. A bit is 1 if either of the input bits, or
18243 both, are 1: @samp{or(2#1100, 2#1010) = 2#1110}.
18248 The @kbd{b x} (@code{calc-xor}) [@code{xor}] command computes the bitwise
18249 exclusive OR of two numbers. A bit is 1 if exactly one of the input bits
18250 is 1: @samp{xor(2#1100, 2#1010) = 2#0110}.
18255 The @kbd{b d} (@code{calc-diff}) [@code{diff}] command computes the bitwise
18256 difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))},
18257 so that @samp{diff(2#1100, 2#1010) = 2#0100}.
18262 The @kbd{b n} (@code{calc-not}) [@code{not}] command computes the bitwise
18263 NOT of a number. A bit is 1 if the input bit is 0 and vice-versa.
18266 @pindex calc-lshift-binary
18268 The @kbd{b l} (@code{calc-lshift-binary}) [@code{lsh}] command shifts a
18269 number left by one bit, or by the number of bits specified in the numeric
18270 prefix argument. A negative prefix argument performs a logical right shift,
18271 in which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)}
18272 is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}.
18273 Bits shifted ``off the end,'' according to the current word size, are lost.
18289 The @kbd{H b l} command also does a left shift, but it takes two arguments
18290 from the stack (the value to shift, and, at top-of-stack, the number of
18291 bits to shift). This version interprets the prefix argument just like
18292 the regular binary operations, i.e., as a word size. The Hyperbolic flag
18293 has a similar effect on the rest of the binary shift and rotate commands.
18296 @pindex calc-rshift-binary
18298 The @kbd{b r} (@code{calc-rshift-binary}) [@code{rsh}] command shifts a
18299 number right by one bit, or by the number of bits specified in the numeric
18300 prefix argument: @samp{rsh(a,n) = lsh(a,-n)}.
18303 @pindex calc-lshift-arith
18305 The @kbd{b L} (@code{calc-lshift-arith}) [@code{ash}] command shifts a
18306 number left. It is analogous to @code{lsh}, except that if the shift
18307 is rightward (the prefix argument is negative), an arithmetic shift
18308 is performed as described below.
18311 @pindex calc-rshift-arith
18313 The @kbd{b R} (@code{calc-rshift-arith}) [@code{rash}] command performs
18314 an ``arithmetic'' shift to the right, in which the leftmost bit (according
18315 to the current word size) is duplicated rather than shifting in zeros.
18316 This corresponds to dividing by a power of two where the input is interpreted
18317 as a signed, twos-complement number. (The distinction between the @samp{rsh}
18318 and @samp{rash} operations is totally independent from whether the word
18319 size is positive or negative.) With a negative prefix argument, this
18320 performs a standard left shift.
18323 @pindex calc-rotate-binary
18325 The @kbd{b t} (@code{calc-rotate-binary}) [@code{rot}] command rotates a
18326 number one bit to the left. The leftmost bit (according to the current
18327 word size) is dropped off the left and shifted in on the right. With a
18328 numeric prefix argument, the number is rotated that many bits to the left
18331 @xref{Set Operations}, for the @kbd{b p} and @kbd{b u} commands that
18332 pack and unpack binary integers into sets. (For example, @kbd{b u}
18333 unpacks the number @samp{2#11001} to the set of bit-numbers
18334 @samp{[0, 3, 4]}.) Type @kbd{b u V #} to count the number of ``1''
18335 bits in a binary integer.
18337 Another interesting use of the set representation of binary integers
18338 is to reverse the bits in, say, a 32-bit integer. Type @kbd{b u} to
18339 unpack; type @kbd{31 @key{TAB} -} to replace each bit-number in the set
18340 with 31 minus that bit-number; type @kbd{b p} to pack the set back
18341 into a binary integer.
18343 @node Scientific Functions, Matrix Functions, Arithmetic, Top
18344 @chapter Scientific Functions
18347 The functions described here perform trigonometric and other transcendental
18348 calculations. They generally produce floating-point answers correct to the
18349 full current precision. The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse)
18350 flag keys must be used to get some of these functions from the keyboard.
18354 @cindex @code{pi} variable
18357 @cindex @code{e} variable
18360 @cindex @code{gamma} variable
18362 @cindex Gamma constant, Euler's
18363 @cindex Euler's gamma constant
18365 @cindex @code{phi} variable
18366 @cindex Phi, golden ratio
18367 @cindex Golden ratio
18368 One miscellaneous command is shift-@kbd{P} (@code{calc-pi}), which pushes
18369 the value of @cpi{} (at the current precision) onto the stack. With the
18370 Hyperbolic flag, it pushes the value @expr{e}, the base of natural logarithms.
18371 With the Inverse flag, it pushes Euler's constant
18372 @texline @math{\gamma}
18373 @infoline @expr{gamma}
18374 (about 0.5772). With both Inverse and Hyperbolic, it
18375 pushes the ``golden ratio''
18376 @texline @math{\phi}
18377 @infoline @expr{phi}
18378 (about 1.618). (At present, Euler's constant is not available
18379 to unlimited precision; Calc knows only the first 100 digits.)
18380 In Symbolic mode, these commands push the
18381 actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi},
18382 respectively, instead of their values; @pxref{Symbolic Mode}.
18392 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] function is described elsewhere;
18393 @pxref{Basic Arithmetic}. With the Inverse flag [@code{sqr}], this command
18394 computes the square of the argument.
18396 @xref{Prefix Arguments}, for a discussion of the effect of numeric
18397 prefix arguments on commands in this chapter which do not otherwise
18398 interpret a prefix argument.
18401 * Logarithmic Functions::
18402 * Trigonometric and Hyperbolic Functions::
18403 * Advanced Math Functions::
18406 * Combinatorial Functions::
18407 * Probability Distribution Functions::
18410 @node Logarithmic Functions, Trigonometric and Hyperbolic Functions, Scientific Functions, Scientific Functions
18411 @section Logarithmic Functions
18421 The shift-@kbd{L} (@code{calc-ln}) [@code{ln}] command computes the natural
18422 logarithm of the real or complex number on the top of the stack. With
18423 the Inverse flag it computes the exponential function instead, although
18424 this is redundant with the @kbd{E} command.
18433 The shift-@kbd{E} (@code{calc-exp}) [@code{exp}] command computes the
18434 exponential, i.e., @expr{e} raised to the power of the number on the stack.
18435 The meanings of the Inverse and Hyperbolic flags follow from those for
18436 the @code{calc-ln} command.
18451 The @kbd{H L} (@code{calc-log10}) [@code{log10}] command computes the common
18452 (base-10) logarithm of a number. (With the Inverse flag [@code{exp10}],
18453 it raises ten to a given power.) Note that the common logarithm of a
18454 complex number is computed by taking the natural logarithm and dividing
18456 @texline @math{\ln10}.
18457 @infoline @expr{ln(10)}.
18464 The @kbd{B} (@code{calc-log}) [@code{log}] command computes a logarithm
18465 to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since
18466 @texline @math{2^{10} = 1024}.
18467 @infoline @expr{2^10 = 1024}.
18468 In certain cases like @samp{log(3,9)}, the result
18469 will be either @expr{1:2} or @expr{0.5} depending on the current Fraction
18470 mode setting. With the Inverse flag [@code{alog}], this command is
18471 similar to @kbd{^} except that the order of the arguments is reversed.
18476 The @kbd{f I} (@code{calc-ilog}) [@code{ilog}] command computes the
18477 integer logarithm of a number to any base. The number and the base must
18478 themselves be positive integers. This is the true logarithm, rounded
18479 down to an integer. Thus @kbd{ilog(x,10)} is 3 for all @expr{x} in the
18480 range from 1000 to 9999. If both arguments are positive integers, exact
18481 integer arithmetic is used; otherwise, this is equivalent to
18482 @samp{floor(log(x,b))}.
18487 The @kbd{f E} (@code{calc-expm1}) [@code{expm1}] command computes
18488 @texline @math{e^x - 1},
18489 @infoline @expr{exp(x)-1},
18490 but using an algorithm that produces a more accurate
18491 answer when the result is close to zero, i.e., when
18492 @texline @math{e^x}
18493 @infoline @expr{exp(x)}
18499 The @kbd{f L} (@code{calc-lnp1}) [@code{lnp1}] command computes
18500 @texline @math{\ln(x+1)},
18501 @infoline @expr{ln(x+1)},
18502 producing a more accurate answer when @expr{x} is close to zero.
18504 @node Trigonometric and Hyperbolic Functions, Advanced Math Functions, Logarithmic Functions, Scientific Functions
18505 @section Trigonometric/Hyperbolic Functions
18511 The shift-@kbd{S} (@code{calc-sin}) [@code{sin}] command computes the sine
18512 of an angle or complex number. If the input is an HMS form, it is interpreted
18513 as degrees-minutes-seconds; otherwise, the input is interpreted according
18514 to the current angular mode. It is best to use Radians mode when operating
18515 on complex numbers.
18517 Calc's ``units'' mechanism includes angular units like @code{deg},
18518 @code{rad}, and @code{grad}. While @samp{sin(45 deg)} is not evaluated
18519 all the time, the @kbd{u s} (@code{calc-simplify-units}) command will
18520 simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless
18521 of the current angular mode. @xref{Basic Operations on Units}.
18523 Also, the symbolic variable @code{pi} is not ordinarily recognized in
18524 arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but
18525 the @kbd{a s} (@code{calc-simplify}) command recognizes many such
18526 formulas when the current angular mode is Radians @emph{and} Symbolic
18527 mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}.
18528 @xref{Symbolic Mode}. Beware, this simplification occurs even if you
18529 have stored a different value in the variable @samp{pi}; this is one
18530 reason why changing built-in variables is a bad idea. Arguments of
18531 the form @expr{x} plus a multiple of @cpiover{2} are also simplified.
18532 Calc includes similar formulas for @code{cos} and @code{tan}.
18534 The @kbd{a s} command knows all angles which are integer multiples of
18535 @cpiover{12}, @cpiover{10}, or @cpiover{8} radians. In Degrees mode,
18536 analogous simplifications occur for integer multiples of 15 or 18
18537 degrees, and for arguments plus multiples of 90 degrees.
18540 @pindex calc-arcsin
18542 With the Inverse flag, @code{calc-sin} computes an arcsine. This is also
18543 available as the @code{calc-arcsin} command or @code{arcsin} algebraic
18544 function. The returned argument is converted to degrees, radians, or HMS
18545 notation depending on the current angular mode.
18551 @pindex calc-arcsinh
18553 With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic
18554 sine, also available as @code{calc-sinh} [@code{sinh}]. With the
18555 Hyperbolic and Inverse flags, it computes the hyperbolic arcsine
18556 (@code{calc-arcsinh}) [@code{arcsinh}].
18565 @pindex calc-arccos
18583 @pindex calc-arccosh
18601 @pindex calc-arctan
18619 @pindex calc-arctanh
18624 The shift-@kbd{C} (@code{calc-cos}) [@code{cos}] command computes the cosine
18625 of an angle or complex number, and shift-@kbd{T} (@code{calc-tan}) [@code{tan}]
18626 computes the tangent, along with all the various inverse and hyperbolic
18627 variants of these functions.
18630 @pindex calc-arctan2
18632 The @kbd{f T} (@code{calc-arctan2}) [@code{arctan2}] command takes two
18633 numbers from the stack and computes the arc tangent of their ratio. The
18634 result is in the full range from @mathit{-180} (exclusive) to @mathit{+180}
18635 (inclusive) degrees, or the analogous range in radians. A similar
18636 result would be obtained with @kbd{/} followed by @kbd{I T}, but the
18637 value would only be in the range from @mathit{-90} to @mathit{+90} degrees
18638 since the division loses information about the signs of the two
18639 components, and an error might result from an explicit division by zero
18640 which @code{arctan2} would avoid. By (arbitrary) definition,
18641 @samp{arctan2(0,0)=0}.
18643 @pindex calc-sincos
18655 The @code{calc-sincos} [@code{sincos}] command computes the sine and
18656 cosine of a number, returning them as a vector of the form
18657 @samp{[@var{cos}, @var{sin}]}.
18658 With the Inverse flag [@code{arcsincos}], this command takes a two-element
18659 vector as an argument and computes @code{arctan2} of the elements.
18660 (This command does not accept the Hyperbolic flag.)
18674 The remaining trigonometric functions, @code{calc-sec} [@code{sec}],
18675 @code{calc-csc} [@code{csc}] and @code{calc-sec} [@code{sec}], are also
18676 available. With the Hyperbolic flag, these compute their hyperbolic
18677 counterparts, which are also available separately as @code{calc-sech}
18678 [@code{sech}], @code{calc-csch} [@code{csch}] and @code{calc-sech}
18679 [@code{sech}]. (These commmands do not accept the Inverse flag.)
18681 @node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions
18682 @section Advanced Mathematical Functions
18685 Calc can compute a variety of less common functions that arise in
18686 various branches of mathematics. All of the functions described in
18687 this section allow arbitrary complex arguments and, except as noted,
18688 will work to arbitrarily large precisions. They can not at present
18689 handle error forms or intervals as arguments.
18691 NOTE: These functions are still experimental. In particular, their
18692 accuracy is not guaranteed in all domains. It is advisable to set the
18693 current precision comfortably higher than you actually need when
18694 using these functions. Also, these functions may be impractically
18695 slow for some values of the arguments.
18700 The @kbd{f g} (@code{calc-gamma}) [@code{gamma}] command computes the Euler
18701 gamma function. For positive integer arguments, this is related to the
18702 factorial function: @samp{gamma(n+1) = fact(n)}. For general complex
18703 arguments the gamma function can be defined by the following definite
18705 @texline @math{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt}.
18706 @infoline @expr{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}.
18707 (The actual implementation uses far more efficient computational methods.)
18723 @pindex calc-inc-gamma
18736 The @kbd{f G} (@code{calc-inc-gamma}) [@code{gammaP}] command computes
18737 the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by
18739 @texline @math{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)}.
18740 @infoline @expr{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}.
18741 This implies that @samp{gammaP(a,inf) = 1} for any @expr{a} (see the
18742 definition of the normal gamma function).
18744 Several other varieties of incomplete gamma function are defined.
18745 The complement of @expr{P(a,x)}, called @expr{Q(a,x) = 1-P(a,x)} by
18746 some authors, is computed by the @kbd{I f G} [@code{gammaQ}] command.
18747 You can think of this as taking the other half of the integral, from
18748 @expr{x} to infinity.
18751 The functions corresponding to the integrals that define @expr{P(a,x)}
18752 and @expr{Q(a,x)} but without the normalizing @expr{1/gamma(a)}
18753 factor are called @expr{g(a,x)} and @expr{G(a,x)}, respectively
18754 (where @expr{g} and @expr{G} represent the lower- and upper-case Greek
18755 letter gamma). You can obtain these using the @kbd{H f G} [@code{gammag}]
18756 and @kbd{H I f G} [@code{gammaG}] commands.
18760 The functions corresponding to the integrals that define $P(a,x)$
18761 and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
18762 factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
18763 You can obtain these using the \kbd{H f G} [\code{gammag}] and
18764 \kbd{I H f G} [\code{gammaG}] commands.
18770 The @kbd{f b} (@code{calc-beta}) [@code{beta}] command computes the
18771 Euler beta function, which is defined in terms of the gamma function as
18772 @texline @math{B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)},
18773 @infoline @expr{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)},
18775 @texline @math{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}.
18776 @infoline @expr{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}.
18780 @pindex calc-inc-beta
18783 The @kbd{f B} (@code{calc-inc-beta}) [@code{betaI}] command computes
18784 the incomplete beta function @expr{I(x,a,b)}. It is defined by
18785 @texline @math{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) / B(a,b)}.
18786 @infoline @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)}.
18787 Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding
18788 un-normalized version [@code{betaB}].
18795 The @kbd{f e} (@code{calc-erf}) [@code{erf}] command computes the
18797 @texline @math{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt}.
18798 @infoline @expr{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}.
18799 The complementary error function @kbd{I f e} (@code{calc-erfc}) [@code{erfc}]
18800 is the corresponding integral from @samp{x} to infinity; the sum
18801 @texline @math{\hbox{erf}(x) + \hbox{erfc}(x) = 1}.
18802 @infoline @expr{erf(x) + erfc(x) = 1}.
18806 @pindex calc-bessel-J
18807 @pindex calc-bessel-Y
18810 The @kbd{f j} (@code{calc-bessel-J}) [@code{besJ}] and @kbd{f y}
18811 (@code{calc-bessel-Y}) [@code{besY}] commands compute the Bessel
18812 functions of the first and second kinds, respectively.
18813 In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter
18814 @expr{n} is often an integer, but is not required to be one.
18815 Calc's implementation of the Bessel functions currently limits the
18816 precision to 8 digits, and may not be exact even to that precision.
18819 @node Branch Cuts, Random Numbers, Advanced Math Functions, Scientific Functions
18820 @section Branch Cuts and Principal Values
18823 @cindex Branch cuts
18824 @cindex Principal values
18825 All of the logarithmic, trigonometric, and other scientific functions are
18826 defined for complex numbers as well as for reals.
18827 This section describes the values
18828 returned in cases where the general result is a family of possible values.
18829 Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language},
18830 second edition, in these matters. This section will describe each
18831 function briefly; for a more detailed discussion (including some nifty
18832 diagrams), consult Steele's book.
18834 Note that the branch cuts for @code{arctan} and @code{arctanh} were
18835 changed between the first and second editions of Steele. Versions of
18836 Calc starting with 2.00 follow the second edition.
18838 The new branch cuts exactly match those of the HP-28/48 calculators.
18839 They also match those of Mathematica 1.2, except that Mathematica's
18840 @code{arctan} cut is always in the right half of the complex plane,
18841 and its @code{arctanh} cut is always in the top half of the plane.
18842 Calc's cuts are continuous with quadrants I and III for @code{arctan},
18843 or II and IV for @code{arctanh}.
18845 Note: The current implementations of these functions with complex arguments
18846 are designed with proper behavior around the branch cuts in mind, @emph{not}
18847 efficiency or accuracy. You may need to increase the floating precision
18848 and wait a while to get suitable answers from them.
18850 For @samp{sqrt(a+bi)}: When @expr{a<0} and @expr{b} is small but positive
18851 or zero, the result is close to the @expr{+i} axis. For @expr{b} small and
18852 negative, the result is close to the @expr{-i} axis. The result always lies
18853 in the right half of the complex plane.
18855 For @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}.
18856 The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}.
18857 Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the
18858 negative real axis.
18860 The following table describes these branch cuts in another way.
18861 If the real and imaginary parts of @expr{z} are as shown, then
18862 the real and imaginary parts of @expr{f(z)} will be as shown.
18863 Here @code{eps} stands for a small positive value; each
18864 occurrence of @code{eps} may stand for a different small value.
18868 ----------------------------------------
18871 -, +eps +eps, + +eps, +
18872 -, -eps +eps, - +eps, -
18875 For @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}.
18876 One interesting consequence of this is that @samp{(-8)^1:3} does
18877 not evaluate to @mathit{-2} as you might expect, but to the complex
18878 number @expr{(1., 1.732)}. Both of these are valid cube roots
18879 of @mathit{-8} (as is @expr{(1., -1.732)}); Calc chooses a perhaps
18880 less-obvious root for the sake of mathematical consistency.
18882 For @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}.
18883 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18885 For @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))},
18886 or equivalently by @samp{pi/2 - arcsin(z)}. The branch cuts are on
18887 the real axis, less than @mathit{-1} and greater than 1.
18889 For @samp{arctan(z)}: This is defined by
18890 @samp{(ln(1+i*z) - ln(1-i*z)) / (2*i)}. The branch cuts are on the
18891 imaginary axis, below @expr{-i} and above @expr{i}.
18893 For @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}.
18894 The branch cuts are on the imaginary axis, below @expr{-i} and
18897 For @samp{arccosh(z)}: This is defined by
18898 @samp{ln(z + (z+1)*sqrt((z-1)/(z+1)))}. The branch cut is on the
18899 real axis less than 1.
18901 For @samp{arctanh(z)}: This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}.
18902 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18904 The following tables for @code{arcsin}, @code{arccos}, and
18905 @code{arctan} assume the current angular mode is Radians. The
18906 hyperbolic functions operate independently of the angular mode.
18909 z arcsin(z) arccos(z)
18910 -------------------------------------------------------
18911 (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0
18912 (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps
18913 (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps
18914 <-1, 0 -pi/2, + pi, -
18915 <-1, +eps -pi/2 + eps, + pi - eps, -
18916 <-1, -eps -pi/2 + eps, - pi - eps, +
18918 >1, +eps pi/2 - eps, + +eps, -
18919 >1, -eps pi/2 - eps, - +eps, +
18923 z arccosh(z) arctanh(z)
18924 -----------------------------------------------------
18925 (-1..1), 0 0, (0..pi) any, 0
18926 (-1..1), +eps +eps, (0..pi) any, +eps
18927 (-1..1), -eps +eps, (-pi..0) any, -eps
18928 <-1, 0 +, pi -, pi/2
18929 <-1, +eps +, pi - eps -, pi/2 - eps
18930 <-1, -eps +, -pi + eps -, -pi/2 + eps
18931 >1, 0 +, 0 +, -pi/2
18932 >1, +eps +, +eps +, pi/2 - eps
18933 >1, -eps +, -eps +, -pi/2 + eps
18937 z arcsinh(z) arctan(z)
18938 -----------------------------------------------------
18939 0, (-1..1) 0, (-pi/2..pi/2) 0, any
18940 0, <-1 -, -pi/2 -pi/2, -
18941 +eps, <-1 +, -pi/2 + eps pi/2 - eps, -
18942 -eps, <-1 -, -pi/2 + eps -pi/2 + eps, -
18943 0, >1 +, pi/2 pi/2, +
18944 +eps, >1 +, pi/2 - eps pi/2 - eps, +
18945 -eps, >1 -, pi/2 - eps -pi/2 + eps, +
18948 Finally, the following identities help to illustrate the relationship
18949 between the complex trigonometric and hyperbolic functions. They
18950 are valid everywhere, including on the branch cuts.
18953 sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z)
18954 cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z)
18955 tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z)
18956 sinh(i*z) = i*sin(z) cosh(i*z) = cos(z)
18959 The ``advanced math'' functions (gamma, Bessel, etc.@:) are also defined
18960 for general complex arguments, but their branch cuts and principal values
18961 are not rigorously specified at present.
18963 @node Random Numbers, Combinatorial Functions, Branch Cuts, Scientific Functions
18964 @section Random Numbers
18968 @pindex calc-random
18970 The @kbd{k r} (@code{calc-random}) [@code{random}] command produces
18971 random numbers of various sorts.
18973 Given a positive numeric prefix argument @expr{M}, it produces a random
18974 integer @expr{N} in the range
18975 @texline @math{0 \le N < M}.
18976 @infoline @expr{0 <= N < M}.
18977 Each of the @expr{M} values appears with equal probability.
18979 With no numeric prefix argument, the @kbd{k r} command takes its argument
18980 from the stack instead. Once again, if this is a positive integer @expr{M}
18981 the result is a random integer less than @expr{M}. However, note that
18982 while numeric prefix arguments are limited to six digits or so, an @expr{M}
18983 taken from the stack can be arbitrarily large. If @expr{M} is negative,
18984 the result is a random integer in the range
18985 @texline @math{M < N \le 0}.
18986 @infoline @expr{M < N <= 0}.
18988 If the value on the stack is a floating-point number @expr{M}, the result
18989 is a random floating-point number @expr{N} in the range
18990 @texline @math{0 \le N < M}
18991 @infoline @expr{0 <= N < M}
18993 @texline @math{M < N \le 0},
18994 @infoline @expr{M < N <= 0},
18995 according to the sign of @expr{M}.
18997 If @expr{M} is zero, the result is a Gaussian-distributed random real
18998 number; the distribution has a mean of zero and a standard deviation
18999 of one. The algorithm used generates random numbers in pairs; thus,
19000 every other call to this function will be especially fast.
19002 If @expr{M} is an error form
19003 @texline @math{m} @code{+/-} @math{\sigma}
19004 @infoline @samp{m +/- s}
19006 @texline @math{\sigma}
19008 are both real numbers, the result uses a Gaussian distribution with mean
19009 @var{m} and standard deviation
19010 @texline @math{\sigma}.
19013 If @expr{M} is an interval form, the lower and upper bounds specify the
19014 acceptable limits of the random numbers. If both bounds are integers,
19015 the result is a random integer in the specified range. If either bound
19016 is floating-point, the result is a random real number in the specified
19017 range. If the interval is open at either end, the result will be sure
19018 not to equal that end value. (This makes a big difference for integer
19019 intervals, but for floating-point intervals it's relatively minor:
19020 with a precision of 6, @samp{random([1.0..2.0))} will return any of one
19021 million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may
19022 additionally return 2.00000, but the probability of this happening is
19025 If @expr{M} is a vector, the result is one element taken at random from
19026 the vector. All elements of the vector are given equal probabilities.
19029 The sequence of numbers produced by @kbd{k r} is completely random by
19030 default, i.e., the sequence is seeded each time you start Calc using
19031 the current time and other information. You can get a reproducible
19032 sequence by storing a particular ``seed value'' in the Calc variable
19033 @code{RandSeed}. Any integer will do for a seed; integers of from 1
19034 to 12 digits are good. If you later store a different integer into
19035 @code{RandSeed}, Calc will switch to a different pseudo-random
19036 sequence. If you ``unstore'' @code{RandSeed}, Calc will re-seed itself
19037 from the current time. If you store the same integer that you used
19038 before back into @code{RandSeed}, you will get the exact same sequence
19039 of random numbers as before.
19041 @pindex calc-rrandom
19042 The @code{calc-rrandom} command (not on any key) produces a random real
19043 number between zero and one. It is equivalent to @samp{random(1.0)}.
19046 @pindex calc-random-again
19047 The @kbd{k a} (@code{calc-random-again}) command produces another random
19048 number, re-using the most recent value of @expr{M}. With a numeric
19049 prefix argument @var{n}, it produces @var{n} more random numbers using
19050 that value of @expr{M}.
19053 @pindex calc-shuffle
19055 The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several
19056 random values with no duplicates. The value on the top of the stack
19057 specifies the set from which the random values are drawn, and may be any
19058 of the @expr{M} formats described above. The numeric prefix argument
19059 gives the length of the desired list. (If you do not provide a numeric
19060 prefix argument, the length of the list is taken from the top of the
19061 stack, and @expr{M} from second-to-top.)
19063 If @expr{M} is a floating-point number, zero, or an error form (so
19064 that the random values are being drawn from the set of real numbers)
19065 there is little practical difference between using @kbd{k h} and using
19066 @kbd{k r} several times. But if the set of possible values consists
19067 of just a few integers, or the elements of a vector, then there is
19068 a very real chance that multiple @kbd{k r}'s will produce the same
19069 number more than once. The @kbd{k h} command produces a vector whose
19070 elements are always distinct. (Actually, there is a slight exception:
19071 If @expr{M} is a vector, no given vector element will be drawn more
19072 than once, but if several elements of @expr{M} are equal, they may
19073 each make it into the result vector.)
19075 One use of @kbd{k h} is to rearrange a list at random. This happens
19076 if the prefix argument is equal to the number of values in the list:
19077 @kbd{[1, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list
19078 @samp{[2.5, 1, 1.5, 3, 2]}. As a convenient feature, if the argument
19079 @var{n} is negative it is replaced by the size of the set represented
19080 by @expr{M}. Naturally, this is allowed only when @expr{M} specifies
19081 a small discrete set of possibilities.
19083 To do the equivalent of @kbd{k h} but with duplications allowed,
19084 given @expr{M} on the stack and with @var{n} just entered as a numeric
19085 prefix, use @kbd{v b} to build a vector of copies of @expr{M}, then use
19086 @kbd{V M k r} to ``map'' the normal @kbd{k r} function over the
19087 elements of this vector. @xref{Matrix Functions}.
19090 * Random Number Generator:: (Complete description of Calc's algorithm)
19093 @node Random Number Generator, , Random Numbers, Random Numbers
19094 @subsection Random Number Generator
19096 Calc's random number generator uses several methods to ensure that
19097 the numbers it produces are highly random. Knuth's @emph{Art of
19098 Computer Programming}, Volume II, contains a thorough description
19099 of the theory of random number generators and their measurement and
19102 If @code{RandSeed} has no stored value, Calc calls Emacs' built-in
19103 @code{random} function to get a stream of random numbers, which it
19104 then treats in various ways to avoid problems inherent in the simple
19105 random number generators that many systems use to implement @code{random}.
19107 When Calc's random number generator is first invoked, it ``seeds''
19108 the low-level random sequence using the time of day, so that the
19109 random number sequence will be different every time you use Calc.
19111 Since Emacs Lisp doesn't specify the range of values that will be
19112 returned by its @code{random} function, Calc exercises the function
19113 several times to estimate the range. When Calc subsequently uses
19114 the @code{random} function, it takes only 10 bits of the result
19115 near the most-significant end. (It avoids at least the bottom
19116 four bits, preferably more, and also tries to avoid the top two
19117 bits.) This strategy works well with the linear congruential
19118 generators that are typically used to implement @code{random}.
19120 If @code{RandSeed} contains an integer, Calc uses this integer to
19121 seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A,
19123 @texline @math{X_{n-55} - X_{n-24}}.
19124 @infoline @expr{X_n-55 - X_n-24}).
19125 This method expands the seed
19126 value into a large table which is maintained internally; the variable
19127 @code{RandSeed} is changed from, e.g., 42 to the vector @expr{[42]}
19128 to indicate that the seed has been absorbed into this table. When
19129 @code{RandSeed} contains a vector, @kbd{k r} and related commands
19130 continue to use the same internal table as last time. There is no
19131 way to extract the complete state of the random number generator
19132 so that you can restart it from any point; you can only restart it
19133 from the same initial seed value. A simple way to restart from the
19134 same seed is to type @kbd{s r RandSeed} to get the seed vector,
19135 @kbd{v u} to unpack it back into a number, then @kbd{s t RandSeed}
19136 to reseed the generator with that number.
19138 Calc uses a ``shuffling'' method as described in algorithm 3.2.2B
19139 of Knuth. It fills a table with 13 random 10-bit numbers. Then,
19140 to generate a new random number, it uses the previous number to
19141 index into the table, picks the value it finds there as the new
19142 random number, then replaces that table entry with a new value
19143 obtained from a call to the base random number generator (either
19144 the additive congruential generator or the @code{random} function
19145 supplied by the system). If there are any flaws in the base
19146 generator, shuffling will tend to even them out. But if the system
19147 provides an excellent @code{random} function, shuffling will not
19148 damage its randomness.
19150 To create a random integer of a certain number of digits, Calc
19151 builds the integer three decimal digits at a time. For each group
19152 of three digits, Calc calls its 10-bit shuffling random number generator
19153 (which returns a value from 0 to 1023); if the random value is 1000
19154 or more, Calc throws it out and tries again until it gets a suitable
19157 To create a random floating-point number with precision @var{p}, Calc
19158 simply creates a random @var{p}-digit integer and multiplies by
19159 @texline @math{10^{-p}}.
19160 @infoline @expr{10^-p}.
19161 The resulting random numbers should be very clean, but note
19162 that relatively small numbers will have few significant random digits.
19163 In other words, with a precision of 12, you will occasionally get
19164 numbers on the order of
19165 @texline @math{10^{-9}}
19166 @infoline @expr{10^-9}
19168 @texline @math{10^{-10}},
19169 @infoline @expr{10^-10},
19170 but those numbers will only have two or three random digits since they
19171 correspond to small integers times
19172 @texline @math{10^{-12}}.
19173 @infoline @expr{10^-12}.
19175 To create a random integer in the interval @samp{[0 .. @var{m})}, Calc
19176 counts the digits in @var{m}, creates a random integer with three
19177 additional digits, then reduces modulo @var{m}. Unless @var{m} is a
19178 power of ten the resulting values will be very slightly biased toward
19179 the lower numbers, but this bias will be less than 0.1%. (For example,
19180 if @var{m} is 42, Calc will reduce a random integer less than 100000
19181 modulo 42 to get a result less than 42. It is easy to show that the
19182 numbers 40 and 41 will be only 2380/2381 as likely to result from this
19183 modulo operation as numbers 39 and below.) If @var{m} is a power of
19184 ten, however, the numbers should be completely unbiased.
19186 The Gaussian random numbers generated by @samp{random(0.0)} use the
19187 ``polar'' method described in Knuth section 3.4.1C. This method
19188 generates a pair of Gaussian random numbers at a time, so only every
19189 other call to @samp{random(0.0)} will require significant calculations.
19191 @node Combinatorial Functions, Probability Distribution Functions, Random Numbers, Scientific Functions
19192 @section Combinatorial Functions
19195 Commands relating to combinatorics and number theory begin with the
19196 @kbd{k} key prefix.
19201 The @kbd{k g} (@code{calc-gcd}) [@code{gcd}] command computes the
19202 Greatest Common Divisor of two integers. It also accepts fractions;
19203 the GCD of two fractions is defined by taking the GCD of the
19204 numerators, and the LCM of the denominators. This definition is
19205 consistent with the idea that @samp{a / gcd(a,x)} should yield an
19206 integer for any @samp{a} and @samp{x}. For other types of arguments,
19207 the operation is left in symbolic form.
19212 The @kbd{k l} (@code{calc-lcm}) [@code{lcm}] command computes the
19213 Least Common Multiple of two integers or fractions. The product of
19214 the LCM and GCD of two numbers is equal to the product of the
19218 @pindex calc-extended-gcd
19220 The @kbd{k E} (@code{calc-extended-gcd}) [@code{egcd}] command computes
19221 the GCD of two integers @expr{x} and @expr{y} and returns a vector
19222 @expr{[g, a, b]} where
19223 @texline @math{g = \gcd(x,y) = a x + b y}.
19224 @infoline @expr{g = gcd(x,y) = a x + b y}.
19227 @pindex calc-factorial
19233 The @kbd{!} (@code{calc-factorial}) [@code{fact}] command computes the
19234 factorial of the number at the top of the stack. If the number is an
19235 integer, the result is an exact integer. If the number is an
19236 integer-valued float, the result is a floating-point approximation. If
19237 the number is a non-integral real number, the generalized factorial is used,
19238 as defined by the Euler Gamma function. Please note that computation of
19239 large factorials can be slow; using floating-point format will help
19240 since fewer digits must be maintained. The same is true of many of
19241 the commands in this section.
19244 @pindex calc-double-factorial
19250 The @kbd{k d} (@code{calc-double-factorial}) [@code{dfact}] command
19251 computes the ``double factorial'' of an integer. For an even integer,
19252 this is the product of even integers from 2 to @expr{N}. For an odd
19253 integer, this is the product of odd integers from 3 to @expr{N}. If
19254 the argument is an integer-valued float, the result is a floating-point
19255 approximation. This function is undefined for negative even integers.
19256 The notation @expr{N!!} is also recognized for double factorials.
19259 @pindex calc-choose
19261 The @kbd{k c} (@code{calc-choose}) [@code{choose}] command computes the
19262 binomial coefficient @expr{N}-choose-@expr{M}, where @expr{M} is the number
19263 on the top of the stack and @expr{N} is second-to-top. If both arguments
19264 are integers, the result is an exact integer. Otherwise, the result is a
19265 floating-point approximation. The binomial coefficient is defined for all
19267 @texline @math{N! \over M! (N-M)!\,}.
19268 @infoline @expr{N! / M! (N-M)!}.
19274 The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the
19275 number-of-permutations function @expr{N! / (N-M)!}.
19278 The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
19279 number-of-perm\-utations function $N! \over (N-M)!\,$.
19284 @pindex calc-bernoulli-number
19286 The @kbd{k b} (@code{calc-bernoulli-number}) [@code{bern}] command
19287 computes a given Bernoulli number. The value at the top of the stack
19288 is a nonnegative integer @expr{n} that specifies which Bernoulli number
19289 is desired. The @kbd{H k b} command computes a Bernoulli polynomial,
19290 taking @expr{n} from the second-to-top position and @expr{x} from the
19291 top of the stack. If @expr{x} is a variable or formula the result is
19292 a polynomial in @expr{x}; if @expr{x} is a number the result is a number.
19296 @pindex calc-euler-number
19298 The @kbd{k e} (@code{calc-euler-number}) [@code{euler}] command similarly
19299 computes an Euler number, and @w{@kbd{H k e}} computes an Euler polynomial.
19300 Bernoulli and Euler numbers occur in the Taylor expansions of several
19305 @pindex calc-stirling-number
19308 The @kbd{k s} (@code{calc-stirling-number}) [@code{stir1}] command
19309 computes a Stirling number of the first
19310 @texline kind@tie{}@math{n \brack m},
19312 given two integers @expr{n} and @expr{m} on the stack. The @kbd{H k s}
19313 [@code{stir2}] command computes a Stirling number of the second
19314 @texline kind@tie{}@math{n \brace m}.
19316 These are the number of @expr{m}-cycle permutations of @expr{n} objects,
19317 and the number of ways to partition @expr{n} objects into @expr{m}
19318 non-empty sets, respectively.
19321 @pindex calc-prime-test
19323 The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on
19324 the top of the stack is prime. For integers less than eight million, the
19325 answer is always exact and reasonably fast. For larger integers, a
19326 probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P).
19327 The number is first checked against small prime factors (up to 13). Then,
19328 any number of iterations of the algorithm are performed. Each step either
19329 discovers that the number is non-prime, or substantially increases the
19330 certainty that the number is prime. After a few steps, the chance that
19331 a number was mistakenly described as prime will be less than one percent.
19332 (Indeed, this is a worst-case estimate of the probability; in practice
19333 even a single iteration is quite reliable.) After the @kbd{k p} command,
19334 the number will be reported as definitely prime or non-prime if possible,
19335 or otherwise ``probably'' prime with a certain probability of error.
19341 The normal @kbd{k p} command performs one iteration of the primality
19342 test. Pressing @kbd{k p} repeatedly for the same integer will perform
19343 additional iterations. Also, @kbd{k p} with a numeric prefix performs
19344 the specified number of iterations. There is also an algebraic function
19345 @samp{prime(n)} or @samp{prime(n,iters)} which returns 1 if @expr{n}
19346 is (probably) prime and 0 if not.
19349 @pindex calc-prime-factors
19351 The @kbd{k f} (@code{calc-prime-factors}) [@code{prfac}] command
19352 attempts to decompose an integer into its prime factors. For numbers up
19353 to 25 million, the answer is exact although it may take some time. The
19354 result is a vector of the prime factors in increasing order. For larger
19355 inputs, prime factors above 5000 may not be found, in which case the
19356 last number in the vector will be an unfactored integer greater than 25
19357 million (with a warning message). For negative integers, the first
19358 element of the list will be @mathit{-1}. For inputs @mathit{-1}, @mathit{0}, and
19359 @mathit{1}, the result is a list of the same number.
19362 @pindex calc-next-prime
19364 @mindex nextpr@idots
19367 The @kbd{k n} (@code{calc-next-prime}) [@code{nextprime}] command finds
19368 the next prime above a given number. Essentially, it searches by calling
19369 @code{calc-prime-test} on successive integers until it finds one that
19370 passes the test. This is quite fast for integers less than eight million,
19371 but once the probabilistic test comes into play the search may be rather
19372 slow. Ordinarily this command stops for any prime that passes one iteration
19373 of the primality test. With a numeric prefix argument, a number must pass
19374 the specified number of iterations before the search stops. (This only
19375 matters when searching above eight million.) You can always use additional
19376 @kbd{k p} commands to increase your certainty that the number is indeed
19380 @pindex calc-prev-prime
19382 @mindex prevpr@idots
19385 The @kbd{I k n} (@code{calc-prev-prime}) [@code{prevprime}] command
19386 analogously finds the next prime less than a given number.
19389 @pindex calc-totient
19391 The @kbd{k t} (@code{calc-totient}) [@code{totient}] command computes the
19393 @texline function@tie{}@math{\phi(n)},
19394 @infoline function,
19395 the number of integers less than @expr{n} which
19396 are relatively prime to @expr{n}.
19399 @pindex calc-moebius
19401 The @kbd{k m} (@code{calc-moebius}) [@code{moebius}] command computes the
19402 @texline M@"obius @math{\mu}
19403 @infoline Moebius ``mu''
19404 function. If the input number is a product of @expr{k}
19405 distinct factors, this is @expr{(-1)^k}. If the input number has any
19406 duplicate factors (i.e., can be divided by the same prime more than once),
19407 the result is zero.
19409 @node Probability Distribution Functions, , Combinatorial Functions, Scientific Functions
19410 @section Probability Distribution Functions
19413 The functions in this section compute various probability distributions.
19414 For continuous distributions, this is the integral of the probability
19415 density function from @expr{x} to infinity. (These are the ``upper
19416 tail'' distribution functions; there are also corresponding ``lower
19417 tail'' functions which integrate from minus infinity to @expr{x}.)
19418 For discrete distributions, the upper tail function gives the sum
19419 from @expr{x} to infinity; the lower tail function gives the sum
19420 from minus infinity up to, but not including,@w{ }@expr{x}.
19422 To integrate from @expr{x} to @expr{y}, just use the distribution
19423 function twice and subtract. For example, the probability that a
19424 Gaussian random variable with mean 2 and standard deviation 1 will
19425 lie in the range from 2.5 to 2.8 is @samp{utpn(2.5,2,1) - utpn(2.8,2,1)}
19426 (``the probability that it is greater than 2.5, but not greater than 2.8''),
19427 or equivalently @samp{ltpn(2.8,2,1) - ltpn(2.5,2,1)}.
19434 The @kbd{k B} (@code{calc-utpb}) [@code{utpb}] function uses the
19435 binomial distribution. Push the parameters @var{n}, @var{p}, and
19436 then @var{x} onto the stack; the result (@samp{utpb(x,n,p)}) is the
19437 probability that an event will occur @var{x} or more times out
19438 of @var{n} trials, if its probability of occurring in any given
19439 trial is @var{p}. The @kbd{I k B} [@code{ltpb}] function is
19440 the probability that the event will occur fewer than @var{x} times.
19442 The other probability distribution functions similarly take the
19443 form @kbd{k @var{X}} (@code{calc-utp@var{x}}) [@code{utp@var{x}}]
19444 and @kbd{I k @var{X}} [@code{ltp@var{x}}], for various letters
19445 @var{x}. The arguments to the algebraic functions are the value of
19446 the random variable first, then whatever other parameters define the
19447 distribution. Note these are among the few Calc functions where the
19448 order of the arguments in algebraic form differs from the order of
19449 arguments as found on the stack. (The random variable comes last on
19450 the stack, so that you can type, e.g., @kbd{2 @key{RET} 1 @key{RET} 2.5
19451 k N M-@key{RET} @key{DEL} 2.8 k N -}, using @kbd{M-@key{RET} @key{DEL}} to
19452 recover the original arguments but substitute a new value for @expr{x}.)
19465 The @samp{utpc(x,v)} function uses the chi-square distribution with
19466 @texline @math{\nu}
19468 degrees of freedom. It is the probability that a model is
19469 correct if its chi-square statistic is @expr{x}.
19482 The @samp{utpf(F,v1,v2)} function uses the F distribution, used in
19483 various statistical tests. The parameters
19484 @texline @math{\nu_1}
19485 @infoline @expr{v1}
19487 @texline @math{\nu_2}
19488 @infoline @expr{v2}
19489 are the degrees of freedom in the numerator and denominator,
19490 respectively, used in computing the statistic @expr{F}.
19503 The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution
19504 with mean @expr{m} and standard deviation
19505 @texline @math{\sigma}.
19506 @infoline @expr{s}.
19507 It is the probability that such a normal-distributed random variable
19508 would exceed @expr{x}.
19521 The @samp{utpp(n,x)} function uses a Poisson distribution with
19522 mean @expr{x}. It is the probability that @expr{n} or more such
19523 Poisson random events will occur.
19536 The @samp{utpt(t,v)} function uses the Student's ``t'' distribution
19538 @texline @math{\nu}
19540 degrees of freedom. It is the probability that a
19541 t-distributed random variable will be greater than @expr{t}.
19542 (Note: This computes the distribution function
19543 @texline @math{A(t|\nu)}
19544 @infoline @expr{A(t|v)}
19546 @texline @math{A(0|\nu) = 1}
19547 @infoline @expr{A(0|v) = 1}
19549 @texline @math{A(\infty|\nu) \to 0}.
19550 @infoline @expr{A(inf|v) -> 0}.
19551 The @code{UTPT} operation on the HP-48 uses a different definition which
19552 returns half of Calc's value: @samp{UTPT(t,v) = .5*utpt(t,v)}.)
19554 While Calc does not provide inverses of the probability distribution
19555 functions, the @kbd{a R} command can be used to solve for the inverse.
19556 Since the distribution functions are monotonic, @kbd{a R} is guaranteed
19557 to be able to find a solution given any initial guess.
19558 @xref{Numerical Solutions}.
19560 @node Matrix Functions, Algebra, Scientific Functions, Top
19561 @chapter Vector/Matrix Functions
19564 Many of the commands described here begin with the @kbd{v} prefix.
19565 (For convenience, the shift-@kbd{V} prefix is equivalent to @kbd{v}.)
19566 The commands usually apply to both plain vectors and matrices; some
19567 apply only to matrices or only to square matrices. If the argument
19568 has the wrong dimensions the operation is left in symbolic form.
19570 Vectors are entered and displayed using @samp{[a,b,c]} notation.
19571 Matrices are vectors of which all elements are vectors of equal length.
19572 (Though none of the standard Calc commands use this concept, a
19573 three-dimensional matrix or rank-3 tensor could be defined as a
19574 vector of matrices, and so on.)
19577 * Packing and Unpacking::
19578 * Building Vectors::
19579 * Extracting Elements::
19580 * Manipulating Vectors::
19581 * Vector and Matrix Arithmetic::
19583 * Statistical Operations::
19584 * Reducing and Mapping::
19585 * Vector and Matrix Formats::
19588 @node Packing and Unpacking, Building Vectors, Matrix Functions, Matrix Functions
19589 @section Packing and Unpacking
19592 Calc's ``pack'' and ``unpack'' commands collect stack entries to build
19593 composite objects such as vectors and complex numbers. They are
19594 described in this chapter because they are most often used to build
19599 The @kbd{v p} (@code{calc-pack}) [@code{pack}] command collects several
19600 elements from the stack into a matrix, complex number, HMS form, error
19601 form, etc. It uses a numeric prefix argument to specify the kind of
19602 object to be built; this argument is referred to as the ``packing mode.''
19603 If the packing mode is a nonnegative integer, a vector of that
19604 length is created. For example, @kbd{C-u 5 v p} will pop the top
19605 five stack elements and push back a single vector of those five
19606 elements. (@kbd{C-u 0 v p} simply creates an empty vector.)
19608 The same effect can be had by pressing @kbd{[} to push an incomplete
19609 vector on the stack, using @key{TAB} (@code{calc-roll-down}) to sneak
19610 the incomplete object up past a certain number of elements, and
19611 then pressing @kbd{]} to complete the vector.
19613 Negative packing modes create other kinds of composite objects:
19617 Two values are collected to build a complex number. For example,
19618 @kbd{5 @key{RET} 7 C-u -1 v p} creates the complex number
19619 @expr{(5, 7)}. The result is always a rectangular complex
19620 number. The two input values must both be real numbers,
19621 i.e., integers, fractions, or floats. If they are not, Calc
19622 will instead build a formula like @samp{a + (0, 1) b}. (The
19623 other packing modes also create a symbolic answer if the
19624 components are not suitable.)
19627 Two values are collected to build a polar complex number.
19628 The first is the magnitude; the second is the phase expressed
19629 in either degrees or radians according to the current angular
19633 Three values are collected into an HMS form. The first
19634 two values (hours and minutes) must be integers or
19635 integer-valued floats. The third value may be any real
19639 Two values are collected into an error form. The inputs
19640 may be real numbers or formulas.
19643 Two values are collected into a modulo form. The inputs
19644 must be real numbers.
19647 Two values are collected into the interval @samp{[a .. b]}.
19648 The inputs may be real numbers, HMS or date forms, or formulas.
19651 Two values are collected into the interval @samp{[a .. b)}.
19654 Two values are collected into the interval @samp{(a .. b]}.
19657 Two values are collected into the interval @samp{(a .. b)}.
19660 Two integer values are collected into a fraction.
19663 Two values are collected into a floating-point number.
19664 The first is the mantissa; the second, which must be an
19665 integer, is the exponent. The result is the mantissa
19666 times ten to the power of the exponent.
19669 This is treated the same as @mathit{-11} by the @kbd{v p} command.
19670 When unpacking, @mathit{-12} specifies that a floating-point mantissa
19674 A real number is converted into a date form.
19677 Three numbers (year, month, day) are packed into a pure date form.
19680 Six numbers are packed into a date/time form.
19683 With any of the two-input negative packing modes, either or both
19684 of the inputs may be vectors. If both are vectors of the same
19685 length, the result is another vector made by packing corresponding
19686 elements of the input vectors. If one input is a vector and the
19687 other is a plain number, the number is packed along with each vector
19688 element to produce a new vector. For example, @kbd{C-u -4 v p}
19689 could be used to convert a vector of numbers and a vector of errors
19690 into a single vector of error forms; @kbd{C-u -5 v p} could convert
19691 a vector of numbers and a single number @var{M} into a vector of
19692 numbers modulo @var{M}.
19694 If you don't give a prefix argument to @kbd{v p}, it takes
19695 the packing mode from the top of the stack. The elements to
19696 be packed then begin at stack level 2. Thus
19697 @kbd{1 @key{RET} 2 @key{RET} 4 n v p} is another way to
19698 enter the error form @samp{1 +/- 2}.
19700 If the packing mode taken from the stack is a vector, the result is a
19701 matrix with the dimensions specified by the elements of the vector,
19702 which must each be integers. For example, if the packing mode is
19703 @samp{[2, 3]}, then six numbers will be taken from the stack and
19704 returned in the form @samp{[@w{[a, b, c]}, [d, e, f]]}.
19706 If any elements of the vector are negative, other kinds of
19707 packing are done at that level as described above. For
19708 example, @samp{[2, 3, -4]} takes 12 objects and creates a
19709 @texline @math{2\times3}
19711 matrix of error forms: @samp{[[a +/- b, c +/- d ... ]]}.
19712 Also, @samp{[-4, -10]} will convert four integers into an
19713 error form consisting of two fractions: @samp{a:b +/- c:d}.
19719 There is an equivalent algebraic function,
19720 @samp{pack(@var{mode}, @var{items})} where @var{mode} is a
19721 packing mode (an integer or a vector of integers) and @var{items}
19722 is a vector of objects to be packed (re-packed, really) according
19723 to that mode. For example, @samp{pack([3, -4], [a,b,c,d,e,f])}
19724 yields @samp{[a +/- b, @w{c +/- d}, e +/- f]}. The function is
19725 left in symbolic form if the packing mode is invalid, or if the
19726 number of data items does not match the number of items required
19730 @pindex calc-unpack
19731 The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
19732 number, HMS form, or other composite object on the top of the stack and
19733 ``unpacks'' it, pushing each of its elements onto the stack as separate
19734 objects. Thus, it is the ``inverse'' of @kbd{v p}. If the value
19735 at the top of the stack is a formula, @kbd{v u} unpacks it by pushing
19736 each of the arguments of the top-level operator onto the stack.
19738 You can optionally give a numeric prefix argument to @kbd{v u}
19739 to specify an explicit (un)packing mode. If the packing mode is
19740 negative and the input is actually a vector or matrix, the result
19741 will be two or more similar vectors or matrices of the elements.
19742 For example, given the vector @samp{[@w{a +/- b}, c^2, d +/- 7]},
19743 the result of @kbd{C-u -4 v u} will be the two vectors
19744 @samp{[a, c^2, d]} and @w{@samp{[b, 0, 7]}}.
19746 Note that the prefix argument can have an effect even when the input is
19747 not a vector. For example, if the input is the number @mathit{-5}, then
19748 @kbd{c-u -1 v u} yields @mathit{-5} and 0 (the components of @mathit{-5}
19749 when viewed as a rectangular complex number); @kbd{C-u -2 v u} yields 5
19750 and 180 (assuming Degrees mode); and @kbd{C-u -10 v u} yields @mathit{-5}
19751 and 1 (the numerator and denominator of @mathit{-5}, viewed as a rational
19752 number). Plain @kbd{v u} with this input would complain that the input
19753 is not a composite object.
19755 Unpacking mode @mathit{-11} converts a float into an integer mantissa and
19756 an integer exponent, where the mantissa is not divisible by 10
19757 (except that 0.0 is represented by a mantissa and exponent of 0).
19758 Unpacking mode @mathit{-12} converts a float into a floating-point mantissa
19759 and integer exponent, where the mantissa (for non-zero numbers)
19760 is guaranteed to lie in the range [1 .. 10). In both cases,
19761 the mantissa is shifted left or right (and the exponent adjusted
19762 to compensate) in order to satisfy these constraints.
19764 Positive unpacking modes are treated differently than for @kbd{v p}.
19765 A mode of 1 is much like plain @kbd{v u} with no prefix argument,
19766 except that in addition to the components of the input object,
19767 a suitable packing mode to re-pack the object is also pushed.
19768 Thus, @kbd{C-u 1 v u} followed by @kbd{v p} will re-build the
19771 A mode of 2 unpacks two levels of the object; the resulting
19772 re-packing mode will be a vector of length 2. This might be used
19773 to unpack a matrix, say, or a vector of error forms. Higher
19774 unpacking modes unpack the input even more deeply.
19780 There are two algebraic functions analogous to @kbd{v u}.
19781 The @samp{unpack(@var{mode}, @var{item})} function unpacks the
19782 @var{item} using the given @var{mode}, returning the result as
19783 a vector of components. Here the @var{mode} must be an
19784 integer, not a vector. For example, @samp{unpack(-4, a +/- b)}
19785 returns @samp{[a, b]}, as does @samp{unpack(1, a +/- b)}.
19791 The @code{unpackt} function is like @code{unpack} but instead
19792 of returning a simple vector of items, it returns a vector of
19793 two things: The mode, and the vector of items. For example,
19794 @samp{unpackt(1, 2:3 +/- 1:4)} returns @samp{[-4, [2:3, 1:4]]},
19795 and @samp{unpackt(2, 2:3 +/- 1:4)} returns @samp{[[-4, -10], [2, 3, 1, 4]]}.
19796 The identity for re-building the original object is
19797 @samp{apply(pack, unpackt(@var{n}, @var{x})) = @var{x}}. (The
19798 @code{apply} function builds a function call given the function
19799 name and a vector of arguments.)
19801 @cindex Numerator of a fraction, extracting
19802 Subscript notation is a useful way to extract a particular part
19803 of an object. For example, to get the numerator of a rational
19804 number, you can use @samp{unpack(-10, @var{x})_1}.
19806 @node Building Vectors, Extracting Elements, Packing and Unpacking, Matrix Functions
19807 @section Building Vectors
19810 Vectors and matrices can be added,
19811 subtracted, multiplied, and divided; @pxref{Basic Arithmetic}.
19814 @pindex calc-concat
19819 The @kbd{|} (@code{calc-concat}) [@code{vconcat}] command ``concatenates'' two vectors
19820 into one. For example, after @kbd{@w{[ 1 , 2 ]} [ 3 , 4 ] |}, the stack
19821 will contain the single vector @samp{[1, 2, 3, 4]}. If the arguments
19822 are matrices, the rows of the first matrix are concatenated with the
19823 rows of the second. (In other words, two matrices are just two vectors
19824 of row-vectors as far as @kbd{|} is concerned.)
19826 If either argument to @kbd{|} is a scalar (a non-vector), it is treated
19827 like a one-element vector for purposes of concatenation: @kbd{1 [ 2 , 3 ] |}
19828 produces the vector @samp{[1, 2, 3]}. Likewise, if one argument is a
19829 matrix and the other is a plain vector, the vector is treated as a
19834 The @kbd{H |} (@code{calc-append}) [@code{append}] command concatenates
19835 two vectors without any special cases. Both inputs must be vectors.
19836 Whether or not they are matrices is not taken into account. If either
19837 argument is a scalar, the @code{append} function is left in symbolic form.
19838 See also @code{cons} and @code{rcons} below.
19842 The @kbd{I |} and @kbd{H I |} commands are similar, but they use their
19843 two stack arguments in the opposite order. Thus @kbd{I |} is equivalent
19844 to @kbd{@key{TAB} |}, but possibly more convenient and also a bit faster.
19849 The @kbd{v d} (@code{calc-diag}) [@code{diag}] function builds a diagonal
19850 square matrix. The optional numeric prefix gives the number of rows
19851 and columns in the matrix. If the value at the top of the stack is a
19852 vector, the elements of the vector are used as the diagonal elements; the
19853 prefix, if specified, must match the size of the vector. If the value on
19854 the stack is a scalar, it is used for each element on the diagonal, and
19855 the prefix argument is required.
19857 To build a constant square matrix, e.g., a
19858 @texline @math{3\times3}
19860 matrix filled with ones, use @kbd{0 M-3 v d 1 +}, i.e., build a zero
19861 matrix first and then add a constant value to that matrix. (Another
19862 alternative would be to use @kbd{v b} and @kbd{v a}; see below.)
19867 The @kbd{v i} (@code{calc-ident}) [@code{idn}] function builds an identity
19868 matrix of the specified size. It is a convenient form of @kbd{v d}
19869 where the diagonal element is always one. If no prefix argument is given,
19870 this command prompts for one.
19872 In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)},
19873 except that @expr{a} is required to be a scalar (non-vector) quantity.
19874 If @expr{n} is omitted, @samp{idn(a)} represents @expr{a} times an
19875 identity matrix of unknown size. Calc can operate algebraically on
19876 such generic identity matrices, and if one is combined with a matrix
19877 whose size is known, it is converted automatically to an identity
19878 matrix of a suitable matching size. The @kbd{v i} command with an
19879 argument of zero creates a generic identity matrix, @samp{idn(1)}.
19880 Note that in dimensioned Matrix mode (@pxref{Matrix Mode}), generic
19881 identity matrices are immediately expanded to the current default
19887 The @kbd{v x} (@code{calc-index}) [@code{index}] function builds a vector
19888 of consecutive integers from 1 to @var{n}, where @var{n} is the numeric
19889 prefix argument. If you do not provide a prefix argument, you will be
19890 prompted to enter a suitable number. If @var{n} is negative, the result
19891 is a vector of negative integers from @var{n} to @mathit{-1}.
19893 With a prefix argument of just @kbd{C-u}, the @kbd{v x} command takes
19894 three values from the stack: @var{n}, @var{start}, and @var{incr} (with
19895 @var{incr} at top-of-stack). Counting starts at @var{start} and increases
19896 by @var{incr} for successive vector elements. If @var{start} or @var{n}
19897 is in floating-point format, the resulting vector elements will also be
19898 floats. Note that @var{start} and @var{incr} may in fact be any kind
19899 of numbers or formulas.
19901 When @var{start} and @var{incr} are specified, a negative @var{n} has a
19902 different interpretation: It causes a geometric instead of arithmetic
19903 sequence to be generated. For example, @samp{index(-3, a, b)} produces
19904 @samp{[a, a b, a b^2]}. If you omit @var{incr} in the algebraic form,
19905 @samp{index(@var{n}, @var{start})}, the default value for @var{incr}
19906 is one for positive @var{n} or two for negative @var{n}.
19909 @pindex calc-build-vector
19911 The @kbd{v b} (@code{calc-build-vector}) [@code{cvec}] function builds a
19912 vector of @var{n} copies of the value on the top of the stack, where @var{n}
19913 is the numeric prefix argument. In algebraic formulas, @samp{cvec(x,n,m)}
19914 can also be used to build an @var{n}-by-@var{m} matrix of copies of @var{x}.
19915 (Interactively, just use @kbd{v b} twice: once to build a row, then again
19916 to build a matrix of copies of that row.)
19924 The @kbd{v h} (@code{calc-head}) [@code{head}] function returns the first
19925 element of a vector. The @kbd{I v h} (@code{calc-tail}) [@code{tail}]
19926 function returns the vector with its first element removed. In both
19927 cases, the argument must be a non-empty vector.
19932 The @kbd{v k} (@code{calc-cons}) [@code{cons}] function takes a value @var{h}
19933 and a vector @var{t} from the stack, and produces the vector whose head is
19934 @var{h} and whose tail is @var{t}. This is similar to @kbd{|}, except
19935 if @var{h} is itself a vector, @kbd{|} will concatenate the two vectors
19936 whereas @code{cons} will insert @var{h} at the front of the vector @var{t}.
19956 Each of these three functions also accepts the Hyperbolic flag [@code{rhead},
19957 @code{rtail}, @code{rcons}] in which case @var{t} instead represents
19958 the @emph{last} single element of the vector, with @var{h}
19959 representing the remainder of the vector. Thus the vector
19960 @samp{[a, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)}.
19961 Also, @samp{head([a, b, c, d]) = a}, @samp{tail([a, b, c, d]) = [b, c, d]},
19962 @samp{rhead([a, b, c, d]) = [a, b, c]}, and @samp{rtail([a, b, c, d]) = d}.
19964 @node Extracting Elements, Manipulating Vectors, Building Vectors, Matrix Functions
19965 @section Extracting Vector Elements
19971 The @kbd{v r} (@code{calc-mrow}) [@code{mrow}] command extracts one row of
19972 the matrix on the top of the stack, or one element of the plain vector on
19973 the top of the stack. The row or element is specified by the numeric
19974 prefix argument; the default is to prompt for the row or element number.
19975 The matrix or vector is replaced by the specified row or element in the
19976 form of a vector or scalar, respectively.
19978 @cindex Permutations, applying
19979 With a prefix argument of @kbd{C-u} only, @kbd{v r} takes the index of
19980 the element or row from the top of the stack, and the vector or matrix
19981 from the second-to-top position. If the index is itself a vector of
19982 integers, the result is a vector of the corresponding elements of the
19983 input vector, or a matrix of the corresponding rows of the input matrix.
19984 This command can be used to obtain any permutation of a vector.
19986 With @kbd{C-u}, if the index is an interval form with integer components,
19987 it is interpreted as a range of indices and the corresponding subvector or
19988 submatrix is returned.
19990 @cindex Subscript notation
19992 @pindex calc-subscript
19995 Subscript notation in algebraic formulas (@samp{a_b}) stands for the
19996 Calc function @code{subscr}, which is synonymous with @code{mrow}.
19997 Thus, @samp{[x, y, z]_k} produces @expr{x}, @expr{y}, or @expr{z} if
19998 @expr{k} is one, two, or three, respectively. A double subscript
19999 (@samp{M_i_j}, equivalent to @samp{subscr(subscr(M, i), j)}) will
20000 access the element at row @expr{i}, column @expr{j} of a matrix.
20001 The @kbd{a _} (@code{calc-subscript}) command creates a subscript
20002 formula @samp{a_b} out of two stack entries. (It is on the @kbd{a}
20003 ``algebra'' prefix because subscripted variables are often used
20004 purely as an algebraic notation.)
20007 Given a negative prefix argument, @kbd{v r} instead deletes one row or
20008 element from the matrix or vector on the top of the stack. Thus
20009 @kbd{C-u 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v r}
20010 replaces the matrix with the same matrix with its second row removed.
20011 In algebraic form this function is called @code{mrrow}.
20014 Given a prefix argument of zero, @kbd{v r} extracts the diagonal elements
20015 of a square matrix in the form of a vector. In algebraic form this
20016 function is called @code{getdiag}.
20022 The @kbd{v c} (@code{calc-mcol}) [@code{mcol} or @code{mrcol}] command is
20023 the analogous operation on columns of a matrix. Given a plain vector
20024 it extracts (or removes) one element, just like @kbd{v r}. If the
20025 index in @kbd{C-u v c} is an interval or vector and the argument is a
20026 matrix, the result is a submatrix with only the specified columns
20027 retained (and possibly permuted in the case of a vector index).
20029 To extract a matrix element at a given row and column, use @kbd{v r} to
20030 extract the row as a vector, then @kbd{v c} to extract the column element
20031 from that vector. In algebraic formulas, it is often more convenient to
20032 use subscript notation: @samp{m_i_j} gives row @expr{i}, column @expr{j}
20033 of matrix @expr{m}.
20036 @pindex calc-subvector
20038 The @kbd{v s} (@code{calc-subvector}) [@code{subvec}] command extracts
20039 a subvector of a vector. The arguments are the vector, the starting
20040 index, and the ending index, with the ending index in the top-of-stack
20041 position. The starting index indicates the first element of the vector
20042 to take. The ending index indicates the first element @emph{past} the
20043 range to be taken. Thus, @samp{subvec([a, b, c, d, e], 2, 4)} produces
20044 the subvector @samp{[b, c]}. You could get the same result using
20045 @samp{mrow([a, b, c, d, e], @w{[2 .. 4)})}.
20047 If either the start or the end index is zero or negative, it is
20048 interpreted as relative to the end of the vector. Thus
20049 @samp{subvec([a, b, c, d, e], 2, -2)} also produces @samp{[b, c]}. In
20050 the algebraic form, the end index can be omitted in which case it
20051 is taken as zero, i.e., elements from the starting element to the
20052 end of the vector are used. The infinity symbol, @code{inf}, also
20053 has this effect when used as the ending index.
20057 With the Inverse flag, @kbd{I v s} [@code{rsubvec}] removes a subvector
20058 from a vector. The arguments are interpreted the same as for the
20059 normal @kbd{v s} command. Thus, @samp{rsubvec([a, b, c, d, e], 2, 4)}
20060 produces @samp{[a, d, e]}. It is always true that @code{subvec} and
20061 @code{rsubvec} return complementary parts of the input vector.
20063 @xref{Selecting Subformulas}, for an alternative way to operate on
20064 vectors one element at a time.
20066 @node Manipulating Vectors, Vector and Matrix Arithmetic, Extracting Elements, Matrix Functions
20067 @section Manipulating Vectors
20071 @pindex calc-vlength
20073 The @kbd{v l} (@code{calc-vlength}) [@code{vlen}] command computes the
20074 length of a vector. The length of a non-vector is considered to be zero.
20075 Note that matrices are just vectors of vectors for the purposes of this
20080 With the Hyperbolic flag, @kbd{H v l} [@code{mdims}] computes a vector
20081 of the dimensions of a vector, matrix, or higher-order object. For
20082 example, @samp{mdims([[a,b,c],[d,e,f]])} returns @samp{[2, 3]} since
20084 @texline @math{2\times3}
20089 @pindex calc-vector-find
20091 The @kbd{v f} (@code{calc-vector-find}) [@code{find}] command searches
20092 along a vector for the first element equal to a given target. The target
20093 is on the top of the stack; the vector is in the second-to-top position.
20094 If a match is found, the result is the index of the matching element.
20095 Otherwise, the result is zero. The numeric prefix argument, if given,
20096 allows you to select any starting index for the search.
20099 @pindex calc-arrange-vector
20101 @cindex Arranging a matrix
20102 @cindex Reshaping a matrix
20103 @cindex Flattening a matrix
20104 The @kbd{v a} (@code{calc-arrange-vector}) [@code{arrange}] command
20105 rearranges a vector to have a certain number of columns and rows. The
20106 numeric prefix argument specifies the number of columns; if you do not
20107 provide an argument, you will be prompted for the number of columns.
20108 The vector or matrix on the top of the stack is @dfn{flattened} into a
20109 plain vector. If the number of columns is nonzero, this vector is
20110 then formed into a matrix by taking successive groups of @var{n} elements.
20111 If the number of columns does not evenly divide the number of elements
20112 in the vector, the last row will be short and the result will not be
20113 suitable for use as a matrix. For example, with the matrix
20114 @samp{[[1, 2], @w{[3, 4]}]} on the stack, @kbd{v a 4} produces
20115 @samp{[[1, 2, 3, 4]]} (a
20116 @texline @math{1\times4}
20118 matrix), @kbd{v a 1} produces @samp{[[1], [2], [3], [4]]} (a
20119 @texline @math{4\times1}
20121 matrix), @kbd{v a 2} produces @samp{[[1, 2], [3, 4]]} (the original
20122 @texline @math{2\times2}
20124 matrix), @w{@kbd{v a 3}} produces @samp{[[1, 2, 3], [4]]} (not a
20125 matrix), and @kbd{v a 0} produces the flattened list
20126 @samp{[1, 2, @w{3, 4}]}.
20128 @cindex Sorting data
20134 The @kbd{V S} (@code{calc-sort}) [@code{sort}] command sorts the elements of
20135 a vector into increasing order. Real numbers, real infinities, and
20136 constant interval forms come first in this ordering; next come other
20137 kinds of numbers, then variables (in alphabetical order), then finally
20138 come formulas and other kinds of objects; these are sorted according
20139 to a kind of lexicographic ordering with the useful property that
20140 one vector is less or greater than another if the first corresponding
20141 unequal elements are less or greater, respectively. Since quoted strings
20142 are stored by Calc internally as vectors of ASCII character codes
20143 (@pxref{Strings}), this means vectors of strings are also sorted into
20144 alphabetical order by this command.
20146 The @kbd{I V S} [@code{rsort}] command sorts a vector into decreasing order.
20148 @cindex Permutation, inverse of
20149 @cindex Inverse of permutation
20150 @cindex Index tables
20151 @cindex Rank tables
20157 The @kbd{V G} (@code{calc-grade}) [@code{grade}, @code{rgrade}] command
20158 produces an index table or permutation vector which, if applied to the
20159 input vector (as the index of @kbd{C-u v r}, say), would sort the vector.
20160 A permutation vector is just a vector of integers from 1 to @var{n}, where
20161 each integer occurs exactly once. One application of this is to sort a
20162 matrix of data rows using one column as the sort key; extract that column,
20163 grade it with @kbd{V G}, then use the result to reorder the original matrix
20164 with @kbd{C-u v r}. Another interesting property of the @code{V G} command
20165 is that, if the input is itself a permutation vector, the result will
20166 be the inverse of the permutation. The inverse of an index table is
20167 a rank table, whose @var{k}th element says where the @var{k}th original
20168 vector element will rest when the vector is sorted. To get a rank
20169 table, just use @kbd{V G V G}.
20171 With the Inverse flag, @kbd{I V G} produces an index table that would
20172 sort the input into decreasing order. Note that @kbd{V S} and @kbd{V G}
20173 use a ``stable'' sorting algorithm, i.e., any two elements which are equal
20174 will not be moved out of their original order. Generally there is no way
20175 to tell with @kbd{V S}, since two elements which are equal look the same,
20176 but with @kbd{V G} this can be an important issue. In the matrix-of-rows
20177 example, suppose you have names and telephone numbers as two columns and
20178 you wish to sort by phone number primarily, and by name when the numbers
20179 are equal. You can sort the data matrix by names first, and then again
20180 by phone numbers. Because the sort is stable, any two rows with equal
20181 phone numbers will remain sorted by name even after the second sort.
20185 @pindex calc-histogram
20187 @mindex histo@idots
20190 The @kbd{V H} (@code{calc-histogram}) [@code{histogram}] command builds a
20191 histogram of a vector of numbers. Vector elements are assumed to be
20192 integers or real numbers in the range [0..@var{n}) for some ``number of
20193 bins'' @var{n}, which is the numeric prefix argument given to the
20194 command. The result is a vector of @var{n} counts of how many times
20195 each value appeared in the original vector. Non-integers in the input
20196 are rounded down to integers. Any vector elements outside the specified
20197 range are ignored. (You can tell if elements have been ignored by noting
20198 that the counts in the result vector don't add up to the length of the
20202 With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack.
20203 The second-to-top vector is the list of numbers as before. The top
20204 vector is an equal-sized list of ``weights'' to attach to the elements
20205 of the data vector. For example, if the first data element is 4.2 and
20206 the first weight is 10, then 10 will be added to bin 4 of the result
20207 vector. Without the hyperbolic flag, every element has a weight of one.
20210 @pindex calc-transpose
20212 The @kbd{v t} (@code{calc-transpose}) [@code{trn}] command computes
20213 the transpose of the matrix at the top of the stack. If the argument
20214 is a plain vector, it is treated as a row vector and transposed into
20215 a one-column matrix.
20218 @pindex calc-reverse-vector
20220 The @kbd{v v} (@code{calc-reverse-vector}) [@code{rev}] command reverses
20221 a vector end-for-end. Given a matrix, it reverses the order of the rows.
20222 (To reverse the columns instead, just use @kbd{v t v v v t}. The same
20223 principle can be used to apply other vector commands to the columns of
20227 @pindex calc-mask-vector
20229 The @kbd{v m} (@code{calc-mask-vector}) [@code{vmask}] command uses
20230 one vector as a mask to extract elements of another vector. The mask
20231 is in the second-to-top position; the target vector is on the top of
20232 the stack. These vectors must have the same length. The result is
20233 the same as the target vector, but with all elements which correspond
20234 to zeros in the mask vector deleted. Thus, for example,
20235 @samp{vmask([1, 0, 1, 0, 1], [a, b, c, d, e])} produces @samp{[a, c, e]}.
20236 @xref{Logical Operations}.
20239 @pindex calc-expand-vector
20241 The @kbd{v e} (@code{calc-expand-vector}) [@code{vexp}] command
20242 expands a vector according to another mask vector. The result is a
20243 vector the same length as the mask, but with nonzero elements replaced
20244 by successive elements from the target vector. The length of the target
20245 vector is normally the number of nonzero elements in the mask. If the
20246 target vector is longer, its last few elements are lost. If the target
20247 vector is shorter, the last few nonzero mask elements are left
20248 unreplaced in the result. Thus @samp{vexp([2, 0, 3, 0, 7], [a, b])}
20249 produces @samp{[a, 0, b, 0, 7]}.
20252 With the Hyperbolic flag, @kbd{H v e} takes a filler value from the
20253 top of the stack; the mask and target vectors come from the third and
20254 second elements of the stack. This filler is used where the mask is
20255 zero: @samp{vexp([2, 0, 3, 0, 7], [a, b], z)} produces
20256 @samp{[a, z, c, z, 7]}. If the filler value is itself a vector,
20257 then successive values are taken from it, so that the effect is to
20258 interleave two vectors according to the mask:
20259 @samp{vexp([2, 0, 3, 7, 0, 0], [a, b], [x, y])} produces
20260 @samp{[a, x, b, 7, y, 0]}.
20262 Another variation on the masking idea is to combine @samp{[a, b, c, d, e]}
20263 with the mask @samp{[1, 0, 1, 0, 1]} to produce @samp{[a, 0, c, 0, e]}.
20264 You can accomplish this with @kbd{V M a &}, mapping the logical ``and''
20265 operation across the two vectors. @xref{Logical Operations}. Note that
20266 the @code{? :} operation also discussed there allows other types of
20267 masking using vectors.
20269 @node Vector and Matrix Arithmetic, Set Operations, Manipulating Vectors, Matrix Functions
20270 @section Vector and Matrix Arithmetic
20273 Basic arithmetic operations like addition and multiplication are defined
20274 for vectors and matrices as well as for numbers. Division of matrices, in
20275 the sense of multiplying by the inverse, is supported. (Division by a
20276 matrix actually uses LU-decomposition for greater accuracy and speed.)
20277 @xref{Basic Arithmetic}.
20279 The following functions are applied element-wise if their arguments are
20280 vectors or matrices: @code{change-sign}, @code{conj}, @code{arg},
20281 @code{re}, @code{im}, @code{polar}, @code{rect}, @code{clean},
20282 @code{float}, @code{frac}. @xref{Function Index}.
20285 @pindex calc-conj-transpose
20287 The @kbd{V J} (@code{calc-conj-transpose}) [@code{ctrn}] command computes
20288 the conjugate transpose of its argument, i.e., @samp{conj(trn(x))}.
20293 @kindex A (vectors)
20294 @pindex calc-abs (vectors)
20298 @tindex abs (vectors)
20299 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the
20300 Frobenius norm of a vector or matrix argument. This is the square
20301 root of the sum of the squares of the absolute values of the
20302 elements of the vector or matrix. If the vector is interpreted as
20303 a point in two- or three-dimensional space, this is the distance
20304 from that point to the origin.
20309 The @kbd{v n} (@code{calc-rnorm}) [@code{rnorm}] command computes
20310 the row norm, or infinity-norm, of a vector or matrix. For a plain
20311 vector, this is the maximum of the absolute values of the elements.
20312 For a matrix, this is the maximum of the row-absolute-value-sums,
20313 i.e., of the sums of the absolute values of the elements along the
20319 The @kbd{V N} (@code{calc-cnorm}) [@code{cnorm}] command computes
20320 the column norm, or one-norm, of a vector or matrix. For a plain
20321 vector, this is the sum of the absolute values of the elements.
20322 For a matrix, this is the maximum of the column-absolute-value-sums.
20323 General @expr{k}-norms for @expr{k} other than one or infinity are
20329 The @kbd{V C} (@code{calc-cross}) [@code{cross}] command computes the
20330 right-handed cross product of two vectors, each of which must have
20331 exactly three elements.
20336 @kindex & (matrices)
20337 @pindex calc-inv (matrices)
20341 @tindex inv (matrices)
20342 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
20343 inverse of a square matrix. If the matrix is singular, the inverse
20344 operation is left in symbolic form. Matrix inverses are recorded so
20345 that once an inverse (or determinant) of a particular matrix has been
20346 computed, the inverse and determinant of the matrix can be recomputed
20347 quickly in the future.
20349 If the argument to @kbd{&} is a plain number @expr{x}, this
20350 command simply computes @expr{1/x}. This is okay, because the
20351 @samp{/} operator also does a matrix inversion when dividing one
20357 The @kbd{V D} (@code{calc-mdet}) [@code{det}] command computes the
20358 determinant of a square matrix.
20363 The @kbd{V L} (@code{calc-mlud}) [@code{lud}] command computes the
20364 LU decomposition of a matrix. The result is a list of three matrices
20365 which, when multiplied together left-to-right, form the original matrix.
20366 The first is a permutation matrix that arises from pivoting in the
20367 algorithm, the second is lower-triangular with ones on the diagonal,
20368 and the third is upper-triangular.
20371 @pindex calc-mtrace
20373 The @kbd{V T} (@code{calc-mtrace}) [@code{tr}] command computes the
20374 trace of a square matrix. This is defined as the sum of the diagonal
20375 elements of the matrix.
20377 @node Set Operations, Statistical Operations, Vector and Matrix Arithmetic, Matrix Functions
20378 @section Set Operations using Vectors
20381 @cindex Sets, as vectors
20382 Calc includes several commands which interpret vectors as @dfn{sets} of
20383 objects. A set is a collection of objects; any given object can appear
20384 only once in the set. Calc stores sets as vectors of objects in
20385 sorted order. Objects in a Calc set can be any of the usual things,
20386 such as numbers, variables, or formulas. Two set elements are considered
20387 equal if they are identical, except that numerically equal numbers like
20388 the integer 4 and the float 4.0 are considered equal even though they
20389 are not ``identical.'' Variables are treated like plain symbols without
20390 attached values by the set operations; subtracting the set @samp{[b]}
20391 from @samp{[a, b]} always yields the set @samp{[a]} even though if
20392 the variables @samp{a} and @samp{b} both equaled 17, you might
20393 expect the answer @samp{[]}.
20395 If a set contains interval forms, then it is assumed to be a set of
20396 real numbers. In this case, all set operations require the elements
20397 of the set to be only things that are allowed in intervals: Real
20398 numbers, plus and minus infinity, HMS forms, and date forms. If
20399 there are variables or other non-real objects present in a real set,
20400 all set operations on it will be left in unevaluated form.
20402 If the input to a set operation is a plain number or interval form
20403 @var{a}, it is treated like the one-element vector @samp{[@var{a}]}.
20404 The result is always a vector, except that if the set consists of a
20405 single interval, the interval itself is returned instead.
20407 @xref{Logical Operations}, for the @code{in} function which tests if
20408 a certain value is a member of a given set. To test if the set @expr{A}
20409 is a subset of the set @expr{B}, use @samp{vdiff(A, B) = []}.
20412 @pindex calc-remove-duplicates
20414 The @kbd{V +} (@code{calc-remove-duplicates}) [@code{rdup}] command
20415 converts an arbitrary vector into set notation. It works by sorting
20416 the vector as if by @kbd{V S}, then removing duplicates. (For example,
20417 @kbd{[a, 5, 4, a, 4.0]} is sorted to @samp{[4, 4.0, 5, a, a]} and then
20418 reduced to @samp{[4, 5, a]}). Overlapping intervals are merged as
20419 necessary. You rarely need to use @kbd{V +} explicitly, since all the
20420 other set-based commands apply @kbd{V +} to their inputs before using
20424 @pindex calc-set-union
20426 The @kbd{V V} (@code{calc-set-union}) [@code{vunion}] command computes
20427 the union of two sets. An object is in the union of two sets if and
20428 only if it is in either (or both) of the input sets. (You could
20429 accomplish the same thing by concatenating the sets with @kbd{|},
20430 then using @kbd{V +}.)
20433 @pindex calc-set-intersect
20435 The @kbd{V ^} (@code{calc-set-intersect}) [@code{vint}] command computes
20436 the intersection of two sets. An object is in the intersection if
20437 and only if it is in both of the input sets. Thus if the input
20438 sets are disjoint, i.e., if they share no common elements, the result
20439 will be the empty vector @samp{[]}. Note that the characters @kbd{V}
20440 and @kbd{^} were chosen to be close to the conventional mathematical
20442 @texline union@tie{}(@math{A \cup B})
20445 @texline intersection@tie{}(@math{A \cap B}).
20446 @infoline intersection.
20449 @pindex calc-set-difference
20451 The @kbd{V -} (@code{calc-set-difference}) [@code{vdiff}] command computes
20452 the difference between two sets. An object is in the difference
20453 @expr{A - B} if and only if it is in @expr{A} but not in @expr{B}.
20454 Thus subtracting @samp{[y,z]} from a set will remove the elements
20455 @samp{y} and @samp{z} if they are present. You can also think of this
20456 as a general @dfn{set complement} operator; if @expr{A} is the set of
20457 all possible values, then @expr{A - B} is the ``complement'' of @expr{B}.
20458 Obviously this is only practical if the set of all possible values in
20459 your problem is small enough to list in a Calc vector (or simple
20460 enough to express in a few intervals).
20463 @pindex calc-set-xor
20465 The @kbd{V X} (@code{calc-set-xor}) [@code{vxor}] command computes
20466 the ``exclusive-or,'' or ``symmetric difference'' of two sets.
20467 An object is in the symmetric difference of two sets if and only
20468 if it is in one, but @emph{not} both, of the sets. Objects that
20469 occur in both sets ``cancel out.''
20472 @pindex calc-set-complement
20474 The @kbd{V ~} (@code{calc-set-complement}) [@code{vcompl}] command
20475 computes the complement of a set with respect to the real numbers.
20476 Thus @samp{vcompl(x)} is equivalent to @samp{vdiff([-inf .. inf], x)}.
20477 For example, @samp{vcompl([2, (3 .. 4]])} evaluates to
20478 @samp{[[-inf .. 2), (2 .. 3], (4 .. inf]]}.
20481 @pindex calc-set-floor
20483 The @kbd{V F} (@code{calc-set-floor}) [@code{vfloor}] command
20484 reinterprets a set as a set of integers. Any non-integer values,
20485 and intervals that do not enclose any integers, are removed. Open
20486 intervals are converted to equivalent closed intervals. Successive
20487 integers are converted into intervals of integers. For example, the
20488 complement of the set @samp{[2, 6, 7, 8]} is messy, but if you wanted
20489 the complement with respect to the set of integers you could type
20490 @kbd{V ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}.
20493 @pindex calc-set-enumerate
20495 The @kbd{V E} (@code{calc-set-enumerate}) [@code{venum}] command
20496 converts a set of integers into an explicit vector. Intervals in
20497 the set are expanded out to lists of all integers encompassed by
20498 the intervals. This only works for finite sets (i.e., sets which
20499 do not involve @samp{-inf} or @samp{inf}).
20502 @pindex calc-set-span
20504 The @kbd{V :} (@code{calc-set-span}) [@code{vspan}] command converts any
20505 set of reals into an interval form that encompasses all its elements.
20506 The lower limit will be the smallest element in the set; the upper
20507 limit will be the largest element. For an empty set, @samp{vspan([])}
20508 returns the empty interval @w{@samp{[0 .. 0)}}.
20511 @pindex calc-set-cardinality
20513 The @kbd{V #} (@code{calc-set-cardinality}) [@code{vcard}] command counts
20514 the number of integers in a set. The result is the length of the vector
20515 that would be produced by @kbd{V E}, although the computation is much
20516 more efficient than actually producing that vector.
20518 @cindex Sets, as binary numbers
20519 Another representation for sets that may be more appropriate in some
20520 cases is binary numbers. If you are dealing with sets of integers
20521 in the range 0 to 49, you can use a 50-bit binary number where a
20522 particular bit is 1 if the corresponding element is in the set.
20523 @xref{Binary Functions}, for a list of commands that operate on
20524 binary numbers. Note that many of the above set operations have
20525 direct equivalents in binary arithmetic: @kbd{b o} (@code{calc-or}),
20526 @kbd{b a} (@code{calc-and}), @kbd{b d} (@code{calc-diff}),
20527 @kbd{b x} (@code{calc-xor}), and @kbd{b n} (@code{calc-not}),
20528 respectively. You can use whatever representation for sets is most
20533 @pindex calc-pack-bits
20534 @pindex calc-unpack-bits
20537 The @kbd{b u} (@code{calc-unpack-bits}) [@code{vunpack}] command
20538 converts an integer that represents a set in binary into a set
20539 in vector/interval notation. For example, @samp{vunpack(67)}
20540 returns @samp{[[0 .. 1], 6]}. If the input is negative, the set
20541 it represents is semi-infinite: @samp{vunpack(-4) = [2 .. inf)}.
20542 Use @kbd{V E} afterwards to expand intervals to individual
20543 values if you wish. Note that this command uses the @kbd{b}
20544 (binary) prefix key.
20546 The @kbd{b p} (@code{calc-pack-bits}) [@code{vpack}] command
20547 converts the other way, from a vector or interval representing
20548 a set of nonnegative integers into a binary integer describing
20549 the same set. The set may include positive infinity, but must
20550 not include any negative numbers. The input is interpreted as a
20551 set of integers in the sense of @kbd{V F} (@code{vfloor}). Beware
20552 that a simple input like @samp{[100]} can result in a huge integer
20554 @texline (@math{2^{100}}, a 31-digit integer, in this case).
20555 @infoline (@expr{2^100}, a 31-digit integer, in this case).
20557 @node Statistical Operations, Reducing and Mapping, Set Operations, Matrix Functions
20558 @section Statistical Operations on Vectors
20561 @cindex Statistical functions
20562 The commands in this section take vectors as arguments and compute
20563 various statistical measures on the data stored in the vectors. The
20564 references used in the definitions of these functions are Bevington's
20565 @emph{Data Reduction and Error Analysis for the Physical Sciences},
20566 and @emph{Numerical Recipes} by Press, Flannery, Teukolsky and
20569 The statistical commands use the @kbd{u} prefix key followed by
20570 a shifted letter or other character.
20572 @xref{Manipulating Vectors}, for a description of @kbd{V H}
20573 (@code{calc-histogram}).
20575 @xref{Curve Fitting}, for the @kbd{a F} command for doing
20576 least-squares fits to statistical data.
20578 @xref{Probability Distribution Functions}, for several common
20579 probability distribution functions.
20582 * Single-Variable Statistics::
20583 * Paired-Sample Statistics::
20586 @node Single-Variable Statistics, Paired-Sample Statistics, Statistical Operations, Statistical Operations
20587 @subsection Single-Variable Statistics
20590 These functions do various statistical computations on single
20591 vectors. Given a numeric prefix argument, they actually pop
20592 @var{n} objects from the stack and combine them into a data
20593 vector. Each object may be either a number or a vector; if a
20594 vector, any sub-vectors inside it are ``flattened'' as if by
20595 @kbd{v a 0}; @pxref{Manipulating Vectors}. By default one object
20596 is popped, which (in order to be useful) is usually a vector.
20598 If an argument is a variable name, and the value stored in that
20599 variable is a vector, then the stored vector is used. This method
20600 has the advantage that if your data vector is large, you can avoid
20601 the slow process of manipulating it directly on the stack.
20603 These functions are left in symbolic form if any of their arguments
20604 are not numbers or vectors, e.g., if an argument is a formula, or
20605 a non-vector variable. However, formulas embedded within vector
20606 arguments are accepted; the result is a symbolic representation
20607 of the computation, based on the assumption that the formula does
20608 not itself represent a vector. All varieties of numbers such as
20609 error forms and interval forms are acceptable.
20611 Some of the functions in this section also accept a single error form
20612 or interval as an argument. They then describe a property of the
20613 normal or uniform (respectively) statistical distribution described
20614 by the argument. The arguments are interpreted in the same way as
20615 the @var{M} argument of the random number function @kbd{k r}. In
20616 particular, an interval with integer limits is considered an integer
20617 distribution, so that @samp{[2 .. 6)} is the same as @samp{[2 .. 5]}.
20618 An interval with at least one floating-point limit is a continuous
20619 distribution: @samp{[2.0 .. 6.0)} is @emph{not} the same as
20620 @samp{[2.0 .. 5.0]}!
20623 @pindex calc-vector-count
20625 The @kbd{u #} (@code{calc-vector-count}) [@code{vcount}] command
20626 computes the number of data values represented by the inputs.
20627 For example, @samp{vcount(1, [2, 3], [[4, 5], [], x, y])} returns 7.
20628 If the argument is a single vector with no sub-vectors, this
20629 simply computes the length of the vector.
20633 @pindex calc-vector-sum
20634 @pindex calc-vector-prod
20637 @cindex Summations (statistical)
20638 The @kbd{u +} (@code{calc-vector-sum}) [@code{vsum}] command
20639 computes the sum of the data values. The @kbd{u *}
20640 (@code{calc-vector-prod}) [@code{vprod}] command computes the
20641 product of the data values. If the input is a single flat vector,
20642 these are the same as @kbd{V R +} and @kbd{V R *}
20643 (@pxref{Reducing and Mapping}).
20647 @pindex calc-vector-max
20648 @pindex calc-vector-min
20651 The @kbd{u X} (@code{calc-vector-max}) [@code{vmax}] command
20652 computes the maximum of the data values, and the @kbd{u N}
20653 (@code{calc-vector-min}) [@code{vmin}] command computes the minimum.
20654 If the argument is an interval, this finds the minimum or maximum
20655 value in the interval. (Note that @samp{vmax([2..6)) = 5} as
20656 described above.) If the argument is an error form, this returns
20657 plus or minus infinity.
20660 @pindex calc-vector-mean
20662 @cindex Mean of data values
20663 The @kbd{u M} (@code{calc-vector-mean}) [@code{vmean}] command
20664 computes the average (arithmetic mean) of the data values.
20665 If the inputs are error forms
20666 @texline @math{x \pm \sigma},
20667 @infoline @samp{x +/- s},
20668 this is the weighted mean of the @expr{x} values with weights
20669 @texline @math{1 /\sigma^2}.
20670 @infoline @expr{1 / s^2}.
20673 $$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
20674 \displaystyle \sum { 1 \over \sigma_i^2 } } $$
20676 If the inputs are not error forms, this is simply the sum of the
20677 values divided by the count of the values.
20679 Note that a plain number can be considered an error form with
20681 @texline @math{\sigma = 0}.
20682 @infoline @expr{s = 0}.
20683 If the input to @kbd{u M} is a mixture of
20684 plain numbers and error forms, the result is the mean of the
20685 plain numbers, ignoring all values with non-zero errors. (By the
20686 above definitions it's clear that a plain number effectively
20687 has an infinite weight, next to which an error form with a finite
20688 weight is completely negligible.)
20690 This function also works for distributions (error forms or
20691 intervals). The mean of an error form `@var{a} @tfn{+/-} @var{b}' is simply
20692 @expr{a}. The mean of an interval is the mean of the minimum
20693 and maximum values of the interval.
20696 @pindex calc-vector-mean-error
20698 The @kbd{I u M} (@code{calc-vector-mean-error}) [@code{vmeane}]
20699 command computes the mean of the data points expressed as an
20700 error form. This includes the estimated error associated with
20701 the mean. If the inputs are error forms, the error is the square
20702 root of the reciprocal of the sum of the reciprocals of the squares
20703 of the input errors. (I.e., the variance is the reciprocal of the
20704 sum of the reciprocals of the variances.)
20707 $$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
20709 If the inputs are plain
20710 numbers, the error is equal to the standard deviation of the values
20711 divided by the square root of the number of values. (This works
20712 out to be equivalent to calculating the standard deviation and
20713 then assuming each value's error is equal to this standard
20717 $$ \sigma_\mu^2 = {\sigma^2 \over N} $$
20721 @pindex calc-vector-median
20723 @cindex Median of data values
20724 The @kbd{H u M} (@code{calc-vector-median}) [@code{vmedian}]
20725 command computes the median of the data values. The values are
20726 first sorted into numerical order; the median is the middle
20727 value after sorting. (If the number of data values is even,
20728 the median is taken to be the average of the two middle values.)
20729 The median function is different from the other functions in
20730 this section in that the arguments must all be real numbers;
20731 variables are not accepted even when nested inside vectors.
20732 (Otherwise it is not possible to sort the data values.) If
20733 any of the input values are error forms, their error parts are
20736 The median function also accepts distributions. For both normal
20737 (error form) and uniform (interval) distributions, the median is
20738 the same as the mean.
20741 @pindex calc-vector-harmonic-mean
20743 @cindex Harmonic mean
20744 The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) [@code{vhmean}]
20745 command computes the harmonic mean of the data values. This is
20746 defined as the reciprocal of the arithmetic mean of the reciprocals
20750 $$ { N \over \displaystyle \sum {1 \over x_i} } $$
20754 @pindex calc-vector-geometric-mean
20756 @cindex Geometric mean
20757 The @kbd{u G} (@code{calc-vector-geometric-mean}) [@code{vgmean}]
20758 command computes the geometric mean of the data values. This
20759 is the @var{n}th root of the product of the values. This is also
20760 equal to the @code{exp} of the arithmetic mean of the logarithms
20761 of the data values.
20764 $$ \exp \left ( \sum { \ln x_i } \right ) =
20765 \left ( \prod { x_i } \right)^{1 / N} $$
20770 The @kbd{H u G} [@code{agmean}] command computes the ``arithmetic-geometric
20771 mean'' of two numbers taken from the stack. This is computed by
20772 replacing the two numbers with their arithmetic mean and geometric
20773 mean, then repeating until the two values converge.
20776 $$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
20779 @cindex Root-mean-square
20780 Another commonly used mean, the RMS (root-mean-square), can be computed
20781 for a vector of numbers simply by using the @kbd{A} command.
20784 @pindex calc-vector-sdev
20786 @cindex Standard deviation
20787 @cindex Sample statistics
20788 The @kbd{u S} (@code{calc-vector-sdev}) [@code{vsdev}] command
20789 computes the standard
20790 @texline deviation@tie{}@math{\sigma}
20791 @infoline deviation
20792 of the data values. If the values are error forms, the errors are used
20793 as weights just as for @kbd{u M}. This is the @emph{sample} standard
20794 deviation, whose value is the square root of the sum of the squares of
20795 the differences between the values and the mean of the @expr{N} values,
20796 divided by @expr{N-1}.
20799 $$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
20802 This function also applies to distributions. The standard deviation
20803 of a single error form is simply the error part. The standard deviation
20804 of a continuous interval happens to equal the difference between the
20806 @texline @math{\sqrt{12}}.
20807 @infoline @expr{sqrt(12)}.
20808 The standard deviation of an integer interval is the same as the
20809 standard deviation of a vector of those integers.
20812 @pindex calc-vector-pop-sdev
20814 @cindex Population statistics
20815 The @kbd{I u S} (@code{calc-vector-pop-sdev}) [@code{vpsdev}]
20816 command computes the @emph{population} standard deviation.
20817 It is defined by the same formula as above but dividing
20818 by @expr{N} instead of by @expr{N-1}. The population standard
20819 deviation is used when the input represents the entire set of
20820 data values in the distribution; the sample standard deviation
20821 is used when the input represents a sample of the set of all
20822 data values, so that the mean computed from the input is itself
20823 only an estimate of the true mean.
20826 $$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
20829 For error forms and continuous intervals, @code{vpsdev} works
20830 exactly like @code{vsdev}. For integer intervals, it computes the
20831 population standard deviation of the equivalent vector of integers.
20835 @pindex calc-vector-variance
20836 @pindex calc-vector-pop-variance
20839 @cindex Variance of data values
20840 The @kbd{H u S} (@code{calc-vector-variance}) [@code{vvar}] and
20841 @kbd{H I u S} (@code{calc-vector-pop-variance}) [@code{vpvar}]
20842 commands compute the variance of the data values. The variance
20844 @texline square@tie{}@math{\sigma^2}
20846 of the standard deviation, i.e., the sum of the
20847 squares of the deviations of the data values from the mean.
20848 (This definition also applies when the argument is a distribution.)
20854 The @code{vflat} algebraic function returns a vector of its
20855 arguments, interpreted in the same way as the other functions
20856 in this section. For example, @samp{vflat(1, [2, [3, 4]], 5)}
20857 returns @samp{[1, 2, 3, 4, 5]}.
20859 @node Paired-Sample Statistics, , Single-Variable Statistics, Statistical Operations
20860 @subsection Paired-Sample Statistics
20863 The functions in this section take two arguments, which must be
20864 vectors of equal size. The vectors are each flattened in the same
20865 way as by the single-variable statistical functions. Given a numeric
20866 prefix argument of 1, these functions instead take one object from
20867 the stack, which must be an
20868 @texline @math{N\times2}
20870 matrix of data values. Once again, variable names can be used in place
20871 of actual vectors and matrices.
20874 @pindex calc-vector-covariance
20877 The @kbd{u C} (@code{calc-vector-covariance}) [@code{vcov}] command
20878 computes the sample covariance of two vectors. The covariance
20879 of vectors @var{x} and @var{y} is the sum of the products of the
20880 differences between the elements of @var{x} and the mean of @var{x}
20881 times the differences between the corresponding elements of @var{y}
20882 and the mean of @var{y}, all divided by @expr{N-1}. Note that
20883 the variance of a vector is just the covariance of the vector
20884 with itself. Once again, if the inputs are error forms the
20885 errors are used as weight factors. If both @var{x} and @var{y}
20886 are composed of error forms, the error for a given data point
20887 is taken as the square root of the sum of the squares of the two
20891 $$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
20892 $$ \sigma_{x\!y}^2 =
20893 {\displaystyle {1 \over N-1}
20894 \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2}
20895 \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}}
20900 @pindex calc-vector-pop-covariance
20902 The @kbd{I u C} (@code{calc-vector-pop-covariance}) [@code{vpcov}]
20903 command computes the population covariance, which is the same as the
20904 sample covariance computed by @kbd{u C} except dividing by @expr{N}
20905 instead of @expr{N-1}.
20908 @pindex calc-vector-correlation
20910 @cindex Correlation coefficient
20911 @cindex Linear correlation
20912 The @kbd{H u C} (@code{calc-vector-correlation}) [@code{vcorr}]
20913 command computes the linear correlation coefficient of two vectors.
20914 This is defined by the covariance of the vectors divided by the
20915 product of their standard deviations. (There is no difference
20916 between sample or population statistics here.)
20919 $$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
20922 @node Reducing and Mapping, Vector and Matrix Formats, Statistical Operations, Matrix Functions
20923 @section Reducing and Mapping Vectors
20926 The commands in this section allow for more general operations on the
20927 elements of vectors.
20932 The simplest of these operations is @kbd{V A} (@code{calc-apply})
20933 [@code{apply}], which applies a given operator to the elements of a vector.
20934 For example, applying the hypothetical function @code{f} to the vector
20935 @w{@samp{[1, 2, 3]}} would produce the function call @samp{f(1, 2, 3)}.
20936 Applying the @code{+} function to the vector @samp{[a, b]} gives
20937 @samp{a + b}. Applying @code{+} to the vector @samp{[a, b, c]} is an
20938 error, since the @code{+} function expects exactly two arguments.
20940 While @kbd{V A} is useful in some cases, you will usually find that either
20941 @kbd{V R} or @kbd{V M}, described below, is closer to what you want.
20944 * Specifying Operators::
20947 * Nesting and Fixed Points::
20948 * Generalized Products::
20951 @node Specifying Operators, Mapping, Reducing and Mapping, Reducing and Mapping
20952 @subsection Specifying Operators
20955 Commands in this section (like @kbd{V A}) prompt you to press the key
20956 corresponding to the desired operator. Press @kbd{?} for a partial
20957 list of the available operators. Generally, an operator is any key or
20958 sequence of keys that would normally take one or more arguments from
20959 the stack and replace them with a result. For example, @kbd{V A H C}
20960 uses the hyperbolic cosine operator, @code{cosh}. (Since @code{cosh}
20961 expects one argument, @kbd{V A H C} requires a vector with a single
20962 element as its argument.)
20964 You can press @kbd{x} at the operator prompt to select any algebraic
20965 function by name to use as the operator. This includes functions you
20966 have defined yourself using the @kbd{Z F} command. (@xref{Algebraic
20967 Definitions}.) If you give a name for which no function has been
20968 defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}.
20969 Calc will prompt for the number of arguments the function takes if it
20970 can't figure it out on its own (say, because you named a function that
20971 is currently undefined). It is also possible to type a digit key before
20972 the function name to specify the number of arguments, e.g.,
20973 @kbd{V M 3 x f @key{RET}} calls @code{f} with three arguments even if it
20974 looks like it ought to have only two. This technique may be necessary
20975 if the function allows a variable number of arguments. For example,
20976 the @kbd{v e} [@code{vexp}] function accepts two or three arguments;
20977 if you want to map with the three-argument version, you will have to
20978 type @kbd{V M 3 v e}.
20980 It is also possible to apply any formula to a vector by treating that
20981 formula as a function. When prompted for the operator to use, press
20982 @kbd{'} (the apostrophe) and type your formula as an algebraic entry.
20983 You will then be prompted for the argument list, which defaults to a
20984 list of all variables that appear in the formula, sorted into alphabetic
20985 order. For example, suppose you enter the formula @w{@samp{x + 2y^x}}.
20986 The default argument list would be @samp{(x y)}, which means that if
20987 this function is applied to the arguments @samp{[3, 10]} the result will
20988 be @samp{3 + 2*10^3}. (If you plan to use a certain formula in this
20989 way often, you might consider defining it as a function with @kbd{Z F}.)
20991 Another way to specify the arguments to the formula you enter is with
20992 @kbd{$}, @kbd{$$}, and so on. For example, @kbd{V A ' $$ + 2$^$$}
20993 has the same effect as the previous example. The argument list is
20994 automatically taken to be @samp{($$ $)}. (The order of the arguments
20995 may seem backwards, but it is analogous to the way normal algebraic
20996 entry interacts with the stack.)
20998 If you press @kbd{$} at the operator prompt, the effect is similar to
20999 the apostrophe except that the relevant formula is taken from top-of-stack
21000 instead. The actual vector arguments of the @kbd{V A $} or related command
21001 then start at the second-to-top stack position. You will still be
21002 prompted for an argument list.
21004 @cindex Nameless functions
21005 @cindex Generic functions
21006 A function can be written without a name using the notation @samp{<#1 - #2>},
21007 which means ``a function of two arguments that computes the first
21008 argument minus the second argument.'' The symbols @samp{#1} and @samp{#2}
21009 are placeholders for the arguments. You can use any names for these
21010 placeholders if you wish, by including an argument list followed by a
21011 colon: @samp{<x, y : x - y>}. When you type @kbd{V A ' $$ + 2$^$$ @key{RET}},
21012 Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function
21013 to map across the vectors. When you type @kbd{V A ' x + 2y^x @key{RET} @key{RET}},
21014 Calc builds the nameless function @w{@samp{<x, y : x + 2 y^x>}}. In both
21015 cases, Calc also writes the nameless function to the Trail so that you
21016 can get it back later if you wish.
21018 If there is only one argument, you can write @samp{#} in place of @samp{#1}.
21019 (Note that @samp{< >} notation is also used for date forms. Calc tells
21020 that @samp{<@var{stuff}>} is a nameless function by the presence of
21021 @samp{#} signs inside @var{stuff}, or by the fact that @var{stuff}
21022 begins with a list of variables followed by a colon.)
21024 You can type a nameless function directly to @kbd{V A '}, or put one on
21025 the stack and use it with @w{@kbd{V A $}}. Calc will not prompt for an
21026 argument list in this case, since the nameless function specifies the
21027 argument list as well as the function itself. In @kbd{V A '}, you can
21028 omit the @samp{< >} marks if you use @samp{#} notation for the arguments,
21029 so that @kbd{V A ' #1+#2 @key{RET}} is the same as @kbd{V A ' <#1+#2> @key{RET}},
21030 which in turn is the same as @kbd{V A ' $$+$ @key{RET}}.
21032 @cindex Lambda expressions
21037 The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}.
21038 (The word @code{lambda} derives from Lisp notation and the theory of
21039 functions.) The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA,
21040 ArgB, ArgA + ArgB)}. Note that there is no actual Calc function called
21041 @code{lambda}; the whole point is that the @code{lambda} expression is
21042 used in its symbolic form, not evaluated for an answer until it is applied
21043 to specific arguments by a command like @kbd{V A} or @kbd{V M}.
21045 (Actually, @code{lambda} does have one special property: Its arguments
21046 are never evaluated; for example, putting @samp{<(2/3) #>} on the stack
21047 will not simplify the @samp{2/3} until the nameless function is actually
21076 As usual, commands like @kbd{V A} have algebraic function name equivalents.
21077 For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to
21078 @samp{apply(gcd, v)}. The first argument specifies the operator name,
21079 and is either a variable whose name is the same as the function name,
21080 or a nameless function like @samp{<#^3+1>}. Operators that are normally
21081 written as algebraic symbols have the names @code{add}, @code{sub},
21082 @code{mul}, @code{div}, @code{pow}, @code{neg}, @code{mod}, and
21089 The @code{call} function builds a function call out of several arguments:
21090 @samp{call(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which
21091 in turn is the same as @samp{gcd(x, y)}. The first argument of @code{call},
21092 like the other functions described here, may be either a variable naming a
21093 function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same
21096 (Experts will notice that it's not quite proper to use a variable to name
21097 a function, since the name @code{gcd} corresponds to the Lisp variable
21098 @code{var-gcd} but to the Lisp function @code{calcFunc-gcd}. Calc
21099 automatically makes this translation, so you don't have to worry
21102 @node Mapping, Reducing, Specifying Operators, Reducing and Mapping
21103 @subsection Mapping
21109 The @kbd{V M} (@code{calc-map}) [@code{map}] command applies a given
21110 operator elementwise to one or more vectors. For example, mapping
21111 @code{A} [@code{abs}] produces a vector of the absolute values of the
21112 elements in the input vector. Mapping @code{+} pops two vectors from
21113 the stack, which must be of equal length, and produces a vector of the
21114 pairwise sums of the elements. If either argument is a non-vector, it
21115 is duplicated for each element of the other vector. For example,
21116 @kbd{[1,2,3] 2 V M ^} squares the elements of the specified vector.
21117 With the 2 listed first, it would have computed a vector of powers of
21118 two. Mapping a user-defined function pops as many arguments from the
21119 stack as the function requires. If you give an undefined name, you will
21120 be prompted for the number of arguments to use.
21122 If any argument to @kbd{V M} is a matrix, the operator is normally mapped
21123 across all elements of the matrix. For example, given the matrix
21124 @expr{[[1, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to
21126 @texline @math{3\times2}
21128 matrix, @expr{[[1, 2, 3], [4, 5, 6]]}.
21131 The command @kbd{V M _} [@code{mapr}] (i.e., type an underscore at the
21132 operator prompt) maps by rows instead. For example, @kbd{V M _ A} views
21133 the above matrix as a vector of two 3-element row vectors. It produces
21134 a new vector which contains the absolute values of those row vectors,
21135 namely @expr{[3.74, 8.77]}. (Recall, the absolute value of a vector is
21136 defined as the square root of the sum of the squares of the elements.)
21137 Some operators accept vectors and return new vectors; for example,
21138 @kbd{v v} reverses a vector, so @kbd{V M _ v v} would reverse each row
21139 of the matrix to get a new matrix, @expr{[[3, -2, 1], [-6, 5, -4]]}.
21141 Sometimes a vector of vectors (representing, say, strings, sets, or lists)
21142 happens to look like a matrix. If so, remember to use @kbd{V M _} if you
21143 want to map a function across the whole strings or sets rather than across
21144 their individual elements.
21147 The command @kbd{V M :} [@code{mapc}] maps by columns. Basically, it
21148 transposes the input matrix, maps by rows, and then, if the result is a
21149 matrix, transposes again. For example, @kbd{V M : A} takes the absolute
21150 values of the three columns of the matrix, treating each as a 2-vector,
21151 and @kbd{V M : v v} reverses the columns to get the matrix
21152 @expr{[[-4, 5, -6], [1, -2, 3]]}.
21154 (The symbols @kbd{_} and @kbd{:} were chosen because they had row-like
21155 and column-like appearances, and were not already taken by useful
21156 operators. Also, they appear shifted on most keyboards so they are easy
21157 to type after @kbd{V M}.)
21159 The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are
21160 not matrices (so if none of the arguments are matrices, they have no
21161 effect at all). If some of the arguments are matrices and others are
21162 plain numbers, the plain numbers are held constant for all rows of the
21163 matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring
21164 a vector takes a dot product of the vector with itself).
21166 If some of the arguments are vectors with the same lengths as the
21167 rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix
21168 arguments, those vectors are also held constant for every row or
21171 Sometimes it is useful to specify another mapping command as the operator
21172 to use with @kbd{V M}. For example, @kbd{V M _ V A +} applies @kbd{V A +}
21173 to each row of the input matrix, which in turn adds the two values on that
21174 row. If you give another vector-operator command as the operator for
21175 @kbd{V M}, it automatically uses map-by-rows mode if you don't specify
21176 otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}. (If
21177 you really want to map-by-elements another mapping command, you can use
21178 a triple-nested mapping command: @kbd{V M V M V A +} means to map
21179 @kbd{V M V A +} over the rows of the matrix; in turn, @kbd{V A +} is
21180 mapped over the elements of each row.)
21184 Previous versions of Calc had ``map across'' and ``map down'' modes
21185 that are now considered obsolete; the old ``map across'' is now simply
21186 @kbd{V M V A}, and ``map down'' is now @kbd{V M : V A}. The algebraic
21187 functions @code{mapa} and @code{mapd} are still supported, though.
21188 Note also that, while the old mapping modes were persistent (once you
21189 set the mode, it would apply to later mapping commands until you reset
21190 it), the new @kbd{:} and @kbd{_} modifiers apply only to the current
21191 mapping command. The default @kbd{V M} always means map-by-elements.
21193 @xref{Algebraic Manipulation}, for the @kbd{a M} command, which is like
21194 @kbd{V M} but for equations and inequalities instead of vectors.
21195 @xref{Storing Variables}, for the @kbd{s m} command which modifies a
21196 variable's stored value using a @kbd{V M}-like operator.
21198 @node Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping
21199 @subsection Reducing
21203 @pindex calc-reduce
21205 The @kbd{V R} (@code{calc-reduce}) [@code{reduce}] command applies a given
21206 binary operator across all the elements of a vector. A binary operator is
21207 a function such as @code{+} or @code{max} which takes two arguments. For
21208 example, reducing @code{+} over a vector computes the sum of the elements
21209 of the vector. Reducing @code{-} computes the first element minus each of
21210 the remaining elements. Reducing @code{max} computes the maximum element
21211 and so on. In general, reducing @code{f} over the vector @samp{[a, b, c, d]}
21212 produces @samp{f(f(f(a, b), c), d)}.
21216 The @kbd{I V R} [@code{rreduce}] command is similar to @kbd{V R} except
21217 that works from right to left through the vector. For example, plain
21218 @kbd{V R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c - d}
21219 but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))},
21220 or @samp{a - b + c - d}. This ``alternating sum'' occurs frequently
21221 in power series expansions.
21225 The @kbd{V U} (@code{calc-accumulate}) [@code{accum}] command does an
21226 accumulation operation. Here Calc does the corresponding reduction
21227 operation, but instead of producing only the final result, it produces
21228 a vector of all the intermediate results. Accumulating @code{+} over
21229 the vector @samp{[a, b, c, d]} produces the vector
21230 @samp{[a, a + b, a + b + c, a + b + c + d]}.
21234 The @kbd{I V U} [@code{raccum}] command does a right-to-left accumulation.
21235 For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the
21236 vector @samp{[a - b + c - d, b - c + d, c - d, d]}.
21242 As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise. For
21243 example, given the matrix @expr{[[a, b, c], [d, e, f]]}, @kbd{V R +} will
21244 compute @expr{a + b + c + d + e + f}. You can type @kbd{V R _} or
21245 @kbd{V R :} to modify this behavior. The @kbd{V R _} [@code{reducea}]
21246 command reduces ``across'' the matrix; it reduces each row of the matrix
21247 as a vector, then collects the results. Thus @kbd{V R _ +} of this
21248 matrix would produce @expr{[a + b + c, d + e + f]}. Similarly, @kbd{V R :}
21249 [@code{reduced}] reduces down; @kbd{V R : +} would produce @expr{[a + d,
21254 There is a third ``by rows'' mode for reduction that is occasionally
21255 useful; @kbd{V R =} [@code{reducer}] simply reduces the operator over
21256 the rows of the matrix themselves. Thus @kbd{V R = +} on the above
21257 matrix would get the same result as @kbd{V R : +}, since adding two
21258 row vectors is equivalent to adding their elements. But @kbd{V R = *}
21259 would multiply the two rows (to get a single number, their dot product),
21260 while @kbd{V R : *} would produce a vector of the products of the columns.
21262 These three matrix reduction modes work with @kbd{V R} and @kbd{I V R},
21263 but they are not currently supported with @kbd{V U} or @kbd{I V U}.
21267 The obsolete reduce-by-columns function, @code{reducec}, is still
21268 supported but there is no way to get it through the @kbd{V R} command.
21270 The commands @kbd{M-# :} and @kbd{M-# _} are equivalent to typing
21271 @kbd{M-# r} to grab a rectangle of data into Calc, and then typing
21272 @kbd{V R : +} or @kbd{V R _ +}, respectively, to sum the columns or
21273 rows of the matrix. @xref{Grabbing From Buffers}.
21275 @node Nesting and Fixed Points, Generalized Products, Reducing, Reducing and Mapping
21276 @subsection Nesting and Fixed Points
21281 The @kbd{H V R} [@code{nest}] command applies a function to a given
21282 argument repeatedly. It takes two values, @samp{a} and @samp{n}, from
21283 the stack, where @samp{n} must be an integer. It then applies the
21284 function nested @samp{n} times; if the function is @samp{f} and @samp{n}
21285 is 3, the result is @samp{f(f(f(a)))}. The number @samp{n} may be
21286 negative if Calc knows an inverse for the function @samp{f}; for
21287 example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.
21291 The @kbd{H V U} [@code{anest}] command is an accumulating version of
21292 @code{nest}: It returns a vector of @samp{n+1} values, e.g.,
21293 @samp{[a, f(a), f(f(a)), f(f(f(a)))]}. If @samp{n} is negative and
21294 @samp{F} is the inverse of @samp{f}, then the result is of the
21295 form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}.
21299 @cindex Fixed points
21300 The @kbd{H I V R} [@code{fixp}] command is like @kbd{H V R}, except
21301 that it takes only an @samp{a} value from the stack; the function is
21302 applied until it reaches a ``fixed point,'' i.e., until the result
21307 The @kbd{H I V U} [@code{afixp}] command is an accumulating @code{fixp}.
21308 The first element of the return vector will be the initial value @samp{a};
21309 the last element will be the final result that would have been returned
21312 For example, 0.739085 is a fixed point of the cosine function (in radians):
21313 @samp{cos(0.739085) = 0.739085}. You can find this value by putting, say,
21314 1.0 on the stack and typing @kbd{H I V U C}. (We use the accumulating
21315 version so we can see the intermediate results: @samp{[1, 0.540302, 0.857553,
21316 0.65329, ...]}. With a precision of six, this command will take 36 steps
21317 to converge to 0.739085.)
21319 Newton's method for finding roots is a classic example of iteration
21320 to a fixed point. To find the square root of five starting with an
21321 initial guess, Newton's method would look for a fixed point of the
21322 function @samp{(x + 5/x) / 2}. Putting a guess of 1 on the stack
21323 and typing @kbd{H I V R ' ($ + 5/$)/2 @key{RET}} quickly yields the result
21324 2.23607. This is equivalent to using the @kbd{a R} (@code{calc-find-root})
21325 command to find a root of the equation @samp{x^2 = 5}.
21327 These examples used numbers for @samp{a} values. Calc keeps applying
21328 the function until two successive results are equal to within the
21329 current precision. For complex numbers, both the real parts and the
21330 imaginary parts must be equal to within the current precision. If
21331 @samp{a} is a formula (say, a variable name), then the function is
21332 applied until two successive results are exactly the same formula.
21333 It is up to you to ensure that the function will eventually converge;
21334 if it doesn't, you may have to press @kbd{C-g} to stop the Calculator.
21336 The algebraic @code{fixp} function takes two optional arguments, @samp{n}
21337 and @samp{tol}. The first is the maximum number of steps to be allowed,
21338 and must be either an integer or the symbol @samp{inf} (infinity, the
21339 default). The second is a convergence tolerance. If a tolerance is
21340 specified, all results during the calculation must be numbers, not
21341 formulas, and the iteration stops when the magnitude of the difference
21342 between two successive results is less than or equal to the tolerance.
21343 (This implies that a tolerance of zero iterates until the results are
21346 Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)}
21347 computes the square root of @samp{A} given the initial guess @samp{B},
21348 stopping when the result is correct within the specified tolerance, or
21349 when 20 steps have been taken, whichever is sooner.
21351 @node Generalized Products, , Nesting and Fixed Points, Reducing and Mapping
21352 @subsection Generalized Products
21355 @pindex calc-outer-product
21357 The @kbd{V O} (@code{calc-outer-product}) [@code{outer}] command applies
21358 a given binary operator to all possible pairs of elements from two
21359 vectors, to produce a matrix. For example, @kbd{V O *} with @samp{[a, b]}
21360 and @samp{[x, y, z]} on the stack produces a multiplication table:
21361 @samp{[[a x, a y, a z], [b x, b y, b z]]}. Element @var{r},@var{c} of
21362 the result matrix is obtained by applying the operator to element @var{r}
21363 of the lefthand vector and element @var{c} of the righthand vector.
21366 @pindex calc-inner-product
21368 The @kbd{V I} (@code{calc-inner-product}) [@code{inner}] command computes
21369 the generalized inner product of two vectors or matrices, given a
21370 ``multiplicative'' operator and an ``additive'' operator. These can each
21371 actually be any binary operators; if they are @samp{*} and @samp{+},
21372 respectively, the result is a standard matrix multiplication. Element
21373 @var{r},@var{c} of the result matrix is obtained by mapping the
21374 multiplicative operator across row @var{r} of the lefthand matrix and
21375 column @var{c} of the righthand matrix, and then reducing with the additive
21376 operator. Just as for the standard @kbd{*} command, this can also do a
21377 vector-matrix or matrix-vector inner product, or a vector-vector
21378 generalized dot product.
21380 Since @kbd{V I} requires two operators, it prompts twice. In each case,
21381 you can use any of the usual methods for entering the operator. If you
21382 use @kbd{$} twice to take both operator formulas from the stack, the
21383 first (multiplicative) operator is taken from the top of the stack
21384 and the second (additive) operator is taken from second-to-top.
21386 @node Vector and Matrix Formats, , Reducing and Mapping, Matrix Functions
21387 @section Vector and Matrix Display Formats
21390 Commands for controlling vector and matrix display use the @kbd{v} prefix
21391 instead of the usual @kbd{d} prefix. But they are display modes; in
21392 particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys
21393 in the same way (@pxref{Display Modes}). Matrix display is also
21394 influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
21395 @pxref{Normal Language Modes}.
21398 @pindex calc-matrix-left-justify
21400 @pindex calc-matrix-center-justify
21402 @pindex calc-matrix-right-justify
21403 The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
21404 (@code{calc-matrix-right-justify}), and @w{@kbd{v =}}
21405 (@code{calc-matrix-center-justify}) control whether matrix elements
21406 are justified to the left, right, or center of their columns.
21409 @pindex calc-vector-brackets
21411 @pindex calc-vector-braces
21413 @pindex calc-vector-parens
21414 The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
21415 brackets that surround vectors and matrices displayed in the stack on
21416 and off. The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
21417 (@code{calc-vector-parens}) commands use curly braces or parentheses,
21418 respectively, instead of square brackets. For example, @kbd{v @{} might
21419 be used in preparation for yanking a matrix into a buffer running
21420 Mathematica. (In fact, the Mathematica language mode uses this mode;
21421 @pxref{Mathematica Language Mode}.) Note that, regardless of the
21422 display mode, either brackets or braces may be used to enter vectors,
21423 and parentheses may never be used for this purpose.
21426 @pindex calc-matrix-brackets
21427 The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the
21428 ``big'' style display of matrices. It prompts for a string of code
21429 letters; currently implemented letters are @code{R}, which enables
21430 brackets on each row of the matrix; @code{O}, which enables outer
21431 brackets in opposite corners of the matrix; and @code{C}, which
21432 enables commas or semicolons at the ends of all rows but the last.
21433 The default format is @samp{RO}. (Before Calc 2.00, the format
21434 was fixed at @samp{ROC}.) Here are some example matrices:
21438 [ [ 123, 0, 0 ] [ [ 123, 0, 0 ],
21439 [ 0, 123, 0 ] [ 0, 123, 0 ],
21440 [ 0, 0, 123 ] ] [ 0, 0, 123 ] ]
21449 [ 123, 0, 0 [ 123, 0, 0 ;
21450 0, 123, 0 0, 123, 0 ;
21451 0, 0, 123 ] 0, 0, 123 ]
21460 [ 123, 0, 0 ] 123, 0, 0
21461 [ 0, 123, 0 ] 0, 123, 0
21462 [ 0, 0, 123 ] 0, 0, 123
21469 Note that of the formats shown here, @samp{RO}, @samp{ROC}, and
21470 @samp{OC} are all recognized as matrices during reading, while
21471 the others are useful for display only.
21474 @pindex calc-vector-commas
21475 The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
21476 off in vector and matrix display.
21478 In vectors of length one, and in all vectors when commas have been
21479 turned off, Calc adds extra parentheses around formulas that might
21480 otherwise be ambiguous. For example, @samp{[a b]} could be a vector
21481 of the one formula @samp{a b}, or it could be a vector of two
21482 variables with commas turned off. Calc will display the former
21483 case as @samp{[(a b)]}. You can disable these extra parentheses
21484 (to make the output less cluttered at the expense of allowing some
21485 ambiguity) by adding the letter @code{P} to the control string you
21486 give to @kbd{v ]} (as described above).
21489 @pindex calc-full-vectors
21490 The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
21491 display of long vectors on and off. In this mode, vectors of six
21492 or more elements, or matrices of six or more rows or columns, will
21493 be displayed in an abbreviated form that displays only the first
21494 three elements and the last element: @samp{[a, b, c, ..., z]}.
21495 When very large vectors are involved this will substantially
21496 improve Calc's display speed.
21499 @pindex calc-full-trail-vectors
21500 The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a
21501 similar mode for recording vectors in the Trail. If you turn on
21502 this mode, vectors of six or more elements and matrices of six or
21503 more rows or columns will be abbreviated when they are put in the
21504 Trail. The @kbd{t y} (@code{calc-trail-yank}) command will be
21505 unable to recover those vectors. If you are working with very
21506 large vectors, this mode will improve the speed of all operations
21507 that involve the trail.
21510 @pindex calc-break-vectors
21511 The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
21512 vector display on and off. Normally, matrices are displayed with one
21513 row per line but all other types of vectors are displayed in a single
21514 line. This mode causes all vectors, whether matrices or not, to be
21515 displayed with a single element per line. Sub-vectors within the
21516 vectors will still use the normal linear form.
21518 @node Algebra, Units, Matrix Functions, Top
21522 This section covers the Calc features that help you work with
21523 algebraic formulas. First, the general sub-formula selection
21524 mechanism is described; this works in conjunction with any Calc
21525 commands. Then, commands for specific algebraic operations are
21526 described. Finally, the flexible @dfn{rewrite rule} mechanism
21529 The algebraic commands use the @kbd{a} key prefix; selection
21530 commands use the @kbd{j} (for ``just a letter that wasn't used
21531 for anything else'') prefix.
21533 @xref{Editing Stack Entries}, to see how to manipulate formulas
21534 using regular Emacs editing commands.
21536 When doing algebraic work, you may find several of the Calculator's
21537 modes to be helpful, including Algebraic Simplification mode (@kbd{m A})
21538 or No-Simplification mode (@kbd{m O}),
21539 Algebraic entry mode (@kbd{m a}), Fraction mode (@kbd{m f}), and
21540 Symbolic mode (@kbd{m s}). @xref{Mode Settings}, for discussions
21541 of these modes. You may also wish to select Big display mode (@kbd{d B}).
21542 @xref{Normal Language Modes}.
21545 * Selecting Subformulas::
21546 * Algebraic Manipulation::
21547 * Simplifying Formulas::
21550 * Solving Equations::
21551 * Numerical Solutions::
21554 * Logical Operations::
21558 @node Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra
21559 @section Selecting Sub-Formulas
21563 @cindex Sub-formulas
21564 @cindex Parts of formulas
21565 When working with an algebraic formula it is often necessary to
21566 manipulate a portion of the formula rather than the formula as a
21567 whole. Calc allows you to ``select'' a portion of any formula on
21568 the stack. Commands which would normally operate on that stack
21569 entry will now operate only on the sub-formula, leaving the
21570 surrounding part of the stack entry alone.
21572 One common non-algebraic use for selection involves vectors. To work
21573 on one element of a vector in-place, simply select that element as a
21574 ``sub-formula'' of the vector.
21577 * Making Selections::
21578 * Changing Selections::
21579 * Displaying Selections::
21580 * Operating on Selections::
21581 * Rearranging with Selections::
21584 @node Making Selections, Changing Selections, Selecting Subformulas, Selecting Subformulas
21585 @subsection Making Selections
21589 @pindex calc-select-here
21590 To select a sub-formula, move the Emacs cursor to any character in that
21591 sub-formula, and press @w{@kbd{j s}} (@code{calc-select-here}). Calc will
21592 highlight the smallest portion of the formula that contains that
21593 character. By default the sub-formula is highlighted by blanking out
21594 all of the rest of the formula with dots. Selection works in any
21595 display mode but is perhaps easiest in Big mode (@kbd{d B}).
21596 Suppose you enter the following formula:
21608 (by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}). If you move the
21609 cursor to the letter @samp{b} and press @w{@kbd{j s}}, the display changes
21622 Every character not part of the sub-formula @samp{b} has been changed
21623 to a dot. The @samp{*} next to the line number is to remind you that
21624 the formula has a portion of it selected. (In this case, it's very
21625 obvious, but it might not always be. If Embedded mode is enabled,
21626 the word @samp{Sel} also appears in the mode line because the stack
21627 may not be visible. @pxref{Embedded Mode}.)
21629 If you had instead placed the cursor on the parenthesis immediately to
21630 the right of the @samp{b}, the selection would have been:
21642 The portion selected is always large enough to be considered a complete
21643 formula all by itself, so selecting the parenthesis selects the whole
21644 formula that it encloses. Putting the cursor on the @samp{+} sign
21645 would have had the same effect.
21647 (Strictly speaking, the Emacs cursor is really the manifestation of
21648 the Emacs ``point,'' which is a position @emph{between} two characters
21649 in the buffer. So purists would say that Calc selects the smallest
21650 sub-formula which contains the character to the right of ``point.'')
21652 If you supply a numeric prefix argument @var{n}, the selection is
21653 expanded to the @var{n}th enclosing sub-formula. Thus, positioning
21654 the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select
21655 @samp{a + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3},
21658 If the cursor is not on any part of the formula, or if you give a
21659 numeric prefix that is too large, the entire formula is selected.
21661 If the cursor is on the @samp{.} line that marks the top of the stack
21662 (i.e., its normal ``rest position''), this command selects the entire
21663 formula at stack level 1. Most selection commands similarly operate
21664 on the formula at the top of the stack if you haven't positioned the
21665 cursor on any stack entry.
21668 @pindex calc-select-additional
21669 The @kbd{j a} (@code{calc-select-additional}) command enlarges the
21670 current selection to encompass the cursor. To select the smallest
21671 sub-formula defined by two different points, move to the first and
21672 press @kbd{j s}, then move to the other and press @kbd{j a}. This
21673 is roughly analogous to using @kbd{C-@@} (@code{set-mark-command}) to
21674 select the two ends of a region of text during normal Emacs editing.
21677 @pindex calc-select-once
21678 The @kbd{j o} (@code{calc-select-once}) command selects a formula in
21679 exactly the same way as @kbd{j s}, except that the selection will
21680 last only as long as the next command that uses it. For example,
21681 @kbd{j o 1 +} is a handy way to add one to the sub-formula indicated
21684 (A somewhat more precise definition: The @kbd{j o} command sets a flag
21685 such that the next command involving selected stack entries will clear
21686 the selections on those stack entries afterwards. All other selection
21687 commands except @kbd{j a} and @kbd{j O} clear this flag.)
21691 @pindex calc-select-here-maybe
21692 @pindex calc-select-once-maybe
21693 The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O}
21694 (@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s}
21695 and @kbd{j o}, respectively, except that if the formula already
21696 has a selection they have no effect. This is analogous to the
21697 behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection};
21698 @pxref{Selections with Rewrite Rules}) and is mainly intended to be
21699 used in keyboard macros that implement your own selection-oriented
21702 Selection of sub-formulas normally treats associative terms like
21703 @samp{a + b - c + d} and @samp{x * y * z} as single levels of the formula.
21704 If you place the cursor anywhere inside @samp{a + b - c + d} except
21705 on one of the variable names and use @kbd{j s}, you will select the
21706 entire four-term sum.
21709 @pindex calc-break-selections
21710 The @kbd{j b} (@code{calc-break-selections}) command controls a mode
21711 in which the ``deep structure'' of these associative formulas shows
21712 through. Calc actually stores the above formulas as @samp{((a + b) - c) + d}
21713 and @samp{x * (y * z)}. (Note that for certain obscure reasons, Calc
21714 treats multiplication as right-associative.) Once you have enabled
21715 @kbd{j b} mode, selecting with the cursor on the @samp{-} sign would
21716 only select the @samp{a + b - c} portion, which makes sense when the
21717 deep structure of the sum is considered. There is no way to select
21718 the @samp{b - c + d} portion; although this might initially look
21719 like just as legitimate a sub-formula as @samp{a + b - c}, the deep
21720 structure shows that it isn't. The @kbd{d U} command can be used
21721 to view the deep structure of any formula (@pxref{Normal Language Modes}).
21723 When @kbd{j b} mode has not been enabled, the deep structure is
21724 generally hidden by the selection commands---what you see is what
21728 @pindex calc-unselect
21729 The @kbd{j u} (@code{calc-unselect}) command unselects the formula
21730 that the cursor is on. If there was no selection in the formula,
21731 this command has no effect. With a numeric prefix argument, it
21732 unselects the @var{n}th stack element rather than using the cursor
21736 @pindex calc-clear-selections
21737 The @kbd{j c} (@code{calc-clear-selections}) command unselects all
21740 @node Changing Selections, Displaying Selections, Making Selections, Selecting Subformulas
21741 @subsection Changing Selections
21745 @pindex calc-select-more
21746 Once you have selected a sub-formula, you can expand it using the
21747 @w{@kbd{j m}} (@code{calc-select-more}) command. If @samp{a + b} is
21748 selected, pressing @w{@kbd{j m}} repeatedly works as follows:
21753 (a + b) . . . (a + b) + V c (a + b) + V c
21754 1* ............... 1* ............... 1* ---------------
21755 . . . . . . . . 2 x + 1
21760 In the last example, the entire formula is selected. This is roughly
21761 the same as having no selection at all, but because there are subtle
21762 differences the @samp{*} character is still there on the line number.
21764 With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n}
21765 times (or until the entire formula is selected). Note that @kbd{j s}
21766 with argument @var{n} is equivalent to plain @kbd{j s} followed by
21767 @kbd{j m} with argument @var{n}. If @w{@kbd{j m}} is used when there
21768 is no current selection, it is equivalent to @w{@kbd{j s}}.
21770 Even though @kbd{j m} does not explicitly use the location of the
21771 cursor within the formula, it nevertheless uses the cursor to determine
21772 which stack element to operate on. As usual, @kbd{j m} when the cursor
21773 is not on any stack element operates on the top stack element.
21776 @pindex calc-select-less
21777 The @kbd{j l} (@code{calc-select-less}) command reduces the current
21778 selection around the cursor position. That is, it selects the
21779 immediate sub-formula of the current selection which contains the
21780 cursor, the opposite of @kbd{j m}. If the cursor is not inside the
21781 current selection, the command de-selects the formula.
21784 @pindex calc-select-part
21785 The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands
21786 select the @var{n}th sub-formula of the current selection. They are
21787 like @kbd{j l} (@code{calc-select-less}) except they use counting
21788 rather than the cursor position to decide which sub-formula to select.
21789 For example, if the current selection is @kbd{a + b + c} or
21790 @kbd{f(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a},
21791 @kbd{j 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of
21792 these cases, @kbd{j 4} through @kbd{j 9} would be errors.
21794 If there is no current selection, @kbd{j 1} through @kbd{j 9} select
21795 the @var{n}th top-level sub-formula. (In other words, they act as if
21796 the entire stack entry were selected first.) To select the @var{n}th
21797 sub-formula where @var{n} is greater than nine, you must instead invoke
21798 @w{@kbd{j 1}} with @var{n} as a numeric prefix argument.
21802 @pindex calc-select-next
21803 @pindex calc-select-previous
21804 The @kbd{j n} (@code{calc-select-next}) and @kbd{j p}
21805 (@code{calc-select-previous}) commands change the current selection
21806 to the next or previous sub-formula at the same level. For example,
21807 if @samp{b} is selected in @w{@samp{2 + a*b*c + x}}, then @kbd{j n}
21808 selects @samp{c}. Further @kbd{j n} commands would be in error because,
21809 even though there is something to the right of @samp{c} (namely, @samp{x}),
21810 it is not at the same level; in this case, it is not a term of the
21811 same product as @samp{b} and @samp{c}. However, @kbd{j m} (to select
21812 the whole product @samp{a*b*c} as a term of the sum) followed by
21813 @w{@kbd{j n}} would successfully select the @samp{x}.
21815 Similarly, @kbd{j p} moves the selection from the @samp{b} in this
21816 sample formula to the @samp{a}. Both commands accept numeric prefix
21817 arguments to move several steps at a time.
21819 It is interesting to compare Calc's selection commands with the
21820 Emacs Info system's commands for navigating through hierarchically
21821 organized documentation. Calc's @kbd{j n} command is completely
21822 analogous to Info's @kbd{n} command. Likewise, @kbd{j p} maps to
21823 @kbd{p}, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}.
21824 (Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.)
21825 The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and
21826 @kbd{j l}; in each case, you can jump directly to a sub-component
21827 of the hierarchy simply by pointing to it with the cursor.
21829 @node Displaying Selections, Operating on Selections, Changing Selections, Selecting Subformulas
21830 @subsection Displaying Selections
21834 @pindex calc-show-selections
21835 The @kbd{j d} (@code{calc-show-selections}) command controls how
21836 selected sub-formulas are displayed. One of the alternatives is
21837 illustrated in the above examples; if we press @kbd{j d} we switch
21838 to the other style in which the selected portion itself is obscured
21844 (a + b) . . . ## # ## + V c
21845 1* ............... 1* ---------------
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 cancelled 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 simplifies each
22165 side with @kbd{a s} (@code{calc-simplify}) before re-forming the
22166 quotient or equation. You can suppress this simplification by
22167 providing any numeric prefix argument. 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 As a special feature, if the numerator of the quotient is 1, then
22172 the denominator is expanded at the top level using the distributive
22173 law (i.e., using the @kbd{C-u -1 a x} command). Suppose the
22174 formula on the stack is @samp{1 / (sqrt(a) + 1)}, and you wish
22175 to eliminate the square root in the denominator by multiplying both
22176 sides by @samp{sqrt(a) - 1}. Calc's default simplifications would
22177 change the result @samp{(sqrt(a) - 1) / (sqrt(a) - 1) (sqrt(a) + 1)}
22178 right back to the original form by cancellation; Calc expands the
22179 denominator to @samp{sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1} to prevent
22180 this. (You would now want to use an @kbd{a x} command to expand
22181 the rest of the way, whereupon the denominator would cancel out to
22182 the desired form, @samp{a - 1}.) When the numerator is not 1, this
22183 initial expansion is not necessary because Calc's default
22184 simplifications will not notice the potential cancellation.
22186 If the selection is an inequality, @kbd{j *} and @kbd{j /} will
22187 accept any factor, but will warn unless they can prove the factor
22188 is either positive or negative. (In the latter case the direction
22189 of the inequality will be switched appropriately.) @xref{Declarations},
22190 for ways to inform Calc that a given variable is positive or
22191 negative. If Calc can't tell for sure what the sign of the factor
22192 will be, it will assume it is positive and display a warning
22195 For selections that are not quotients, equations, or inequalities,
22196 these commands pull out a multiplicative factor: They divide (or
22197 multiply) by the entered formula, simplify, then multiply (or divide)
22198 back by the formula.
22202 @pindex calc-sel-add-both-sides
22203 @pindex calc-sel-sub-both-sides
22204 The @kbd{j +} (@code{calc-sel-add-both-sides}) and @kbd{j -}
22205 (@code{calc-sel-sub-both-sides}) commands analogously add to or
22206 subtract from both sides of an equation or inequality. For other
22207 types of selections, they extract an additive factor. A numeric
22208 prefix argument suppresses simplification of the intermediate
22212 @pindex calc-sel-unpack
22213 The @kbd{j U} (@code{calc-sel-unpack}) command replaces the
22214 selected function call with its argument. For example, given
22215 @samp{a + sin(x^2)} with @samp{sin(x^2)} selected, the result
22216 is @samp{a + x^2}. (The @samp{x^2} will remain selected; if you
22217 wanted to change the @code{sin} to @code{cos}, just press @kbd{C}
22218 now to take the cosine of the selected part.)
22221 @pindex calc-sel-evaluate
22222 The @kbd{j v} (@code{calc-sel-evaluate}) command performs the
22223 normal default simplifications on the selected sub-formula.
22224 These are the simplifications that are normally done automatically
22225 on all results, but which may have been partially inhibited by
22226 previous selection-related operations, or turned off altogether
22227 by the @kbd{m O} command. This command is just an auto-selecting
22228 version of the @w{@kbd{a v}} command (@pxref{Algebraic Manipulation}).
22230 With a numeric prefix argument of 2, @kbd{C-u 2 j v} applies
22231 the @kbd{a s} (@code{calc-simplify}) command to the selected
22232 sub-formula. With a prefix argument of 3 or more, e.g., @kbd{C-u j v}
22233 applies the @kbd{a e} (@code{calc-simplify-extended}) command.
22234 @xref{Simplifying Formulas}. With a negative prefix argument
22235 it simplifies at the top level only, just as with @kbd{a v}.
22236 Here the ``top'' level refers to the top level of the selected
22240 @pindex calc-sel-expand-formula
22241 The @kbd{j "} (@code{calc-sel-expand-formula}) command is to @kbd{a "}
22242 (@pxref{Algebraic Manipulation}) what @kbd{j v} is to @kbd{a v}.
22244 You can use the @kbd{j r} (@code{calc-rewrite-selection}) command
22245 to define other algebraic operations on sub-formulas. @xref{Rewrite Rules}.
22247 @node Algebraic Manipulation, Simplifying Formulas, Selecting Subformulas, Algebra
22248 @section Algebraic Manipulation
22251 The commands in this section perform general-purpose algebraic
22252 manipulations. They work on the whole formula at the top of the
22253 stack (unless, of course, you have made a selection in that
22256 Many algebra commands prompt for a variable name or formula. If you
22257 answer the prompt with a blank line, the variable or formula is taken
22258 from top-of-stack, and the normal argument for the command is taken
22259 from the second-to-top stack level.
22262 @pindex calc-alg-evaluate
22263 The @kbd{a v} (@code{calc-alg-evaluate}) command performs the normal
22264 default simplifications on a formula; for example, @samp{a - -b} is
22265 changed to @samp{a + b}. These simplifications are normally done
22266 automatically on all Calc results, so this command is useful only if
22267 you have turned default simplifications off with an @kbd{m O}
22268 command. @xref{Simplification Modes}.
22270 It is often more convenient to type @kbd{=}, which is like @kbd{a v}
22271 but which also substitutes stored values for variables in the formula.
22272 Use @kbd{a v} if you want the variables to ignore their stored values.
22274 If you give a numeric prefix argument of 2 to @kbd{a v}, it simplifies
22275 as if in Algebraic Simplification mode. This is equivalent to typing
22276 @kbd{a s}; @pxref{Simplifying Formulas}. If you give a numeric prefix
22277 of 3 or more, it uses Extended Simplification mode (@kbd{a e}).
22279 If you give a negative prefix argument @mathit{-1}, @mathit{-2}, or @mathit{-3},
22280 it simplifies in the corresponding mode but only works on the top-level
22281 function call of the formula. For example, @samp{(2 + 3) * (2 + 3)} will
22282 simplify to @samp{(2 + 3)^2}, without simplifying the sub-formulas
22283 @samp{2 + 3}. As another example, typing @kbd{V R +} to sum the vector
22284 @samp{[1, 2, 3, 4]} produces the formula @samp{reduce(add, [1, 2, 3, 4])}
22285 in No-Simplify mode. Using @kbd{a v} will evaluate this all the way to
22286 10; using @kbd{C-u - a v} will evaluate it only to @samp{1 + 2 + 3 + 4}.
22287 (@xref{Reducing and Mapping}.)
22291 The @kbd{=} command corresponds to the @code{evalv} function, and
22292 the related @kbd{N} command, which is like @kbd{=} but temporarily
22293 disables Symbolic mode (@kbd{m s}) during the evaluation, corresponds
22294 to the @code{evalvn} function. (These commands interpret their prefix
22295 arguments differently than @kbd{a v}; @kbd{=} treats the prefix as
22296 the number of stack elements to evaluate at once, and @kbd{N} treats
22297 it as a temporary different working precision.)
22299 The @code{evalvn} function can take an alternate working precision
22300 as an optional second argument. This argument can be either an
22301 integer, to set the precision absolutely, or a vector containing
22302 a single integer, to adjust the precision relative to the current
22303 precision. Note that @code{evalvn} with a larger than current
22304 precision will do the calculation at this higher precision, but the
22305 result will as usual be rounded back down to the current precision
22306 afterward. For example, @samp{evalvn(pi - 3.1415)} at a precision
22307 of 12 will return @samp{9.265359e-5}; @samp{evalvn(pi - 3.1415, 30)}
22308 will return @samp{9.26535897932e-5} (computing a 25-digit result which
22309 is then rounded down to 12); and @samp{evalvn(pi - 3.1415, [-2])}
22310 will return @samp{9.2654e-5}.
22313 @pindex calc-expand-formula
22314 The @kbd{a "} (@code{calc-expand-formula}) command expands functions
22315 into their defining formulas wherever possible. For example,
22316 @samp{deg(x^2)} is changed to @samp{180 x^2 / pi}. Most functions,
22317 like @code{sin} and @code{gcd}, are not defined by simple formulas
22318 and so are unaffected by this command. One important class of
22319 functions which @emph{can} be expanded is the user-defined functions
22320 created by the @kbd{Z F} command. @xref{Algebraic Definitions}.
22321 Other functions which @kbd{a "} can expand include the probability
22322 distribution functions, most of the financial functions, and the
22323 hyperbolic and inverse hyperbolic functions. A numeric prefix argument
22324 affects @kbd{a "} in the same way as it does @kbd{a v}: A positive
22325 argument expands all functions in the formula and then simplifies in
22326 various ways; a negative argument expands and simplifies only the
22327 top-level function call.
22330 @pindex calc-map-equation
22332 The @kbd{a M} (@code{calc-map-equation}) [@code{mapeq}] command applies
22333 a given function or operator to one or more equations. It is analogous
22334 to @kbd{V M}, which operates on vectors instead of equations.
22335 @pxref{Reducing and Mapping}. For example, @kbd{a M S} changes
22336 @samp{x = y+1} to @samp{sin(x) = sin(y+1)}, and @kbd{a M +} with
22337 @samp{x = y+1} and @expr{6} on the stack produces @samp{x+6 = y+7}.
22338 With two equations on the stack, @kbd{a M +} would add the lefthand
22339 sides together and the righthand sides together to get the two
22340 respective sides of a new equation.
22342 Mapping also works on inequalities. Mapping two similar inequalities
22343 produces another inequality of the same type. Mapping an inequality
22344 with an equation produces an inequality of the same type. Mapping a
22345 @samp{<=} with a @samp{<} or @samp{!=} (not-equal) produces a @samp{<}.
22346 If inequalities with opposite direction (e.g., @samp{<} and @samp{>})
22347 are mapped, the direction of the second inequality is reversed to
22348 match the first: Using @kbd{a M +} on @samp{a < b} and @samp{a > 2}
22349 reverses the latter to get @samp{2 < a}, which then allows the
22350 combination @samp{a + 2 < b + a}, which the @kbd{a s} command can
22351 then simplify to get @samp{2 < b}.
22353 Using @kbd{a M *}, @kbd{a M /}, @kbd{a M n}, or @kbd{a M &} to negate
22354 or invert an inequality will reverse the direction of the inequality.
22355 Other adjustments to inequalities are @emph{not} done automatically;
22356 @kbd{a M S} will change @w{@samp{x < y}} to @samp{sin(x) < sin(y)} even
22357 though this is not true for all values of the variables.
22361 With the Hyperbolic flag, @kbd{H a M} [@code{mapeqp}] does a plain
22362 mapping operation without reversing the direction of any inequalities.
22363 Thus, @kbd{H a M &} would change @kbd{x > 2} to @kbd{1/x > 0.5}.
22364 (This change is mathematically incorrect, but perhaps you were
22365 fixing an inequality which was already incorrect.)
22369 With the Inverse flag, @kbd{I a M} [@code{mapeqr}] always reverses
22370 the direction of the inequality. You might use @kbd{I a M C} to
22371 change @samp{x < y} to @samp{cos(x) > cos(y)} if you know you are
22372 working with small positive angles.
22375 @pindex calc-substitute
22377 The @kbd{a b} (@code{calc-substitute}) [@code{subst}] command substitutes
22379 of some variable or sub-expression of an expression with a new
22380 sub-expression. For example, substituting @samp{sin(x)} with @samp{cos(y)}
22381 in @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces
22382 @samp{2 cos(y)^2 + x cos(y) + @w{sin(2 x)}}.
22383 Note that this is a purely structural substitution; the lone @samp{x} and
22384 the @samp{sin(2 x)} stayed the same because they did not look like
22385 @samp{sin(x)}. @xref{Rewrite Rules}, for a more general method for
22386 doing substitutions.
22388 The @kbd{a b} command normally prompts for two formulas, the old
22389 one and the new one. If you enter a blank line for the first
22390 prompt, all three arguments are taken from the stack (new, then old,
22391 then target expression). If you type an old formula but then enter a
22392 blank line for the new one, the new formula is taken from top-of-stack
22393 and the target from second-to-top. If you answer both prompts, the
22394 target is taken from top-of-stack as usual.
22396 Note that @kbd{a b} has no understanding of commutativity or
22397 associativity. The pattern @samp{x+y} will not match the formula
22398 @samp{y+x}. Also, @samp{y+z} will not match inside the formula @samp{x+y+z}
22399 because the @samp{+} operator is left-associative, so the ``deep
22400 structure'' of that formula is @samp{(x+y) + z}. Use @kbd{d U}
22401 (@code{calc-unformatted-language}) mode to see the true structure of
22402 a formula. The rewrite rule mechanism, discussed later, does not have
22405 As an algebraic function, @code{subst} takes three arguments:
22406 Target expression, old, new. Note that @code{subst} is always
22407 evaluated immediately, even if its arguments are variables, so if
22408 you wish to put a call to @code{subst} onto the stack you must
22409 turn the default simplifications off first (with @kbd{m O}).
22411 @node Simplifying Formulas, Polynomials, Algebraic Manipulation, Algebra
22412 @section Simplifying Formulas
22416 @pindex calc-simplify
22418 The @kbd{a s} (@code{calc-simplify}) [@code{simplify}] command applies
22419 various algebraic rules to simplify a formula. This includes rules which
22420 are not part of the default simplifications because they may be too slow
22421 to apply all the time, or may not be desirable all of the time. For
22422 example, non-adjacent terms of sums are combined, as in @samp{a + b + 2 a}
22423 to @samp{b + 3 a}, and some formulas like @samp{sin(arcsin(x))} are
22424 simplified to @samp{x}.
22426 The sections below describe all the various kinds of algebraic
22427 simplifications Calc provides in full detail. None of Calc's
22428 simplification commands are designed to pull rabbits out of hats;
22429 they simply apply certain specific rules to put formulas into
22430 less redundant or more pleasing forms. Serious algebra in Calc
22431 must be done manually, usually with a combination of selections
22432 and rewrite rules. @xref{Rearranging with Selections}.
22433 @xref{Rewrite Rules}.
22435 @xref{Simplification Modes}, for commands to control what level of
22436 simplification occurs automatically. Normally only the ``default
22437 simplifications'' occur.
22440 * Default Simplifications::
22441 * Algebraic Simplifications::
22442 * Unsafe Simplifications::
22443 * Simplification of Units::
22446 @node Default Simplifications, Algebraic Simplifications, Simplifying Formulas, Simplifying Formulas
22447 @subsection Default Simplifications
22450 @cindex Default simplifications
22451 This section describes the ``default simplifications,'' those which are
22452 normally applied to all results. For example, if you enter the variable
22453 @expr{x} on the stack twice and push @kbd{+}, Calc's default
22454 simplifications automatically change @expr{x + x} to @expr{2 x}.
22456 The @kbd{m O} command turns off the default simplifications, so that
22457 @expr{x + x} will remain in this form unless you give an explicit
22458 ``simplify'' command like @kbd{=} or @kbd{a v}. @xref{Algebraic
22459 Manipulation}. The @kbd{m D} command turns the default simplifications
22462 The most basic default simplification is the evaluation of functions.
22463 For example, @expr{2 + 3} is evaluated to @expr{5}, and @expr{@tfn{sqrt}(9)}
22464 is evaluated to @expr{3}. Evaluation does not occur if the arguments
22465 to a function are somehow of the wrong type @expr{@tfn{tan}([2,3,4])}),
22466 range (@expr{@tfn{tan}(90)}), or number (@expr{@tfn{tan}(3,5)}),
22467 or if the function name is not recognized (@expr{@tfn{f}(5)}), or if
22468 Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation
22469 (@expr{@tfn{sqrt}(2)}).
22471 Calc simplifies (evaluates) the arguments to a function before it
22472 simplifies the function itself. Thus @expr{@tfn{sqrt}(5+4)} is
22473 simplified to @expr{@tfn{sqrt}(9)} before the @code{sqrt} function
22474 itself is applied. There are very few exceptions to this rule:
22475 @code{quote}, @code{lambda}, and @code{condition} (the @code{::}
22476 operator) do not evaluate their arguments, @code{if} (the @code{? :}
22477 operator) does not evaluate all of its arguments, and @code{evalto}
22478 does not evaluate its lefthand argument.
22480 Most commands apply the default simplifications to all arguments they
22481 take from the stack, perform a particular operation, then simplify
22482 the result before pushing it back on the stack. In the common special
22483 case of regular arithmetic commands like @kbd{+} and @kbd{Q} [@code{sqrt}],
22484 the arguments are simply popped from the stack and collected into a
22485 suitable function call, which is then simplified (the arguments being
22486 simplified first as part of the process, as described above).
22488 The default simplifications are too numerous to describe completely
22489 here, but this section will describe the ones that apply to the
22490 major arithmetic operators. This list will be rather technical in
22491 nature, and will probably be interesting to you only if you are
22492 a serious user of Calc's algebra facilities.
22498 As well as the simplifications described here, if you have stored
22499 any rewrite rules in the variable @code{EvalRules} then these rules
22500 will also be applied before any built-in default simplifications.
22501 @xref{Automatic Rewrites}, for details.
22507 And now, on with the default simplifications:
22509 Arithmetic operators like @kbd{+} and @kbd{*} always take two
22510 arguments in Calc's internal form. Sums and products of three or
22511 more terms are arranged by the associative law of algebra into
22512 a left-associative form for sums, @expr{((a + b) + c) + d}, and
22513 a right-associative form for products, @expr{a * (b * (c * d))}.
22514 Formulas like @expr{(a + b) + (c + d)} are rearranged to
22515 left-associative form, though this rarely matters since Calc's
22516 algebra commands are designed to hide the inner structure of
22517 sums and products as much as possible. Sums and products in
22518 their proper associative form will be written without parentheses
22519 in the examples below.
22521 Sums and products are @emph{not} rearranged according to the
22522 commutative law (@expr{a + b} to @expr{b + a}) except in a few
22523 special cases described below. Some algebra programs always
22524 rearrange terms into a canonical order, which enables them to
22525 see that @expr{a b + b a} can be simplified to @expr{2 a b}.
22526 Calc assumes you have put the terms into the order you want
22527 and generally leaves that order alone, with the consequence
22528 that formulas like the above will only be simplified if you
22529 explicitly give the @kbd{a s} command. @xref{Algebraic
22532 Differences @expr{a - b} are treated like sums @expr{a + (-b)}
22533 for purposes of simplification; one of the default simplifications
22534 is to rewrite @expr{a + (-b)} or @expr{(-b) + a}, where @expr{-b}
22535 represents a ``negative-looking'' term, into @expr{a - b} form.
22536 ``Negative-looking'' means negative numbers, negated formulas like
22537 @expr{-x}, and products or quotients in which either term is
22540 Other simplifications involving negation are @expr{-(-x)} to @expr{x};
22541 @expr{-(a b)} or @expr{-(a/b)} where either @expr{a} or @expr{b} is
22542 negative-looking, simplified by negating that term, or else where
22543 @expr{a} or @expr{b} is any number, by negating that number;
22544 @expr{-(a + b)} to @expr{-a - b}, and @expr{-(b - a)} to @expr{a - b}.
22545 (This, and rewriting @expr{(-b) + a} to @expr{a - b}, are the only
22546 cases where the order of terms in a sum is changed by the default
22549 The distributive law is used to simplify sums in some cases:
22550 @expr{a x + b x} to @expr{(a + b) x}, where @expr{a} represents
22551 a number or an implicit 1 or @mathit{-1} (as in @expr{x} or @expr{-x})
22552 and similarly for @expr{b}. Use the @kbd{a c}, @w{@kbd{a f}}, or
22553 @kbd{j M} commands to merge sums with non-numeric coefficients
22554 using the distributive law.
22556 The distributive law is only used for sums of two terms, or
22557 for adjacent terms in a larger sum. Thus @expr{a + b + b + c}
22558 is simplified to @expr{a + 2 b + c}, but @expr{a + b + c + b}
22559 is not simplified. The reason is that comparing all terms of a
22560 sum with one another would require time proportional to the
22561 square of the number of terms; Calc relegates potentially slow
22562 operations like this to commands that have to be invoked
22563 explicitly, like @kbd{a s}.
22565 Finally, @expr{a + 0} and @expr{0 + a} are simplified to @expr{a}.
22566 A consequence of the above rules is that @expr{0 - a} is simplified
22573 The products @expr{1 a} and @expr{a 1} are simplified to @expr{a};
22574 @expr{(-1) a} and @expr{a (-1)} are simplified to @expr{-a};
22575 @expr{0 a} and @expr{a 0} are simplified to @expr{0}, except that
22576 in Matrix mode where @expr{a} is not provably scalar the result
22577 is the generic zero matrix @samp{idn(0)}, and that if @expr{a} is
22578 infinite the result is @samp{nan}.
22580 Also, @expr{(-a) b} and @expr{a (-b)} are simplified to @expr{-(a b)},
22581 where this occurs for negated formulas but not for regular negative
22584 Products are commuted only to move numbers to the front:
22585 @expr{a b 2} is commuted to @expr{2 a b}.
22587 The product @expr{a (b + c)} is distributed over the sum only if
22588 @expr{a} and at least one of @expr{b} and @expr{c} are numbers:
22589 @expr{2 (x + 3)} goes to @expr{2 x + 6}. The formula
22590 @expr{(-a) (b - c)}, where @expr{-a} is a negative number, is
22591 rewritten to @expr{a (c - b)}.
22593 The distributive law of products and powers is used for adjacent
22594 terms of the product: @expr{x^a x^b} goes to
22595 @texline @math{x^{a+b}}
22596 @infoline @expr{x^(a+b)}
22597 where @expr{a} is a number, or an implicit 1 (as in @expr{x}),
22598 or the implicit one-half of @expr{@tfn{sqrt}(x)}, and similarly for
22599 @expr{b}. The result is written using @samp{sqrt} or @samp{1/sqrt}
22600 if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively.
22601 If the sum of the powers is zero, the product is simplified to
22602 @expr{1} or to @samp{idn(1)} if Matrix mode is enabled.
22604 The product of a negative power times anything but another negative
22605 power is changed to use division:
22606 @texline @math{x^{-2} y}
22607 @infoline @expr{x^(-2) y}
22608 goes to @expr{y / x^2} unless Matrix mode is
22609 in effect and neither @expr{x} nor @expr{y} are scalar (in which
22610 case it is considered unsafe to rearrange the order of the terms).
22612 Finally, @expr{a (b/c)} is rewritten to @expr{(a b)/c}, and also
22613 @expr{(a/b) c} is changed to @expr{(a c)/b} unless in Matrix mode.
22619 Simplifications for quotients are analogous to those for products.
22620 The quotient @expr{0 / x} is simplified to @expr{0}, with the same
22621 exceptions that were noted for @expr{0 x}. Likewise, @expr{x / 1}
22622 and @expr{x / (-1)} are simplified to @expr{x} and @expr{-x},
22625 The quotient @expr{x / 0} is left unsimplified or changed to an
22626 infinite quantity, as directed by the current infinite mode.
22627 @xref{Infinite Mode}.
22630 @texline @math{a / b^{-c}}
22631 @infoline @expr{a / b^(-c)}
22632 is changed to @expr{a b^c}, where @expr{-c} is any negative-looking
22633 power. Also, @expr{1 / b^c} is changed to
22634 @texline @math{b^{-c}}
22635 @infoline @expr{b^(-c)}
22636 for any power @expr{c}.
22638 Also, @expr{(-a) / b} and @expr{a / (-b)} go to @expr{-(a/b)};
22639 @expr{(a/b) / c} goes to @expr{a / (b c)}; and @expr{a / (b/c)}
22640 goes to @expr{(a c) / b} unless Matrix mode prevents this
22641 rearrangement. Similarly, @expr{a / (b:c)} is simplified to
22642 @expr{(c:b) a} for any fraction @expr{b:c}.
22644 The distributive law is applied to @expr{(a + b) / c} only if
22645 @expr{c} and at least one of @expr{a} and @expr{b} are numbers.
22646 Quotients of powers and square roots are distributed just as
22647 described for multiplication.
22649 Quotients of products cancel only in the leading terms of the
22650 numerator and denominator. In other words, @expr{a x b / a y b}
22651 is cancelled to @expr{x b / y b} but not to @expr{x / y}. Once
22652 again this is because full cancellation can be slow; use @kbd{a s}
22653 to cancel all terms of the quotient.
22655 Quotients of negative-looking values are simplified according
22656 to @expr{(-a) / (-b)} to @expr{a / b}, @expr{(-a) / (b - c)}
22657 to @expr{a / (c - b)}, and @expr{(a - b) / (-c)} to @expr{(b - a) / c}.
22663 The formula @expr{x^0} is simplified to @expr{1}, or to @samp{idn(1)}
22664 in Matrix mode. The formula @expr{0^x} is simplified to @expr{0}
22665 unless @expr{x} is a negative number, complex number or zero.
22666 If @expr{x} is negative, complex or @expr{0.0}, @expr{0^x} is an
22667 infinity or an unsimplified formula according to the current infinite
22668 mode. The expression @expr{0^0} is simplified to @expr{1}.
22670 Powers of products or quotients @expr{(a b)^c}, @expr{(a/b)^c}
22671 are distributed to @expr{a^c b^c}, @expr{a^c / b^c} only if @expr{c}
22672 is an integer, or if either @expr{a} or @expr{b} are nonnegative
22673 real numbers. Powers of powers @expr{(a^b)^c} are simplified to
22674 @texline @math{a^{b c}}
22675 @infoline @expr{a^(b c)}
22676 only when @expr{c} is an integer and @expr{b c} also
22677 evaluates to an integer. Without these restrictions these simplifications
22678 would not be safe because of problems with principal values.
22680 @texline @math{((-3)^{1/2})^2}
22681 @infoline @expr{((-3)^1:2)^2}
22682 is safe to simplify, but
22683 @texline @math{((-3)^2)^{1/2}}
22684 @infoline @expr{((-3)^2)^1:2}
22685 is not.) @xref{Declarations}, for ways to inform Calc that your
22686 variables satisfy these requirements.
22688 As a special case of this rule, @expr{@tfn{sqrt}(x)^n} is simplified to
22689 @texline @math{x^{n/2}}
22690 @infoline @expr{x^(n/2)}
22691 only for even integers @expr{n}.
22693 If @expr{a} is known to be real, @expr{b} is an even integer, and
22694 @expr{c} is a half- or quarter-integer, then @expr{(a^b)^c} is
22695 simplified to @expr{@tfn{abs}(a^(b c))}.
22697 Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an
22698 even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer,
22699 for any negative-looking expression @expr{-a}.
22701 Square roots @expr{@tfn{sqrt}(x)} generally act like one-half powers
22702 @texline @math{x^{1:2}}
22703 @infoline @expr{x^1:2}
22704 for the purposes of the above-listed simplifications.
22707 @texline @math{1 / x^{1:2}}
22708 @infoline @expr{1 / x^1:2}
22710 @texline @math{x^{-1:2}},
22711 @infoline @expr{x^(-1:2)},
22712 but @expr{1 / @tfn{sqrt}(x)} is left alone.
22718 Generic identity matrices (@pxref{Matrix Mode}) are simplified by the
22719 following rules: @expr{@tfn{idn}(a) + b} to @expr{a + b} if @expr{b}
22720 is provably scalar, or expanded out if @expr{b} is a matrix;
22721 @expr{@tfn{idn}(a) + @tfn{idn}(b)} to @expr{@tfn{idn}(a + b)};
22722 @expr{-@tfn{idn}(a)} to @expr{@tfn{idn}(-a)}; @expr{a @tfn{idn}(b)} to
22723 @expr{@tfn{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b}
22724 if @expr{a} is provably non-scalar; @expr{@tfn{idn}(a) @tfn{idn}(b)} to
22725 @expr{@tfn{idn}(a b)}; analogous simplifications for quotients involving
22726 @code{idn}; and @expr{@tfn{idn}(a)^n} to @expr{@tfn{idn}(a^n)} where
22727 @expr{n} is an integer.
22733 The @code{floor} function and other integer truncation functions
22734 vanish if the argument is provably integer-valued, so that
22735 @expr{@tfn{floor}(@tfn{round}(x))} simplifies to @expr{@tfn{round}(x)}.
22736 Also, combinations of @code{float}, @code{floor} and its friends,
22737 and @code{ffloor} and its friends, are simplified in appropriate
22738 ways. @xref{Integer Truncation}.
22740 The expression @expr{@tfn{abs}(-x)} changes to @expr{@tfn{abs}(x)}.
22741 The expression @expr{@tfn{abs}(@tfn{abs}(x))} changes to
22742 @expr{@tfn{abs}(x)}; in fact, @expr{@tfn{abs}(x)} changes to @expr{x} or
22743 @expr{-x} if @expr{x} is provably nonnegative or nonpositive
22744 (@pxref{Declarations}).
22746 While most functions do not recognize the variable @code{i} as an
22747 imaginary number, the @code{arg} function does handle the two cases
22748 @expr{@tfn{arg}(@tfn{i})} and @expr{@tfn{arg}(-@tfn{i})} just for convenience.
22750 The expression @expr{@tfn{conj}(@tfn{conj}(x))} simplifies to @expr{x}.
22751 Various other expressions involving @code{conj}, @code{re}, and
22752 @code{im} are simplified, especially if some of the arguments are
22753 provably real or involve the constant @code{i}. For example,
22754 @expr{@tfn{conj}(a + b i)} is changed to
22755 @expr{@tfn{conj}(a) - @tfn{conj}(b) i}, or to @expr{a - b i} if @expr{a}
22756 and @expr{b} are known to be real.
22758 Functions like @code{sin} and @code{arctan} generally don't have
22759 any default simplifications beyond simply evaluating the functions
22760 for suitable numeric arguments and infinity. The @kbd{a s} command
22761 described in the next section does provide some simplifications for
22762 these functions, though.
22764 One important simplification that does occur is that
22765 @expr{@tfn{ln}(@tfn{e})} is simplified to 1, and @expr{@tfn{ln}(@tfn{e}^x)} is
22766 simplified to @expr{x} for any @expr{x}. This occurs even if you have
22767 stored a different value in the Calc variable @samp{e}; but this would
22768 be a bad idea in any case if you were also using natural logarithms!
22770 Among the logical functions, @tfn{!(@var{a} <= @var{b})} changes to
22771 @tfn{@var{a} > @var{b}} and so on. Equations and inequalities where both sides
22772 are either negative-looking or zero are simplified by negating both sides
22773 and reversing the inequality. While it might seem reasonable to simplify
22774 @expr{!!x} to @expr{x}, this would not be valid in general because
22775 @expr{!!2} is 1, not 2.
22777 Most other Calc functions have few if any default simplifications
22778 defined, aside of course from evaluation when the arguments are
22781 @node Algebraic Simplifications, Unsafe Simplifications, Default Simplifications, Simplifying Formulas
22782 @subsection Algebraic Simplifications
22785 @cindex Algebraic simplifications
22786 The @kbd{a s} command makes simplifications that may be too slow to
22787 do all the time, or that may not be desirable all of the time.
22788 If you find these simplifications are worthwhile, you can type
22789 @kbd{m A} to have Calc apply them automatically.
22791 This section describes all simplifications that are performed by
22792 the @kbd{a s} command. Note that these occur in addition to the
22793 default simplifications; even if the default simplifications have
22794 been turned off by an @kbd{m O} command, @kbd{a s} will turn them
22795 back on temporarily while it simplifies the formula.
22797 There is a variable, @code{AlgSimpRules}, in which you can put rewrites
22798 to be applied by @kbd{a s}. Its use is analogous to @code{EvalRules},
22799 but without the special restrictions. Basically, the simplifier does
22800 @samp{@w{a r} AlgSimpRules} with an infinite repeat count on the whole
22801 expression being simplified, then it traverses the expression applying
22802 the built-in rules described below. If the result is different from
22803 the original expression, the process repeats with the default
22804 simplifications (including @code{EvalRules}), then @code{AlgSimpRules},
22805 then the built-in simplifications, and so on.
22811 Sums are simplified in two ways. Constant terms are commuted to the
22812 end of the sum, so that @expr{a + 2 + b} changes to @expr{a + b + 2}.
22813 The only exception is that a constant will not be commuted away
22814 from the first position of a difference, i.e., @expr{2 - x} is not
22815 commuted to @expr{-x + 2}.
22817 Also, terms of sums are combined by the distributive law, as in
22818 @expr{x + y + 2 x} to @expr{y + 3 x}. This always occurs for
22819 adjacent terms, but @kbd{a s} compares all pairs of terms including
22826 Products are sorted into a canonical order using the commutative
22827 law. For example, @expr{b c a} is commuted to @expr{a b c}.
22828 This allows easier comparison of products; for example, the default
22829 simplifications will not change @expr{x y + y x} to @expr{2 x y},
22830 but @kbd{a s} will; it first rewrites the sum to @expr{x y + x y},
22831 and then the default simplifications are able to recognize a sum
22832 of identical terms.
22834 The canonical ordering used to sort terms of products has the
22835 property that real-valued numbers, interval forms and infinities
22836 come first, and are sorted into increasing order. The @kbd{V S}
22837 command uses the same ordering when sorting a vector.
22839 Sorting of terms of products is inhibited when Matrix mode is
22840 turned on; in this case, Calc will never exchange the order of
22841 two terms unless it knows at least one of the terms is a scalar.
22843 Products of powers are distributed by comparing all pairs of
22844 terms, using the same method that the default simplifications
22845 use for adjacent terms of products.
22847 Even though sums are not sorted, the commutative law is still
22848 taken into account when terms of a product are being compared.
22849 Thus @expr{(x + y) (y + x)} will be simplified to @expr{(x + y)^2}.
22850 A subtle point is that @expr{(x - y) (y - x)} will @emph{not}
22851 be simplified to @expr{-(x - y)^2}; Calc does not notice that
22852 one term can be written as a constant times the other, even if
22853 that constant is @mathit{-1}.
22855 A fraction times any expression, @expr{(a:b) x}, is changed to
22856 a quotient involving integers: @expr{a x / b}. This is not
22857 done for floating-point numbers like @expr{0.5}, however. This
22858 is one reason why you may find it convenient to turn Fraction mode
22859 on while doing algebra; @pxref{Fraction Mode}.
22865 Quotients are simplified by comparing all terms in the numerator
22866 with all terms in the denominator for possible cancellation using
22867 the distributive law. For example, @expr{a x^2 b / c x^3 d} will
22868 cancel @expr{x^2} from the top and bottom to get @expr{a b / c x d}.
22869 (The terms in the denominator will then be rearranged to @expr{c d x}
22870 as described above.) If there is any common integer or fractional
22871 factor in the numerator and denominator, it is cancelled out;
22872 for example, @expr{(4 x + 6) / 8 x} simplifies to @expr{(2 x + 3) / 4 x}.
22874 Non-constant common factors are not found even by @kbd{a s}. To
22875 cancel the factor @expr{a} in @expr{(a x + a) / a^2} you could first
22876 use @kbd{j M} on the product @expr{a x} to Merge the numerator to
22877 @expr{a (1+x)}, which can then be simplified successfully.
22883 Integer powers of the variable @code{i} are simplified according
22884 to the identity @expr{i^2 = -1}. If you store a new value other
22885 than the complex number @expr{(0,1)} in @code{i}, this simplification
22886 will no longer occur. This is done by @kbd{a s} instead of by default
22887 in case someone (unwisely) uses the name @code{i} for a variable
22888 unrelated to complex numbers; it would be unfortunate if Calc
22889 quietly and automatically changed this formula for reasons the
22890 user might not have been thinking of.
22892 Square roots of integer or rational arguments are simplified in
22893 several ways. (Note that these will be left unevaluated only in
22894 Symbolic mode.) First, square integer or rational factors are
22895 pulled out so that @expr{@tfn{sqrt}(8)} is rewritten as
22896 @texline @math{2\,@tfn{sqrt}(2)}.
22897 @infoline @expr{2 sqrt(2)}.
22898 Conceptually speaking this implies factoring the argument into primes
22899 and moving pairs of primes out of the square root, but for reasons of
22900 efficiency Calc only looks for primes up to 29.
22902 Square roots in the denominator of a quotient are moved to the
22903 numerator: @expr{1 / @tfn{sqrt}(3)} changes to @expr{@tfn{sqrt}(3) / 3}.
22904 The same effect occurs for the square root of a fraction:
22905 @expr{@tfn{sqrt}(2:3)} changes to @expr{@tfn{sqrt}(6) / 3}.
22911 The @code{%} (modulo) operator is simplified in several ways
22912 when the modulus @expr{M} is a positive real number. First, if
22913 the argument is of the form @expr{x + n} for some real number
22914 @expr{n}, then @expr{n} is itself reduced modulo @expr{M}. For
22915 example, @samp{(x - 23) % 10} is simplified to @samp{(x + 7) % 10}.
22917 If the argument is multiplied by a constant, and this constant
22918 has a common integer divisor with the modulus, then this factor is
22919 cancelled out. For example, @samp{12 x % 15} is changed to
22920 @samp{3 (4 x % 5)} by factoring out 3. Also, @samp{(12 x + 1) % 15}
22921 is changed to @samp{3 ((4 x + 1:3) % 5)}. While these forms may
22922 not seem ``simpler,'' they allow Calc to discover useful information
22923 about modulo forms in the presence of declarations.
22925 If the modulus is 1, then Calc can use @code{int} declarations to
22926 evaluate the expression. For example, the idiom @samp{x % 2} is
22927 often used to check whether a number is odd or even. As described
22928 above, @w{@samp{2 n % 2}} and @samp{(2 n + 1) % 2} are simplified to
22929 @samp{2 (n % 1)} and @samp{2 ((n + 1:2) % 1)}, respectively; Calc
22930 can simplify these to 0 and 1 (respectively) if @code{n} has been
22931 declared to be an integer.
22937 Trigonometric functions are simplified in several ways. Whenever a
22938 products of two trigonometric functions can be replaced by a single
22939 function, the replacement is made; for example,
22940 @expr{@tfn{tan}(x) @tfn{cos}(x)} is simplified to @expr{@tfn{sin}(x)}.
22941 Reciprocals of trigonometric functions are replaced by their reciprocal
22942 function; for example, @expr{1/@tfn{sec}(x)} is simplified to
22943 @expr{@tfn{cos}(x)}. The corresponding simplifications for the
22944 hyperbolic functions are also handled.
22946 Trigonometric functions of their inverse functions are
22947 simplified. The expression @expr{@tfn{sin}(@tfn{arcsin}(x))} is
22948 simplified to @expr{x}, and similarly for @code{cos} and @code{tan}.
22949 Trigonometric functions of inverses of different trigonometric
22950 functions can also be simplified, as in @expr{@tfn{sin}(@tfn{arccos}(x))}
22951 to @expr{@tfn{sqrt}(1 - x^2)}.
22953 If the argument to @code{sin} is negative-looking, it is simplified to
22954 @expr{-@tfn{sin}(x)}, and similarly for @code{cos} and @code{tan}.
22955 Finally, certain special values of the argument are recognized;
22956 @pxref{Trigonometric and Hyperbolic Functions}.
22958 Hyperbolic functions of their inverses and of negative-looking
22959 arguments are also handled, as are exponentials of inverse
22960 hyperbolic functions.
22962 No simplifications for inverse trigonometric and hyperbolic
22963 functions are known, except for negative arguments of @code{arcsin},
22964 @code{arctan}, @code{arcsinh}, and @code{arctanh}. Note that
22965 @expr{@tfn{arcsin}(@tfn{sin}(x))} can @emph{not} safely change to
22966 @expr{x}, since this only correct within an integer multiple of
22967 @texline @math{2 \pi}
22968 @infoline @expr{2 pi}
22969 radians or 360 degrees. However, @expr{@tfn{arcsinh}(@tfn{sinh}(x))} is
22970 simplified to @expr{x} if @expr{x} is known to be real.
22972 Several simplifications that apply to logarithms and exponentials
22973 are that @expr{@tfn{exp}(@tfn{ln}(x))},
22974 @texline @tfn{e}@math{^{\ln(x)}},
22975 @infoline @expr{e^@tfn{ln}(x)},
22977 @texline @math{10^{{\rm log10}(x)}}
22978 @infoline @expr{10^@tfn{log10}(x)}
22979 all reduce to @expr{x}. Also, @expr{@tfn{ln}(@tfn{exp}(x))}, etc., can
22980 reduce to @expr{x} if @expr{x} is provably real. The form
22981 @expr{@tfn{exp}(x)^y} is simplified to @expr{@tfn{exp}(x y)}. If @expr{x}
22982 is a suitable multiple of
22983 @texline @math{\pi i}
22984 @infoline @expr{pi i}
22985 (as described above for the trigonometric functions), then
22986 @expr{@tfn{exp}(x)} or @expr{e^x} will be expanded. Finally,
22987 @expr{@tfn{ln}(x)} is simplified to a form involving @code{pi} and
22988 @code{i} where @expr{x} is provably negative, positive imaginary, or
22989 negative imaginary.
22991 The error functions @code{erf} and @code{erfc} are simplified when
22992 their arguments are negative-looking or are calls to the @code{conj}
22999 Equations and inequalities are simplified by cancelling factors
23000 of products, quotients, or sums on both sides. Inequalities
23001 change sign if a negative multiplicative factor is cancelled.
23002 Non-constant multiplicative factors as in @expr{a b = a c} are
23003 cancelled from equations only if they are provably nonzero (generally
23004 because they were declared so; @pxref{Declarations}). Factors
23005 are cancelled from inequalities only if they are nonzero and their
23008 Simplification also replaces an equation or inequality with
23009 1 or 0 (``true'' or ``false'') if it can through the use of
23010 declarations. If @expr{x} is declared to be an integer greater
23011 than 5, then @expr{x < 3}, @expr{x = 3}, and @expr{x = 7.5} are
23012 all simplified to 0, but @expr{x > 3} is simplified to 1.
23013 By a similar analysis, @expr{abs(x) >= 0} is simplified to 1,
23014 as is @expr{x^2 >= 0} if @expr{x} is known to be real.
23016 @node Unsafe Simplifications, Simplification of Units, Algebraic Simplifications, Simplifying Formulas
23017 @subsection ``Unsafe'' Simplifications
23020 @cindex Unsafe simplifications
23021 @cindex Extended simplification
23023 @pindex calc-simplify-extended
23025 @mindex esimpl@idots
23028 The @kbd{a e} (@code{calc-simplify-extended}) [@code{esimplify}] command
23030 except that it applies some additional simplifications which are not
23031 ``safe'' in all cases. Use this only if you know the values in your
23032 formula lie in the restricted ranges for which these simplifications
23033 are valid. The symbolic integrator uses @kbd{a e};
23034 one effect of this is that the integrator's results must be used with
23035 caution. Where an integral table will often attach conditions like
23036 ``for positive @expr{a} only,'' Calc (like most other symbolic
23037 integration programs) will simply produce an unqualified result.
23039 Because @kbd{a e}'s simplifications are unsafe, it is sometimes better
23040 to type @kbd{C-u -3 a v}, which does extended simplification only
23041 on the top level of the formula without affecting the sub-formulas.
23042 In fact, @kbd{C-u -3 j v} allows you to target extended simplification
23043 to any specific part of a formula.
23045 The variable @code{ExtSimpRules} contains rewrites to be applied by
23046 the @kbd{a e} command. These are applied in addition to
23047 @code{EvalRules} and @code{AlgSimpRules}. (The @kbd{a r AlgSimpRules}
23048 step described above is simply followed by an @kbd{a r ExtSimpRules} step.)
23050 Following is a complete list of ``unsafe'' simplifications performed
23057 Inverse trigonometric or hyperbolic functions, called with their
23058 corresponding non-inverse functions as arguments, are simplified
23059 by @kbd{a e}. For example, @expr{@tfn{arcsin}(@tfn{sin}(x))} changes
23060 to @expr{x}. Also, @expr{@tfn{arcsin}(@tfn{cos}(x))} and
23061 @expr{@tfn{arccos}(@tfn{sin}(x))} both change to @expr{@tfn{pi}/2 - x}.
23062 These simplifications are unsafe because they are valid only for
23063 values of @expr{x} in a certain range; outside that range, values
23064 are folded down to the 360-degree range that the inverse trigonometric
23065 functions always produce.
23067 Powers of powers @expr{(x^a)^b} are simplified to
23068 @texline @math{x^{a b}}
23069 @infoline @expr{x^(a b)}
23070 for all @expr{a} and @expr{b}. These results will be valid only
23071 in a restricted range of @expr{x}; for example, in
23072 @texline @math{(x^2)^{1:2}}
23073 @infoline @expr{(x^2)^1:2}
23074 the powers cancel to get @expr{x}, which is valid for positive values
23075 of @expr{x} but not for negative or complex values.
23077 Similarly, @expr{@tfn{sqrt}(x^a)} and @expr{@tfn{sqrt}(x)^a} are both
23078 simplified (possibly unsafely) to
23079 @texline @math{x^{a/2}}.
23080 @infoline @expr{x^(a/2)}.
23082 Forms like @expr{@tfn{sqrt}(1 - sin(x)^2)} are simplified to, e.g.,
23083 @expr{@tfn{cos}(x)}. Calc has identities of this sort for @code{sin},
23084 @code{cos}, @code{tan}, @code{sinh}, and @code{cosh}.
23086 Arguments of square roots are partially factored to look for
23087 squared terms that can be extracted. For example,
23088 @expr{@tfn{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to
23089 @expr{a b @tfn{sqrt}(a+b)}.
23091 The simplifications of @expr{@tfn{ln}(@tfn{exp}(x))},
23092 @expr{@tfn{ln}(@tfn{e}^x)}, and @expr{@tfn{log10}(10^x)} to @expr{x} are also
23093 unsafe because of problems with principal values (although these
23094 simplifications are safe if @expr{x} is known to be real).
23096 Common factors are cancelled from products on both sides of an
23097 equation, even if those factors may be zero: @expr{a x / b x}
23098 to @expr{a / b}. Such factors are never cancelled from
23099 inequalities: Even @kbd{a e} is not bold enough to reduce
23100 @expr{a x < b x} to @expr{a < b} (or @expr{a > b}, depending
23101 on whether you believe @expr{x} is positive or negative).
23102 The @kbd{a M /} command can be used to divide a factor out of
23103 both sides of an inequality.
23105 @node Simplification of Units, , Unsafe Simplifications, Simplifying Formulas
23106 @subsection Simplification of Units
23109 The simplifications described in this section are applied by the
23110 @kbd{u s} (@code{calc-simplify-units}) command. These are in addition
23111 to the regular @kbd{a s} (but not @kbd{a e}) simplifications described
23112 earlier. @xref{Basic Operations on Units}.
23114 The variable @code{UnitSimpRules} contains rewrites to be applied by
23115 the @kbd{u s} command. These are applied in addition to @code{EvalRules}
23116 and @code{AlgSimpRules}.
23118 Scalar mode is automatically put into effect when simplifying units.
23119 @xref{Matrix Mode}.
23121 Sums @expr{a + b} involving units are simplified by extracting the
23122 units of @expr{a} as if by the @kbd{u x} command (call the result
23123 @expr{u_a}), then simplifying the expression @expr{b / u_a}
23124 using @kbd{u b} and @kbd{u s}. If the result has units then the sum
23125 is inconsistent and is left alone. Otherwise, it is rewritten
23126 in terms of the units @expr{u_a}.
23128 If units auto-ranging mode is enabled, products or quotients in
23129 which the first argument is a number which is out of range for the
23130 leading unit are modified accordingly.
23132 When cancelling and combining units in products and quotients,
23133 Calc accounts for unit names that differ only in the prefix letter.
23134 For example, @samp{2 km m} is simplified to @samp{2000 m^2}.
23135 However, compatible but different units like @code{ft} and @code{in}
23136 are not combined in this way.
23138 Quotients @expr{a / b} are simplified in three additional ways. First,
23139 if @expr{b} is a number or a product beginning with a number, Calc
23140 computes the reciprocal of this number and moves it to the numerator.
23142 Second, for each pair of unit names from the numerator and denominator
23143 of a quotient, if the units are compatible (e.g., they are both
23144 units of area) then they are replaced by the ratio between those
23145 units. For example, in @samp{3 s in N / kg cm} the units
23146 @samp{in / cm} will be replaced by @expr{2.54}.
23148 Third, if the units in the quotient exactly cancel out, so that
23149 a @kbd{u b} command on the quotient would produce a dimensionless
23150 number for an answer, then the quotient simplifies to that number.
23152 For powers and square roots, the ``unsafe'' simplifications
23153 @expr{(a b)^c} to @expr{a^c b^c}, @expr{(a/b)^c} to @expr{a^c / b^c},
23154 and @expr{(a^b)^c} to
23155 @texline @math{a^{b c}}
23156 @infoline @expr{a^(b c)}
23157 are done if the powers are real numbers. (These are safe in the context
23158 of units because all numbers involved can reasonably be assumed to be
23161 Also, if a unit name is raised to a fractional power, and the
23162 base units in that unit name all occur to powers which are a
23163 multiple of the denominator of the power, then the unit name
23164 is expanded out into its base units, which can then be simplified
23165 according to the previous paragraph. For example, @samp{acre^1.5}
23166 is simplified by noting that @expr{1.5 = 3:2}, that @samp{acre}
23167 is defined in terms of @samp{m^2}, and that the 2 in the power of
23168 @code{m} is a multiple of 2 in @expr{3:2}. Thus, @code{acre^1.5} is
23169 replaced by approximately
23170 @texline @math{(4046 m^2)^{1.5}}
23171 @infoline @expr{(4046 m^2)^1.5},
23172 which is then changed to
23173 @texline @math{4046^{1.5} \, (m^2)^{1.5}},
23174 @infoline @expr{4046^1.5 (m^2)^1.5},
23175 then to @expr{257440 m^3}.
23177 The functions @code{float}, @code{frac}, @code{clean}, @code{abs},
23178 as well as @code{floor} and the other integer truncation functions,
23179 applied to unit names or products or quotients involving units, are
23180 simplified. For example, @samp{round(1.6 in)} is changed to
23181 @samp{round(1.6) round(in)}; the lefthand term evaluates to 2,
23182 and the righthand term simplifies to @code{in}.
23184 The functions @code{sin}, @code{cos}, and @code{tan} with arguments
23185 that have angular units like @code{rad} or @code{arcmin} are
23186 simplified by converting to base units (radians), then evaluating
23187 with the angular mode temporarily set to radians.
23189 @node Polynomials, Calculus, Simplifying Formulas, Algebra
23190 @section Polynomials
23192 A @dfn{polynomial} is a sum of terms which are coefficients times
23193 various powers of a ``base'' variable. For example, @expr{2 x^2 + 3 x - 4}
23194 is a polynomial in @expr{x}. Some formulas can be considered
23195 polynomials in several different variables: @expr{1 + 2 x + 3 y + 4 x y^2}
23196 is a polynomial in both @expr{x} and @expr{y}. Polynomial coefficients
23197 are often numbers, but they may in general be any formulas not
23198 involving the base variable.
23201 @pindex calc-factor
23203 The @kbd{a f} (@code{calc-factor}) [@code{factor}] command factors a
23204 polynomial into a product of terms. For example, the polynomial
23205 @expr{x^3 + 2 x^2 + x} is factored into @samp{x*(x+1)^2}. As another
23206 example, @expr{a c + b d + b c + a d} is factored into the product
23207 @expr{(a + b) (c + d)}.
23209 Calc currently has three algorithms for factoring. Formulas which are
23210 linear in several variables, such as the second example above, are
23211 merged according to the distributive law. Formulas which are
23212 polynomials in a single variable, with constant integer or fractional
23213 coefficients, are factored into irreducible linear and/or quadratic
23214 terms. The first example above factors into three linear terms
23215 (@expr{x}, @expr{x+1}, and @expr{x+1} again). Finally, formulas
23216 which do not fit the above criteria are handled by the algebraic
23219 Calc's polynomial factorization algorithm works by using the general
23220 root-finding command (@w{@kbd{a P}}) to solve for the roots of the
23221 polynomial. It then looks for roots which are rational numbers
23222 or complex-conjugate pairs, and converts these into linear and
23223 quadratic terms, respectively. Because it uses floating-point
23224 arithmetic, it may be unable to find terms that involve large
23225 integers (whose number of digits approaches the current precision).
23226 Also, irreducible factors of degree higher than quadratic are not
23227 found, and polynomials in more than one variable are not treated.
23228 (A more robust factorization algorithm may be included in a future
23231 @vindex FactorRules
23243 The rewrite-based factorization method uses rules stored in the variable
23244 @code{FactorRules}. @xref{Rewrite Rules}, for a discussion of the
23245 operation of rewrite rules. The default @code{FactorRules} are able
23246 to factor quadratic forms symbolically into two linear terms,
23247 @expr{(a x + b) (c x + d)}. You can edit these rules to include other
23248 cases if you wish. To use the rules, Calc builds the formula
23249 @samp{thecoefs(x, [a, b, c, ...])} where @code{x} is the polynomial
23250 base variable and @code{a}, @code{b}, etc., are polynomial coefficients
23251 (which may be numbers or formulas). The constant term is written first,
23252 i.e., in the @code{a} position. When the rules complete, they should have
23253 changed the formula into the form @samp{thefactors(x, [f1, f2, f3, ...])}
23254 where each @code{fi} should be a factored term, e.g., @samp{x - ai}.
23255 Calc then multiplies these terms together to get the complete
23256 factored form of the polynomial. If the rules do not change the
23257 @code{thecoefs} call to a @code{thefactors} call, @kbd{a f} leaves the
23258 polynomial alone on the assumption that it is unfactorable. (Note that
23259 the function names @code{thecoefs} and @code{thefactors} are used only
23260 as placeholders; there are no actual Calc functions by those names.)
23264 The @kbd{H a f} [@code{factors}] command also factors a polynomial,
23265 but it returns a list of factors instead of an expression which is the
23266 product of the factors. Each factor is represented by a sub-vector
23267 of the factor, and the power with which it appears. For example,
23268 @expr{x^5 + x^4 - 33 x^3 + 63 x^2} factors to @expr{(x + 7) x^2 (x - 3)^2}
23269 in @kbd{a f}, or to @expr{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}.
23270 If there is an overall numeric factor, it always comes first in the list.
23271 The functions @code{factor} and @code{factors} allow a second argument
23272 when written in algebraic form; @samp{factor(x,v)} factors @expr{x} with
23273 respect to the specific variable @expr{v}. The default is to factor with
23274 respect to all the variables that appear in @expr{x}.
23277 @pindex calc-collect
23279 The @kbd{a c} (@code{calc-collect}) [@code{collect}] command rearranges a
23281 polynomial in a given variable, ordered in decreasing powers of that
23282 variable. For example, given @expr{1 + 2 x + 3 y + 4 x y^2} on
23283 the stack, @kbd{a c x} would produce @expr{(2 + 4 y^2) x + (1 + 3 y)},
23284 and @kbd{a c y} would produce @expr{(4 x) y^2 + 3 y + (1 + 2 x)}.
23285 The polynomial will be expanded out using the distributive law as
23286 necessary: Collecting @expr{x} in @expr{(x - 1)^3} produces
23287 @expr{x^3 - 3 x^2 + 3 x - 1}. Terms not involving @expr{x} will
23290 The ``variable'' you specify at the prompt can actually be any
23291 expression: @kbd{a c ln(x+1)} will collect together all terms multiplied
23292 by @samp{ln(x+1)} or integer powers thereof. If @samp{x} also appears
23293 in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will
23294 treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants.
23297 @pindex calc-expand
23299 The @kbd{a x} (@code{calc-expand}) [@code{expand}] command expands an
23300 expression by applying the distributive law everywhere. It applies to
23301 products, quotients, and powers involving sums. By default, it fully
23302 distributes all parts of the expression. With a numeric prefix argument,
23303 the distributive law is applied only the specified number of times, then
23304 the partially expanded expression is left on the stack.
23306 The @kbd{a x} and @kbd{j D} commands are somewhat redundant. Use
23307 @kbd{a x} if you want to expand all products of sums in your formula.
23308 Use @kbd{j D} if you want to expand a particular specified term of
23309 the formula. There is an exactly analogous correspondence between
23310 @kbd{a f} and @kbd{j M}. (The @kbd{j D} and @kbd{j M} commands
23311 also know many other kinds of expansions, such as
23312 @samp{exp(a + b) = exp(a) exp(b)}, which @kbd{a x} and @kbd{a f}
23315 Calc's automatic simplifications will sometimes reverse a partial
23316 expansion. For example, the first step in expanding @expr{(x+1)^3} is
23317 to write @expr{(x+1) (x+1)^2}. If @kbd{a x} stops there and tries
23318 to put this formula onto the stack, though, Calc will automatically
23319 simplify it back to @expr{(x+1)^3} form. The solution is to turn
23320 simplification off first (@pxref{Simplification Modes}), or to run
23321 @kbd{a x} without a numeric prefix argument so that it expands all
23322 the way in one step.
23327 The @kbd{a a} (@code{calc-apart}) [@code{apart}] command expands a
23328 rational function by partial fractions. A rational function is the
23329 quotient of two polynomials; @code{apart} pulls this apart into a
23330 sum of rational functions with simple denominators. In algebraic
23331 notation, the @code{apart} function allows a second argument that
23332 specifies which variable to use as the ``base''; by default, Calc
23333 chooses the base variable automatically.
23336 @pindex calc-normalize-rat
23338 The @kbd{a n} (@code{calc-normalize-rat}) [@code{nrat}] command
23339 attempts to arrange a formula into a quotient of two polynomials.
23340 For example, given @expr{1 + (a + b/c) / d}, the result would be
23341 @expr{(b + a c + c d) / c d}. The quotient is reduced, so that
23342 @kbd{a n} will simplify @expr{(x^2 + 2x + 1) / (x^2 - 1)} by dividing
23343 out the common factor @expr{x + 1}, yielding @expr{(x + 1) / (x - 1)}.
23346 @pindex calc-poly-div
23348 The @kbd{a \} (@code{calc-poly-div}) [@code{pdiv}] command divides
23349 two polynomials @expr{u} and @expr{v}, yielding a new polynomial
23350 @expr{q}. If several variables occur in the inputs, the inputs are
23351 considered multivariate polynomials. (Calc divides by the variable
23352 with the largest power in @expr{u} first, or, in the case of equal
23353 powers, chooses the variables in alphabetical order.) For example,
23354 dividing @expr{x^2 + 3 x + 2} by @expr{x + 2} yields @expr{x + 1}.
23355 The remainder from the division, if any, is reported at the bottom
23356 of the screen and is also placed in the Trail along with the quotient.
23358 Using @code{pdiv} in algebraic notation, you can specify the particular
23359 variable to be used as the base: @code{pdiv(@var{a},@var{b},@var{x})}.
23360 If @code{pdiv} is given only two arguments (as is always the case with
23361 the @kbd{a \} command), then it does a multivariate division as outlined
23365 @pindex calc-poly-rem
23367 The @kbd{a %} (@code{calc-poly-rem}) [@code{prem}] command divides
23368 two polynomials and keeps the remainder @expr{r}. The quotient
23369 @expr{q} is discarded. For any formulas @expr{a} and @expr{b}, the
23370 results of @kbd{a \} and @kbd{a %} satisfy @expr{a = q b + r}.
23371 (This is analogous to plain @kbd{\} and @kbd{%}, which compute the
23372 integer quotient and remainder from dividing two numbers.)
23376 @pindex calc-poly-div-rem
23379 The @kbd{a /} (@code{calc-poly-div-rem}) [@code{pdivrem}] command
23380 divides two polynomials and reports both the quotient and the
23381 remainder as a vector @expr{[q, r]}. The @kbd{H a /} [@code{pdivide}]
23382 command divides two polynomials and constructs the formula
23383 @expr{q + r/b} on the stack. (Naturally if the remainder is zero,
23384 this will immediately simplify to @expr{q}.)
23387 @pindex calc-poly-gcd
23389 The @kbd{a g} (@code{calc-poly-gcd}) [@code{pgcd}] command computes
23390 the greatest common divisor of two polynomials. (The GCD actually
23391 is unique only to within a constant multiplier; Calc attempts to
23392 choose a GCD which will be unsurprising.) For example, the @kbd{a n}
23393 command uses @kbd{a g} to take the GCD of the numerator and denominator
23394 of a quotient, then divides each by the result using @kbd{a \}. (The
23395 definition of GCD ensures that this division can take place without
23396 leaving a remainder.)
23398 While the polynomials used in operations like @kbd{a /} and @kbd{a g}
23399 often have integer coefficients, this is not required. Calc can also
23400 deal with polynomials over the rationals or floating-point reals.
23401 Polynomials with modulo-form coefficients are also useful in many
23402 applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc
23403 automatically transforms this into a polynomial over the field of
23404 integers mod 5: @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}.
23406 Congratulations and thanks go to Ove Ewerlid
23407 (@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the
23408 polynomial routines used in the above commands.
23410 @xref{Decomposing Polynomials}, for several useful functions for
23411 extracting the individual coefficients of a polynomial.
23413 @node Calculus, Solving Equations, Polynomials, Algebra
23417 The following calculus commands do not automatically simplify their
23418 inputs or outputs using @code{calc-simplify}. You may find it helps
23419 to do this by hand by typing @kbd{a s} or @kbd{a e}. It may also help
23420 to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most
23424 * Differentiation::
23426 * Customizing the Integrator::
23427 * Numerical Integration::
23431 @node Differentiation, Integration, Calculus, Calculus
23432 @subsection Differentiation
23437 @pindex calc-derivative
23440 The @kbd{a d} (@code{calc-derivative}) [@code{deriv}] command computes
23441 the derivative of the expression on the top of the stack with respect to
23442 some variable, which it will prompt you to enter. Normally, variables
23443 in the formula other than the specified differentiation variable are
23444 considered constant, i.e., @samp{deriv(y,x)} is reduced to zero. With
23445 the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used
23446 instead, in which derivatives of variables are not reduced to zero
23447 unless those variables are known to be ``constant,'' i.e., independent
23448 of any other variables. (The built-in special variables like @code{pi}
23449 are considered constant, as are variables that have been declared
23450 @code{const}; @pxref{Declarations}.)
23452 With a numeric prefix argument @var{n}, this command computes the
23453 @var{n}th derivative.
23455 When working with trigonometric functions, it is best to switch to
23456 Radians mode first (with @w{@kbd{m r}}). The derivative of @samp{sin(x)}
23457 in degrees is @samp{(pi/180) cos(x)}, probably not the expected
23460 If you use the @code{deriv} function directly in an algebraic formula,
23461 you can write @samp{deriv(f,x,x0)} which represents the derivative
23462 of @expr{f} with respect to @expr{x}, evaluated at the point
23463 @texline @math{x=x_0}.
23464 @infoline @expr{x=x0}.
23466 If the formula being differentiated contains functions which Calc does
23467 not know, the derivatives of those functions are produced by adding
23468 primes (apostrophe characters). For example, @samp{deriv(f(2x), x)}
23469 produces @samp{2 f'(2 x)}, where the function @code{f'} represents the
23470 derivative of @code{f}.
23472 For functions you have defined with the @kbd{Z F} command, Calc expands
23473 the functions according to their defining formulas unless you have
23474 also defined @code{f'} suitably. For example, suppose we define
23475 @samp{sinc(x) = sin(x)/x} using @kbd{Z F}. If we then differentiate
23476 the formula @samp{sinc(2 x)}, the formula will be expanded to
23477 @samp{sin(2 x) / (2 x)} and differentiated. However, if we also
23478 define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the
23479 result as @samp{2 dsinc(2 x)}. @xref{Algebraic Definitions}.
23481 For multi-argument functions @samp{f(x,y,z)}, the derivative with respect
23482 to the first argument is written @samp{f'(x,y,z)}; derivatives with
23483 respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}.
23484 Various higher-order derivatives can be formed in the obvious way, e.g.,
23485 @samp{f'@var{}'(x)} (the second derivative of @code{f}) or
23486 @samp{f'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each
23489 @node Integration, Customizing the Integrator, Differentiation, Calculus
23490 @subsection Integration
23494 @pindex calc-integral
23496 The @kbd{a i} (@code{calc-integral}) [@code{integ}] command computes the
23497 indefinite integral of the expression on the top of the stack with
23498 respect to a variable. The integrator is not guaranteed to work for
23499 all integrable functions, but it is able to integrate several large
23500 classes of formulas. In particular, any polynomial or rational function
23501 (a polynomial divided by a polynomial) is acceptable. (Rational functions
23502 don't have to be in explicit quotient form, however;
23503 @texline @math{x/(1+x^{-2})}
23504 @infoline @expr{x/(1+x^-2)}
23505 is not strictly a quotient of polynomials, but it is equivalent to
23506 @expr{x^3/(x^2+1)}, which is.) Also, square roots of terms involving
23507 @expr{x} and @expr{x^2} may appear in rational functions being
23508 integrated. Finally, rational functions involving trigonometric or
23509 hyperbolic functions can be integrated.
23512 If you use the @code{integ} function directly in an algebraic formula,
23513 you can also write @samp{integ(f,x,v)} which expresses the resulting
23514 indefinite integral in terms of variable @code{v} instead of @code{x}.
23515 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23516 integral from @code{a} to @code{b}.
23519 If you use the @code{integ} function directly in an algebraic formula,
23520 you can also write @samp{integ(f,x,v)} which expresses the resulting
23521 indefinite integral in terms of variable @code{v} instead of @code{x}.
23522 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23523 integral $\int_a^b f(x) \, dx$.
23526 Please note that the current implementation of Calc's integrator sometimes
23527 produces results that are significantly more complex than they need to
23528 be. For example, the integral Calc finds for
23529 @texline @math{1/(x+\sqrt{x^2+1})}
23530 @infoline @expr{1/(x+sqrt(x^2+1))}
23531 is several times more complicated than the answer Mathematica
23532 returns for the same input, although the two forms are numerically
23533 equivalent. Also, any indefinite integral should be considered to have
23534 an arbitrary constant of integration added to it, although Calc does not
23535 write an explicit constant of integration in its result. For example,
23536 Calc's solution for
23537 @texline @math{1/(1+\tan x)}
23538 @infoline @expr{1/(1+tan(x))}
23539 differs from the solution given in the @emph{CRC Math Tables} by a
23541 @texline @math{\pi i / 2}
23542 @infoline @expr{pi i / 2},
23543 due to a different choice of constant of integration.
23545 The Calculator remembers all the integrals it has done. If conditions
23546 change in a way that would invalidate the old integrals, say, a switch
23547 from Degrees to Radians mode, then they will be thrown out. If you
23548 suspect this is not happening when it should, use the
23549 @code{calc-flush-caches} command; @pxref{Caches}.
23552 Calc normally will pursue integration by substitution or integration by
23553 parts up to 3 nested times before abandoning an approach as fruitless.
23554 If the integrator is taking too long, you can lower this limit by storing
23555 a number (like 2) in the variable @code{IntegLimit}. (The @kbd{s I}
23556 command is a convenient way to edit @code{IntegLimit}.) If this variable
23557 has no stored value or does not contain a nonnegative integer, a limit
23558 of 3 is used. The lower this limit is, the greater the chance that Calc
23559 will be unable to integrate a function it could otherwise handle. Raising
23560 this limit allows the Calculator to solve more integrals, though the time
23561 it takes may grow exponentially. You can monitor the integrator's actions
23562 by creating an Emacs buffer called @code{*Trace*}. If such a buffer
23563 exists, the @kbd{a i} command will write a log of its actions there.
23565 If you want to manipulate integrals in a purely symbolic way, you can
23566 set the integration nesting limit to 0 to prevent all but fast
23567 table-lookup solutions of integrals. You might then wish to define
23568 rewrite rules for integration by parts, various kinds of substitutions,
23569 and so on. @xref{Rewrite Rules}.
23571 @node Customizing the Integrator, Numerical Integration, Integration, Calculus
23572 @subsection Customizing the Integrator
23576 Calc has two built-in rewrite rules called @code{IntegRules} and
23577 @code{IntegAfterRules} which you can edit to define new integration
23578 methods. @xref{Rewrite Rules}. At each step of the integration process,
23579 Calc wraps the current integrand in a call to the fictitious function
23580 @samp{integtry(@var{expr},@var{var})}, where @var{expr} is the
23581 integrand and @var{var} is the integration variable. If your rules
23582 rewrite this to be a plain formula (not a call to @code{integtry}), then
23583 Calc will use this formula as the integral of @var{expr}. For example,
23584 the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to
23585 integrate a function @code{mysin} that acts like the sine function.
23586 Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y}
23587 will produce the integral @samp{-2 mycos(2y+1)}. Note that Calc has
23588 automatically made various transformations on the integral to allow it
23589 to use your rule; integral tables generally give rules for
23590 @samp{mysin(a x + b)}, but you don't need to use this much generality
23591 in your @code{IntegRules}.
23593 @cindex Exponential integral Ei(x)
23598 As a more serious example, the expression @samp{exp(x)/x} cannot be
23599 integrated in terms of the standard functions, so the ``exponential
23600 integral'' function
23601 @texline @math{{\rm Ei}(x)}
23602 @infoline @expr{Ei(x)}
23603 was invented to describe it.
23604 We can get Calc to do this integral in terms of a made-up @code{Ei}
23605 function by adding the rule @samp{[integtry(exp(x)/x, x) := Ei(x)]}
23606 to @code{IntegRules}. Now entering @samp{exp(2x)/x} on the stack
23607 and typing @kbd{a i x} yields @samp{Ei(2 x)}. This new rule will
23608 work with Calc's various built-in integration methods (such as
23609 integration by substitution) to solve a variety of other problems
23610 involving @code{Ei}: For example, now Calc will also be able to
23611 integrate @samp{exp(exp(x))} and @samp{ln(ln(x))} (to get @samp{Ei(exp(x))}
23612 and @samp{x ln(ln(x)) - Ei(ln(x))}, respectively).
23614 Your rule may do further integration by calling @code{integ}. For
23615 example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc
23616 to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}.
23617 Note that @code{integ} was called with only one argument. This notation
23618 is allowed only within @code{IntegRules}; it means ``integrate this
23619 with respect to the same integration variable.'' If Calc is unable
23620 to integrate @code{u}, the integration that invoked @code{IntegRules}
23621 also fails. Thus integrating @samp{twice(f(x))} fails, returning the
23622 unevaluated integral @samp{integ(twice(f(x)), x)}. It is still valid
23623 to call @code{integ} with two or more arguments, however; in this case,
23624 if @code{u} is not integrable, @code{twice} itself will still be
23625 integrated: If the above rule is changed to @samp{... := twice(integ(u,x))},
23626 then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}.
23628 If a rule instead produces the formula @samp{integsubst(@var{sexpr},
23629 @var{svar})}, either replacing the top-level @code{integtry} call or
23630 nested anywhere inside the expression, then Calc will apply the
23631 substitution @samp{@var{u} = @var{sexpr}(@var{svar})} to try to
23632 integrate the original @var{expr}. For example, the rule
23633 @samp{sqrt(a) := integsubst(sqrt(x),x)} says that if Calc ever finds
23634 a square root in the integrand, it should attempt the substitution
23635 @samp{u = sqrt(x)}. (This particular rule is unnecessary because
23636 Calc always tries ``obvious'' substitutions where @var{sexpr} actually
23637 appears in the integrand.) The variable @var{svar} may be the same
23638 as the @var{var} that appeared in the call to @code{integtry}, but
23641 When integrating according to an @code{integsubst}, Calc uses the
23642 equation solver to find the inverse of @var{sexpr} (if the integrand
23643 refers to @var{var} anywhere except in subexpressions that exactly
23644 match @var{sexpr}). It uses the differentiator to find the derivative
23645 of @var{sexpr} and/or its inverse (it has two methods that use one
23646 derivative or the other). You can also specify these items by adding
23647 extra arguments to the @code{integsubst} your rules construct; the
23648 general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv},
23649 @var{sprime})}, where @var{sinv} is the inverse of @var{sexpr} (still
23650 written as a function of @var{svar}), and @var{sprime} is the
23651 derivative of @var{sexpr} with respect to @var{svar}. If you don't
23652 specify these things, and Calc is not able to work them out on its
23653 own with the information it knows, then your substitution rule will
23654 work only in very specific, simple cases.
23656 Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules};
23657 in other words, Calc stops rewriting as soon as any rule in your rule
23658 set succeeds. (If it weren't for this, the @samp{integsubst(sqrt(x),x)}
23659 example above would keep on adding layers of @code{integsubst} calls
23662 @vindex IntegSimpRules
23663 Another set of rules, stored in @code{IntegSimpRules}, are applied
23664 every time the integrator uses @kbd{a s} to simplify an intermediate
23665 result. For example, putting the rule @samp{twice(x) := 2 x} into
23666 @code{IntegSimpRules} would tell Calc to convert the @code{twice}
23667 function into a form it knows whenever integration is attempted.
23669 One more way to influence the integrator is to define a function with
23670 the @kbd{Z F} command (@pxref{Algebraic Definitions}). Calc's
23671 integrator automatically expands such functions according to their
23672 defining formulas, even if you originally asked for the function to
23673 be left unevaluated for symbolic arguments. (Certain other Calc
23674 systems, such as the differentiator and the equation solver, also
23677 @vindex IntegAfterRules
23678 Sometimes Calc is able to find a solution to your integral, but it
23679 expresses the result in a way that is unnecessarily complicated. If
23680 this happens, you can either use @code{integsubst} as described
23681 above to try to hint at a more direct path to the desired result, or
23682 you can use @code{IntegAfterRules}. This is an extra rule set that
23683 runs after the main integrator returns its result; basically, Calc does
23684 an @kbd{a r IntegAfterRules} on the result before showing it to you.
23685 (It also does an @kbd{a s}, without @code{IntegSimpRules}, after that
23686 to further simplify the result.) For example, Calc's integrator
23687 sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)};
23688 the default @code{IntegAfterRules} rewrite this into the more readable
23689 form @samp{2 arctanh(x)}. Note that, unlike @code{IntegRules},
23690 @code{IntegSimpRules} and @code{IntegAfterRules} are applied any number
23691 of times until no further changes are possible. Rewriting by
23692 @code{IntegAfterRules} occurs only after the main integrator has
23693 finished, not at every step as for @code{IntegRules} and
23694 @code{IntegSimpRules}.
23696 @node Numerical Integration, Taylor Series, Customizing the Integrator, Calculus
23697 @subsection Numerical Integration
23701 @pindex calc-num-integral
23703 If you want a purely numerical answer to an integration problem, you can
23704 use the @kbd{a I} (@code{calc-num-integral}) [@code{ninteg}] command. This
23705 command prompts for an integration variable, a lower limit, and an
23706 upper limit. Except for the integration variable, all other variables
23707 that appear in the integrand formula must have stored values. (A stored
23708 value, if any, for the integration variable itself is ignored.)
23710 Numerical integration works by evaluating your formula at many points in
23711 the specified interval. Calc uses an ``open Romberg'' method; this means
23712 that it does not evaluate the formula actually at the endpoints (so that
23713 it is safe to integrate @samp{sin(x)/x} from zero, for example). Also,
23714 the Romberg method works especially well when the function being
23715 integrated is fairly smooth. If the function is not smooth, Calc will
23716 have to evaluate it at quite a few points before it can accurately
23717 determine the value of the integral.
23719 Integration is much faster when the current precision is small. It is
23720 best to set the precision to the smallest acceptable number of digits
23721 before you use @kbd{a I}. If Calc appears to be taking too long, press
23722 @kbd{C-g} to halt it and try a lower precision. If Calc still appears
23723 to need hundreds of evaluations, check to make sure your function is
23724 well-behaved in the specified interval.
23726 It is possible for the lower integration limit to be @samp{-inf} (minus
23727 infinity). Likewise, the upper limit may be plus infinity. Calc
23728 internally transforms the integral into an equivalent one with finite
23729 limits. However, integration to or across singularities is not supported:
23730 The integral of @samp{1/sqrt(x)} from 0 to 1 exists (it can be found
23731 by Calc's symbolic integrator, for example), but @kbd{a I} will fail
23732 because the integrand goes to infinity at one of the endpoints.
23734 @node Taylor Series, , Numerical Integration, Calculus
23735 @subsection Taylor Series
23739 @pindex calc-taylor
23741 The @kbd{a t} (@code{calc-taylor}) [@code{taylor}] command computes a
23742 power series expansion or Taylor series of a function. You specify the
23743 variable and the desired number of terms. You may give an expression of
23744 the form @samp{@var{var} = @var{a}} or @samp{@var{var} - @var{a}} instead
23745 of just a variable to produce a Taylor expansion about the point @var{a}.
23746 You may specify the number of terms with a numeric prefix argument;
23747 otherwise the command will prompt you for the number of terms. Note that
23748 many series expansions have coefficients of zero for some terms, so you
23749 may appear to get fewer terms than you asked for.
23751 If the @kbd{a i} command is unable to find a symbolic integral for a
23752 function, you can get an approximation by integrating the function's
23755 @node Solving Equations, Numerical Solutions, Calculus, Algebra
23756 @section Solving Equations
23760 @pindex calc-solve-for
23762 @cindex Equations, solving
23763 @cindex Solving equations
23764 The @kbd{a S} (@code{calc-solve-for}) [@code{solve}] command rearranges
23765 an equation to solve for a specific variable. An equation is an
23766 expression of the form @expr{L = R}. For example, the command @kbd{a S x}
23767 will rearrange @expr{y = 3x + 6} to the form, @expr{x = y/3 - 2}. If the
23768 input is not an equation, it is treated like an equation of the
23771 This command also works for inequalities, as in @expr{y < 3x + 6}.
23772 Some inequalities cannot be solved where the analogous equation could
23773 be; for example, solving
23774 @texline @math{a < b \, c}
23775 @infoline @expr{a < b c}
23776 for @expr{b} is impossible
23777 without knowing the sign of @expr{c}. In this case, @kbd{a S} will
23779 @texline @math{b \mathbin{\hbox{\code{!=}}} a/c}
23780 @infoline @expr{b != a/c}
23781 (using the not-equal-to operator) to signify that the direction of the
23782 inequality is now unknown. The inequality
23783 @texline @math{a \le b \, c}
23784 @infoline @expr{a <= b c}
23785 is not even partially solved. @xref{Declarations}, for a way to tell
23786 Calc that the signs of the variables in a formula are in fact known.
23788 Two useful commands for working with the result of @kbd{a S} are
23789 @kbd{a .} (@pxref{Logical Operations}), which converts @expr{x = y/3 - 2}
23790 to @expr{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates
23791 another formula with @expr{x} set equal to @expr{y/3 - 2}.
23794 * Multiple Solutions::
23795 * Solving Systems of Equations::
23796 * Decomposing Polynomials::
23799 @node Multiple Solutions, Solving Systems of Equations, Solving Equations, Solving Equations
23800 @subsection Multiple Solutions
23805 Some equations have more than one solution. The Hyperbolic flag
23806 (@code{H a S}) [@code{fsolve}] tells the solver to report the fully
23807 general family of solutions. It will invent variables @code{n1},
23808 @code{n2}, @dots{}, which represent independent arbitrary integers, and
23809 @code{s1}, @code{s2}, @dots{}, which represent independent arbitrary
23810 signs (either @mathit{+1} or @mathit{-1}). If you don't use the Hyperbolic
23811 flag, Calc will use zero in place of all arbitrary integers, and plus
23812 one in place of all arbitrary signs. Note that variables like @code{n1}
23813 and @code{s1} are not given any special interpretation in Calc except by
23814 the equation solver itself. As usual, you can use the @w{@kbd{s l}}
23815 (@code{calc-let}) command to obtain solutions for various actual values
23816 of these variables.
23818 For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to
23819 get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the
23820 equation are @samp{sqrt(y)} and @samp{-sqrt(y)}. Another way to
23821 think about it is that the square-root operation is really a
23822 two-valued function; since every Calc function must return a
23823 single result, @code{sqrt} chooses to return the positive result.
23824 Then @kbd{H a S} doctors this result using @code{s1} to indicate
23825 the full set of possible values of the mathematical square-root.
23827 There is a similar phenomenon going the other direction: Suppose
23828 we solve @samp{sqrt(y) = x} for @code{y}. Calc squares both sides
23829 to get @samp{y = x^2}. This is correct, except that it introduces
23830 some dubious solutions. Consider solving @samp{sqrt(y) = -3}:
23831 Calc will report @expr{y = 9} as a valid solution, which is true
23832 in the mathematical sense of square-root, but false (there is no
23833 solution) for the actual Calc positive-valued @code{sqrt}. This
23834 happens for both @kbd{a S} and @kbd{H a S}.
23836 @cindex @code{GenCount} variable
23846 If you store a positive integer in the Calc variable @code{GenCount},
23847 then Calc will generate formulas of the form @samp{as(@var{n})} for
23848 arbitrary signs, and @samp{an(@var{n})} for arbitrary integers,
23849 where @var{n} represents successive values taken by incrementing
23850 @code{GenCount} by one. While the normal arbitrary sign and
23851 integer symbols start over at @code{s1} and @code{n1} with each
23852 new Calc command, the @code{GenCount} approach will give each
23853 arbitrary value a name that is unique throughout the entire Calc
23854 session. Also, the arbitrary values are function calls instead
23855 of variables, which is advantageous in some cases. For example,
23856 you can make a rewrite rule that recognizes all arbitrary signs
23857 using a pattern like @samp{as(n)}. The @kbd{s l} command only works
23858 on variables, but you can use the @kbd{a b} (@code{calc-substitute})
23859 command to substitute actual values for function calls like @samp{as(3)}.
23861 The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient
23862 way to create or edit this variable. Press @kbd{C-c C-c} to finish.
23864 If you have not stored a value in @code{GenCount}, or if the value
23865 in that variable is not a positive integer, the regular
23866 @code{s1}/@code{n1} notation is used.
23872 With the Inverse flag, @kbd{I a S} [@code{finv}] treats the expression
23873 on top of the stack as a function of the specified variable and solves
23874 to find the inverse function, written in terms of the same variable.
23875 For example, @kbd{I a S x} inverts @expr{2x + 6} to @expr{x/2 - 3}.
23876 You can use both Inverse and Hyperbolic [@code{ffinv}] to obtain a
23877 fully general inverse, as described above.
23880 @pindex calc-poly-roots
23882 Some equations, specifically polynomials, have a known, finite number
23883 of solutions. The @kbd{a P} (@code{calc-poly-roots}) [@code{roots}]
23884 command uses @kbd{H a S} to solve an equation in general form, then, for
23885 all arbitrary-sign variables like @code{s1}, and all arbitrary-integer
23886 variables like @code{n1} for which @code{n1} only usefully varies over
23887 a finite range, it expands these variables out to all their possible
23888 values. The results are collected into a vector, which is returned.
23889 For example, @samp{roots(x^4 = 1, x)} returns the four solutions
23890 @samp{[1, -1, (0, 1), (0, -1)]}. Generally an @var{n}th degree
23891 polynomial will always have @var{n} roots on the complex plane.
23892 (If you have given a @code{real} declaration for the solution
23893 variable, then only the real-valued solutions, if any, will be
23894 reported; @pxref{Declarations}.)
23896 Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver
23897 symbolic solutions if the polynomial has symbolic coefficients. Also
23898 note that Calc's solver is not able to get exact symbolic solutions
23899 to all polynomials. Polynomials containing powers up to @expr{x^4}
23900 can always be solved exactly; polynomials of higher degree sometimes
23901 can be: @expr{x^6 + x^3 + 1} is converted to @expr{(x^3)^2 + (x^3) + 1},
23902 which can be solved for @expr{x^3} using the quadratic equation, and then
23903 for @expr{x} by taking cube roots. But in many cases, like
23904 @expr{x^6 + x + 1}, Calc does not know how to rewrite the polynomial
23905 into a form it can solve. The @kbd{a P} command can still deliver a
23906 list of numerical roots, however, provided that Symbolic mode (@kbd{m s})
23907 is not turned on. (If you work with Symbolic mode on, recall that the
23908 @kbd{N} (@code{calc-eval-num}) key is a handy way to reevaluate the
23909 formula on the stack with Symbolic mode temporarily off.) Naturally,
23910 @kbd{a P} can only provide numerical roots if the polynomial coefficients
23911 are all numbers (real or complex).
23913 @node Solving Systems of Equations, Decomposing Polynomials, Multiple Solutions, Solving Equations
23914 @subsection Solving Systems of Equations
23917 @cindex Systems of equations, symbolic
23918 You can also use the commands described above to solve systems of
23919 simultaneous equations. Just create a vector of equations, then
23920 specify a vector of variables for which to solve. (You can omit
23921 the surrounding brackets when entering the vector of variables
23924 For example, putting @samp{[x + y = a, x - y = b]} on the stack
23925 and typing @kbd{a S x,y @key{RET}} produces the vector of solutions
23926 @samp{[x = a - (a-b)/2, y = (a-b)/2]}. The result vector will
23927 have the same length as the variables vector, and the variables
23928 will be listed in the same order there. Note that the solutions
23929 are not always simplified as far as possible; the solution for
23930 @expr{x} here could be improved by an application of the @kbd{a n}
23933 Calc's algorithm works by trying to eliminate one variable at a
23934 time by solving one of the equations for that variable and then
23935 substituting into the other equations. Calc will try all the
23936 possibilities, but you can speed things up by noting that Calc
23937 first tries to eliminate the first variable with the first
23938 equation, then the second variable with the second equation,
23939 and so on. It also helps to put the simpler (e.g., more linear)
23940 equations toward the front of the list. Calc's algorithm will
23941 solve any system of linear equations, and also many kinds of
23948 Normally there will be as many variables as equations. If you
23949 give fewer variables than equations (an ``over-determined'' system
23950 of equations), Calc will find a partial solution. For example,
23951 typing @kbd{a S y @key{RET}} with the above system of equations
23952 would produce @samp{[y = a - x]}. There are now several ways to
23953 express this solution in terms of the original variables; Calc uses
23954 the first one that it finds. You can control the choice by adding
23955 variable specifiers of the form @samp{elim(@var{v})} to the
23956 variables list. This says that @var{v} should be eliminated from
23957 the equations; the variable will not appear at all in the solution.
23958 For example, typing @kbd{a S y,elim(x)} would yield
23959 @samp{[y = a - (b+a)/2]}.
23961 If the variables list contains only @code{elim} specifiers,
23962 Calc simply eliminates those variables from the equations
23963 and then returns the resulting set of equations. For example,
23964 @kbd{a S elim(x)} produces @samp{[a - 2 y = b]}. Every variable
23965 eliminated will reduce the number of equations in the system
23968 Again, @kbd{a S} gives you one solution to the system of
23969 equations. If there are several solutions, you can use @kbd{H a S}
23970 to get a general family of solutions, or, if there is a finite
23971 number of solutions, you can use @kbd{a P} to get a list. (In
23972 the latter case, the result will take the form of a matrix where
23973 the rows are different solutions and the columns correspond to the
23974 variables you requested.)
23976 Another way to deal with certain kinds of overdetermined systems of
23977 equations is the @kbd{a F} command, which does least-squares fitting
23978 to satisfy the equations. @xref{Curve Fitting}.
23980 @node Decomposing Polynomials, , Solving Systems of Equations, Solving Equations
23981 @subsection Decomposing Polynomials
23988 The @code{poly} function takes a polynomial and a variable as
23989 arguments, and returns a vector of polynomial coefficients (constant
23990 coefficient first). For example, @samp{poly(x^3 + 2 x, x)} returns
23991 @expr{[0, 2, 0, 1]}. If the input is not a polynomial in @expr{x},
23992 the call to @code{poly} is left in symbolic form. If the input does
23993 not involve the variable @expr{x}, the input is returned in a list
23994 of length one, representing a polynomial with only a constant
23995 coefficient. The call @samp{poly(x, x)} returns the vector @expr{[0, 1]}.
23996 The last element of the returned vector is guaranteed to be nonzero;
23997 note that @samp{poly(0, x)} returns the empty vector @expr{[]}.
23998 Note also that @expr{x} may actually be any formula; for example,
23999 @samp{poly(sin(x)^2 - sin(x) + 3, sin(x))} returns @expr{[3, -1, 1]}.
24001 @cindex Coefficients of polynomial
24002 @cindex Degree of polynomial
24003 To get the @expr{x^k} coefficient of polynomial @expr{p}, use
24004 @samp{poly(p, x)_(k+1)}. To get the degree of polynomial @expr{p},
24005 use @samp{vlen(poly(p, x)) - 1}. For example, @samp{poly((x+1)^4, x)}
24006 returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)}
24007 gives the @expr{x^2} coefficient of this polynomial, 6.
24013 One important feature of the solver is its ability to recognize
24014 formulas which are ``essentially'' polynomials. This ability is
24015 made available to the user through the @code{gpoly} function, which
24016 is used just like @code{poly}: @samp{gpoly(@var{expr}, @var{var})}.
24017 If @var{expr} is a polynomial in some term which includes @var{var}, then
24018 this function will return a vector @samp{[@var{x}, @var{c}, @var{a}]}
24019 where @var{x} is the term that depends on @var{var}, @var{c} is a
24020 vector of polynomial coefficients (like the one returned by @code{poly}),
24021 and @var{a} is a multiplier which is usually 1. Basically,
24022 @samp{@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} +
24023 @var{c}_3 @var{x}^2 + ...)}. The last element of @var{c} is
24024 guaranteed to be non-zero, and @var{c} will not equal @samp{[1]}
24025 (i.e., the trivial decomposition @var{expr} = @var{x} is not
24026 considered a polynomial). One side effect is that @samp{gpoly(x, x)}
24027 and @samp{gpoly(6, x)}, both of which might be expected to recognize
24028 their arguments as polynomials, will not because the decomposition
24029 is considered trivial.
24031 For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]},
24032 since the expanded form of this polynomial is @expr{4 - 4 x + x^2}.
24034 The term @var{x} may itself be a polynomial in @var{var}. This is
24035 done to reduce the size of the @var{c} vector. For example,
24036 @samp{gpoly(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]},
24037 since a quadratic polynomial in @expr{x^2} is easier to solve than
24038 a quartic polynomial in @expr{x}.
24040 A few more examples of the kinds of polynomials @code{gpoly} can
24044 sin(x) - 1 [sin(x), [-1, 1], 1]
24045 x + 1/x - 1 [x, [1, -1, 1], 1/x]
24046 x + 1/x [x^2, [1, 1], 1/x]
24047 x^3 + 2 x [x^2, [2, 1], x]
24048 x + x^2:3 + sqrt(x) [x^1:6, [1, 1, 0, 1], x^1:2]
24049 x^(2a) + 2 x^a + 5 [x^a, [5, 2, 1], 1]
24050 (exp(-x) + exp(x)) / 2 [e^(2 x), [0.5, 0.5], e^-x]
24053 The @code{poly} and @code{gpoly} functions accept a third integer argument
24054 which specifies the largest degree of polynomial that is acceptable.
24055 If this is @expr{n}, then only @var{c} vectors of length @expr{n+1}
24056 or less will be returned. Otherwise, the @code{poly} or @code{gpoly}
24057 call will remain in symbolic form. For example, the equation solver
24058 can handle quartics and smaller polynomials, so it calls
24059 @samp{gpoly(@var{expr}, @var{var}, 4)} to discover whether @var{expr}
24060 can be treated by its linear, quadratic, cubic, or quartic formulas.
24066 The @code{pdeg} function computes the degree of a polynomial;
24067 @samp{pdeg(p,x)} is the highest power of @code{x} that appears in
24068 @code{p}. This is the same as @samp{vlen(poly(p,x))-1}, but is
24069 much more efficient. If @code{p} is constant with respect to @code{x},
24070 then @samp{pdeg(p,x) = 0}. If @code{p} is not a polynomial in @code{x}
24071 (e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated.
24072 It is possible to omit the second argument @code{x}, in which case
24073 @samp{pdeg(p)} returns the highest total degree of any term of the
24074 polynomial, counting all variables that appear in @code{p}. Note
24075 that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c};
24076 the degree of the constant zero is considered to be @code{-inf}
24083 The @code{plead} function finds the leading term of a polynomial.
24084 Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))},
24085 though again more efficient. In particular, @samp{plead((2x+1)^10, x)}
24086 returns 1024 without expanding out the list of coefficients. The
24087 value of @code{plead(p,x)} will be zero only if @expr{p = 0}.
24093 The @code{pcont} function finds the @dfn{content} of a polynomial. This
24094 is the greatest common divisor of all the coefficients of the polynomial.
24095 With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)}
24096 to get a list of coefficients, then uses @code{pgcd} (the polynomial
24097 GCD function) to combine these into an answer. For example,
24098 @samp{pcont(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}. The content is
24099 basically the ``biggest'' polynomial that can be divided into @code{p}
24100 exactly. The sign of the content is the same as the sign of the leading
24103 With only one argument, @samp{pcont(p)} computes the numerical
24104 content of the polynomial, i.e., the @code{gcd} of the numerical
24105 coefficients of all the terms in the formula. Note that @code{gcd}
24106 is defined on rational numbers as well as integers; it computes
24107 the @code{gcd} of the numerators and the @code{lcm} of the
24108 denominators. Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3.
24109 Dividing the polynomial by this number will clear all the
24110 denominators, as well as dividing by any common content in the
24111 numerators. The numerical content of a polynomial is negative only
24112 if all the coefficients in the polynomial are negative.
24118 The @code{pprim} function finds the @dfn{primitive part} of a
24119 polynomial, which is simply the polynomial divided (using @code{pdiv}
24120 if necessary) by its content. If the input polynomial has rational
24121 coefficients, the result will have integer coefficients in simplest
24124 @node Numerical Solutions, Curve Fitting, Solving Equations, Algebra
24125 @section Numerical Solutions
24128 Not all equations can be solved symbolically. The commands in this
24129 section use numerical algorithms that can find a solution to a specific
24130 instance of an equation to any desired accuracy. Note that the
24131 numerical commands are slower than their algebraic cousins; it is a
24132 good idea to try @kbd{a S} before resorting to these commands.
24134 (@xref{Curve Fitting}, for some other, more specialized, operations
24135 on numerical data.)
24140 * Numerical Systems of Equations::
24143 @node Root Finding, Minimization, Numerical Solutions, Numerical Solutions
24144 @subsection Root Finding
24148 @pindex calc-find-root
24150 @cindex Newton's method
24151 @cindex Roots of equations
24152 @cindex Numerical root-finding
24153 The @kbd{a R} (@code{calc-find-root}) [@code{root}] command finds a
24154 numerical solution (or @dfn{root}) of an equation. (This command treats
24155 inequalities the same as equations. If the input is any other kind
24156 of formula, it is interpreted as an equation of the form @expr{X = 0}.)
24158 The @kbd{a R} command requires an initial guess on the top of the
24159 stack, and a formula in the second-to-top position. It prompts for a
24160 solution variable, which must appear in the formula. All other variables
24161 that appear in the formula must have assigned values, i.e., when
24162 a value is assigned to the solution variable and the formula is
24163 evaluated with @kbd{=}, it should evaluate to a number. Any assigned
24164 value for the solution variable itself is ignored and unaffected by
24167 When the command completes, the initial guess is replaced on the stack
24168 by a vector of two numbers: The value of the solution variable that
24169 solves the equation, and the difference between the lefthand and
24170 righthand sides of the equation at that value. Ordinarily, the second
24171 number will be zero or very nearly zero. (Note that Calc uses a
24172 slightly higher precision while finding the root, and thus the second
24173 number may be slightly different from the value you would compute from
24174 the equation yourself.)
24176 The @kbd{v h} (@code{calc-head}) command is a handy way to extract
24177 the first element of the result vector, discarding the error term.
24179 The initial guess can be a real number, in which case Calc searches
24180 for a real solution near that number, or a complex number, in which
24181 case Calc searches the whole complex plane near that number for a
24182 solution, or it can be an interval form which restricts the search
24183 to real numbers inside that interval.
24185 Calc tries to use @kbd{a d} to take the derivative of the equation.
24186 If this succeeds, it uses Newton's method. If the equation is not
24187 differentiable Calc uses a bisection method. (If Newton's method
24188 appears to be going astray, Calc switches over to bisection if it
24189 can, or otherwise gives up. In this case it may help to try again
24190 with a slightly different initial guess.) If the initial guess is a
24191 complex number, the function must be differentiable.
24193 If the formula (or the difference between the sides of an equation)
24194 is negative at one end of the interval you specify and positive at
24195 the other end, the root finder is guaranteed to find a root.
24196 Otherwise, Calc subdivides the interval into small parts looking for
24197 positive and negative values to bracket the root. When your guess is
24198 an interval, Calc will not look outside that interval for a root.
24202 The @kbd{H a R} [@code{wroot}] command is similar to @kbd{a R}, except
24203 that if the initial guess is an interval for which the function has
24204 the same sign at both ends, then rather than subdividing the interval
24205 Calc attempts to widen it to enclose a root. Use this mode if
24206 you are not sure if the function has a root in your interval.
24208 If the function is not differentiable, and you give a simple number
24209 instead of an interval as your initial guess, Calc uses this widening
24210 process even if you did not type the Hyperbolic flag. (If the function
24211 @emph{is} differentiable, Calc uses Newton's method which does not
24212 require a bounding interval in order to work.)
24214 If Calc leaves the @code{root} or @code{wroot} function in symbolic
24215 form on the stack, it will normally display an explanation for why
24216 no root was found. If you miss this explanation, press @kbd{w}
24217 (@code{calc-why}) to get it back.
24219 @node Minimization, Numerical Systems of Equations, Root Finding, Numerical Solutions
24220 @subsection Minimization
24227 @pindex calc-find-minimum
24228 @pindex calc-find-maximum
24231 @cindex Minimization, numerical
24232 The @kbd{a N} (@code{calc-find-minimum}) [@code{minimize}] command
24233 finds a minimum value for a formula. It is very similar in operation
24234 to @kbd{a R} (@code{calc-find-root}): You give the formula and an initial
24235 guess on the stack, and are prompted for the name of a variable. The guess
24236 may be either a number near the desired minimum, or an interval enclosing
24237 the desired minimum. The function returns a vector containing the
24238 value of the variable which minimizes the formula's value, along
24239 with the minimum value itself.
24241 Note that this command looks for a @emph{local} minimum. Many functions
24242 have more than one minimum; some, like
24243 @texline @math{x \sin x},
24244 @infoline @expr{x sin(x)},
24245 have infinitely many. In fact, there is no easy way to define the
24246 ``global'' minimum of
24247 @texline @math{x \sin x}
24248 @infoline @expr{x sin(x)}
24249 but Calc can still locate any particular local minimum
24250 for you. Calc basically goes downhill from the initial guess until it
24251 finds a point at which the function's value is greater both to the left
24252 and to the right. Calc does not use derivatives when minimizing a function.
24254 If your initial guess is an interval and it looks like the minimum
24255 occurs at one or the other endpoint of the interval, Calc will return
24256 that endpoint only if that endpoint is closed; thus, minimizing @expr{17 x}
24257 over @expr{[2..3]} will return @expr{[2, 38]}, but minimizing over
24258 @expr{(2..3]} would report no minimum found. In general, you should
24259 use closed intervals to find literally the minimum value in that
24260 range of @expr{x}, or open intervals to find the local minimum, if
24261 any, that happens to lie in that range.
24263 Most functions are smooth and flat near their minimum values. Because
24264 of this flatness, if the current precision is, say, 12 digits, the
24265 variable can only be determined meaningfully to about six digits. Thus
24266 you should set the precision to twice as many digits as you need in your
24277 The @kbd{H a N} [@code{wminimize}] command, analogously to @kbd{H a R},
24278 expands the guess interval to enclose a minimum rather than requiring
24279 that the minimum lie inside the interval you supply.
24281 The @kbd{a X} (@code{calc-find-maximum}) [@code{maximize}] and
24282 @kbd{H a X} [@code{wmaximize}] commands effectively minimize the
24283 negative of the formula you supply.
24285 The formula must evaluate to a real number at all points inside the
24286 interval (or near the initial guess if the guess is a number). If
24287 the initial guess is a complex number the variable will be minimized
24288 over the complex numbers; if it is real or an interval it will
24289 be minimized over the reals.
24291 @node Numerical Systems of Equations, , Minimization, Numerical Solutions
24292 @subsection Systems of Equations
24295 @cindex Systems of equations, numerical
24296 The @kbd{a R} command can also solve systems of equations. In this
24297 case, the equation should instead be a vector of equations, the
24298 guess should instead be a vector of numbers (intervals are not
24299 supported), and the variable should be a vector of variables. You
24300 can omit the brackets while entering the list of variables. Each
24301 equation must be differentiable by each variable for this mode to
24302 work. The result will be a vector of two vectors: The variable
24303 values that solved the system of equations, and the differences
24304 between the sides of the equations with those variable values.
24305 There must be the same number of equations as variables. Since
24306 only plain numbers are allowed as guesses, the Hyperbolic flag has
24307 no effect when solving a system of equations.
24309 It is also possible to minimize over many variables with @kbd{a N}
24310 (or maximize with @kbd{a X}). Once again the variable name should
24311 be replaced by a vector of variables, and the initial guess should
24312 be an equal-sized vector of initial guesses. But, unlike the case of
24313 multidimensional @kbd{a R}, the formula being minimized should
24314 still be a single formula, @emph{not} a vector. Beware that
24315 multidimensional minimization is currently @emph{very} slow.
24317 @node Curve Fitting, Summations, Numerical Solutions, Algebra
24318 @section Curve Fitting
24321 The @kbd{a F} command fits a set of data to a @dfn{model formula},
24322 such as @expr{y = m x + b} where @expr{m} and @expr{b} are parameters
24323 to be determined. For a typical set of measured data there will be
24324 no single @expr{m} and @expr{b} that exactly fit the data; in this
24325 case, Calc chooses values of the parameters that provide the closest
24330 * Polynomial and Multilinear Fits::
24331 * Error Estimates for Fits::
24332 * Standard Nonlinear Models::
24333 * Curve Fitting Details::
24337 @node Linear Fits, Polynomial and Multilinear Fits, Curve Fitting, Curve Fitting
24338 @subsection Linear Fits
24342 @pindex calc-curve-fit
24344 @cindex Linear regression
24345 @cindex Least-squares fits
24346 The @kbd{a F} (@code{calc-curve-fit}) [@code{fit}] command attempts
24347 to fit a set of data (@expr{x} and @expr{y} vectors of numbers) to a
24348 straight line, polynomial, or other function of @expr{x}. For the
24349 moment we will consider only the case of fitting to a line, and we
24350 will ignore the issue of whether or not the model was in fact a good
24353 In a standard linear least-squares fit, we have a set of @expr{(x,y)}
24354 data points that we wish to fit to the model @expr{y = m x + b}
24355 by adjusting the parameters @expr{m} and @expr{b} to make the @expr{y}
24356 values calculated from the formula be as close as possible to the actual
24357 @expr{y} values in the data set. (In a polynomial fit, the model is
24358 instead, say, @expr{y = a x^3 + b x^2 + c x + d}. In a multilinear fit,
24359 we have data points of the form @expr{(x_1,x_2,x_3,y)} and our model is
24360 @expr{y = a x_1 + b x_2 + c x_3 + d}. These will be discussed later.)
24362 In the model formula, variables like @expr{x} and @expr{x_2} are called
24363 the @dfn{independent variables}, and @expr{y} is the @dfn{dependent
24364 variable}. Variables like @expr{m}, @expr{a}, and @expr{b} are called
24365 the @dfn{parameters} of the model.
24367 The @kbd{a F} command takes the data set to be fitted from the stack.
24368 By default, it expects the data in the form of a matrix. For example,
24369 for a linear or polynomial fit, this would be a
24370 @texline @math{2\times N}
24372 matrix where the first row is a list of @expr{x} values and the second
24373 row has the corresponding @expr{y} values. For the multilinear fit
24374 shown above, the matrix would have four rows (@expr{x_1}, @expr{x_2},
24375 @expr{x_3}, and @expr{y}, respectively).
24377 If you happen to have an
24378 @texline @math{N\times2}
24380 matrix instead of a
24381 @texline @math{2\times N}
24383 matrix, just press @kbd{v t} first to transpose the matrix.
24385 After you type @kbd{a F}, Calc prompts you to select a model. For a
24386 linear fit, press the digit @kbd{1}.
24388 Calc then prompts for you to name the variables. By default it chooses
24389 high letters like @expr{x} and @expr{y} for independent variables and
24390 low letters like @expr{a} and @expr{b} for parameters. (The dependent
24391 variable doesn't need a name.) The two kinds of variables are separated
24392 by a semicolon. Since you generally care more about the names of the
24393 independent variables than of the parameters, Calc also allows you to
24394 name only those and let the parameters use default names.
24396 For example, suppose the data matrix
24401 [ [ 1, 2, 3, 4, 5 ]
24402 [ 5, 7, 9, 11, 13 ] ]
24410 $$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr
24411 5 & 7 & 9 & 11 & 13 }
24417 is on the stack and we wish to do a simple linear fit. Type
24418 @kbd{a F}, then @kbd{1} for the model, then @key{RET} to use
24419 the default names. The result will be the formula @expr{3 + 2 x}
24420 on the stack. Calc has created the model expression @kbd{a + b x},
24421 then found the optimal values of @expr{a} and @expr{b} to fit the
24422 data. (In this case, it was able to find an exact fit.) Calc then
24423 substituted those values for @expr{a} and @expr{b} in the model
24426 The @kbd{a F} command puts two entries in the trail. One is, as
24427 always, a copy of the result that went to the stack; the other is
24428 a vector of the actual parameter values, written as equations:
24429 @expr{[a = 3, b = 2]}, in case you'd rather read them in a list
24430 than pick them out of the formula. (You can type @kbd{t y}
24431 to move this vector to the stack; see @ref{Trail Commands}.
24433 Specifying a different independent variable name will affect the
24434 resulting formula: @kbd{a F 1 k @key{RET}} produces @kbd{3 + 2 k}.
24435 Changing the parameter names (say, @kbd{a F 1 k;b,m @key{RET}}) will affect
24436 the equations that go into the trail.
24442 To see what happens when the fit is not exact, we could change
24443 the number 13 in the data matrix to 14 and try the fit again.
24450 Evaluating this formula, say with @kbd{v x 5 @key{RET} @key{TAB} V M $ @key{RET}}, shows
24451 a reasonably close match to the y-values in the data.
24454 [4.8, 7., 9.2, 11.4, 13.6]
24457 Since there is no line which passes through all the @var{n} data points,
24458 Calc has chosen a line that best approximates the data points using
24459 the method of least squares. The idea is to define the @dfn{chi-square}
24464 chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
24470 $$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
24475 which is clearly zero if @expr{a + b x} exactly fits all data points,
24476 and increases as various @expr{a + b x_i} values fail to match the
24477 corresponding @expr{y_i} values. There are several reasons why the
24478 summand is squared, one of them being to ensure that
24479 @texline @math{\chi^2 \ge 0}.
24480 @infoline @expr{chi^2 >= 0}.
24481 Least-squares fitting simply chooses the values of @expr{a} and @expr{b}
24482 for which the error
24483 @texline @math{\chi^2}
24484 @infoline @expr{chi^2}
24485 is as small as possible.
24487 Other kinds of models do the same thing but with a different model
24488 formula in place of @expr{a + b x_i}.
24494 A numeric prefix argument causes the @kbd{a F} command to take the
24495 data in some other form than one big matrix. A positive argument @var{n}
24496 will take @var{N} items from the stack, corresponding to the @var{n} rows
24497 of a data matrix. In the linear case, @var{n} must be 2 since there
24498 is always one independent variable and one dependent variable.
24500 A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two
24501 items from the stack, an @var{n}-row matrix of @expr{x} values, and a
24502 vector of @expr{y} values. If there is only one independent variable,
24503 the @expr{x} values can be either a one-row matrix or a plain vector,
24504 in which case the @kbd{C-u} prefix is the same as a @w{@kbd{C-u 2}} prefix.
24506 @node Polynomial and Multilinear Fits, Error Estimates for Fits, Linear Fits, Curve Fitting
24507 @subsection Polynomial and Multilinear Fits
24510 To fit the data to higher-order polynomials, just type one of the
24511 digits @kbd{2} through @kbd{9} when prompted for a model. For example,
24512 we could fit the original data matrix from the previous section
24513 (with 13, not 14) to a parabola instead of a line by typing
24514 @kbd{a F 2 @key{RET}}.
24517 2.00000000001 x - 1.5e-12 x^2 + 2.99999999999
24520 Note that since the constant and linear terms are enough to fit the
24521 data exactly, it's no surprise that Calc chose a tiny contribution
24522 for @expr{x^2}. (The fact that it's not exactly zero is due only
24523 to roundoff error. Since our data are exact integers, we could get
24524 an exact answer by typing @kbd{m f} first to get Fraction mode.
24525 Then the @expr{x^2} term would vanish altogether. Usually, though,
24526 the data being fitted will be approximate floats so Fraction mode
24529 Doing the @kbd{a F 2} fit on the data set with 14 instead of 13
24530 gives a much larger @expr{x^2} contribution, as Calc bends the
24531 line slightly to improve the fit.
24534 0.142857142855 x^2 + 1.34285714287 x + 3.59999999998
24537 An important result from the theory of polynomial fitting is that it
24538 is always possible to fit @var{n} data points exactly using a polynomial
24539 of degree @mathit{@var{n}-1}, sometimes called an @dfn{interpolating polynomial}.
24540 Using the modified (14) data matrix, a model number of 4 gives
24541 a polynomial that exactly matches all five data points:
24544 0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.
24547 The actual coefficients we get with a precision of 12, like
24548 @expr{0.0416666663588}, clearly suffer from loss of precision.
24549 It is a good idea to increase the working precision to several
24550 digits beyond what you need when you do a fitting operation.
24551 Or, if your data are exact, use Fraction mode to get exact
24554 You can type @kbd{i} instead of a digit at the model prompt to fit
24555 the data exactly to a polynomial. This just counts the number of
24556 columns of the data matrix to choose the degree of the polynomial
24559 Fitting data ``exactly'' to high-degree polynomials is not always
24560 a good idea, though. High-degree polynomials have a tendency to
24561 wiggle uncontrollably in between the fitting data points. Also,
24562 if the exact-fit polynomial is going to be used to interpolate or
24563 extrapolate the data, it is numerically better to use the @kbd{a p}
24564 command described below. @xref{Interpolation}.
24570 Another generalization of the linear model is to assume the
24571 @expr{y} values are a sum of linear contributions from several
24572 @expr{x} values. This is a @dfn{multilinear} fit, and it is also
24573 selected by the @kbd{1} digit key. (Calc decides whether the fit
24574 is linear or multilinear by counting the rows in the data matrix.)
24576 Given the data matrix,
24580 [ [ 1, 2, 3, 4, 5 ]
24582 [ 14.5, 15, 18.5, 22.5, 24 ] ]
24587 the command @kbd{a F 1 @key{RET}} will call the first row @expr{x} and the
24588 second row @expr{y}, and will fit the values in the third row to the
24589 model @expr{a + b x + c y}.
24595 Calc can do multilinear fits with any number of independent variables
24596 (i.e., with any number of data rows).
24602 Yet another variation is @dfn{homogeneous} linear models, in which
24603 the constant term is known to be zero. In the linear case, this
24604 means the model formula is simply @expr{a x}; in the multilinear
24605 case, the model might be @expr{a x + b y + c z}; and in the polynomial
24606 case, the model could be @expr{a x + b x^2 + c x^3}. You can get
24607 a homogeneous linear or multilinear model by pressing the letter
24608 @kbd{h} followed by a regular model key, like @kbd{1} or @kbd{2}.
24610 It is certainly possible to have other constrained linear models,
24611 like @expr{2.3 + a x} or @expr{a - 4 x}. While there is no single
24612 key to select models like these, a later section shows how to enter
24613 any desired model by hand. In the first case, for example, you
24614 would enter @kbd{a F ' 2.3 + a x}.
24616 Another class of models that will work but must be entered by hand
24617 are multinomial fits, e.g., @expr{a + b x + c y + d x^2 + e y^2 + f x y}.
24619 @node Error Estimates for Fits, Standard Nonlinear Models, Polynomial and Multilinear Fits, Curve Fitting
24620 @subsection Error Estimates for Fits
24625 With the Hyperbolic flag, @kbd{H a F} [@code{efit}] performs the same
24626 fitting operation as @kbd{a F}, but reports the coefficients as error
24627 forms instead of plain numbers. Fitting our two data matrices (first
24628 with 13, then with 14) to a line with @kbd{H a F} gives the results,
24632 2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x
24635 In the first case the estimated errors are zero because the linear
24636 fit is perfect. In the second case, the errors are nonzero but
24637 moderately small, because the data are still very close to linear.
24639 It is also possible for the @emph{input} to a fitting operation to
24640 contain error forms. The data values must either all include errors
24641 or all be plain numbers. Error forms can go anywhere but generally
24642 go on the numbers in the last row of the data matrix. If the last
24643 row contains error forms
24644 @texline `@var{y_i}@w{ @tfn{+/-} }@math{\sigma_i}',
24645 @infoline `@var{y_i}@w{ @tfn{+/-} }@var{sigma_i}',
24647 @texline @math{\chi^2}
24648 @infoline @expr{chi^2}
24653 chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
24659 $$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
24664 so that data points with larger error estimates contribute less to
24665 the fitting operation.
24667 If there are error forms on other rows of the data matrix, all the
24668 errors for a given data point are combined; the square root of the
24669 sum of the squares of the errors forms the
24670 @texline @math{\sigma_i}
24671 @infoline @expr{sigma_i}
24672 used for the data point.
24674 Both @kbd{a F} and @kbd{H a F} can accept error forms in the input
24675 matrix, although if you are concerned about error analysis you will
24676 probably use @kbd{H a F} so that the output also contains error
24679 If the input contains error forms but all the
24680 @texline @math{\sigma_i}
24681 @infoline @expr{sigma_i}
24682 values are the same, it is easy to see that the resulting fitted model
24683 will be the same as if the input did not have error forms at all
24684 @texline (@math{\chi^2}
24685 @infoline (@expr{chi^2}
24686 is simply scaled uniformly by
24687 @texline @math{1 / \sigma^2},
24688 @infoline @expr{1 / sigma^2},
24689 which doesn't affect where it has a minimum). But there @emph{will} be
24690 a difference in the estimated errors of the coefficients reported by
24693 Consult any text on statistical modeling of data for a discussion
24694 of where these error estimates come from and how they should be
24703 With the Inverse flag, @kbd{I a F} [@code{xfit}] produces even more
24704 information. The result is a vector of six items:
24708 The model formula with error forms for its coefficients or
24709 parameters. This is the result that @kbd{H a F} would have
24713 A vector of ``raw'' parameter values for the model. These are the
24714 polynomial coefficients or other parameters as plain numbers, in the
24715 same order as the parameters appeared in the final prompt of the
24716 @kbd{I a F} command. For polynomials of degree @expr{d}, this vector
24717 will have length @expr{M = d+1} with the constant term first.
24720 The covariance matrix @expr{C} computed from the fit. This is
24721 an @var{m}x@var{m} symmetric matrix; the diagonal elements
24722 @texline @math{C_{jj}}
24723 @infoline @expr{C_j_j}
24725 @texline @math{\sigma_j^2}
24726 @infoline @expr{sigma_j^2}
24727 of the parameters. The other elements are covariances
24728 @texline @math{\sigma_{ij}^2}
24729 @infoline @expr{sigma_i_j^2}
24730 that describe the correlation between pairs of parameters. (A related
24731 set of numbers, the @dfn{linear correlation coefficients}
24732 @texline @math{r_{ij}},
24733 @infoline @expr{r_i_j},
24735 @texline @math{\sigma_{ij}^2 / \sigma_i \, \sigma_j}.)
24736 @infoline @expr{sigma_i_j^2 / sigma_i sigma_j}.)
24739 A vector of @expr{M} ``parameter filter'' functions whose
24740 meanings are described below. If no filters are necessary this
24741 will instead be an empty vector; this is always the case for the
24742 polynomial and multilinear fits described so far.
24746 @texline @math{\chi^2}
24747 @infoline @expr{chi^2}
24748 for the fit, calculated by the formulas shown above. This gives a
24749 measure of the quality of the fit; statisticians consider
24750 @texline @math{\chi^2 \approx N - M}
24751 @infoline @expr{chi^2 = N - M}
24752 to indicate a moderately good fit (where again @expr{N} is the number of
24753 data points and @expr{M} is the number of parameters).
24756 A measure of goodness of fit expressed as a probability @expr{Q}.
24757 This is computed from the @code{utpc} probability distribution
24759 @texline @math{\chi^2}
24760 @infoline @expr{chi^2}
24761 with @expr{N - M} degrees of freedom. A
24762 value of 0.5 implies a good fit; some texts recommend that often
24763 @expr{Q = 0.1} or even 0.001 can signify an acceptable fit. In
24765 @texline @math{\chi^2}
24766 @infoline @expr{chi^2}
24767 statistics assume the errors in your inputs
24768 follow a normal (Gaussian) distribution; if they don't, you may
24769 have to accept smaller values of @expr{Q}.
24771 The @expr{Q} value is computed only if the input included error
24772 estimates. Otherwise, Calc will report the symbol @code{nan}
24773 for @expr{Q}. The reason is that in this case the
24774 @texline @math{\chi^2}
24775 @infoline @expr{chi^2}
24776 value has effectively been used to estimate the original errors
24777 in the input, and thus there is no redundant information left
24778 over to use for a confidence test.
24781 @node Standard Nonlinear Models, Curve Fitting Details, Error Estimates for Fits, Curve Fitting
24782 @subsection Standard Nonlinear Models
24785 The @kbd{a F} command also accepts other kinds of models besides
24786 lines and polynomials. Some common models have quick single-key
24787 abbreviations; others must be entered by hand as algebraic formulas.
24789 Here is a complete list of the standard models recognized by @kbd{a F}:
24793 Linear or multilinear. @mathit{a + b x + c y + d z}.
24795 Polynomials. @mathit{a + b x + c x^2 + d x^3}.
24797 Exponential. @mathit{a} @tfn{exp}@mathit{(b x)} @tfn{exp}@mathit{(c y)}.
24799 Base-10 exponential. @mathit{a} @tfn{10^}@mathit{(b x)} @tfn{10^}@mathit{(c y)}.
24801 Exponential (alternate notation). @tfn{exp}@mathit{(a + b x + c y)}.
24803 Base-10 exponential (alternate). @tfn{10^}@mathit{(a + b x + c y)}.
24805 Logarithmic. @mathit{a + b} @tfn{ln}@mathit{(x) + c} @tfn{ln}@mathit{(y)}.
24807 Base-10 logarithmic. @mathit{a + b} @tfn{log10}@mathit{(x) + c} @tfn{log10}@mathit{(y)}.
24809 General exponential. @mathit{a b^x c^y}.
24811 Power law. @mathit{a x^b y^c}.
24813 Quadratic. @mathit{a + b (x-c)^2 + d (x-e)^2}.
24816 @texline @math{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)}.
24817 @infoline @mathit{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}.
24820 All of these models are used in the usual way; just press the appropriate
24821 letter at the model prompt, and choose variable names if you wish. The
24822 result will be a formula as shown in the above table, with the best-fit
24823 values of the parameters substituted. (You may find it easier to read
24824 the parameter values from the vector that is placed in the trail.)
24826 All models except Gaussian and polynomials can generalize as shown to any
24827 number of independent variables. Also, all the built-in models have an
24828 additive or multiplicative parameter shown as @expr{a} in the above table
24829 which can be replaced by zero or one, as appropriate, by typing @kbd{h}
24830 before the model key.
24832 Note that many of these models are essentially equivalent, but express
24833 the parameters slightly differently. For example, @expr{a b^x} and
24834 the other two exponential models are all algebraic rearrangements of
24835 each other. Also, the ``quadratic'' model is just a degree-2 polynomial
24836 with the parameters expressed differently. Use whichever form best
24837 matches the problem.
24839 The HP-28/48 calculators support four different models for curve
24840 fitting, called @code{LIN}, @code{LOG}, @code{EXP}, and @code{PWR}.
24841 These correspond to Calc models @samp{a + b x}, @samp{a + b ln(x)},
24842 @samp{a exp(b x)}, and @samp{a x^b}, respectively. In each case,
24843 @expr{a} is what the HP-48 identifies as the ``intercept,'' and
24844 @expr{b} is what it calls the ``slope.''
24850 If the model you want doesn't appear on this list, press @kbd{'}
24851 (the apostrophe key) at the model prompt to enter any algebraic
24852 formula, such as @kbd{m x - b}, as the model. (Not all models
24853 will work, though---see the next section for details.)
24855 The model can also be an equation like @expr{y = m x + b}.
24856 In this case, Calc thinks of all the rows of the data matrix on
24857 equal terms; this model effectively has two parameters
24858 (@expr{m} and @expr{b}) and two independent variables (@expr{x}
24859 and @expr{y}), with no ``dependent'' variables. Model equations
24860 do not need to take this @expr{y =} form. For example, the
24861 implicit line equation @expr{a x + b y = 1} works fine as a
24864 When you enter a model, Calc makes an alphabetical list of all
24865 the variables that appear in the model. These are used for the
24866 default parameters, independent variables, and dependent variable
24867 (in that order). If you enter a plain formula (not an equation),
24868 Calc assumes the dependent variable does not appear in the formula
24869 and thus does not need a name.
24871 For example, if the model formula has the variables @expr{a,mu,sigma,t,x},
24872 and the data matrix has three rows (meaning two independent variables),
24873 Calc will use @expr{a,mu,sigma} as the default parameters, and the
24874 data rows will be named @expr{t} and @expr{x}, respectively. If you
24875 enter an equation instead of a plain formula, Calc will use @expr{a,mu}
24876 as the parameters, and @expr{sigma,t,x} as the three independent
24879 You can, of course, override these choices by entering something
24880 different at the prompt. If you leave some variables out of the list,
24881 those variables must have stored values and those stored values will
24882 be used as constants in the model. (Stored values for the parameters
24883 and independent variables are ignored by the @kbd{a F} command.)
24884 If you list only independent variables, all the remaining variables
24885 in the model formula will become parameters.
24887 If there are @kbd{$} signs in the model you type, they will stand
24888 for parameters and all other variables (in alphabetical order)
24889 will be independent. Use @kbd{$} for one parameter, @kbd{$$} for
24890 another, and so on. Thus @kbd{$ x + $$} is another way to describe
24893 If you type a @kbd{$} instead of @kbd{'} at the model prompt itself,
24894 Calc will take the model formula from the stack. (The data must then
24895 appear at the second stack level.) The same conventions are used to
24896 choose which variables in the formula are independent by default and
24897 which are parameters.
24899 Models taken from the stack can also be expressed as vectors of
24900 two or three elements, @expr{[@var{model}, @var{vars}]} or
24901 @expr{[@var{model}, @var{vars}, @var{params}]}. Each of @var{vars}
24902 and @var{params} may be either a variable or a vector of variables.
24903 (If @var{params} is omitted, all variables in @var{model} except
24904 those listed as @var{vars} are parameters.)
24906 When you enter a model manually with @kbd{'}, Calc puts a 3-vector
24907 describing the model in the trail so you can get it back if you wish.
24915 Finally, you can store a model in one of the Calc variables
24916 @code{Model1} or @code{Model2}, then use this model by typing
24917 @kbd{a F u} or @kbd{a F U} (respectively). The value stored in
24918 the variable can be any of the formats that @kbd{a F $} would
24919 accept for a model on the stack.
24925 Calc uses the principal values of inverse functions like @code{ln}
24926 and @code{arcsin} when doing fits. For example, when you enter
24927 the model @samp{y = sin(a t + b)} Calc actually uses the easier
24928 form @samp{arcsin(y) = a t + b}. The @code{arcsin} function always
24929 returns results in the range from @mathit{-90} to 90 degrees (or the
24930 equivalent range in radians). Suppose you had data that you
24931 believed to represent roughly three oscillations of a sine wave,
24932 so that the argument of the sine might go from zero to
24933 @texline @math{3\times360}
24934 @infoline @mathit{3*360}
24936 The above model would appear to be a good way to determine the
24937 true frequency and phase of the sine wave, but in practice it
24938 would fail utterly. The righthand side of the actual model
24939 @samp{arcsin(y) = a t + b} will grow smoothly with @expr{t}, but
24940 the lefthand side will bounce back and forth between @mathit{-90} and 90.
24941 No values of @expr{a} and @expr{b} can make the two sides match,
24942 even approximately.
24944 There is no good solution to this problem at present. You could
24945 restrict your data to small enough ranges so that the above problem
24946 doesn't occur (i.e., not straddling any peaks in the sine wave).
24947 Or, in this case, you could use a totally different method such as
24948 Fourier analysis, which is beyond the scope of the @kbd{a F} command.
24949 (Unfortunately, Calc does not currently have any facilities for
24950 taking Fourier and related transforms.)
24952 @node Curve Fitting Details, Interpolation, Standard Nonlinear Models, Curve Fitting
24953 @subsection Curve Fitting Details
24956 Calc's internal least-squares fitter can only handle multilinear
24957 models. More precisely, it can handle any model of the form
24958 @expr{a f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @expr{a,b,c}
24959 are the parameters and @expr{x,y,z} are the independent variables
24960 (of course there can be any number of each, not just three).
24962 In a simple multilinear or polynomial fit, it is easy to see how
24963 to convert the model into this form. For example, if the model
24964 is @expr{a + b x + c x^2}, then @expr{f(x) = 1}, @expr{g(x) = x},
24965 and @expr{h(x) = x^2} are suitable functions.
24967 For other models, Calc uses a variety of algebraic manipulations
24968 to try to put the problem into the form
24971 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)
24975 where @expr{Y,A,B,C,F,G,H} are arbitrary functions. It computes
24976 @expr{Y}, @expr{F}, @expr{G}, and @expr{H} for all the data points,
24977 does a standard linear fit to find the values of @expr{A}, @expr{B},
24978 and @expr{C}, then uses the equation solver to solve for @expr{a,b,c}
24979 in terms of @expr{A,B,C}.
24981 A remarkable number of models can be cast into this general form.
24982 We'll look at two examples here to see how it works. The power-law
24983 model @expr{y = a x^b} with two independent variables and two parameters
24984 can be rewritten as follows:
24989 y = exp(ln(a) + b ln(x))
24990 ln(y) = ln(a) + b ln(x)
24994 which matches the desired form with
24995 @texline @math{Y = \ln(y)},
24996 @infoline @expr{Y = ln(y)},
24997 @texline @math{A = \ln(a)},
24998 @infoline @expr{A = ln(a)},
24999 @expr{F = 1}, @expr{B = b}, and
25000 @texline @math{G = \ln(x)}.
25001 @infoline @expr{G = ln(x)}.
25002 Calc thus computes the logarithms of your @expr{y} and @expr{x} values,
25003 does a linear fit for @expr{A} and @expr{B}, then solves to get
25004 @texline @math{a = \exp(A)}
25005 @infoline @expr{a = exp(A)}
25008 Another interesting example is the ``quadratic'' model, which can
25009 be handled by expanding according to the distributive law.
25012 y = a + b*(x - c)^2
25013 y = a + b c^2 - 2 b c x + b x^2
25017 which matches with @expr{Y = y}, @expr{A = a + b c^2}, @expr{F = 1},
25018 @expr{B = -2 b c}, @expr{G = x} (the @mathit{-2} factor could just as easily
25019 have been put into @expr{G} instead of @expr{B}), @expr{C = b}, and
25022 The Gaussian model looks quite complicated, but a closer examination
25023 shows that it's actually similar to the quadratic model but with an
25024 exponential that can be brought to the top and moved into @expr{Y}.
25026 An example of a model that cannot be put into general linear
25027 form is a Gaussian with a constant background added on, i.e.,
25028 @expr{d} + the regular Gaussian formula. If you have a model like
25029 this, your best bet is to replace enough of your parameters with
25030 constants to make the model linearizable, then adjust the constants
25031 manually by doing a series of fits. You can compare the fits by
25032 graphing them, by examining the goodness-of-fit measures returned by
25033 @kbd{I a F}, or by some other method suitable to your application.
25034 Note that some models can be linearized in several ways. The
25035 Gaussian-plus-@var{d} model can be linearized by setting @expr{d}
25036 (the background) to a constant, or by setting @expr{b} (the standard
25037 deviation) and @expr{c} (the mean) to constants.
25039 To fit a model with constants substituted for some parameters, just
25040 store suitable values in those parameter variables, then omit them
25041 from the list of parameters when you answer the variables prompt.
25047 A last desperate step would be to use the general-purpose
25048 @code{minimize} function rather than @code{fit}. After all, both
25049 functions solve the problem of minimizing an expression (the
25050 @texline @math{\chi^2}
25051 @infoline @expr{chi^2}
25052 sum) by adjusting certain parameters in the expression. The @kbd{a F}
25053 command is able to use a vastly more efficient algorithm due to its
25054 special knowledge about linear chi-square sums, but the @kbd{a N}
25055 command can do the same thing by brute force.
25057 A compromise would be to pick out a few parameters without which the
25058 fit is linearizable, and use @code{minimize} on a call to @code{fit}
25059 which efficiently takes care of the rest of the parameters. The thing
25060 to be minimized would be the value of
25061 @texline @math{\chi^2}
25062 @infoline @expr{chi^2}
25063 returned as the fifth result of the @code{xfit} function:
25066 minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)
25070 where @code{gaus} represents the Gaussian model with background,
25071 @code{data} represents the data matrix, and @code{guess} represents
25072 the initial guess for @expr{d} that @code{minimize} requires.
25073 This operation will only be, shall we say, extraordinarily slow
25074 rather than astronomically slow (as would be the case if @code{minimize}
25075 were used by itself to solve the problem).
25081 The @kbd{I a F} [@code{xfit}] command is somewhat trickier when
25082 nonlinear models are used. The second item in the result is the
25083 vector of ``raw'' parameters @expr{A}, @expr{B}, @expr{C}. The
25084 covariance matrix is written in terms of those raw parameters.
25085 The fifth item is a vector of @dfn{filter} expressions. This
25086 is the empty vector @samp{[]} if the raw parameters were the same
25087 as the requested parameters, i.e., if @expr{A = a}, @expr{B = b},
25088 and so on (which is always true if the model is already linear
25089 in the parameters as written, e.g., for polynomial fits). If the
25090 parameters had to be rearranged, the fifth item is instead a vector
25091 of one formula per parameter in the original model. The raw
25092 parameters are expressed in these ``filter'' formulas as
25093 @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)} for @expr{B},
25096 When Calc needs to modify the model to return the result, it replaces
25097 @samp{fitdummy(1)} in all the filters with the first item in the raw
25098 parameters list, and so on for the other raw parameters, then
25099 evaluates the resulting filter formulas to get the actual parameter
25100 values to be substituted into the original model. In the case of
25101 @kbd{H a F} and @kbd{I a F} where the parameters must be error forms,
25102 Calc uses the square roots of the diagonal entries of the covariance
25103 matrix as error values for the raw parameters, then lets Calc's
25104 standard error-form arithmetic take it from there.
25106 If you use @kbd{I a F} with a nonlinear model, be sure to remember
25107 that the covariance matrix is in terms of the raw parameters,
25108 @emph{not} the actual requested parameters. It's up to you to
25109 figure out how to interpret the covariances in the presence of
25110 nontrivial filter functions.
25112 Things are also complicated when the input contains error forms.
25113 Suppose there are three independent and dependent variables, @expr{x},
25114 @expr{y}, and @expr{z}, one or more of which are error forms in the
25115 data. Calc combines all the error values by taking the square root
25116 of the sum of the squares of the errors. It then changes @expr{x}
25117 and @expr{y} to be plain numbers, and makes @expr{z} into an error
25118 form with this combined error. The @expr{Y(x,y,z)} part of the
25119 linearized model is evaluated, and the result should be an error
25120 form. The error part of that result is used for
25121 @texline @math{\sigma_i}
25122 @infoline @expr{sigma_i}
25123 for the data point. If for some reason @expr{Y(x,y,z)} does not return
25124 an error form, the combined error from @expr{z} is used directly for
25125 @texline @math{\sigma_i}.
25126 @infoline @expr{sigma_i}.
25127 Finally, @expr{z} is also stripped of its error
25128 for use in computing @expr{F(x,y,z)}, @expr{G(x,y,z)} and so on;
25129 the righthand side of the linearized model is computed in regular
25130 arithmetic with no error forms.
25132 (While these rules may seem complicated, they are designed to do
25133 the most reasonable thing in the typical case that @expr{Y(x,y,z)}
25134 depends only on the dependent variable @expr{z}, and in fact is
25135 often simply equal to @expr{z}. For common cases like polynomials
25136 and multilinear models, the combined error is simply used as the
25137 @texline @math{\sigma}
25138 @infoline @expr{sigma}
25139 for the data point with no further ado.)
25146 It may be the case that the model you wish to use is linearizable,
25147 but Calc's built-in rules are unable to figure it out. Calc uses
25148 its algebraic rewrite mechanism to linearize a model. The rewrite
25149 rules are kept in the variable @code{FitRules}. You can edit this
25150 variable using the @kbd{s e FitRules} command; in fact, there is
25151 a special @kbd{s F} command just for editing @code{FitRules}.
25152 @xref{Operations on Variables}.
25154 @xref{Rewrite Rules}, for a discussion of rewrite rules.
25188 Calc uses @code{FitRules} as follows. First, it converts the model
25189 to an equation if necessary and encloses the model equation in a
25190 call to the function @code{fitmodel} (which is not actually a defined
25191 function in Calc; it is only used as a placeholder by the rewrite rules).
25192 Parameter variables are renamed to function calls @samp{fitparam(1)},
25193 @samp{fitparam(2)}, and so on, and independent variables are renamed
25194 to @samp{fitvar(1)}, @samp{fitvar(2)}, etc. The dependent variable
25195 is the highest-numbered @code{fitvar}. For example, the power law
25196 model @expr{a x^b} is converted to @expr{y = a x^b}, then to
25200 fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))
25204 Calc then applies the rewrites as if by @samp{C-u 0 a r FitRules}.
25205 (The zero prefix means that rewriting should continue until no further
25206 changes are possible.)
25208 When rewriting is complete, the @code{fitmodel} call should have
25209 been replaced by a @code{fitsystem} call that looks like this:
25212 fitsystem(@var{Y}, @var{FGH}, @var{abc})
25216 where @var{Y} is a formula that describes the function @expr{Y(x,y,z)},
25217 @var{FGH} is the vector of formulas @expr{[F(x,y,z), G(x,y,z), H(x,y,z)]},
25218 and @var{abc} is the vector of parameter filters which refer to the
25219 raw parameters as @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)}
25220 for @expr{B}, etc. While the number of raw parameters (the length of
25221 the @var{FGH} vector) is usually the same as the number of original
25222 parameters (the length of the @var{abc} vector), this is not required.
25224 The power law model eventually boils down to
25228 fitsystem(ln(fitvar(2)),
25229 [1, ln(fitvar(1))],
25230 [exp(fitdummy(1)), fitdummy(2)])
25234 The actual implementation of @code{FitRules} is complicated; it
25235 proceeds in four phases. First, common rearrangements are done
25236 to try to bring linear terms together and to isolate functions like
25237 @code{exp} and @code{ln} either all the way ``out'' (so that they
25238 can be put into @var{Y}) or all the way ``in'' (so that they can
25239 be put into @var{abc} or @var{FGH}). In particular, all
25240 non-constant powers are converted to logs-and-exponentials form,
25241 and the distributive law is used to expand products of sums.
25242 Quotients are rewritten to use the @samp{fitinv} function, where
25243 @samp{fitinv(x)} represents @expr{1/x} while the @code{FitRules}
25244 are operating. (The use of @code{fitinv} makes recognition of
25245 linear-looking forms easier.) If you modify @code{FitRules}, you
25246 will probably only need to modify the rules for this phase.
25248 Phase two, whose rules can actually also apply during phases one
25249 and three, first rewrites @code{fitmodel} to a two-argument
25250 form @samp{fitmodel(@var{Y}, @var{model})}, where @var{Y} is
25251 initially zero and @var{model} has been changed from @expr{a=b}
25252 to @expr{a-b} form. It then tries to peel off invertible functions
25253 from the outside of @var{model} and put them into @var{Y} instead,
25254 calling the equation solver to invert the functions. Finally, when
25255 this is no longer possible, the @code{fitmodel} is changed to a
25256 four-argument @code{fitsystem}, where the fourth argument is
25257 @var{model} and the @var{FGH} and @var{abc} vectors are initially
25258 empty. (The last vector is really @var{ABC}, corresponding to
25259 raw parameters, for now.)
25261 Phase three converts a sum of items in the @var{model} to a sum
25262 of @samp{fitpart(@var{a}, @var{b}, @var{c})} terms which represent
25263 terms @samp{@var{a}*@var{b}*@var{c}} of the sum, where @var{a}
25264 is all factors that do not involve any variables, @var{b} is all
25265 factors that involve only parameters, and @var{c} is the factors
25266 that involve only independent variables. (If this decomposition
25267 is not possible, the rule set will not complete and Calc will
25268 complain that the model is too complex.) Then @code{fitpart}s
25269 with equal @var{b} or @var{c} components are merged back together
25270 using the distributive law in order to minimize the number of
25271 raw parameters needed.
25273 Phase four moves the @code{fitpart} terms into the @var{FGH} and
25274 @var{ABC} vectors. Also, some of the algebraic expansions that
25275 were done in phase 1 are undone now to make the formulas more
25276 computationally efficient. Finally, it calls the solver one more
25277 time to convert the @var{ABC} vector to an @var{abc} vector, and
25278 removes the fourth @var{model} argument (which by now will be zero)
25279 to obtain the three-argument @code{fitsystem} that the linear
25280 least-squares solver wants to see.
25286 @mindex hasfit@idots
25288 @tindex hasfitparams
25296 Two functions which are useful in connection with @code{FitRules}
25297 are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check
25298 whether @expr{x} refers to any parameters or independent variables,
25299 respectively. Specifically, these functions return ``true'' if the
25300 argument contains any @code{fitparam} (or @code{fitvar}) function
25301 calls, and ``false'' otherwise. (Recall that ``true'' means a
25302 nonzero number, and ``false'' means zero. The actual nonzero number
25303 returned is the largest @var{n} from all the @samp{fitparam(@var{n})}s
25304 or @samp{fitvar(@var{n})}s, respectively, that appear in the formula.)
25310 The @code{fit} function in algebraic notation normally takes four
25311 arguments, @samp{fit(@var{model}, @var{vars}, @var{params}, @var{data})},
25312 where @var{model} is the model formula as it would be typed after
25313 @kbd{a F '}, @var{vars} is the independent variable or a vector of
25314 independent variables, @var{params} likewise gives the parameter(s),
25315 and @var{data} is the data matrix. Note that the length of @var{vars}
25316 must be equal to the number of rows in @var{data} if @var{model} is
25317 an equation, or one less than the number of rows if @var{model} is
25318 a plain formula. (Actually, a name for the dependent variable is
25319 allowed but will be ignored in the plain-formula case.)
25321 If @var{params} is omitted, the parameters are all variables in
25322 @var{model} except those that appear in @var{vars}. If @var{vars}
25323 is also omitted, Calc sorts all the variables that appear in
25324 @var{model} alphabetically and uses the higher ones for @var{vars}
25325 and the lower ones for @var{params}.
25327 Alternatively, @samp{fit(@var{modelvec}, @var{data})} is allowed
25328 where @var{modelvec} is a 2- or 3-vector describing the model
25329 and variables, as discussed previously.
25331 If Calc is unable to do the fit, the @code{fit} function is left
25332 in symbolic form, ordinarily with an explanatory message. The
25333 message will be ``Model expression is too complex'' if the
25334 linearizer was unable to put the model into the required form.
25336 The @code{efit} (corresponding to @kbd{H a F}) and @code{xfit}
25337 (for @kbd{I a F}) functions are completely analogous.
25339 @node Interpolation, , Curve Fitting Details, Curve Fitting
25340 @subsection Polynomial Interpolation
25343 @pindex calc-poly-interp
25345 The @kbd{a p} (@code{calc-poly-interp}) [@code{polint}] command does
25346 a polynomial interpolation at a particular @expr{x} value. It takes
25347 two arguments from the stack: A data matrix of the sort used by
25348 @kbd{a F}, and a single number which represents the desired @expr{x}
25349 value. Calc effectively does an exact polynomial fit as if by @kbd{a F i},
25350 then substitutes the @expr{x} value into the result in order to get an
25351 approximate @expr{y} value based on the fit. (Calc does not actually
25352 use @kbd{a F i}, however; it uses a direct method which is both more
25353 efficient and more numerically stable.)
25355 The result of @kbd{a p} is actually a vector of two values: The @expr{y}
25356 value approximation, and an error measure @expr{dy} that reflects Calc's
25357 estimation of the probable error of the approximation at that value of
25358 @expr{x}. If the input @expr{x} is equal to any of the @expr{x} values
25359 in the data matrix, the output @expr{y} will be the corresponding @expr{y}
25360 value from the matrix, and the output @expr{dy} will be exactly zero.
25362 A prefix argument of 2 causes @kbd{a p} to take separate x- and
25363 y-vectors from the stack instead of one data matrix.
25365 If @expr{x} is a vector of numbers, @kbd{a p} will return a matrix of
25366 interpolated results for each of those @expr{x} values. (The matrix will
25367 have two columns, the @expr{y} values and the @expr{dy} values.)
25368 If @expr{x} is a formula instead of a number, the @code{polint} function
25369 remains in symbolic form; use the @kbd{a "} command to expand it out to
25370 a formula that describes the fit in symbolic terms.
25372 In all cases, the @kbd{a p} command leaves the data vectors or matrix
25373 on the stack. Only the @expr{x} value is replaced by the result.
25377 The @kbd{H a p} [@code{ratint}] command does a rational function
25378 interpolation. It is used exactly like @kbd{a p}, except that it
25379 uses as its model the quotient of two polynomials. If there are
25380 @expr{N} data points, the numerator and denominator polynomials will
25381 each have degree @expr{N/2} (if @expr{N} is odd, the denominator will
25382 have degree one higher than the numerator).
25384 Rational approximations have the advantage that they can accurately
25385 describe functions that have poles (points at which the function's value
25386 goes to infinity, so that the denominator polynomial of the approximation
25387 goes to zero). If @expr{x} corresponds to a pole of the fitted rational
25388 function, then the result will be a division by zero. If Infinite mode
25389 is enabled, the result will be @samp{[uinf, uinf]}.
25391 There is no way to get the actual coefficients of the rational function
25392 used by @kbd{H a p}. (The algorithm never generates these coefficients
25393 explicitly, and quotients of polynomials are beyond @w{@kbd{a F}}'s
25394 capabilities to fit.)
25396 @node Summations, Logical Operations, Curve Fitting, Algebra
25397 @section Summations
25400 @cindex Summation of a series
25402 @pindex calc-summation
25404 The @kbd{a +} (@code{calc-summation}) [@code{sum}] command computes
25405 the sum of a formula over a certain range of index values. The formula
25406 is taken from the top of the stack; the command prompts for the
25407 name of the summation index variable, the lower limit of the
25408 sum (any formula), and the upper limit of the sum. If you
25409 enter a blank line at any of these prompts, that prompt and
25410 any later ones are answered by reading additional elements from
25411 the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a + @key{RET}}
25412 produces the result 55.
25415 $$ \sum_{k=1}^5 k^2 = 55 $$
25418 The choice of index variable is arbitrary, but it's best not to
25419 use a variable with a stored value. In particular, while
25420 @code{i} is often a favorite index variable, it should be avoided
25421 in Calc because @code{i} has the imaginary constant @expr{(0, 1)}
25422 as a value. If you pressed @kbd{=} on a sum over @code{i}, it would
25423 be changed to a nonsensical sum over the ``variable'' @expr{(0, 1)}!
25424 If you really want to use @code{i} as an index variable, use
25425 @w{@kbd{s u i @key{RET}}} first to ``unstore'' this variable.
25426 (@xref{Storing Variables}.)
25428 A numeric prefix argument steps the index by that amount rather
25429 than by one. Thus @kbd{' a_k @key{RET} C-u -2 a + k @key{RET} 10 @key{RET} 0 @key{RET}}
25430 yields @samp{a_10 + a_8 + a_6 + a_4 + a_2 + a_0}. A prefix
25431 argument of plain @kbd{C-u} causes @kbd{a +} to prompt for the
25432 step value, in which case you can enter any formula or enter
25433 a blank line to take the step value from the stack. With the
25434 @kbd{C-u} prefix, @kbd{a +} can take up to five arguments from
25435 the stack: The formula, the variable, the lower limit, the
25436 upper limit, and (at the top of the stack), the step value.
25438 Calc knows how to do certain sums in closed form. For example,
25439 @samp{sum(6 k^2, k, 1, n) = @w{2 n^3} + 3 n^2 + n}. In particular,
25440 this is possible if the formula being summed is polynomial or
25441 exponential in the index variable. Sums of logarithms are
25442 transformed into logarithms of products. Sums of trigonometric
25443 and hyperbolic functions are transformed to sums of exponentials
25444 and then done in closed form. Also, of course, sums in which the
25445 lower and upper limits are both numbers can always be evaluated
25446 just by grinding them out, although Calc will use closed forms
25447 whenever it can for the sake of efficiency.
25449 The notation for sums in algebraic formulas is
25450 @samp{sum(@var{expr}, @var{var}, @var{low}, @var{high}, @var{step})}.
25451 If @var{step} is omitted, it defaults to one. If @var{high} is
25452 omitted, @var{low} is actually the upper limit and the lower limit
25453 is one. If @var{low} is also omitted, the limits are @samp{-inf}
25454 and @samp{inf}, respectively.
25456 Infinite sums can sometimes be evaluated: @samp{sum(.5^k, k, 1, inf)}
25457 returns @expr{1}. This is done by evaluating the sum in closed
25458 form (to @samp{1. - 0.5^n} in this case), then evaluating this
25459 formula with @code{n} set to @code{inf}. Calc's usual rules
25460 for ``infinite'' arithmetic can find the answer from there. If
25461 infinite arithmetic yields a @samp{nan}, or if the sum cannot be
25462 solved in closed form, Calc leaves the @code{sum} function in
25463 symbolic form. @xref{Infinities}.
25465 As a special feature, if the limits are infinite (or omitted, as
25466 described above) but the formula includes vectors subscripted by
25467 expressions that involve the iteration variable, Calc narrows
25468 the limits to include only the range of integers which result in
25469 valid subscripts for the vector. For example, the sum
25470 @samp{sum(k [a,b,c,d,e,f,g]_(2k),k)} evaluates to @samp{b + 2 d + 3 f}.
25472 The limits of a sum do not need to be integers. For example,
25473 @samp{sum(a_k, k, 0, 2 n, n)} produces @samp{a_0 + a_n + a_(2 n)}.
25474 Calc computes the number of iterations using the formula
25475 @samp{1 + (@var{high} - @var{low}) / @var{step}}, which must,
25476 after simplification as if by @kbd{a s}, evaluate to an integer.
25478 If the number of iterations according to the above formula does
25479 not come out to an integer, the sum is invalid and will be left
25480 in symbolic form. However, closed forms are still supplied, and
25481 you are on your honor not to misuse the resulting formulas by
25482 substituting mismatched bounds into them. For example,
25483 @samp{sum(k, k, 1, 10, 2)} is invalid, but Calc will go ahead and
25484 evaluate the closed form solution for the limits 1 and 10 to get
25485 the rather dubious answer, 29.25.
25487 If the lower limit is greater than the upper limit (assuming a
25488 positive step size), the result is generally zero. However,
25489 Calc only guarantees a zero result when the upper limit is
25490 exactly one step less than the lower limit, i.e., if the number
25491 of iterations is @mathit{-1}. Thus @samp{sum(f(k), k, n, n-1)} is zero
25492 but the sum from @samp{n} to @samp{n-2} may report a nonzero value
25493 if Calc used a closed form solution.
25495 Calc's logical predicates like @expr{a < b} return 1 for ``true''
25496 and 0 for ``false.'' @xref{Logical Operations}. This can be
25497 used to advantage for building conditional sums. For example,
25498 @samp{sum(prime(k)*k^2, k, 1, 20)} is the sum of the squares of all
25499 prime numbers from 1 to 20; the @code{prime} predicate returns 1 if
25500 its argument is prime and 0 otherwise. You can read this expression
25501 as ``the sum of @expr{k^2}, where @expr{k} is prime.'' Indeed,
25502 @samp{sum(prime(k)*k^2, k)} would represent the sum of @emph{all} primes
25503 squared, since the limits default to plus and minus infinity, but
25504 there are no such sums that Calc's built-in rules can do in
25507 As another example, @samp{sum((k != k_0) * f(k), k, 1, n)} is the
25508 sum of @expr{f(k)} for all @expr{k} from 1 to @expr{n}, excluding
25509 one value @expr{k_0}. Slightly more tricky is the summand
25510 @samp{(k != k_0) / (k - k_0)}, which is an attempt to describe
25511 the sum of all @expr{1/(k-k_0)} except at @expr{k = k_0}, where
25512 this would be a division by zero. But at @expr{k = k_0}, this
25513 formula works out to the indeterminate form @expr{0 / 0}, which
25514 Calc will not assume is zero. Better would be to use
25515 @samp{(k != k_0) ? 1/(k-k_0) : 0}; the @samp{? :} operator does
25516 an ``if-then-else'' test: This expression says, ``if
25517 @texline @math{k \ne k_0},
25518 @infoline @expr{k != k_0},
25519 then @expr{1/(k-k_0)}, else zero.'' Now the formula @expr{1/(k-k_0)}
25520 will not even be evaluated by Calc when @expr{k = k_0}.
25522 @cindex Alternating sums
25524 @pindex calc-alt-summation
25526 The @kbd{a -} (@code{calc-alt-summation}) [@code{asum}] command
25527 computes an alternating sum. Successive terms of the sequence
25528 are given alternating signs, with the first term (corresponding
25529 to the lower index value) being positive. Alternating sums
25530 are converted to normal sums with an extra term of the form
25531 @samp{(-1)^(k-@var{low})}. This formula is adjusted appropriately
25532 if the step value is other than one. For example, the Taylor
25533 series for the sine function is @samp{asum(x^k / k!, k, 1, inf, 2)}.
25534 (Calc cannot evaluate this infinite series, but it can approximate
25535 it if you replace @code{inf} with any particular odd number.)
25536 Calc converts this series to a regular sum with a step of one,
25537 namely @samp{sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)}.
25539 @cindex Product of a sequence
25541 @pindex calc-product
25543 The @kbd{a *} (@code{calc-product}) [@code{prod}] command is
25544 the analogous way to take a product of many terms. Calc also knows
25545 some closed forms for products, such as @samp{prod(k, k, 1, n) = n!}.
25546 Conditional products can be written @samp{prod(k^prime(k), k, 1, n)}
25547 or @samp{prod(prime(k) ? k : 1, k, 1, n)}.
25550 @pindex calc-tabulate
25552 The @kbd{a T} (@code{calc-tabulate}) [@code{table}] command
25553 evaluates a formula at a series of iterated index values, just
25554 like @code{sum} and @code{prod}, but its result is simply a
25555 vector of the results. For example, @samp{table(a_i, i, 1, 7, 2)}
25556 produces @samp{[a_1, a_3, a_5, a_7]}.
25558 @node Logical Operations, Rewrite Rules, Summations, Algebra
25559 @section Logical Operations
25562 The following commands and algebraic functions return true/false values,
25563 where 1 represents ``true'' and 0 represents ``false.'' In cases where
25564 a truth value is required (such as for the condition part of a rewrite
25565 rule, or as the condition for a @w{@kbd{Z [ Z ]}} control structure), any
25566 nonzero value is accepted to mean ``true.'' (Specifically, anything
25567 for which @code{dnonzero} returns 1 is ``true,'' and anything for
25568 which @code{dnonzero} returns 0 or cannot decide is assumed ``false.''
25569 Note that this means that @w{@kbd{Z [ Z ]}} will execute the ``then''
25570 portion if its condition is provably true, but it will execute the
25571 ``else'' portion for any condition like @expr{a = b} that is not
25572 provably true, even if it might be true. Algebraic functions that
25573 have conditions as arguments, like @code{? :} and @code{&&}, remain
25574 unevaluated if the condition is neither provably true nor provably
25575 false. @xref{Declarations}.)
25578 @pindex calc-equal-to
25582 The @kbd{a =} (@code{calc-equal-to}) command, or @samp{eq(a,b)} function
25583 (which can also be written @samp{a = b} or @samp{a == b} in an algebraic
25584 formula) is true if @expr{a} and @expr{b} are equal, either because they
25585 are identical expressions, or because they are numbers which are
25586 numerically equal. (Thus the integer 1 is considered equal to the float
25587 1.0.) If the equality of @expr{a} and @expr{b} cannot be determined,
25588 the comparison is left in symbolic form. Note that as a command, this
25589 operation pops two values from the stack and pushes back either a 1 or
25590 a 0, or a formula @samp{a = b} if the values' equality cannot be determined.
25592 Many Calc commands use @samp{=} formulas to represent @dfn{equations}.
25593 For example, the @kbd{a S} (@code{calc-solve-for}) command rearranges
25594 an equation to solve for a given variable. The @kbd{a M}
25595 (@code{calc-map-equation}) command can be used to apply any
25596 function to both sides of an equation; for example, @kbd{2 a M *}
25597 multiplies both sides of the equation by two. Note that just
25598 @kbd{2 *} would not do the same thing; it would produce the formula
25599 @samp{2 (a = b)} which represents 2 if the equality is true or
25602 The @code{eq} function with more than two arguments (e.g., @kbd{C-u 3 a =}
25603 or @samp{a = b = c}) tests if all of its arguments are equal. In
25604 algebraic notation, the @samp{=} operator is unusual in that it is
25605 neither left- nor right-associative: @samp{a = b = c} is not the
25606 same as @samp{(a = b) = c} or @samp{a = (b = c)} (which each compare
25607 one variable with the 1 or 0 that results from comparing two other
25611 @pindex calc-not-equal-to
25614 The @kbd{a #} (@code{calc-not-equal-to}) command, or @samp{neq(a,b)} or
25615 @samp{a != b} function, is true if @expr{a} and @expr{b} are not equal.
25616 This also works with more than two arguments; @samp{a != b != c != d}
25617 tests that all four of @expr{a}, @expr{b}, @expr{c}, and @expr{d} are
25634 @pindex calc-less-than
25635 @pindex calc-greater-than
25636 @pindex calc-less-equal
25637 @pindex calc-greater-equal
25666 The @kbd{a <} (@code{calc-less-than}) [@samp{lt(a,b)} or @samp{a < b}]
25667 operation is true if @expr{a} is less than @expr{b}. Similar functions
25668 are @kbd{a >} (@code{calc-greater-than}) [@samp{gt(a,b)} or @samp{a > b}],
25669 @kbd{a [} (@code{calc-less-equal}) [@samp{leq(a,b)} or @samp{a <= b}], and
25670 @kbd{a ]} (@code{calc-greater-equal}) [@samp{geq(a,b)} or @samp{a >= b}].
25672 While the inequality functions like @code{lt} do not accept more
25673 than two arguments, the syntax @w{@samp{a <= b < c}} is translated to an
25674 equivalent expression involving intervals: @samp{b in [a .. c)}.
25675 (See the description of @code{in} below.) All four combinations
25676 of @samp{<} and @samp{<=} are allowed, or any of the four combinations
25677 of @samp{>} and @samp{>=}. Four-argument constructions like
25678 @samp{a < b < c < d}, and mixtures like @w{@samp{a < b = c}} that
25679 involve both equalities and inequalities, are not allowed.
25682 @pindex calc-remove-equal
25684 The @kbd{a .} (@code{calc-remove-equal}) [@code{rmeq}] command extracts
25685 the righthand side of the equation or inequality on the top of the
25686 stack. It also works elementwise on vectors. For example, if
25687 @samp{[x = 2.34, y = z / 2]} is on the stack, then @kbd{a .} produces
25688 @samp{[2.34, z / 2]}. As a special case, if the righthand side is a
25689 variable and the lefthand side is a number (as in @samp{2.34 = x}), then
25690 Calc keeps the lefthand side instead. Finally, this command works with
25691 assignments @samp{x := 2.34} as well as equations, always taking the
25692 the righthand side, and for @samp{=>} (evaluates-to) operators, always
25693 taking the lefthand side.
25696 @pindex calc-logical-and
25699 The @kbd{a &} (@code{calc-logical-and}) [@samp{land(a,b)} or @samp{a && b}]
25700 function is true if both of its arguments are true, i.e., are
25701 non-zero numbers. In this case, the result will be either @expr{a} or
25702 @expr{b}, chosen arbitrarily. If either argument is zero, the result is
25703 zero. Otherwise, the formula is left in symbolic form.
25706 @pindex calc-logical-or
25709 The @kbd{a |} (@code{calc-logical-or}) [@samp{lor(a,b)} or @samp{a || b}]
25710 function is true if either or both of its arguments are true (nonzero).
25711 The result is whichever argument was nonzero, choosing arbitrarily if both
25712 are nonzero. If both @expr{a} and @expr{b} are zero, the result is
25716 @pindex calc-logical-not
25719 The @kbd{a !} (@code{calc-logical-not}) [@samp{lnot(a)} or @samp{!@: a}]
25720 function is true if @expr{a} is false (zero), or false if @expr{a} is
25721 true (nonzero). It is left in symbolic form if @expr{a} is not a
25725 @pindex calc-logical-if
25735 @cindex Arguments, not evaluated
25736 The @kbd{a :} (@code{calc-logical-if}) [@samp{if(a,b,c)} or @samp{a ? b :@: c}]
25737 function is equal to either @expr{b} or @expr{c} if @expr{a} is a nonzero
25738 number or zero, respectively. If @expr{a} is not a number, the test is
25739 left in symbolic form and neither @expr{b} nor @expr{c} is evaluated in
25740 any way. In algebraic formulas, this is one of the few Calc functions
25741 whose arguments are not automatically evaluated when the function itself
25742 is evaluated. The others are @code{lambda}, @code{quote}, and
25745 One minor surprise to watch out for is that the formula @samp{a?3:4}
25746 will not work because the @samp{3:4} is parsed as a fraction instead of
25747 as three separate symbols. Type something like @samp{a ? 3 : 4} or
25748 @samp{a?(3):4} instead.
25750 As a special case, if @expr{a} evaluates to a vector, then both @expr{b}
25751 and @expr{c} are evaluated; the result is a vector of the same length
25752 as @expr{a} whose elements are chosen from corresponding elements of
25753 @expr{b} and @expr{c} according to whether each element of @expr{a}
25754 is zero or nonzero. Each of @expr{b} and @expr{c} must be either a
25755 vector of the same length as @expr{a}, or a non-vector which is matched
25756 with all elements of @expr{a}.
25759 @pindex calc-in-set
25761 The @kbd{a @{} (@code{calc-in-set}) [@samp{in(a,b)}] function is true if
25762 the number @expr{a} is in the set of numbers represented by @expr{b}.
25763 If @expr{b} is an interval form, @expr{a} must be one of the values
25764 encompassed by the interval. If @expr{b} is a vector, @expr{a} must be
25765 equal to one of the elements of the vector. (If any vector elements are
25766 intervals, @expr{a} must be in any of the intervals.) If @expr{b} is a
25767 plain number, @expr{a} must be numerically equal to @expr{b}.
25768 @xref{Set Operations}, for a group of commands that manipulate sets
25775 The @samp{typeof(a)} function produces an integer or variable which
25776 characterizes @expr{a}. If @expr{a} is a number, vector, or variable,
25777 the result will be one of the following numbers:
25782 3 Floating-point number
25784 5 Rectangular complex number
25785 6 Polar complex number
25791 12 Infinity (inf, uinf, or nan)
25793 101 Vector (but not a matrix)
25797 Otherwise, @expr{a} is a formula, and the result is a variable which
25798 represents the name of the top-level function call.
25812 The @samp{integer(a)} function returns true if @expr{a} is an integer.
25813 The @samp{real(a)} function
25814 is true if @expr{a} is a real number, either integer, fraction, or
25815 float. The @samp{constant(a)} function returns true if @expr{a} is
25816 any of the objects for which @code{typeof} would produce an integer
25817 code result except for variables, and provided that the components of
25818 an object like a vector or error form are themselves constant.
25819 Note that infinities do not satisfy any of these tests, nor do
25820 special constants like @code{pi} and @code{e}.
25822 @xref{Declarations}, for a set of similar functions that recognize
25823 formulas as well as actual numbers. For example, @samp{dint(floor(x))}
25824 is true because @samp{floor(x)} is provably integer-valued, but
25825 @samp{integer(floor(x))} does not because @samp{floor(x)} is not
25826 literally an integer constant.
25832 The @samp{refers(a,b)} function is true if the variable (or sub-expression)
25833 @expr{b} appears in @expr{a}, or false otherwise. Unlike the other
25834 tests described here, this function returns a definite ``no'' answer
25835 even if its arguments are still in symbolic form. The only case where
25836 @code{refers} will be left unevaluated is if @expr{a} is a plain
25837 variable (different from @expr{b}).
25843 The @samp{negative(a)} function returns true if @expr{a} ``looks'' negative,
25844 because it is a negative number, because it is of the form @expr{-x},
25845 or because it is a product or quotient with a term that looks negative.
25846 This is most useful in rewrite rules. Beware that @samp{negative(a)}
25847 evaluates to 1 or 0 for @emph{any} argument @expr{a}, so it can only
25848 be stored in a formula if the default simplifications are turned off
25849 first with @kbd{m O} (or if it appears in an unevaluated context such
25850 as a rewrite rule condition).
25856 The @samp{variable(a)} function is true if @expr{a} is a variable,
25857 or false if not. If @expr{a} is a function call, this test is left
25858 in symbolic form. Built-in variables like @code{pi} and @code{inf}
25859 are considered variables like any others by this test.
25865 The @samp{nonvar(a)} function is true if @expr{a} is a non-variable.
25866 If its argument is a variable it is left unsimplified; it never
25867 actually returns zero. However, since Calc's condition-testing
25868 commands consider ``false'' anything not provably true, this is
25887 @cindex Linearity testing
25888 The functions @code{lin}, @code{linnt}, @code{islin}, and @code{islinnt}
25889 check if an expression is ``linear,'' i.e., can be written in the form
25890 @expr{a + b x} for some constants @expr{a} and @expr{b}, and some
25891 variable or subformula @expr{x}. The function @samp{islin(f,x)} checks
25892 if formula @expr{f} is linear in @expr{x}, returning 1 if so. For
25893 example, @samp{islin(x,x)}, @samp{islin(-x,x)}, @samp{islin(3,x)}, and
25894 @samp{islin(x y / 3 - 2, x)} all return 1. The @samp{lin(f,x)} function
25895 is similar, except that instead of returning 1 it returns the vector
25896 @expr{[a, b, x]}. For the above examples, this vector would be
25897 @expr{[0, 1, x]}, @expr{[0, -1, x]}, @expr{[3, 0, x]}, and
25898 @expr{[-2, y/3, x]}, respectively. Both @code{lin} and @code{islin}
25899 generally remain unevaluated for expressions which are not linear,
25900 e.g., @samp{lin(2 x^2, x)} and @samp{lin(sin(x), x)}. The second
25901 argument can also be a formula; @samp{islin(2 + 3 sin(x), sin(x))}
25904 The @code{linnt} and @code{islinnt} functions perform a similar check,
25905 but require a ``non-trivial'' linear form, which means that the
25906 @expr{b} coefficient must be non-zero. For example, @samp{lin(2,x)}
25907 returns @expr{[2, 0, x]} and @samp{lin(y,x)} returns @expr{[y, 0, x]},
25908 but @samp{linnt(2,x)} and @samp{linnt(y,x)} are left unevaluated
25909 (in other words, these formulas are considered to be only ``trivially''
25910 linear in @expr{x}).
25912 All four linearity-testing functions allow you to omit the second
25913 argument, in which case the input may be linear in any non-constant
25914 formula. Here, the @expr{a=0}, @expr{b=1} case is also considered
25915 trivial, and only constant values for @expr{a} and @expr{b} are
25916 recognized. Thus, @samp{lin(2 x y)} returns @expr{[0, 2, x y]},
25917 @samp{lin(2 - x y)} returns @expr{[2, -1, x y]}, and @samp{lin(x y)}
25918 returns @expr{[0, 1, x y]}. The @code{linnt} function would allow the
25919 first two cases but not the third. Also, neither @code{lin} nor
25920 @code{linnt} accept plain constants as linear in the one-argument
25921 case: @samp{islin(2,x)} is true, but @samp{islin(2)} is false.
25927 The @samp{istrue(a)} function returns 1 if @expr{a} is a nonzero
25928 number or provably nonzero formula, or 0 if @expr{a} is anything else.
25929 Calls to @code{istrue} can only be manipulated if @kbd{m O} mode is
25930 used to make sure they are not evaluated prematurely. (Note that
25931 declarations are used when deciding whether a formula is true;
25932 @code{istrue} returns 1 when @code{dnonzero} would return 1, and
25933 it returns 0 when @code{dnonzero} would return 0 or leave itself
25936 @node Rewrite Rules, , Logical Operations, Algebra
25937 @section Rewrite Rules
25940 @cindex Rewrite rules
25941 @cindex Transformations
25942 @cindex Pattern matching
25944 @pindex calc-rewrite
25946 The @kbd{a r} (@code{calc-rewrite}) [@code{rewrite}] command makes
25947 substitutions in a formula according to a specified pattern or patterns
25948 known as @dfn{rewrite rules}. Whereas @kbd{a b} (@code{calc-substitute})
25949 matches literally, so that substituting @samp{sin(x)} with @samp{cos(x)}
25950 matches only the @code{sin} function applied to the variable @code{x},
25951 rewrite rules match general kinds of formulas; rewriting using the rule
25952 @samp{sin(x) := cos(x)} matches @code{sin} of any argument and replaces
25953 it with @code{cos} of that same argument. The only significance of the
25954 name @code{x} is that the same name is used on both sides of the rule.
25956 Rewrite rules rearrange formulas already in Calc's memory.
25957 @xref{Syntax Tables}, to read about @dfn{syntax rules}, which are
25958 similar to algebraic rewrite rules but operate when new algebraic
25959 entries are being parsed, converting strings of characters into
25963 * Entering Rewrite Rules::
25964 * Basic Rewrite Rules::
25965 * Conditional Rewrite Rules::
25966 * Algebraic Properties of Rewrite Rules::
25967 * Other Features of Rewrite Rules::
25968 * Composing Patterns in Rewrite Rules::
25969 * Nested Formulas with Rewrite Rules::
25970 * Multi-Phase Rewrite Rules::
25971 * Selections with Rewrite Rules::
25972 * Matching Commands::
25973 * Automatic Rewrites::
25974 * Debugging Rewrites::
25975 * Examples of Rewrite Rules::
25978 @node Entering Rewrite Rules, Basic Rewrite Rules, Rewrite Rules, Rewrite Rules
25979 @subsection Entering Rewrite Rules
25982 Rewrite rules normally use the ``assignment'' operator
25983 @samp{@var{old} := @var{new}}.
25984 This operator is equivalent to the function call @samp{assign(old, new)}.
25985 The @code{assign} function is undefined by itself in Calc, so an
25986 assignment formula such as a rewrite rule will be left alone by ordinary
25987 Calc commands. But certain commands, like the rewrite system, interpret
25988 assignments in special ways.
25990 For example, the rule @samp{sin(x)^2 := 1-cos(x)^2} says to replace
25991 every occurrence of the sine of something, squared, with one minus the
25992 square of the cosine of that same thing. All by itself as a formula
25993 on the stack it does nothing, but when given to the @kbd{a r} command
25994 it turns that command into a sine-squared-to-cosine-squared converter.
25996 To specify a set of rules to be applied all at once, make a vector of
25999 When @kbd{a r} prompts you to enter the rewrite rules, you can answer
26004 With a rule: @kbd{f(x) := g(x) @key{RET}}.
26006 With a vector of rules: @kbd{[f1(x) := g1(x), f2(x) := g2(x)] @key{RET}}.
26007 (You can omit the enclosing square brackets if you wish.)
26009 With the name of a variable that contains the rule or rules vector:
26010 @kbd{myrules @key{RET}}.
26012 With any formula except a rule, a vector, or a variable name; this
26013 will be interpreted as the @var{old} half of a rewrite rule,
26014 and you will be prompted a second time for the @var{new} half:
26015 @kbd{f(x) @key{RET} g(x) @key{RET}}.
26017 With a blank line, in which case the rule, rules vector, or variable
26018 will be taken from the top of the stack (and the formula to be
26019 rewritten will come from the second-to-top position).
26022 If you enter the rules directly (as opposed to using rules stored
26023 in a variable), those rules will be put into the Trail so that you
26024 can retrieve them later. @xref{Trail Commands}.
26026 It is most convenient to store rules you use often in a variable and
26027 invoke them by giving the variable name. The @kbd{s e}
26028 (@code{calc-edit-variable}) command is an easy way to create or edit a
26029 rule set stored in a variable. You may also wish to use @kbd{s p}
26030 (@code{calc-permanent-variable}) to save your rules permanently;
26031 @pxref{Operations on Variables}.
26033 Rewrite rules are compiled into a special internal form for faster
26034 matching. If you enter a rule set directly it must be recompiled
26035 every time. If you store the rules in a variable and refer to them
26036 through that variable, they will be compiled once and saved away
26037 along with the variable for later reference. This is another good
26038 reason to store your rules in a variable.
26040 Calc also accepts an obsolete notation for rules, as vectors
26041 @samp{[@var{old}, @var{new}]}. But because it is easily confused with a
26042 vector of two rules, the use of this notation is no longer recommended.
26044 @node Basic Rewrite Rules, Conditional Rewrite Rules, Entering Rewrite Rules, Rewrite Rules
26045 @subsection Basic Rewrite Rules
26048 To match a particular formula @expr{x} with a particular rewrite rule
26049 @samp{@var{old} := @var{new}}, Calc compares the structure of @expr{x} with
26050 the structure of @var{old}. Variables that appear in @var{old} are
26051 treated as @dfn{meta-variables}; the corresponding positions in @expr{x}
26052 may contain any sub-formulas. For example, the pattern @samp{f(x,y)}
26053 would match the expression @samp{f(12, a+1)} with the meta-variable
26054 @samp{x} corresponding to 12 and with @samp{y} corresponding to
26055 @samp{a+1}. However, this pattern would not match @samp{f(12)} or
26056 @samp{g(12, a+1)}, since there is no assignment of the meta-variables
26057 that will make the pattern match these expressions. Notice that if
26058 the pattern is a single meta-variable, it will match any expression.
26060 If a given meta-variable appears more than once in @var{old}, the
26061 corresponding sub-formulas of @expr{x} must be identical. Thus
26062 the pattern @samp{f(x,x)} would match @samp{f(12, 12)} and
26063 @samp{f(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(a+b, b+a)}.
26064 (@xref{Conditional Rewrite Rules}, for a way to match the latter.)
26066 Things other than variables must match exactly between the pattern
26067 and the target formula. To match a particular variable exactly, use
26068 the pseudo-function @samp{quote(v)} in the pattern. For example, the
26069 pattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or
26072 The special variable names @samp{e}, @samp{pi}, @samp{i}, @samp{phi},
26073 @samp{gamma}, @samp{inf}, @samp{uinf}, and @samp{nan} always match
26074 literally. Thus the pattern @samp{sin(d + e + f)} acts exactly like
26075 @samp{sin(d + quote(e) + f)}.
26077 If the @var{old} pattern is found to match a given formula, that
26078 formula is replaced by @var{new}, where any occurrences in @var{new}
26079 of meta-variables from the pattern are replaced with the sub-formulas
26080 that they matched. Thus, applying the rule @samp{f(x,y) := g(y+x,x)}
26081 to @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}.
26083 The normal @kbd{a r} command applies rewrite rules over and over
26084 throughout the target formula until no further changes are possible
26085 (up to a limit of 100 times). Use @kbd{C-u 1 a r} to make only one
26088 @node Conditional Rewrite Rules, Algebraic Properties of Rewrite Rules, Basic Rewrite Rules, Rewrite Rules
26089 @subsection Conditional Rewrite Rules
26092 A rewrite rule can also be @dfn{conditional}, written in the form
26093 @samp{@var{old} := @var{new} :: @var{cond}}. (There is also the obsolete
26094 form @samp{[@var{old}, @var{new}, @var{cond}]}.) If a @var{cond} part
26096 rule, this is an additional condition that must be satisfied before
26097 the rule is accepted. Once @var{old} has been successfully matched
26098 to the target expression, @var{cond} is evaluated (with all the
26099 meta-variables substituted for the values they matched) and simplified
26100 with @kbd{a s} (@code{calc-simplify}). If the result is a nonzero
26101 number or any other object known to be nonzero (@pxref{Declarations}),
26102 the rule is accepted. If the result is zero or if it is a symbolic
26103 formula that is not known to be nonzero, the rule is rejected.
26104 @xref{Logical Operations}, for a number of functions that return
26105 1 or 0 according to the results of various tests.
26107 For example, the formula @samp{n > 0} simplifies to 1 or 0 if @expr{n}
26108 is replaced by a positive or nonpositive number, respectively (or if
26109 @expr{n} has been declared to be positive or nonpositive). Thus,
26110 the rule @samp{f(x,y) := g(y+x,x) :: x+y > 0} would apply to
26111 @samp{f(0, 4)} but not to @samp{f(-3, 2)} or @samp{f(12, a+1)}
26112 (assuming no outstanding declarations for @expr{a}). In the case of
26113 @samp{f(-3, 2)}, the condition can be shown not to be satisfied; in
26114 the case of @samp{f(12, a+1)}, the condition merely cannot be shown
26115 to be satisfied, but that is enough to reject the rule.
26117 While Calc will use declarations to reason about variables in the
26118 formula being rewritten, declarations do not apply to meta-variables.
26119 For example, the rule @samp{f(a) := g(a+1)} will match for any values
26120 of @samp{a}, such as complex numbers, vectors, or formulas, even if
26121 @samp{a} has been declared to be real or scalar. If you want the
26122 meta-variable @samp{a} to match only literal real numbers, use
26123 @samp{f(a) := g(a+1) :: real(a)}. If you want @samp{a} to match only
26124 reals and formulas which are provably real, use @samp{dreal(a)} as
26127 The @samp{::} operator is a shorthand for the @code{condition}
26128 function; @samp{@var{old} := @var{new} :: @var{cond}} is equivalent to
26129 the formula @samp{condition(assign(@var{old}, @var{new}), @var{cond})}.
26131 If you have several conditions, you can use @samp{... :: c1 :: c2 :: c3}
26132 or @samp{... :: c1 && c2 && c3}. The two are entirely equivalent.
26134 It is also possible to embed conditions inside the pattern:
26135 @samp{f(x :: x>0, y) := g(y+x, x)}. This is purely a notational
26136 convenience, though; where a condition appears in a rule has no
26137 effect on when it is tested. The rewrite-rule compiler automatically
26138 decides when it is best to test each condition while a rule is being
26141 Certain conditions are handled as special cases by the rewrite rule
26142 system and are tested very efficiently: Where @expr{x} is any
26143 meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)},
26144 @samp{constant(x)}, @samp{negative(x)}, @samp{x >= y} where @expr{y}
26145 is either a constant or another meta-variable and @samp{>=} may be
26146 replaced by any of the six relational operators, and @samp{x % a = b}
26147 where @expr{a} and @expr{b} are constants. Other conditions, like
26148 @samp{x >= y+1} or @samp{dreal(x)}, will be less efficient to check
26149 since Calc must bring the whole evaluator and simplifier into play.
26151 An interesting property of @samp{::} is that neither of its arguments
26152 will be touched by Calc's default simplifications. This is important
26153 because conditions often are expressions that cannot safely be
26154 evaluated early. For example, the @code{typeof} function never
26155 remains in symbolic form; entering @samp{typeof(a)} will put the
26156 number 100 (the type code for variables like @samp{a}) on the stack.
26157 But putting the condition @samp{... :: typeof(a) = 6} on the stack
26158 is safe since @samp{::} prevents the @code{typeof} from being
26159 evaluated until the condition is actually used by the rewrite system.
26161 Since @samp{::} protects its lefthand side, too, you can use a dummy
26162 condition to protect a rule that must itself not evaluate early.
26163 For example, it's not safe to put @samp{a(f,x) := apply(f, [x])} on
26164 the stack because it will immediately evaluate to @samp{a(f,x) := f(x)},
26165 where the meta-variable-ness of @code{f} on the righthand side has been
26166 lost. But @samp{a(f,x) := apply(f, [x]) :: 1} is safe, and of course
26167 the condition @samp{1} is always true (nonzero) so it has no effect on
26168 the functioning of the rule. (The rewrite compiler will ensure that
26169 it doesn't even impact the speed of matching the rule.)
26171 @node Algebraic Properties of Rewrite Rules, Other Features of Rewrite Rules, Conditional Rewrite Rules, Rewrite Rules
26172 @subsection Algebraic Properties of Rewrite Rules
26175 The rewrite mechanism understands the algebraic properties of functions
26176 like @samp{+} and @samp{*}. In particular, pattern matching takes
26177 the associativity and commutativity of the following functions into
26181 + - * = != && || and or xor vint vunion vxor gcd lcm max min beta
26184 For example, the rewrite rule:
26187 a x + b x := (a + b) x
26191 will match formulas of the form,
26194 a x + b x, x a + x b, a x + x b, x a + b x
26197 Rewrites also understand the relationship between the @samp{+} and @samp{-}
26198 operators. The above rewrite rule will also match the formulas,
26201 a x - b x, x a - x b, a x - x b, x a - b x
26205 by matching @samp{b} in the pattern to @samp{-b} from the formula.
26207 Applied to a sum of many terms like @samp{r + a x + s + b x + t}, this
26208 pattern will check all pairs of terms for possible matches. The rewrite
26209 will take whichever suitable pair it discovers first.
26211 In general, a pattern using an associative operator like @samp{a + b}
26212 will try @var{2 n} different ways to match a sum of @var{n} terms
26213 like @samp{x + y + z - w}. First, @samp{a} is matched against each
26214 of @samp{x}, @samp{y}, @samp{z}, and @samp{-w} in turn, with @samp{b}
26215 being matched to the remainders @samp{y + z - w}, @samp{x + z - w}, etc.
26216 If none of these succeed, then @samp{b} is matched against each of the
26217 four terms with @samp{a} matching the remainder. Half-and-half matches,
26218 like @samp{(x + y) + (z - w)}, are not tried.
26220 Note that @samp{*} is not commutative when applied to matrices, but
26221 rewrite rules pretend that it is. If you type @kbd{m v} to enable
26222 Matrix mode (@pxref{Matrix Mode}), rewrite rules will match @samp{*}
26223 literally, ignoring its usual commutativity property. (In the
26224 current implementation, the associativity also vanishes---it is as
26225 if the pattern had been enclosed in a @code{plain} marker; see below.)
26226 If you are applying rewrites to formulas with matrices, it's best to
26227 enable Matrix mode first to prevent algebraically incorrect rewrites
26230 The pattern @samp{-x} will actually match any expression. For example,
26238 will rewrite @samp{f(a)} to @samp{-f(-a)}. To avoid this, either use
26239 a @code{plain} marker as described below, or add a @samp{negative(x)}
26240 condition. The @code{negative} function is true if its argument
26241 ``looks'' negative, for example, because it is a negative number or
26242 because it is a formula like @samp{-x}. The new rule using this
26246 f(x) := -f(-x) :: negative(x) @r{or, equivalently,}
26247 f(-x) := -f(x) :: negative(-x)
26250 In the same way, the pattern @samp{x - y} will match the sum @samp{a + b}
26251 by matching @samp{y} to @samp{-b}.
26253 The pattern @samp{a b} will also match the formula @samp{x/y} if
26254 @samp{y} is a number. Thus the rule @samp{a x + @w{b x} := (a+b) x}
26255 will also convert @samp{a x + x / 2} to @samp{(a + 0.5) x} (or
26256 @samp{(a + 1:2) x}, depending on the current fraction mode).
26258 Calc will @emph{not} take other liberties with @samp{*}, @samp{/}, and
26259 @samp{^}. For example, the pattern @samp{f(a b)} will not match
26260 @samp{f(x^2)}, and @samp{f(a + b)} will not match @samp{f(2 x)}, even
26261 though conceivably these patterns could match with @samp{a = b = x}.
26262 Nor will @samp{f(a b)} match @samp{f(x / y)} if @samp{y} is not a
26263 constant, even though it could be considered to match with @samp{a = x}
26264 and @samp{b = 1/y}. The reasons are partly for efficiency, and partly
26265 because while few mathematical operations are substantively different
26266 for addition and subtraction, often it is preferable to treat the cases
26267 of multiplication, division, and integer powers separately.
26269 Even more subtle is the rule set
26272 [ f(a) + f(b) := f(a + b), -f(a) := f(-a) ]
26276 attempting to match @samp{f(x) - f(y)}. You might think that Calc
26277 will view this subtraction as @samp{f(x) + (-f(y))} and then apply
26278 the above two rules in turn, but actually this will not work because
26279 Calc only does this when considering rules for @samp{+} (like the
26280 first rule in this set). So it will see first that @samp{f(x) + (-f(y))}
26281 does not match @samp{f(a) + f(b)} for any assignments of the
26282 meta-variables, and then it will see that @samp{f(x) - f(y)} does
26283 not match @samp{-f(a)} for any assignment of @samp{a}. Because Calc
26284 tries only one rule at a time, it will not be able to rewrite
26285 @samp{f(x) - f(y)} with this rule set. An explicit @samp{f(a) - f(b)}
26286 rule will have to be added.
26288 Another thing patterns will @emph{not} do is break up complex numbers.
26289 The pattern @samp{myconj(a + @w{b i)} := a - b i} will work for formulas
26290 involving the special constant @samp{i} (such as @samp{3 - 4 i}), but
26291 it will not match actual complex numbers like @samp{(3, -4)}. A version
26292 of the above rule for complex numbers would be
26295 myconj(a) := re(a) - im(a) (0,1) :: im(a) != 0
26299 (Because the @code{re} and @code{im} functions understand the properties
26300 of the special constant @samp{i}, this rule will also work for
26301 @samp{3 - 4 i}. In fact, this particular rule would probably be better
26302 without the @samp{im(a) != 0} condition, since if @samp{im(a) = 0} the
26303 righthand side of the rule will still give the correct answer for the
26304 conjugate of a real number.)
26306 It is also possible to specify optional arguments in patterns. The rule
26309 opt(a) x + opt(b) (x^opt(c) + opt(d)) := f(a, b, c, d)
26313 will match the formula
26320 in a fairly straightforward manner, but it will also match reduced
26324 x + x^2, 2(x + 1) - x, x + x
26328 producing, respectively,
26331 f(1, 1, 2, 0), f(-1, 2, 1, 1), f(1, 1, 1, 0)
26334 (The latter two formulas can be entered only if default simplifications
26335 have been turned off with @kbd{m O}.)
26337 The default value for a term of a sum is zero. The default value
26338 for a part of a product, for a power, or for the denominator of a
26339 quotient, is one. Also, @samp{-x} matches the pattern @samp{opt(a) b}
26340 with @samp{a = -1}.
26342 In particular, the distributive-law rule can be refined to
26345 opt(a) x + opt(b) x := (a + b) x
26349 so that it will convert, e.g., @samp{a x - x}, to @samp{(a - 1) x}.
26351 The pattern @samp{opt(a) + opt(b) x} matches almost any formulas which
26352 are linear in @samp{x}. You can also use the @code{lin} and @code{islin}
26353 functions with rewrite conditions to test for this; @pxref{Logical
26354 Operations}. These functions are not as convenient to use in rewrite
26355 rules, but they recognize more kinds of formulas as linear:
26356 @samp{x/z} is considered linear with @expr{b = 1/z} by @code{lin},
26357 but it will not match the above pattern because that pattern calls
26358 for a multiplication, not a division.
26360 As another example, the obvious rule to replace @samp{sin(x)^2 + cos(x)^2}
26364 sin(x)^2 + cos(x)^2 := 1
26368 misses many cases because the sine and cosine may both be multiplied by
26369 an equal factor. Here's a more successful rule:
26372 opt(a) sin(x)^2 + opt(a) cos(x)^2 := a
26375 Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2}
26376 because one @expr{a} would have ``matched'' 1 while the other matched 6.
26378 Calc automatically converts a rule like
26388 f(temp, x) := g(x) :: temp = x-1
26392 (where @code{temp} stands for a new, invented meta-variable that
26393 doesn't actually have a name). This modified rule will successfully
26394 match @samp{f(6, 7)}, binding @samp{temp} and @samp{x} to 6 and 7,
26395 respectively, then verifying that they differ by one even though
26396 @samp{6} does not superficially look like @samp{x-1}.
26398 However, Calc does not solve equations to interpret a rule. The
26402 f(x-1, x+1) := g(x)
26406 will not work. That is, it will match @samp{f(a - 1 + b, a + 1 + b)}
26407 but not @samp{f(6, 8)}. Calc always interprets at least one occurrence
26408 of a variable by literal matching. If the variable appears ``isolated''
26409 then Calc is smart enough to use it for literal matching. But in this
26410 last example, Calc is forced to rewrite the rule to @samp{f(x-1, temp)
26411 := g(x) :: temp = x+1} where the @samp{x-1} term must correspond to an
26412 actual ``something-minus-one'' in the target formula.
26414 A successful way to write this would be @samp{f(x, x+2) := g(x+1)}.
26415 You could make this resemble the original form more closely by using
26416 @code{let} notation, which is described in the next section:
26419 f(xm1, x+1) := g(x) :: let(x := xm1+1)
26422 Calc does this rewriting or ``conditionalizing'' for any sub-pattern
26423 which involves only the functions in the following list, operating
26424 only on constants and meta-variables which have already been matched
26425 elsewhere in the pattern. When matching a function call, Calc is
26426 careful to match arguments which are plain variables before arguments
26427 which are calls to any of the functions below, so that a pattern like
26428 @samp{f(x-1, x)} can be conditionalized even though the isolated
26429 @samp{x} comes after the @samp{x-1}.
26432 + - * / \ % ^ abs sign round rounde roundu trunc floor ceil
26433 max min re im conj arg
26436 You can suppress all of the special treatments described in this
26437 section by surrounding a function call with a @code{plain} marker.
26438 This marker causes the function call which is its argument to be
26439 matched literally, without regard to commutativity, associativity,
26440 negation, or conditionalization. When you use @code{plain}, the
26441 ``deep structure'' of the formula being matched can show through.
26445 plain(a - a b) := f(a, b)
26449 will match only literal subtractions. However, the @code{plain}
26450 marker does not affect its arguments' arguments. In this case,
26451 commutativity and associativity is still considered while matching
26452 the @w{@samp{a b}} sub-pattern, so the whole pattern will match
26453 @samp{x - y x} as well as @samp{x - x y}. We could go still
26457 plain(a - plain(a b)) := f(a, b)
26461 which would do a completely strict match for the pattern.
26463 By contrast, the @code{quote} marker means that not only the
26464 function name but also the arguments must be literally the same.
26465 The above pattern will match @samp{x - x y} but
26468 quote(a - a b) := f(a, b)
26472 will match only the single formula @samp{a - a b}. Also,
26475 quote(a - quote(a b)) := f(a, b)
26479 will match only @samp{a - quote(a b)}---probably not the desired
26482 A certain amount of algebra is also done when substituting the
26483 meta-variables on the righthand side of a rule. For example,
26487 a + f(b) := f(a + b)
26491 matching @samp{f(x) - y} would produce @samp{f((-y) + x)} if
26492 taken literally, but the rewrite mechanism will simplify the
26493 righthand side to @samp{f(x - y)} automatically. (Of course,
26494 the default simplifications would do this anyway, so this
26495 special simplification is only noticeable if you have turned the
26496 default simplifications off.) This rewriting is done only when
26497 a meta-variable expands to a ``negative-looking'' expression.
26498 If this simplification is not desirable, you can use a @code{plain}
26499 marker on the righthand side:
26502 a + f(b) := f(plain(a + b))
26506 In this example, we are still allowing the pattern-matcher to
26507 use all the algebra it can muster, but the righthand side will
26508 always simplify to a literal addition like @samp{f((-y) + x)}.
26510 @node Other Features of Rewrite Rules, Composing Patterns in Rewrite Rules, Algebraic Properties of Rewrite Rules, Rewrite Rules
26511 @subsection Other Features of Rewrite Rules
26514 Certain ``function names'' serve as markers in rewrite rules.
26515 Here is a complete list of these markers. First are listed the
26516 markers that work inside a pattern; then come the markers that
26517 work in the righthand side of a rule.
26523 One kind of marker, @samp{import(x)}, takes the place of a whole
26524 rule. Here @expr{x} is the name of a variable containing another
26525 rule set; those rules are ``spliced into'' the rule set that
26526 imports them. For example, if @samp{[f(a+b) := f(a) + f(b),
26527 f(a b) := a f(b) :: real(a)]} is stored in variable @samp{linearF},
26528 then the rule set @samp{[f(0) := 0, import(linearF)]} will apply
26529 all three rules. It is possible to modify the imported rules
26530 slightly: @samp{import(x, v1, x1, v2, x2, @dots{})} imports
26531 the rule set @expr{x} with all occurrences of
26532 @texline @math{v_1},
26533 @infoline @expr{v1},
26534 as either a variable name or a function name, replaced with
26535 @texline @math{x_1}
26536 @infoline @expr{x1}
26538 @texline @math{v_1}
26539 @infoline @expr{v1}
26540 is used as a function name, then
26541 @texline @math{x_1}
26542 @infoline @expr{x1}
26543 must be either a function name itself or a @w{@samp{< >}} nameless
26544 function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0,
26545 import(linearF, f, g)]} applies the linearity rules to the function
26546 @samp{g} instead of @samp{f}. Imports can be nested, but the
26547 import-with-renaming feature may fail to rename sub-imports properly.
26549 The special functions allowed in patterns are:
26557 This pattern matches exactly @expr{x}; variable names in @expr{x} are
26558 not interpreted as meta-variables. The only flexibility is that
26559 numbers are compared for numeric equality, so that the pattern
26560 @samp{f(quote(12))} will match both @samp{f(12)} and @samp{f(12.0)}.
26561 (Numbers are always treated this way by the rewrite mechanism:
26562 The rule @samp{f(x,x) := g(x)} will match @samp{f(12, 12.0)}.
26563 The rewrite may produce either @samp{g(12)} or @samp{g(12.0)}
26564 as a result in this case.)
26571 Here @expr{x} must be a function call @samp{f(x1,x2,@dots{})}. This
26572 pattern matches a call to function @expr{f} with the specified
26573 argument patterns. No special knowledge of the properties of the
26574 function @expr{f} is used in this case; @samp{+} is not commutative or
26575 associative. Unlike @code{quote}, the arguments @samp{x1,x2,@dots{}}
26576 are treated as patterns. If you wish them to be treated ``plainly''
26577 as well, you must enclose them with more @code{plain} markers:
26578 @samp{plain(plain(@w{-a}) + plain(b c))}.
26585 Here @expr{x} must be a variable name. This must appear as an
26586 argument to a function or an element of a vector; it specifies that
26587 the argument or element is optional.
26588 As an argument to @samp{+}, @samp{-}, @samp{*}, @samp{&&}, or @samp{||},
26589 or as the second argument to @samp{/} or @samp{^}, the value @var{def}
26590 may be omitted. The pattern @samp{x + opt(y)} matches a sum by
26591 binding one summand to @expr{x} and the other to @expr{y}, and it
26592 matches anything else by binding the whole expression to @expr{x} and
26593 zero to @expr{y}. The other operators above work similarly.
26595 For general miscellaneous functions, the default value @code{def}
26596 must be specified. Optional arguments are dropped starting with
26597 the rightmost one during matching. For example, the pattern
26598 @samp{f(opt(a,0), b, opt(c,b))} will match @samp{f(b)}, @samp{f(a,b)},
26599 or @samp{f(a,b,c)}. Default values of zero and @expr{b} are
26600 supplied in this example for the omitted arguments. Note that
26601 the literal variable @expr{b} will be the default in the latter
26602 case, @emph{not} the value that matched the meta-variable @expr{b}.
26603 In other words, the default @var{def} is effectively quoted.
26605 @item condition(x,c)
26611 This matches the pattern @expr{x}, with the attached condition
26612 @expr{c}. It is the same as @samp{x :: c}.
26620 This matches anything that matches both pattern @expr{x} and
26621 pattern @expr{y}. It is the same as @samp{x &&& y}.
26622 @pxref{Composing Patterns in Rewrite Rules}.
26630 This matches anything that matches either pattern @expr{x} or
26631 pattern @expr{y}. It is the same as @w{@samp{x ||| y}}.
26639 This matches anything that does not match pattern @expr{x}.
26640 It is the same as @samp{!!! x}.
26646 @tindex cons (rewrites)
26647 This matches any vector of one or more elements. The first
26648 element is matched to @expr{h}; a vector of the remaining
26649 elements is matched to @expr{t}. Note that vectors of fixed
26650 length can also be matched as actual vectors: The rule
26651 @samp{cons(a,cons(b,[])) := cons(a+b,[])} is equivalent
26652 to the rule @samp{[a,b] := [a+b]}.
26658 @tindex rcons (rewrites)
26659 This is like @code{cons}, except that the @emph{last} element
26660 is matched to @expr{h}, with the remaining elements matched
26663 @item apply(f,args)
26667 @tindex apply (rewrites)
26668 This matches any function call. The name of the function, in
26669 the form of a variable, is matched to @expr{f}. The arguments
26670 of the function, as a vector of zero or more objects, are
26671 matched to @samp{args}. Constants, variables, and vectors
26672 do @emph{not} match an @code{apply} pattern. For example,
26673 @samp{apply(f,x)} matches any function call, @samp{apply(quote(f),x)}
26674 matches any call to the function @samp{f}, @samp{apply(f,[a,b])}
26675 matches any function call with exactly two arguments, and
26676 @samp{apply(quote(f), cons(a,cons(b,x)))} matches any call
26677 to the function @samp{f} with two or more arguments. Another
26678 way to implement the latter, if the rest of the rule does not
26679 need to refer to the first two arguments of @samp{f} by name,
26680 would be @samp{apply(quote(f), x :: vlen(x) >= 2)}.
26681 Here's a more interesting sample use of @code{apply}:
26684 apply(f,[x+n]) := n + apply(f,[x])
26685 :: in(f, [floor,ceil,round,trunc]) :: integer(n)
26688 Note, however, that this will be slower to match than a rule
26689 set with four separate rules. The reason is that Calc sorts
26690 the rules of a rule set according to top-level function name;
26691 if the top-level function is @code{apply}, Calc must try the
26692 rule for every single formula and sub-formula. If the top-level
26693 function in the pattern is, say, @code{floor}, then Calc invokes
26694 the rule only for sub-formulas which are calls to @code{floor}.
26696 Formulas normally written with operators like @code{+} are still
26697 considered function calls: @code{apply(f,x)} matches @samp{a+b}
26698 with @samp{f = add}, @samp{x = [a,b]}.
26700 You must use @code{apply} for meta-variables with function names
26701 on both sides of a rewrite rule: @samp{apply(f, [x]) := f(x+1)}
26702 is @emph{not} correct, because it rewrites @samp{spam(6)} into
26703 @samp{f(7)}. The righthand side should be @samp{apply(f, [x+1])}.
26704 Also note that you will have to use No-Simplify mode (@kbd{m O})
26705 when entering this rule so that the @code{apply} isn't
26706 evaluated immediately to get the new rule @samp{f(x) := f(x+1)}.
26707 Or, use @kbd{s e} to enter the rule without going through the stack,
26708 or enter the rule as @samp{apply(f, [x]) := apply(f, [x+1]) @w{:: 1}}.
26709 @xref{Conditional Rewrite Rules}.
26716 This is used for applying rules to formulas with selections;
26717 @pxref{Selections with Rewrite Rules}.
26720 Special functions for the righthand sides of rules are:
26724 The notation @samp{quote(x)} is changed to @samp{x} when the
26725 righthand side is used. As far as the rewrite rule is concerned,
26726 @code{quote} is invisible. However, @code{quote} has the special
26727 property in Calc that its argument is not evaluated. Thus,
26728 while it will not work to put the rule @samp{t(a) := typeof(a)}
26729 on the stack because @samp{typeof(a)} is evaluated immediately
26730 to produce @samp{t(a) := 100}, you can use @code{quote} to
26731 protect the righthand side: @samp{t(a) := quote(typeof(a))}.
26732 (@xref{Conditional Rewrite Rules}, for another trick for
26733 protecting rules from evaluation.)
26736 Special properties of and simplifications for the function call
26737 @expr{x} are not used. One interesting case where @code{plain}
26738 is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a
26739 shorthand notation for the @code{quote} function. This rule will
26740 not work as shown; instead of replacing @samp{q(foo)} with
26741 @samp{quote(foo)}, it will replace it with @samp{foo}! The correct
26742 rule would be @samp{q(x) := plain(quote(x))}.
26745 Where @expr{t} is a vector, this is converted into an expanded
26746 vector during rewrite processing. Note that @code{cons} is a regular
26747 Calc function which normally does this anyway; the only way @code{cons}
26748 is treated specially by rewrites is that @code{cons} on the righthand
26749 side of a rule will be evaluated even if default simplifications
26750 have been turned off.
26753 Analogous to @code{cons} except putting @expr{h} at the @emph{end} of
26754 the vector @expr{t}.
26756 @item apply(f,args)
26757 Where @expr{f} is a variable and @var{args} is a vector, this
26758 is converted to a function call. Once again, note that @code{apply}
26759 is also a regular Calc function.
26766 The formula @expr{x} is handled in the usual way, then the
26767 default simplifications are applied to it even if they have
26768 been turned off normally. This allows you to treat any function
26769 similarly to the way @code{cons} and @code{apply} are always
26770 treated. However, there is a slight difference: @samp{cons(2+3, [])}
26771 with default simplifications off will be converted to @samp{[2+3]},
26772 whereas @samp{eval(cons(2+3, []))} will be converted to @samp{[5]}.
26779 The formula @expr{x} has meta-variables substituted in the usual
26780 way, then algebraically simplified as if by the @kbd{a s} command.
26782 @item evalextsimp(x)
26786 @tindex evalextsimp
26787 The formula @expr{x} has meta-variables substituted in the normal
26788 way, then ``extendedly'' simplified as if by the @kbd{a e} command.
26791 @xref{Selections with Rewrite Rules}.
26794 There are also some special functions you can use in conditions.
26802 The expression @expr{x} is evaluated with meta-variables substituted.
26803 The @kbd{a s} command's simplifications are @emph{not} applied by
26804 default, but @expr{x} can include calls to @code{evalsimp} or
26805 @code{evalextsimp} as described above to invoke higher levels
26806 of simplification. The
26807 result of @expr{x} is then bound to the meta-variable @expr{v}. As
26808 usual, if this meta-variable has already been matched to something
26809 else the two values must be equal; if the meta-variable is new then
26810 it is bound to the result of the expression. This variable can then
26811 appear in later conditions, and on the righthand side of the rule.
26812 In fact, @expr{v} may be any pattern in which case the result of
26813 evaluating @expr{x} is matched to that pattern, binding any
26814 meta-variables that appear in that pattern. Note that @code{let}
26815 can only appear by itself as a condition, or as one term of an
26816 @samp{&&} which is a whole condition: It cannot be inside
26817 an @samp{||} term or otherwise buried.
26819 The alternate, equivalent form @samp{let(v, x)} is also recognized.
26820 Note that the use of @samp{:=} by @code{let}, while still being
26821 assignment-like in character, is unrelated to the use of @samp{:=}
26822 in the main part of a rewrite rule.
26824 As an example, @samp{f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)}
26825 replaces @samp{f(a)} with @samp{g} of the inverse of @samp{a}, if
26826 that inverse exists and is constant. For example, if @samp{a} is a
26827 singular matrix the operation @samp{1/a} is left unsimplified and
26828 @samp{constant(ia)} fails, but if @samp{a} is an invertible matrix
26829 then the rule succeeds. Without @code{let} there would be no way
26830 to express this rule that didn't have to invert the matrix twice.
26831 Note that, because the meta-variable @samp{ia} is otherwise unbound
26832 in this rule, the @code{let} condition itself always ``succeeds''
26833 because no matter what @samp{1/a} evaluates to, it can successfully
26834 be bound to @code{ia}.
26836 Here's another example, for integrating cosines of linear
26837 terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}.
26838 The @code{lin} function returns a 3-vector if its argument is linear,
26839 or leaves itself unevaluated if not. But an unevaluated @code{lin}
26840 call will not match the 3-vector on the lefthand side of the @code{let},
26841 so this @code{let} both verifies that @code{y} is linear, and binds
26842 the coefficients @code{a} and @code{b} for use elsewhere in the rule.
26843 (It would have been possible to use @samp{sin(a x + b)/b} for the
26844 righthand side instead, but using @samp{sin(y)/b} avoids gratuitous
26845 rearrangement of the argument of the sine.)
26851 Similarly, here is a rule that implements an inverse-@code{erf}
26852 function. It uses @code{root} to search for a solution. If
26853 @code{root} succeeds, it will return a vector of two numbers
26854 where the first number is the desired solution. If no solution
26855 is found, @code{root} remains in symbolic form. So we use
26856 @code{let} to check that the result was indeed a vector.
26859 ierf(x) := y :: let([y,z] := root(erf(a) = x, a, .5))
26863 The meta-variable @var{v}, which must already have been matched
26864 to something elsewhere in the rule, is compared against pattern
26865 @var{p}. Since @code{matches} is a standard Calc function, it
26866 can appear anywhere in a condition. But if it appears alone or
26867 as a term of a top-level @samp{&&}, then you get the special
26868 extra feature that meta-variables which are bound to things
26869 inside @var{p} can be used elsewhere in the surrounding rewrite
26872 The only real difference between @samp{let(p := v)} and
26873 @samp{matches(v, p)} is that the former evaluates @samp{v} using
26874 the default simplifications, while the latter does not.
26878 This is actually a variable, not a function. If @code{remember}
26879 appears as a condition in a rule, then when that rule succeeds
26880 the original expression and rewritten expression are added to the
26881 front of the rule set that contained the rule. If the rule set
26882 was not stored in a variable, @code{remember} is ignored. The
26883 lefthand side is enclosed in @code{quote} in the added rule if it
26884 contains any variables.
26886 For example, the rule @samp{f(n) := n f(n-1) :: remember} applied
26887 to @samp{f(7)} will add the rule @samp{f(7) := 7 f(6)} to the front
26888 of the rule set. The rule set @code{EvalRules} works slightly
26889 differently: There, the evaluation of @samp{f(6)} will complete before
26890 the result is added to the rule set, in this case as @samp{f(7) := 5040}.
26891 Thus @code{remember} is most useful inside @code{EvalRules}.
26893 It is up to you to ensure that the optimization performed by
26894 @code{remember} is safe. For example, the rule @samp{foo(n) := n
26895 :: evalv(eatfoo) > 0 :: remember} is a bad idea (@code{evalv} is
26896 the function equivalent of the @kbd{=} command); if the variable
26897 @code{eatfoo} ever contains 1, rules like @samp{foo(7) := 7} will
26898 be added to the rule set and will continue to operate even if
26899 @code{eatfoo} is later changed to 0.
26906 Remember the match as described above, but only if condition @expr{c}
26907 is true. For example, @samp{remember(n % 4 = 0)} in the above factorial
26908 rule remembers only every fourth result. Note that @samp{remember(1)}
26909 is equivalent to @samp{remember}, and @samp{remember(0)} has no effect.
26912 @node Composing Patterns in Rewrite Rules, Nested Formulas with Rewrite Rules, Other Features of Rewrite Rules, Rewrite Rules
26913 @subsection Composing Patterns in Rewrite Rules
26916 There are three operators, @samp{&&&}, @samp{|||}, and @samp{!!!},
26917 that combine rewrite patterns to make larger patterns. The
26918 combinations are ``and,'' ``or,'' and ``not,'' respectively, and
26919 these operators are the pattern equivalents of @samp{&&}, @samp{||}
26920 and @samp{!} (which operate on zero-or-nonzero logical values).
26922 Note that @samp{&&&}, @samp{|||}, and @samp{!!!} are left in symbolic
26923 form by all regular Calc features; they have special meaning only in
26924 the context of rewrite rule patterns.
26926 The pattern @samp{@var{p1} &&& @var{p2}} matches anything that
26927 matches both @var{p1} and @var{p2}. One especially useful case is
26928 when one of @var{p1} or @var{p2} is a meta-variable. For example,
26929 here is a rule that operates on error forms:
26932 f(x &&& a +/- b, x) := g(x)
26935 This does the same thing, but is arguably simpler than, the rule
26938 f(a +/- b, a +/- b) := g(a +/- b)
26945 Here's another interesting example:
26948 ends(cons(a, x) &&& rcons(y, b)) := [a, b]
26952 which effectively clips out the middle of a vector leaving just
26953 the first and last elements. This rule will change a one-element
26954 vector @samp{[a]} to @samp{[a, a]}. The similar rule
26957 ends(cons(a, rcons(y, b))) := [a, b]
26961 would do the same thing except that it would fail to match a
26962 one-element vector.
26968 The pattern @samp{@var{p1} ||| @var{p2}} matches anything that
26969 matches either @var{p1} or @var{p2}. Calc first tries matching
26970 against @var{p1}; if that fails, it goes on to try @var{p2}.
26976 A simple example of @samp{|||} is
26979 curve(inf ||| -inf) := 0
26983 which converts both @samp{curve(inf)} and @samp{curve(-inf)} to zero.
26985 Here is a larger example:
26988 log(a, b) ||| (ln(a) :: let(b := e)) := mylog(a, b)
26991 This matches both generalized and natural logarithms in a single rule.
26992 Note that the @samp{::} term must be enclosed in parentheses because
26993 that operator has lower precedence than @samp{|||} or @samp{:=}.
26995 (In practice this rule would probably include a third alternative,
26996 omitted here for brevity, to take care of @code{log10}.)
26998 While Calc generally treats interior conditions exactly the same as
26999 conditions on the outside of a rule, it does guarantee that if all the
27000 variables in the condition are special names like @code{e}, or already
27001 bound in the pattern to which the condition is attached (say, if
27002 @samp{a} had appeared in this condition), then Calc will process this
27003 condition right after matching the pattern to the left of the @samp{::}.
27004 Thus, we know that @samp{b} will be bound to @samp{e} only if the
27005 @code{ln} branch of the @samp{|||} was taken.
27007 Note that this rule was careful to bind the same set of meta-variables
27008 on both sides of the @samp{|||}. Calc does not check this, but if
27009 you bind a certain meta-variable only in one branch and then use that
27010 meta-variable elsewhere in the rule, results are unpredictable:
27013 f(a,b) ||| g(b) := h(a,b)
27016 Here if the pattern matches @samp{g(17)}, Calc makes no promises about
27017 the value that will be substituted for @samp{a} on the righthand side.
27023 The pattern @samp{!!! @var{pat}} matches anything that does not
27024 match @var{pat}. Any meta-variables that are bound while matching
27025 @var{pat} remain unbound outside of @var{pat}.
27030 f(x &&& !!! a +/- b, !!![]) := g(x)
27034 converts @code{f} whose first argument is anything @emph{except} an
27035 error form, and whose second argument is not the empty vector, into
27036 a similar call to @code{g} (but without the second argument).
27038 If we know that the second argument will be a vector (empty or not),
27039 then an equivalent rule would be:
27042 f(x, y) := g(x) :: typeof(x) != 7 :: vlen(y) > 0
27046 where of course 7 is the @code{typeof} code for error forms.
27047 Another final condition, that works for any kind of @samp{y},
27048 would be @samp{!istrue(y == [])}. (The @code{istrue} function
27049 returns an explicit 0 if its argument was left in symbolic form;
27050 plain @samp{!(y == [])} or @samp{y != []} would not work to replace
27051 @samp{!!![]} since these would be left unsimplified, and thus cause
27052 the rule to fail, if @samp{y} was something like a variable name.)
27054 It is possible for a @samp{!!!} to refer to meta-variables bound
27055 elsewhere in the pattern. For example,
27062 matches any call to @code{f} with different arguments, changing
27063 this to @code{g} with only the first argument.
27065 If a function call is to be matched and one of the argument patterns
27066 contains a @samp{!!!} somewhere inside it, that argument will be
27074 will be careful to bind @samp{a} to the second argument of @code{f}
27075 before testing the first argument. If Calc had tried to match the
27076 first argument of @code{f} first, the results would have been
27077 disastrous: since @code{a} was unbound so far, the pattern @samp{a}
27078 would have matched anything at all, and the pattern @samp{!!!a}
27079 therefore would @emph{not} have matched anything at all!
27081 @node Nested Formulas with Rewrite Rules, Multi-Phase Rewrite Rules, Composing Patterns in Rewrite Rules, Rewrite Rules
27082 @subsection Nested Formulas with Rewrite Rules
27085 When @kbd{a r} (@code{calc-rewrite}) is used, it takes an expression from
27086 the top of the stack and attempts to match any of the specified rules
27087 to any part of the expression, starting with the whole expression
27088 and then, if that fails, trying deeper and deeper sub-expressions.
27089 For each part of the expression, the rules are tried in the order
27090 they appear in the rules vector. The first rule to match the first
27091 sub-expression wins; it replaces the matched sub-expression according
27092 to the @var{new} part of the rule.
27094 Often, the rule set will match and change the formula several times.
27095 The top-level formula is first matched and substituted repeatedly until
27096 it no longer matches the pattern; then, sub-formulas are tried, and
27097 so on. Once every part of the formula has gotten its chance, the
27098 rewrite mechanism starts over again with the top-level formula
27099 (in case a substitution of one of its arguments has caused it again
27100 to match). This continues until no further matches can be made
27101 anywhere in the formula.
27103 It is possible for a rule set to get into an infinite loop. The
27104 most obvious case, replacing a formula with itself, is not a problem
27105 because a rule is not considered to ``succeed'' unless the righthand
27106 side actually comes out to something different than the original
27107 formula or sub-formula that was matched. But if you accidentally
27108 had both @samp{ln(a b) := ln(a) + ln(b)} and the reverse
27109 @samp{ln(a) + ln(b) := ln(a b)} in your rule set, Calc would
27110 run forever switching a formula back and forth between the two
27113 To avoid disaster, Calc normally stops after 100 changes have been
27114 made to the formula. This will be enough for most multiple rewrites,
27115 but it will keep an endless loop of rewrites from locking up the
27116 computer forever. (On most systems, you can also type @kbd{C-g} to
27117 halt any Emacs command prematurely.)
27119 To change this limit, give a positive numeric prefix argument.
27120 In particular, @kbd{M-1 a r} applies only one rewrite at a time,
27121 useful when you are first testing your rule (or just if repeated
27122 rewriting is not what is called for by your application).
27131 You can also put a ``function call'' @samp{iterations(@var{n})}
27132 in place of a rule anywhere in your rules vector (but usually at
27133 the top). Then, @var{n} will be used instead of 100 as the default
27134 number of iterations for this rule set. You can use
27135 @samp{iterations(inf)} if you want no iteration limit by default.
27136 A prefix argument will override the @code{iterations} limit in the
27144 More precisely, the limit controls the number of ``iterations,''
27145 where each iteration is a successful matching of a rule pattern whose
27146 righthand side, after substituting meta-variables and applying the
27147 default simplifications, is different from the original sub-formula
27150 A prefix argument of zero sets the limit to infinity. Use with caution!
27152 Given a negative numeric prefix argument, @kbd{a r} will match and
27153 substitute the top-level expression up to that many times, but
27154 will not attempt to match the rules to any sub-expressions.
27156 In a formula, @code{rewrite(@var{expr}, @var{rules}, @var{n})}
27157 does a rewriting operation. Here @var{expr} is the expression
27158 being rewritten, @var{rules} is the rule, vector of rules, or
27159 variable containing the rules, and @var{n} is the optional
27160 iteration limit, which may be a positive integer, a negative
27161 integer, or @samp{inf} or @samp{-inf}. If @var{n} is omitted
27162 the @code{iterations} value from the rule set is used; if both
27163 are omitted, 100 is used.
27165 @node Multi-Phase Rewrite Rules, Selections with Rewrite Rules, Nested Formulas with Rewrite Rules, Rewrite Rules
27166 @subsection Multi-Phase Rewrite Rules
27169 It is possible to separate a rewrite rule set into several @dfn{phases}.
27170 During each phase, certain rules will be enabled while certain others
27171 will be disabled. A @dfn{phase schedule} controls the order in which
27172 phases occur during the rewriting process.
27179 If a call to the marker function @code{phase} appears in the rules
27180 vector in place of a rule, all rules following that point will be
27181 members of the phase(s) identified in the arguments to @code{phase}.
27182 Phases are given integer numbers. The markers @samp{phase()} and
27183 @samp{phase(all)} both mean the following rules belong to all phases;
27184 this is the default at the start of the rule set.
27186 If you do not explicitly schedule the phases, Calc sorts all phase
27187 numbers that appear in the rule set and executes the phases in
27188 ascending order. For example, the rule set
27205 has three phases, 1 through 3. Phase 1 consists of the @code{f0},
27206 @code{f1}, and @code{f4} rules (in that order). Phase 2 consists of
27207 @code{f0}, @code{f2}, and @code{f4}. Phase 3 consists of @code{f0}
27210 When Calc rewrites a formula using this rule set, it first rewrites
27211 the formula using only the phase 1 rules until no further changes are
27212 possible. Then it switches to the phase 2 rule set and continues
27213 until no further changes occur, then finally rewrites with phase 3.
27214 When no more phase 3 rules apply, rewriting finishes. (This is
27215 assuming @kbd{a r} with a large enough prefix argument to allow the
27216 rewriting to run to completion; the sequence just described stops
27217 early if the number of iterations specified in the prefix argument,
27218 100 by default, is reached.)
27220 During each phase, Calc descends through the nested levels of the
27221 formula as described previously. (@xref{Nested Formulas with Rewrite
27222 Rules}.) Rewriting starts at the top of the formula, then works its
27223 way down to the parts, then goes back to the top and works down again.
27224 The phase 2 rules do not begin until no phase 1 rules apply anywhere
27231 A @code{schedule} marker appearing in the rule set (anywhere, but
27232 conventionally at the top) changes the default schedule of phases.
27233 In the simplest case, @code{schedule} has a sequence of phase numbers
27234 for arguments; each phase number is invoked in turn until the
27235 arguments to @code{schedule} are exhausted. Thus adding
27236 @samp{schedule(3,2,1)} at the top of the above rule set would
27237 reverse the order of the phases; @samp{schedule(1,2,3)} would have
27238 no effect since this is the default schedule; and @samp{schedule(1,2,1,3)}
27239 would give phase 1 a second chance after phase 2 has completed, before
27240 moving on to phase 3.
27242 Any argument to @code{schedule} can instead be a vector of phase
27243 numbers (or even of sub-vectors). Then the sub-sequence of phases
27244 described by the vector are tried repeatedly until no change occurs
27245 in any phase in the sequence. For example, @samp{schedule([1, 2], 3)}
27246 tries phase 1, then phase 2, then, if either phase made any changes
27247 to the formula, repeats these two phases until they can make no
27248 further progress. Finally, it goes on to phase 3 for finishing
27251 Also, items in @code{schedule} can be variable names as well as
27252 numbers. A variable name is interpreted as the name of a function
27253 to call on the whole formula. For example, @samp{schedule(1, simplify)}
27254 says to apply the phase-1 rules (presumably, all of them), then to
27255 call @code{simplify} which is the function name equivalent of @kbd{a s}.
27256 Likewise, @samp{schedule([1, simplify])} says to alternate between
27257 phase 1 and @kbd{a s} until no further changes occur.
27259 Phases can be used purely to improve efficiency; if it is known that
27260 a certain group of rules will apply only at the beginning of rewriting,
27261 and a certain other group will apply only at the end, then rewriting
27262 will be faster if these groups are identified as separate phases.
27263 Once the phase 1 rules are done, Calc can put them aside and no longer
27264 spend any time on them while it works on phase 2.
27266 There are also some problems that can only be solved with several
27267 rewrite phases. For a real-world example of a multi-phase rule set,
27268 examine the set @code{FitRules}, which is used by the curve-fitting
27269 command to convert a model expression to linear form.
27270 @xref{Curve Fitting Details}. This set is divided into four phases.
27271 The first phase rewrites certain kinds of expressions to be more
27272 easily linearizable, but less computationally efficient. After the
27273 linear components have been picked out, the final phase includes the
27274 opposite rewrites to put each component back into an efficient form.
27275 If both sets of rules were included in one big phase, Calc could get
27276 into an infinite loop going back and forth between the two forms.
27278 Elsewhere in @code{FitRules}, the components are first isolated,
27279 then recombined where possible to reduce the complexity of the linear
27280 fit, then finally packaged one component at a time into vectors.
27281 If the packaging rules were allowed to begin before the recombining
27282 rules were finished, some components might be put away into vectors
27283 before they had a chance to recombine. By putting these rules in
27284 two separate phases, this problem is neatly avoided.
27286 @node Selections with Rewrite Rules, Matching Commands, Multi-Phase Rewrite Rules, Rewrite Rules
27287 @subsection Selections with Rewrite Rules
27290 If a sub-formula of the current formula is selected (as by @kbd{j s};
27291 @pxref{Selecting Subformulas}), the @kbd{a r} (@code{calc-rewrite})
27292 command applies only to that sub-formula. Together with a negative
27293 prefix argument, you can use this fact to apply a rewrite to one
27294 specific part of a formula without affecting any other parts.
27297 @pindex calc-rewrite-selection
27298 The @kbd{j r} (@code{calc-rewrite-selection}) command allows more
27299 sophisticated operations on selections. This command prompts for
27300 the rules in the same way as @kbd{a r}, but it then applies those
27301 rules to the whole formula in question even though a sub-formula
27302 of it has been selected. However, the selected sub-formula will
27303 first have been surrounded by a @samp{select( )} function call.
27304 (Calc's evaluator does not understand the function name @code{select};
27305 this is only a tag used by the @kbd{j r} command.)
27307 For example, suppose the formula on the stack is @samp{2 (a + b)^2}
27308 and the sub-formula @samp{a + b} is selected. This formula will
27309 be rewritten to @samp{2 select(a + b)^2} and then the rewrite
27310 rules will be applied in the usual way. The rewrite rules can
27311 include references to @code{select} to tell where in the pattern
27312 the selected sub-formula should appear.
27314 If there is still exactly one @samp{select( )} function call in
27315 the formula after rewriting is done, it indicates which part of
27316 the formula should be selected afterwards. Otherwise, the
27317 formula will be unselected.
27319 You can make @kbd{j r} act much like @kbd{a r} by enclosing both parts
27320 of the rewrite rule with @samp{select()}. However, @kbd{j r}
27321 allows you to use the current selection in more flexible ways.
27322 Suppose you wished to make a rule which removed the exponent from
27323 the selected term; the rule @samp{select(a)^x := select(a)} would
27324 work. In the above example, it would rewrite @samp{2 select(a + b)^2}
27325 to @samp{2 select(a + b)}. This would then be returned to the
27326 stack as @samp{2 (a + b)} with the @samp{a + b} selected.
27328 The @kbd{j r} command uses one iteration by default, unlike
27329 @kbd{a r} which defaults to 100 iterations. A numeric prefix
27330 argument affects @kbd{j r} in the same way as @kbd{a r}.
27331 @xref{Nested Formulas with Rewrite Rules}.
27333 As with other selection commands, @kbd{j r} operates on the stack
27334 entry that contains the cursor. (If the cursor is on the top-of-stack
27335 @samp{.} marker, it works as if the cursor were on the formula
27338 If you don't specify a set of rules, the rules are taken from the
27339 top of the stack, just as with @kbd{a r}. In this case, the
27340 cursor must indicate stack entry 2 or above as the formula to be
27341 rewritten (otherwise the same formula would be used as both the
27342 target and the rewrite rules).
27344 If the indicated formula has no selection, the cursor position within
27345 the formula temporarily selects a sub-formula for the purposes of this
27346 command. If the cursor is not on any sub-formula (e.g., it is in
27347 the line-number area to the left of the formula), the @samp{select( )}
27348 markers are ignored by the rewrite mechanism and the rules are allowed
27349 to apply anywhere in the formula.
27351 As a special feature, the normal @kbd{a r} command also ignores
27352 @samp{select( )} calls in rewrite rules. For example, if you used the
27353 above rule @samp{select(a)^x := select(a)} with @kbd{a r}, it would apply
27354 the rule as if it were @samp{a^x := a}. Thus, you can write general
27355 purpose rules with @samp{select( )} hints inside them so that they
27356 will ``do the right thing'' in both @kbd{a r} and @kbd{j r},
27357 both with and without selections.
27359 @node Matching Commands, Automatic Rewrites, Selections with Rewrite Rules, Rewrite Rules
27360 @subsection Matching Commands
27366 The @kbd{a m} (@code{calc-match}) [@code{match}] function takes a
27367 vector of formulas and a rewrite-rule-style pattern, and produces
27368 a vector of all formulas which match the pattern. The command
27369 prompts you to enter the pattern; as for @kbd{a r}, you can enter
27370 a single pattern (i.e., a formula with meta-variables), or a
27371 vector of patterns, or a variable which contains patterns, or
27372 you can give a blank response in which case the patterns are taken
27373 from the top of the stack. The pattern set will be compiled once
27374 and saved if it is stored in a variable. If there are several
27375 patterns in the set, vector elements are kept if they match any
27378 For example, @samp{match(a+b, [x, x+y, x-y, 7, x+y+z])}
27379 will return @samp{[x+y, x-y, x+y+z]}.
27381 The @code{import} mechanism is not available for pattern sets.
27383 The @kbd{a m} command can also be used to extract all vector elements
27384 which satisfy any condition: The pattern @samp{x :: x>0} will select
27385 all the positive vector elements.
27389 With the Inverse flag [@code{matchnot}], this command extracts all
27390 vector elements which do @emph{not} match the given pattern.
27396 There is also a function @samp{matches(@var{x}, @var{p})} which
27397 evaluates to 1 if expression @var{x} matches pattern @var{p}, or
27398 to 0 otherwise. This is sometimes useful for including into the
27399 conditional clauses of other rewrite rules.
27405 The function @code{vmatches} is just like @code{matches}, except
27406 that if the match succeeds it returns a vector of assignments to
27407 the meta-variables instead of the number 1. For example,
27408 @samp{vmatches(f(1,2), f(a,b))} returns @samp{[a := 1, b := 2]}.
27409 If the match fails, the function returns the number 0.
27411 @node Automatic Rewrites, Debugging Rewrites, Matching Commands, Rewrite Rules
27412 @subsection Automatic Rewrites
27415 @cindex @code{EvalRules} variable
27417 It is possible to get Calc to apply a set of rewrite rules on all
27418 results, effectively adding to the built-in set of default
27419 simplifications. To do this, simply store your rule set in the
27420 variable @code{EvalRules}. There is a convenient @kbd{s E} command
27421 for editing @code{EvalRules}; @pxref{Operations on Variables}.
27423 For example, suppose you want @samp{sin(a + b)} to be expanded out
27424 to @samp{sin(b) cos(a) + cos(b) sin(a)} wherever it appears, and
27425 similarly for @samp{cos(a + b)}. The corresponding rewrite rule
27430 [ sin(a + b) := cos(a) sin(b) + sin(a) cos(b),
27431 cos(a + b) := cos(a) cos(b) - sin(a) sin(b) ]
27435 To apply these manually, you could put them in a variable called
27436 @code{trigexp} and then use @kbd{a r trigexp} every time you wanted
27437 to expand trig functions. But if instead you store them in the
27438 variable @code{EvalRules}, they will automatically be applied to all
27439 sines and cosines of sums. Then, with @samp{2 x} and @samp{45} on
27440 the stack, typing @kbd{+ S} will (assuming Degrees mode) result in
27441 @samp{0.7071 sin(2 x) + 0.7071 cos(2 x)} automatically.
27443 As each level of a formula is evaluated, the rules from
27444 @code{EvalRules} are applied before the default simplifications.
27445 Rewriting continues until no further @code{EvalRules} apply.
27446 Note that this is different from the usual order of application of
27447 rewrite rules: @code{EvalRules} works from the bottom up, simplifying
27448 the arguments to a function before the function itself, while @kbd{a r}
27449 applies rules from the top down.
27451 Because the @code{EvalRules} are tried first, you can use them to
27452 override the normal behavior of any built-in Calc function.
27454 It is important not to write a rule that will get into an infinite
27455 loop. For example, the rule set @samp{[f(0) := 1, f(n) := n f(n-1)]}
27456 appears to be a good definition of a factorial function, but it is
27457 unsafe. Imagine what happens if @samp{f(2.5)} is simplified. Calc
27458 will continue to subtract 1 from this argument forever without reaching
27459 zero. A safer second rule would be @samp{f(n) := n f(n-1) :: n>0}.
27460 Another dangerous rule is @samp{g(x, y) := g(y, x)}. Rewriting
27461 @samp{g(2, 4)}, this would bounce back and forth between that and
27462 @samp{g(4, 2)} forever. If an infinite loop in @code{EvalRules}
27463 occurs, Emacs will eventually stop with a ``Computation got stuck
27464 or ran too long'' message.
27466 Another subtle difference between @code{EvalRules} and regular rewrites
27467 concerns rules that rewrite a formula into an identical formula. For
27468 example, @samp{f(n) := f(floor(n))} ``fails to match'' when @expr{n} is
27469 already an integer. But in @code{EvalRules} this case is detected only
27470 if the righthand side literally becomes the original formula before any
27471 further simplification. This means that @samp{f(n) := f(floor(n))} will
27472 get into an infinite loop if it occurs in @code{EvalRules}. Calc will
27473 replace @samp{f(6)} with @samp{f(floor(6))}, which is different from
27474 @samp{f(6)}, so it will consider the rule to have matched and will
27475 continue simplifying that formula; first the argument is simplified
27476 to get @samp{f(6)}, then the rule matches again to get @samp{f(floor(6))}
27477 again, ad infinitum. A much safer rule would check its argument first,
27478 say, with @samp{f(n) := f(floor(n)) :: !dint(n)}.
27480 (What really happens is that the rewrite mechanism substitutes the
27481 meta-variables in the righthand side of a rule, compares to see if the
27482 result is the same as the original formula and fails if so, then uses
27483 the default simplifications to simplify the result and compares again
27484 (and again fails if the formula has simplified back to its original
27485 form). The only special wrinkle for the @code{EvalRules} is that the
27486 same rules will come back into play when the default simplifications
27487 are used. What Calc wants to do is build @samp{f(floor(6))}, see that
27488 this is different from the original formula, simplify to @samp{f(6)},
27489 see that this is the same as the original formula, and thus halt the
27490 rewriting. But while simplifying, @samp{f(6)} will again trigger
27491 the same @code{EvalRules} rule and Calc will get into a loop inside
27492 the rewrite mechanism itself.)
27494 The @code{phase}, @code{schedule}, and @code{iterations} markers do
27495 not work in @code{EvalRules}. If the rule set is divided into phases,
27496 only the phase 1 rules are applied, and the schedule is ignored.
27497 The rules are always repeated as many times as possible.
27499 The @code{EvalRules} are applied to all function calls in a formula,
27500 but not to numbers (and other number-like objects like error forms),
27501 nor to vectors or individual variable names. (Though they will apply
27502 to @emph{components} of vectors and error forms when appropriate.) You
27503 might try to make a variable @code{phihat} which automatically expands
27504 to its definition without the need to press @kbd{=} by writing the
27505 rule @samp{quote(phihat) := (1-sqrt(5))/2}, but unfortunately this rule
27506 will not work as part of @code{EvalRules}.
27508 Finally, another limitation is that Calc sometimes calls its built-in
27509 functions directly rather than going through the default simplifications.
27510 When it does this, @code{EvalRules} will not be able to override those
27511 functions. For example, when you take the absolute value of the complex
27512 number @expr{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling
27513 the multiplication, addition, and square root functions directly rather
27514 than applying the default simplifications to this formula. So an
27515 @code{EvalRules} rule that (perversely) rewrites @samp{sqrt(13) := 6}
27516 would not apply. (However, if you put Calc into Symbolic mode so that
27517 @samp{sqrt(13)} will be left in symbolic form by the built-in square
27518 root function, your rule will be able to apply. But if the complex
27519 number were @expr{(3,4)}, so that @samp{sqrt(25)} must be calculated,
27520 then Symbolic mode will not help because @samp{sqrt(25)} can be
27521 evaluated exactly to 5.)
27523 One subtle restriction that normally only manifests itself with
27524 @code{EvalRules} is that while a given rewrite rule is in the process
27525 of being checked, that same rule cannot be recursively applied. Calc
27526 effectively removes the rule from its rule set while checking the rule,
27527 then puts it back once the match succeeds or fails. (The technical
27528 reason for this is that compiled pattern programs are not reentrant.)
27529 For example, consider the rule @samp{foo(x) := x :: foo(x/2) > 0}
27530 attempting to match @samp{foo(8)}. This rule will be inactive while
27531 the condition @samp{foo(4) > 0} is checked, even though it might be
27532 an integral part of evaluating that condition. Note that this is not
27533 a problem for the more usual recursive type of rule, such as
27534 @samp{foo(x) := foo(x/2)}, because there the rule has succeeded and
27535 been reactivated by the time the righthand side is evaluated.
27537 If @code{EvalRules} has no stored value (its default state), or if
27538 anything but a vector is stored in it, then it is ignored.
27540 Even though Calc's rewrite mechanism is designed to compare rewrite
27541 rules to formulas as quickly as possible, storing rules in
27542 @code{EvalRules} may make Calc run substantially slower. This is
27543 particularly true of rules where the top-level call is a commonly used
27544 function, or is not fixed. The rule @samp{f(n) := n f(n-1) :: n>0} will
27545 only activate the rewrite mechanism for calls to the function @code{f},
27546 but @samp{lg(n) + lg(m) := lg(n m)} will check every @samp{+} operator.
27549 apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10])
27553 may seem more ``efficient'' than two separate rules for @code{ln} and
27554 @code{log10}, but actually it is vastly less efficient because rules
27555 with @code{apply} as the top-level pattern must be tested against
27556 @emph{every} function call that is simplified.
27558 @cindex @code{AlgSimpRules} variable
27559 @vindex AlgSimpRules
27560 Suppose you want @samp{sin(a + b)} to be expanded out not all the time,
27561 but only when @kbd{a s} is used to simplify the formula. The variable
27562 @code{AlgSimpRules} holds rules for this purpose. The @kbd{a s} command
27563 will apply @code{EvalRules} and @code{AlgSimpRules} to the formula, as
27564 well as all of its built-in simplifications.
27566 Most of the special limitations for @code{EvalRules} don't apply to
27567 @code{AlgSimpRules}. Calc simply does an @kbd{a r AlgSimpRules}
27568 command with an infinite repeat count as the first step of @kbd{a s}.
27569 It then applies its own built-in simplifications throughout the
27570 formula, and then repeats these two steps (along with applying the
27571 default simplifications) until no further changes are possible.
27573 @cindex @code{ExtSimpRules} variable
27574 @cindex @code{UnitSimpRules} variable
27575 @vindex ExtSimpRules
27576 @vindex UnitSimpRules
27577 There are also @code{ExtSimpRules} and @code{UnitSimpRules} variables
27578 that are used by @kbd{a e} and @kbd{u s}, respectively; these commands
27579 also apply @code{EvalRules} and @code{AlgSimpRules}. The variable
27580 @code{IntegSimpRules} contains simplification rules that are used
27581 only during integration by @kbd{a i}.
27583 @node Debugging Rewrites, Examples of Rewrite Rules, Automatic Rewrites, Rewrite Rules
27584 @subsection Debugging Rewrites
27587 If a buffer named @samp{*Trace*} exists, the rewrite mechanism will
27588 record some useful information there as it operates. The original
27589 formula is written there, as is the result of each successful rewrite,
27590 and the final result of the rewriting. All phase changes are also
27593 Calc always appends to @samp{*Trace*}. You must empty this buffer
27594 yourself periodically if it is in danger of growing unwieldy.
27596 Note that the rewriting mechanism is substantially slower when the
27597 @samp{*Trace*} buffer exists, even if the buffer is not visible on
27598 the screen. Once you are done, you will probably want to kill this
27599 buffer (with @kbd{C-x k *Trace* @key{RET}}). If you leave it in
27600 existence and forget about it, all your future rewrite commands will
27601 be needlessly slow.
27603 @node Examples of Rewrite Rules, , Debugging Rewrites, Rewrite Rules
27604 @subsection Examples of Rewrite Rules
27607 Returning to the example of substituting the pattern
27608 @samp{sin(x)^2 + cos(x)^2} with 1, we saw that the rule
27609 @samp{opt(a) sin(x)^2 + opt(a) cos(x)^2 := a} does a good job of
27610 finding suitable cases. Another solution would be to use the rule
27611 @samp{cos(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification
27612 if necessary. This rule will be the most effective way to do the job,
27613 but at the expense of making some changes that you might not desire.
27615 Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}.
27616 To make this work with the @w{@kbd{j r}} command so that it can be
27617 easily targeted to a particular exponential in a large formula,
27618 you might wish to write the rule as @samp{select(exp(x+y)) :=
27619 select(exp(x) exp(y))}. The @samp{select} markers will be
27620 ignored by the regular @kbd{a r} command
27621 (@pxref{Selections with Rewrite Rules}).
27623 A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}.
27624 This will simplify the formula whenever @expr{b} and/or @expr{c} can
27625 be made simpler by squaring. For example, applying this rule to
27626 @samp{2 / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming
27627 Symbolic mode has been enabled to keep the square root from being
27628 evaluated to a floating-point approximation). This rule is also
27629 useful when working with symbolic complex numbers, e.g.,
27630 @samp{(a + b i) / (c + d i)}.
27632 As another example, we could define our own ``triangular numbers'' function
27633 with the rules @samp{[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]}. Enter
27634 this vector and store it in a variable: @kbd{@w{s t} trirules}. Now, given
27635 a suitable formula like @samp{tri(5)} on the stack, type @samp{a r trirules}
27636 to apply these rules repeatedly. After six applications, @kbd{a r} will
27637 stop with 15 on the stack. Once these rules are debugged, it would probably
27638 be most useful to add them to @code{EvalRules} so that Calc will evaluate
27639 the new @code{tri} function automatically. We could then use @kbd{Z K} on
27640 the keyboard macro @kbd{' tri($) @key{RET}} to make a command that applies
27641 @code{tri} to the value on the top of the stack. @xref{Programming}.
27643 @cindex Quaternions
27644 The following rule set, contributed by
27645 @texline Fran\c cois
27647 Pinard, implements @dfn{quaternions}, a generalization of the concept of
27648 complex numbers. Quaternions have four components, and are here
27649 represented by function calls @samp{quat(@var{w}, [@var{x}, @var{y},
27650 @var{z}])} with ``real part'' @var{w} and the three ``imaginary'' parts
27651 collected into a vector. Various arithmetical operations on quaternions
27652 are supported. To use these rules, either add them to @code{EvalRules},
27653 or create a command based on @kbd{a r} for simplifying quaternion
27654 formulas. A convenient way to enter quaternions would be a command
27655 defined by a keyboard macro containing: @kbd{' quat($$$$, [$$$, $$, $])
27659 [ quat(w, x, y, z) := quat(w, [x, y, z]),
27660 quat(w, [0, 0, 0]) := w,
27661 abs(quat(w, v)) := hypot(w, v),
27662 -quat(w, v) := quat(-w, -v),
27663 r + quat(w, v) := quat(r + w, v) :: real(r),
27664 r - quat(w, v) := quat(r - w, -v) :: real(r),
27665 quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
27666 r * quat(w, v) := quat(r * w, r * v) :: real(r),
27667 plain(quat(w1, v1) * quat(w2, v2))
27668 := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
27669 quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
27670 z / quat(w, v) := z * quatinv(quat(w, v)),
27671 quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
27672 quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
27673 quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
27674 :: integer(k) :: k > 0 :: k % 2 = 0,
27675 quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
27676 :: integer(k) :: k > 2,
27677 quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]
27680 Quaternions, like matrices, have non-commutative multiplication.
27681 In other words, @expr{q1 * q2 = q2 * q1} is not necessarily true if
27682 @expr{q1} and @expr{q2} are @code{quat} forms. The @samp{quat*quat}
27683 rule above uses @code{plain} to prevent Calc from rearranging the
27684 product. It may also be wise to add the line @samp{[quat(), matrix]}
27685 to the @code{Decls} matrix, to ensure that Calc's other algebraic
27686 operations will not rearrange a quaternion product. @xref{Declarations}.
27688 These rules also accept a four-argument @code{quat} form, converting
27689 it to the preferred form in the first rule. If you would rather see
27690 results in the four-argument form, just append the two items
27691 @samp{phase(2), quat(w, [x, y, z]) := quat(w, x, y, z)} to the end
27692 of the rule set. (But remember that multi-phase rule sets don't work
27693 in @code{EvalRules}.)
27695 @node Units, Store and Recall, Algebra, Top
27696 @chapter Operating on Units
27699 One special interpretation of algebraic formulas is as numbers with units.
27700 For example, the formula @samp{5 m / s^2} can be read ``five meters
27701 per second squared.'' The commands in this chapter help you
27702 manipulate units expressions in this form. Units-related commands
27703 begin with the @kbd{u} prefix key.
27706 * Basic Operations on Units::
27707 * The Units Table::
27708 * Predefined Units::
27709 * User-Defined Units::
27712 @node Basic Operations on Units, The Units Table, Units, Units
27713 @section Basic Operations on Units
27716 A @dfn{units expression} is a formula which is basically a number
27717 multiplied and/or divided by one or more @dfn{unit names}, which may
27718 optionally be raised to integer powers. Actually, the value part need not
27719 be a number; any product or quotient involving unit names is a units
27720 expression. Many of the units commands will also accept any formula,
27721 where the command applies to all units expressions which appear in the
27724 A unit name is a variable whose name appears in the @dfn{unit table},
27725 or a variable whose name is a prefix character like @samp{k} (for ``kilo'')
27726 or @samp{u} (for ``micro'') followed by a name in the unit table.
27727 A substantial table of built-in units is provided with Calc;
27728 @pxref{Predefined Units}. You can also define your own unit names;
27729 @pxref{User-Defined Units}.
27731 Note that if the value part of a units expression is exactly @samp{1},
27732 it will be removed by the Calculator's automatic algebra routines: The
27733 formula @samp{1 mm} is ``simplified'' to @samp{mm}. This is only a
27734 display anomaly, however; @samp{mm} will work just fine as a
27735 representation of one millimeter.
27737 You may find that Algebraic mode (@pxref{Algebraic Entry}) makes working
27738 with units expressions easier. Otherwise, you will have to remember
27739 to hit the apostrophe key every time you wish to enter units.
27742 @pindex calc-simplify-units
27744 @mindex usimpl@idots
27747 The @kbd{u s} (@code{calc-simplify-units}) [@code{usimplify}] command
27749 expression. It uses @kbd{a s} (@code{calc-simplify}) to simplify the
27750 expression first as a regular algebraic formula; it then looks for
27751 features that can be further simplified by converting one object's units
27752 to be compatible with another's. For example, @samp{5 m + 23 mm} will
27753 simplify to @samp{5.023 m}. When different but compatible units are
27754 added, the righthand term's units are converted to match those of the
27755 lefthand term. @xref{Simplification Modes}, for a way to have this done
27756 automatically at all times.
27758 Units simplification also handles quotients of two units with the same
27759 dimensionality, as in @w{@samp{2 in s/L cm}} to @samp{5.08 s/L}; fractional
27760 powers of unit expressions, as in @samp{sqrt(9 mm^2)} to @samp{3 mm} and
27761 @samp{sqrt(9 acre)} to a quantity in meters; and @code{floor},
27762 @code{ceil}, @code{round}, @code{rounde}, @code{roundu}, @code{trunc},
27763 @code{float}, @code{frac}, @code{abs}, and @code{clean}
27764 applied to units expressions, in which case
27765 the operation in question is applied only to the numeric part of the
27766 expression. Finally, trigonometric functions of quantities with units
27767 of angle are evaluated, regardless of the current angular mode.
27770 @pindex calc-convert-units
27771 The @kbd{u c} (@code{calc-convert-units}) command converts a units
27772 expression to new, compatible units. For example, given the units
27773 expression @samp{55 mph}, typing @kbd{u c m/s @key{RET}} produces
27774 @samp{24.5872 m/s}. If the units you request are inconsistent with
27775 the original units, the number will be converted into your units
27776 times whatever ``remainder'' units are left over. For example,
27777 converting @samp{55 mph} into acres produces @samp{6.08e-3 acre / m s}.
27778 (Recall that multiplication binds more strongly than division in Calc
27779 formulas, so the units here are acres per meter-second.) Remainder
27780 units are expressed in terms of ``fundamental'' units like @samp{m} and
27781 @samp{s}, regardless of the input units.
27783 One special exception is that if you specify a single unit name, and
27784 a compatible unit appears somewhere in the units expression, then
27785 that compatible unit will be converted to the new unit and the
27786 remaining units in the expression will be left alone. For example,
27787 given the input @samp{980 cm/s^2}, the command @kbd{u c ms} will
27788 change the @samp{s} to @samp{ms} to get @samp{9.8e-4 cm/ms^2}.
27789 The ``remainder unit'' @samp{cm} is left alone rather than being
27790 changed to the base unit @samp{m}.
27792 You can use explicit unit conversion instead of the @kbd{u s} command
27793 to gain more control over the units of the result of an expression.
27794 For example, given @samp{5 m + 23 mm}, you can type @kbd{u c m} or
27795 @kbd{u c mm} to express the result in either meters or millimeters.
27796 (For that matter, you could type @kbd{u c fath} to express the result
27797 in fathoms, if you preferred!)
27799 In place of a specific set of units, you can also enter one of the
27800 units system names @code{si}, @code{mks} (equivalent), or @code{cgs}.
27801 For example, @kbd{u c si @key{RET}} converts the expression into
27802 International System of Units (SI) base units. Also, @kbd{u c base}
27803 converts to Calc's base units, which are the same as @code{si} units
27804 except that @code{base} uses @samp{g} as the fundamental unit of mass
27805 whereas @code{si} uses @samp{kg}.
27807 @cindex Composite units
27808 The @kbd{u c} command also accepts @dfn{composite units}, which
27809 are expressed as the sum of several compatible unit names. For
27810 example, converting @samp{30.5 in} to units @samp{mi+ft+in} (miles,
27811 feet, and inches) produces @samp{2 ft + 6.5 in}. Calc first
27812 sorts the unit names into order of decreasing relative size.
27813 It then accounts for as much of the input quantity as it can
27814 using an integer number times the largest unit, then moves on
27815 to the next smaller unit, and so on. Only the smallest unit
27816 may have a non-integer amount attached in the result. A few
27817 standard unit names exist for common combinations, such as
27818 @code{mfi} for @samp{mi+ft+in}, and @code{tpo} for @samp{ton+lb+oz}.
27819 Composite units are expanded as if by @kbd{a x}, so that
27820 @samp{(ft+in)/hr} is first converted to @samp{ft/hr+in/hr}.
27822 If the value on the stack does not contain any units, @kbd{u c} will
27823 prompt first for the old units which this value should be considered
27824 to have, then for the new units. Assuming the old and new units you
27825 give are consistent with each other, the result also will not contain
27826 any units. For example, @kbd{@w{u c} cm @key{RET} in @key{RET}} converts the number
27827 2 on the stack to 5.08.
27830 @pindex calc-base-units
27831 The @kbd{u b} (@code{calc-base-units}) command is shorthand for
27832 @kbd{u c base}; it converts the units expression on the top of the
27833 stack into @code{base} units. If @kbd{u s} does not simplify a
27834 units expression as far as you would like, try @kbd{u b}.
27836 The @kbd{u c} and @kbd{u b} commands treat temperature units (like
27837 @samp{degC} and @samp{K}) as relative temperatures. For example,
27838 @kbd{u c} converts @samp{10 degC} to @samp{18 degF}: A change of 10
27839 degrees Celsius corresponds to a change of 18 degrees Fahrenheit.
27842 @pindex calc-convert-temperature
27843 @cindex Temperature conversion
27844 The @kbd{u t} (@code{calc-convert-temperature}) command converts
27845 absolute temperatures. The value on the stack must be a simple units
27846 expression with units of temperature only. This command would convert
27847 @samp{10 degC} to @samp{50 degF}, the equivalent temperature on the
27851 @pindex calc-remove-units
27853 @pindex calc-extract-units
27854 The @kbd{u r} (@code{calc-remove-units}) command removes units from the
27855 formula at the top of the stack. The @kbd{u x}
27856 (@code{calc-extract-units}) command extracts only the units portion of a
27857 formula. These commands essentially replace every term of the formula
27858 that does or doesn't (respectively) look like a unit name by the
27859 constant 1, then resimplify the formula.
27862 @pindex calc-autorange-units
27863 The @kbd{u a} (@code{calc-autorange-units}) command turns on and off a
27864 mode in which unit prefixes like @code{k} (``kilo'') are automatically
27865 applied to keep the numeric part of a units expression in a reasonable
27866 range. This mode affects @kbd{u s} and all units conversion commands
27867 except @kbd{u b}. For example, with autoranging on, @samp{12345 Hz}
27868 will be simplified to @samp{12.345 kHz}. Autoranging is useful for
27869 some kinds of units (like @code{Hz} and @code{m}), but is probably
27870 undesirable for non-metric units like @code{ft} and @code{tbsp}.
27871 (Composite units are more appropriate for those; see above.)
27873 Autoranging always applies the prefix to the leftmost unit name.
27874 Calc chooses the largest prefix that causes the number to be greater
27875 than or equal to 1.0. Thus an increasing sequence of adjusted times
27876 would be @samp{1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks}.
27877 Generally the rule of thumb is that the number will be adjusted
27878 to be in the interval @samp{[1 .. 1000)}, although there are several
27879 exceptions to this rule. First, if the unit has a power then this
27880 is not possible; @samp{0.1 s^2} simplifies to @samp{100000 ms^2}.
27881 Second, the ``centi-'' prefix is allowed to form @code{cm} (centimeters),
27882 but will not apply to other units. The ``deci-,'' ``deka-,'' and
27883 ``hecto-'' prefixes are never used. Thus the allowable interval is
27884 @samp{[1 .. 10)} for millimeters and @samp{[1 .. 100)} for centimeters.
27885 Finally, a prefix will not be added to a unit if the resulting name
27886 is also the actual name of another unit; @samp{1e-15 t} would normally
27887 be considered a ``femto-ton,'' but it is written as @samp{1000 at}
27888 (1000 atto-tons) instead because @code{ft} would be confused with feet.
27890 @node The Units Table, Predefined Units, Basic Operations on Units, Units
27891 @section The Units Table
27895 @pindex calc-enter-units-table
27896 The @kbd{u v} (@code{calc-enter-units-table}) command displays the units table
27897 in another buffer called @code{*Units Table*}. Each entry in this table
27898 gives the unit name as it would appear in an expression, the definition
27899 of the unit in terms of simpler units, and a full name or description of
27900 the unit. Fundamental units are defined as themselves; these are the
27901 units produced by the @kbd{u b} command. The fundamental units are
27902 meters, seconds, grams, kelvins, amperes, candelas, moles, radians,
27905 The Units Table buffer also displays the Unit Prefix Table. Note that
27906 two prefixes, ``kilo'' and ``hecto,'' accept either upper- or lower-case
27907 prefix letters. @samp{Meg} is also accepted as a synonym for the @samp{M}
27908 prefix. Whenever a unit name can be interpreted as either a built-in name
27909 or a prefix followed by another built-in name, the former interpretation
27910 wins. For example, @samp{2 pt} means two pints, not two pico-tons.
27912 The Units Table buffer, once created, is not rebuilt unless you define
27913 new units. To force the buffer to be rebuilt, give any numeric prefix
27914 argument to @kbd{u v}.
27917 @pindex calc-view-units-table
27918 The @kbd{u V} (@code{calc-view-units-table}) command is like @kbd{u v} except
27919 that the cursor is not moved into the Units Table buffer. You can
27920 type @kbd{u V} again to remove the Units Table from the display. To
27921 return from the Units Table buffer after a @kbd{u v}, type @kbd{M-# c}
27922 again or use the regular Emacs @w{@kbd{C-x o}} (@code{other-window})
27923 command. You can also kill the buffer with @kbd{C-x k} if you wish;
27924 the actual units table is safely stored inside the Calculator.
27927 @pindex calc-get-unit-definition
27928 The @kbd{u g} (@code{calc-get-unit-definition}) command retrieves a unit's
27929 defining expression and pushes it onto the Calculator stack. For example,
27930 @kbd{u g in} will produce the expression @samp{2.54 cm}. This is the
27931 same definition for the unit that would appear in the Units Table buffer.
27932 Note that this command works only for actual unit names; @kbd{u g km}
27933 will report that no such unit exists, for example, because @code{km} is
27934 really the unit @code{m} with a @code{k} (``kilo'') prefix. To see a
27935 definition of a unit in terms of base units, it is easier to push the
27936 unit name on the stack and then reduce it to base units with @kbd{u b}.
27939 @pindex calc-explain-units
27940 The @kbd{u e} (@code{calc-explain-units}) command displays an English
27941 description of the units of the expression on the stack. For example,
27942 for the expression @samp{62 km^2 g / s^2 mol K}, the description is
27943 ``Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin).'' This
27944 command uses the English descriptions that appear in the righthand
27945 column of the Units Table.
27947 @node Predefined Units, User-Defined Units, The Units Table, Units
27948 @section Predefined Units
27951 Since the exact definitions of many kinds of units have evolved over the
27952 years, and since certain countries sometimes have local differences in
27953 their definitions, it is a good idea to examine Calc's definition of a
27954 unit before depending on its exact value. For example, there are three
27955 different units for gallons, corresponding to the US (@code{gal}),
27956 Canadian (@code{galC}), and British (@code{galUK}) definitions. Also,
27957 note that @code{oz} is a standard ounce of mass, @code{ozt} is a Troy
27958 ounce, and @code{ozfl} is a fluid ounce.
27960 The temperature units corresponding to degrees Kelvin and Centigrade
27961 (Celsius) are the same in this table, since most units commands treat
27962 temperatures as being relative. The @code{calc-convert-temperature}
27963 command has special rules for handling the different absolute magnitudes
27964 of the various temperature scales.
27966 The unit of volume ``liters'' can be referred to by either the lower-case
27967 @code{l} or the upper-case @code{L}.
27969 The unit @code{A} stands for Amperes; the name @code{Ang} is used
27977 The unit @code{pt} stands for pints; the name @code{point} stands for
27978 a typographical point, defined by @samp{72 point = 1 in}. There is
27979 also @code{tpt}, which stands for a printer's point as defined by the
27980 @TeX{} typesetting system: @samp{72.27 tpt = 1 in}.
27982 The unit @code{e} stands for the elementary (electron) unit of charge;
27983 because algebra command could mistake this for the special constant
27984 @expr{e}, Calc provides the alternate unit name @code{ech} which is
27985 preferable to @code{e}.
27987 The name @code{g} stands for one gram of mass; there is also @code{gf},
27988 one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.)
27989 Meanwhile, one ``@expr{g}'' of acceleration is denoted @code{ga}.
27991 The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is
27992 a metric ton of @samp{1000 kg}.
27994 The names @code{s} (or @code{sec}) and @code{min} refer to units of
27995 time; @code{arcsec} and @code{arcmin} are units of angle.
27997 Some ``units'' are really physical constants; for example, @code{c}
27998 represents the speed of light, and @code{h} represents Planck's
27999 constant. You can use these just like other units: converting
28000 @samp{.5 c} to @samp{m/s} expresses one-half the speed of light in
28001 meters per second. You can also use this merely as a handy reference;
28002 the @kbd{u g} command gets the definition of one of these constants
28003 in its normal terms, and @kbd{u b} expresses the definition in base
28006 Two units, @code{pi} and @code{fsc} (the fine structure constant,
28007 approximately @mathit{1/137}) are dimensionless. The units simplification
28008 commands simply treat these names as equivalent to their corresponding
28009 values. However you can, for example, use @kbd{u c} to convert a pure
28010 number into multiples of the fine structure constant, or @kbd{u b} to
28011 convert this back into a pure number. (When @kbd{u c} prompts for the
28012 ``old units,'' just enter a blank line to signify that the value
28013 really is unitless.)
28015 @c Describe angular units, luminosity vs. steradians problem.
28017 @node User-Defined Units, , Predefined Units, Units
28018 @section User-Defined Units
28021 Calc provides ways to get quick access to your selected ``favorite''
28022 units, as well as ways to define your own new units.
28025 @pindex calc-quick-units
28027 @cindex @code{Units} variable
28028 @cindex Quick units
28029 To select your favorite units, store a vector of unit names or
28030 expressions in the Calc variable @code{Units}. The @kbd{u 1}
28031 through @kbd{u 9} commands (@code{calc-quick-units}) provide access
28032 to these units. If the value on the top of the stack is a plain
28033 number (with no units attached), then @kbd{u 1} gives it the
28034 specified units. (Basically, it multiplies the number by the
28035 first item in the @code{Units} vector.) If the number on the
28036 stack @emph{does} have units, then @kbd{u 1} converts that number
28037 to the new units. For example, suppose the vector @samp{[in, ft]}
28038 is stored in @code{Units}. Then @kbd{30 u 1} will create the
28039 expression @samp{30 in}, and @kbd{u 2} will convert that expression
28042 The @kbd{u 0} command accesses the tenth element of @code{Units}.
28043 Only ten quick units may be defined at a time. If the @code{Units}
28044 variable has no stored value (the default), or if its value is not
28045 a vector, then the quick-units commands will not function. The
28046 @kbd{s U} command is a convenient way to edit the @code{Units}
28047 variable; @pxref{Operations on Variables}.
28050 @pindex calc-define-unit
28051 @cindex User-defined units
28052 The @kbd{u d} (@code{calc-define-unit}) command records the units
28053 expression on the top of the stack as the definition for a new,
28054 user-defined unit. For example, putting @samp{16.5 ft} on the stack and
28055 typing @kbd{u d rod} defines the new unit @samp{rod} to be equivalent to
28056 16.5 feet. The unit conversion and simplification commands will now
28057 treat @code{rod} just like any other unit of length. You will also be
28058 prompted for an optional English description of the unit, which will
28059 appear in the Units Table.
28062 @pindex calc-undefine-unit
28063 The @kbd{u u} (@code{calc-undefine-unit}) command removes a user-defined
28064 unit. It is not possible to remove one of the predefined units,
28067 If you define a unit with an existing unit name, your new definition
28068 will replace the original definition of that unit. If the unit was a
28069 predefined unit, the old definition will not be replaced, only
28070 ``shadowed.'' The built-in definition will reappear if you later use
28071 @kbd{u u} to remove the shadowing definition.
28073 To create a new fundamental unit, use either 1 or the unit name itself
28074 as the defining expression. Otherwise the expression can involve any
28075 other units that you like (except for composite units like @samp{mfi}).
28076 You can create a new composite unit with a sum of other units as the
28077 defining expression. The next unit operation like @kbd{u c} or @kbd{u v}
28078 will rebuild the internal unit table incorporating your modifications.
28079 Note that erroneous definitions (such as two units defined in terms of
28080 each other) will not be detected until the unit table is next rebuilt;
28081 @kbd{u v} is a convenient way to force this to happen.
28083 Temperature units are treated specially inside the Calculator; it is not
28084 possible to create user-defined temperature units.
28087 @pindex calc-permanent-units
28088 @cindex Calc init file, user-defined units
28089 The @kbd{u p} (@code{calc-permanent-units}) command stores the user-defined
28090 units in your Calc init file (the file given by the variable
28091 @code{calc-settings-file}, typically @file{~/.calc.el}), so that the
28092 units will still be available in subsequent Emacs sessions. If there
28093 was already a set of user-defined units in your Calc init file, it
28094 is replaced by the new set. (@xref{General Mode Commands}, for a way to
28095 tell Calc to use a different file for the Calc init file.)
28097 @node Store and Recall, Graphics, Units, Top
28098 @chapter Storing and Recalling
28101 Calculator variables are really just Lisp variables that contain numbers
28102 or formulas in a form that Calc can understand. The commands in this
28103 section allow you to manipulate variables conveniently. Commands related
28104 to variables use the @kbd{s} prefix key.
28107 * Storing Variables::
28108 * Recalling Variables::
28109 * Operations on Variables::
28111 * Evaluates-To Operator::
28114 @node Storing Variables, Recalling Variables, Store and Recall, Store and Recall
28115 @section Storing Variables
28120 @cindex Storing variables
28121 @cindex Quick variables
28124 The @kbd{s s} (@code{calc-store}) command stores the value at the top of
28125 the stack into a specified variable. It prompts you to enter the
28126 name of the variable. If you press a single digit, the value is stored
28127 immediately in one of the ``quick'' variables @code{q0} through
28128 @code{q9}. Or you can enter any variable name.
28131 @pindex calc-store-into
28132 The @kbd{s s} command leaves the stored value on the stack. There is
28133 also an @kbd{s t} (@code{calc-store-into}) command, which removes a
28134 value from the stack and stores it in a variable.
28136 If the top of stack value is an equation @samp{a = 7} or assignment
28137 @samp{a := 7} with a variable on the lefthand side, then Calc will
28138 assign that variable with that value by default, i.e., if you type
28139 @kbd{s s @key{RET}} or @kbd{s t @key{RET}}. In this example, the
28140 value 7 would be stored in the variable @samp{a}. (If you do type
28141 a variable name at the prompt, the top-of-stack value is stored in
28142 its entirety, even if it is an equation: @samp{s s b @key{RET}}
28143 with @samp{a := 7} on the stack stores @samp{a := 7} in @code{b}.)
28145 In fact, the top of stack value can be a vector of equations or
28146 assignments with different variables on their lefthand sides; the
28147 default will be to store all the variables with their corresponding
28148 righthand sides simultaneously.
28150 It is also possible to type an equation or assignment directly at
28151 the prompt for the @kbd{s s} or @kbd{s t} command: @kbd{s s foo = 7}.
28152 In this case the expression to the right of the @kbd{=} or @kbd{:=}
28153 symbol is evaluated as if by the @kbd{=} command, and that value is
28154 stored in the variable. No value is taken from the stack; @kbd{s s}
28155 and @kbd{s t} are equivalent when used in this way.
28159 The prefix keys @kbd{s} and @kbd{t} may be followed immediately by a
28160 digit; @kbd{s 9} is equivalent to @kbd{s s 9}, and @kbd{t 9} is
28161 equivalent to @kbd{s t 9}. (The @kbd{t} prefix is otherwise used
28162 for trail and time/date commands.)
28198 @pindex calc-store-plus
28199 @pindex calc-store-minus
28200 @pindex calc-store-times
28201 @pindex calc-store-div
28202 @pindex calc-store-power
28203 @pindex calc-store-concat
28204 @pindex calc-store-neg
28205 @pindex calc-store-inv
28206 @pindex calc-store-decr
28207 @pindex calc-store-incr
28208 There are also several ``arithmetic store'' commands. For example,
28209 @kbd{s +} removes a value from the stack and adds it to the specified
28210 variable. The other arithmetic stores are @kbd{s -}, @kbd{s *}, @kbd{s /},
28211 @kbd{s ^}, and @w{@kbd{s |}} (vector concatenation), plus @kbd{s n} and
28212 @kbd{s &} which negate or invert the value in a variable, and @w{@kbd{s [}}
28213 and @kbd{s ]} which decrease or increase a variable by one.
28215 All the arithmetic stores accept the Inverse prefix to reverse the
28216 order of the operands. If @expr{v} represents the contents of the
28217 variable, and @expr{a} is the value drawn from the stack, then regular
28218 @w{@kbd{s -}} assigns
28219 @texline @math{v \coloneq v - a},
28220 @infoline @expr{v := v - a},
28221 but @kbd{I s -} assigns
28222 @texline @math{v \coloneq a - v}.
28223 @infoline @expr{v := a - v}.
28224 While @kbd{I s *} might seem pointless, it is
28225 useful if matrix multiplication is involved. Actually, all the
28226 arithmetic stores use formulas designed to behave usefully both
28227 forwards and backwards:
28231 s + v := v + a v := a + v
28232 s - v := v - a v := a - v
28233 s * v := v * a v := a * v
28234 s / v := v / a v := a / v
28235 s ^ v := v ^ a v := a ^ v
28236 s | v := v | a v := a | v
28237 s n v := v / (-1) v := (-1) / v
28238 s & v := v ^ (-1) v := (-1) ^ v
28239 s [ v := v - 1 v := 1 - v
28240 s ] v := v - (-1) v := (-1) - v
28244 In the last four cases, a numeric prefix argument will be used in
28245 place of the number one. (For example, @kbd{M-2 s ]} increases
28246 a variable by 2, and @kbd{M-2 I s ]} replaces a variable by
28247 minus-two minus the variable.
28249 The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -},
28250 etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous
28251 arithmetic stores that don't remove the value @expr{a} from the stack.
28253 All arithmetic stores report the new value of the variable in the
28254 Trail for your information. They signal an error if the variable
28255 previously had no stored value. If default simplifications have been
28256 turned off, the arithmetic stores temporarily turn them on for numeric
28257 arguments only (i.e., they temporarily do an @kbd{m N} command).
28258 @xref{Simplification Modes}. Large vectors put in the trail by
28259 these commands always use abbreviated (@kbd{t .}) mode.
28262 @pindex calc-store-map
28263 The @kbd{s m} command is a general way to adjust a variable's value
28264 using any Calc function. It is a ``mapping'' command analogous to
28265 @kbd{V M}, @kbd{V R}, etc. @xref{Reducing and Mapping}, to see
28266 how to specify a function for a mapping command. Basically,
28267 all you do is type the Calc command key that would invoke that
28268 function normally. For example, @kbd{s m n} applies the @kbd{n}
28269 key to negate the contents of the variable, so @kbd{s m n} is
28270 equivalent to @kbd{s n}. Also, @kbd{s m Q} takes the square root
28271 of the value stored in a variable, @kbd{s m v v} uses @kbd{v v} to
28272 reverse the vector stored in the variable, and @kbd{s m H I S}
28273 takes the hyperbolic arcsine of the variable contents.
28275 If the mapping function takes two or more arguments, the additional
28276 arguments are taken from the stack; the old value of the variable
28277 is provided as the first argument. Thus @kbd{s m -} with @expr{a}
28278 on the stack computes @expr{v - a}, just like @kbd{s -}. With the
28279 Inverse prefix, the variable's original value becomes the @emph{last}
28280 argument instead of the first. Thus @kbd{I s m -} is also
28281 equivalent to @kbd{I s -}.
28284 @pindex calc-store-exchange
28285 The @kbd{s x} (@code{calc-store-exchange}) command exchanges the value
28286 of a variable with the value on the top of the stack. Naturally, the
28287 variable must already have a stored value for this to work.
28289 You can type an equation or assignment at the @kbd{s x} prompt. The
28290 command @kbd{s x a=6} takes no values from the stack; instead, it
28291 pushes the old value of @samp{a} on the stack and stores @samp{a = 6}.
28294 @pindex calc-unstore
28295 @cindex Void variables
28296 @cindex Un-storing variables
28297 Until you store something in them, most variables are ``void,'' that is,
28298 they contain no value at all. If they appear in an algebraic formula
28299 they will be left alone even if you press @kbd{=} (@code{calc-evaluate}).
28300 The @kbd{s u} (@code{calc-unstore}) command returns a variable to the
28304 @pindex calc-copy-variable
28305 The @kbd{s c} (@code{calc-copy-variable}) command copies the stored
28306 value of one variable to another. One way it differs from a simple
28307 @kbd{s r} followed by an @kbd{s t} (aside from saving keystrokes) is
28308 that the value never goes on the stack and thus is never rounded,
28309 evaluated, or simplified in any way; it is not even rounded down to the
28312 The only variables with predefined values are the ``special constants''
28313 @code{pi}, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free
28314 to unstore these variables or to store new values into them if you like,
28315 although some of the algebraic-manipulation functions may assume these
28316 variables represent their standard values. Calc displays a warning if
28317 you change the value of one of these variables, or of one of the other
28318 special variables @code{inf}, @code{uinf}, and @code{nan} (which are
28321 Note that @code{pi} doesn't actually have 3.14159265359 stored in it,
28322 but rather a special magic value that evaluates to @cpi{} at the current
28323 precision. Likewise @code{e}, @code{i}, and @code{phi} evaluate
28324 according to the current precision or polar mode. If you recall a value
28325 from @code{pi} and store it back, this magic property will be lost. The
28326 magic property is preserved, however, when a variable is copied with
28330 @pindex calc-copy-special-constant
28331 If one of the ``special constants'' is redefined (or undefined) so that
28332 it no longer has its magic property, the property can be restored with
28333 @kbd{s k} (@code{calc-copy-special-constant}). This command will prompt
28334 for a special constant and a variable to store it in, and so a special
28335 constant can be stored in any variable. Here, the special constant that
28336 you enter doesn't depend on the value of the corresponding variable;
28337 @code{pi} will represent 3.14159@dots{} regardless of what is currently
28338 stored in the Calc variable @code{pi}. If one of the other special
28339 variables, @code{inf}, @code{uinf} or @code{nan}, is given a value, its
28340 original behavior can be restored by voiding it with @kbd{s u}.
28342 @node Recalling Variables, Operations on Variables, Storing Variables, Store and Recall
28343 @section Recalling Variables
28347 @pindex calc-recall
28348 @cindex Recalling variables
28349 The most straightforward way to extract the stored value from a variable
28350 is to use the @kbd{s r} (@code{calc-recall}) command. This command prompts
28351 for a variable name (similarly to @code{calc-store}), looks up the value
28352 of the specified variable, and pushes that value onto the stack. It is
28353 an error to try to recall a void variable.
28355 It is also possible to recall the value from a variable by evaluating a
28356 formula containing that variable. For example, @kbd{' a @key{RET} =} is
28357 the same as @kbd{s r a @key{RET}} except that if the variable is void, the
28358 former will simply leave the formula @samp{a} on the stack whereas the
28359 latter will produce an error message.
28362 The @kbd{r} prefix may be followed by a digit, so that @kbd{r 9} is
28363 equivalent to @kbd{s r 9}. (The @kbd{r} prefix is otherwise unused
28364 in the current version of Calc.)
28366 @node Operations on Variables, Let Command, Recalling Variables, Store and Recall
28367 @section Other Operations on Variables
28371 @pindex calc-edit-variable
28372 The @kbd{s e} (@code{calc-edit-variable}) command edits the stored
28373 value of a variable without ever putting that value on the stack
28374 or simplifying or evaluating the value. It prompts for the name of
28375 the variable to edit. If the variable has no stored value, the
28376 editing buffer will start out empty. If the editing buffer is
28377 empty when you press @kbd{C-c C-c} to finish, the variable will
28378 be made void. @xref{Editing Stack Entries}, for a general
28379 description of editing.
28381 The @kbd{s e} command is especially useful for creating and editing
28382 rewrite rules which are stored in variables. Sometimes these rules
28383 contain formulas which must not be evaluated until the rules are
28384 actually used. (For example, they may refer to @samp{deriv(x,y)},
28385 where @code{x} will someday become some expression involving @code{y};
28386 if you let Calc evaluate the rule while you are defining it, Calc will
28387 replace @samp{deriv(x,y)} with 0 because the formula @code{x} does
28388 not itself refer to @code{y}.) By contrast, recalling the variable,
28389 editing with @kbd{`}, and storing will evaluate the variable's value
28390 as a side effect of putting the value on the stack.
28438 @pindex calc-store-AlgSimpRules
28439 @pindex calc-store-Decls
28440 @pindex calc-store-EvalRules
28441 @pindex calc-store-FitRules
28442 @pindex calc-store-GenCount
28443 @pindex calc-store-Holidays
28444 @pindex calc-store-IntegLimit
28445 @pindex calc-store-LineStyles
28446 @pindex calc-store-PointStyles
28447 @pindex calc-store-PlotRejects
28448 @pindex calc-store-TimeZone
28449 @pindex calc-store-Units
28450 @pindex calc-store-ExtSimpRules
28451 There are several special-purpose variable-editing commands that
28452 use the @kbd{s} prefix followed by a shifted letter:
28456 Edit @code{AlgSimpRules}. @xref{Algebraic Simplifications}.
28458 Edit @code{Decls}. @xref{Declarations}.
28460 Edit @code{EvalRules}. @xref{Default Simplifications}.
28462 Edit @code{FitRules}. @xref{Curve Fitting}.
28464 Edit @code{GenCount}. @xref{Solving Equations}.
28466 Edit @code{Holidays}. @xref{Business Days}.
28468 Edit @code{IntegLimit}. @xref{Calculus}.
28470 Edit @code{LineStyles}. @xref{Graphics}.
28472 Edit @code{PointStyles}. @xref{Graphics}.
28474 Edit @code{PlotRejects}. @xref{Graphics}.
28476 Edit @code{TimeZone}. @xref{Time Zones}.
28478 Edit @code{Units}. @xref{User-Defined Units}.
28480 Edit @code{ExtSimpRules}. @xref{Unsafe Simplifications}.
28483 These commands are just versions of @kbd{s e} that use fixed variable
28484 names rather than prompting for the variable name.
28487 @pindex calc-permanent-variable
28488 @cindex Storing variables
28489 @cindex Permanent variables
28490 @cindex Calc init file, variables
28491 The @kbd{s p} (@code{calc-permanent-variable}) command saves a
28492 variable's value permanently in your Calc init file (the file given by
28493 the variable @code{calc-settings-file}, typically @file{~/.calc.el}), so
28494 that its value will still be available in future Emacs sessions. You
28495 can re-execute @w{@kbd{s p}} later on to update the saved value, but the
28496 only way to remove a saved variable is to edit your calc init file
28497 by hand. (@xref{General Mode Commands}, for a way to tell Calc to
28498 use a different file for the Calc init file.)
28500 If you do not specify the name of a variable to save (i.e.,
28501 @kbd{s p @key{RET}}), all Calc variables with defined values
28502 are saved except for the special constants @code{pi}, @code{e},
28503 @code{i}, @code{phi}, and @code{gamma}; the variables @code{TimeZone}
28504 and @code{PlotRejects};
28505 @code{FitRules}, @code{DistribRules}, and other built-in rewrite
28506 rules; and @code{PlotData@var{n}} variables generated
28507 by the graphics commands. (You can still save these variables by
28508 explicitly naming them in an @kbd{s p} command.)
28511 @pindex calc-insert-variables
28512 The @kbd{s i} (@code{calc-insert-variables}) command writes
28513 the values of all Calc variables into a specified buffer.
28514 The variables are written with the prefix @code{var-} in the form of
28515 Lisp @code{setq} commands
28516 which store the values in string form. You can place these commands
28517 in your Calc init file (or @file{.emacs}) if you wish, though in this case it
28518 would be easier to use @kbd{s p @key{RET}}. (Note that @kbd{s i}
28519 omits the same set of variables as @w{@kbd{s p @key{RET}}}; the difference
28520 is that @kbd{s i} will store the variables in any buffer, and it also
28521 stores in a more human-readable format.)
28523 @node Let Command, Evaluates-To Operator, Operations on Variables, Store and Recall
28524 @section The Let Command
28529 @cindex Variables, temporary assignment
28530 @cindex Temporary assignment to variables
28531 If you have an expression like @samp{a+b^2} on the stack and you wish to
28532 compute its value where @expr{b=3}, you can simply store 3 in @expr{b} and
28533 then press @kbd{=} to reevaluate the formula. This has the side-effect
28534 of leaving the stored value of 3 in @expr{b} for future operations.
28536 The @kbd{s l} (@code{calc-let}) command evaluates a formula under a
28537 @emph{temporary} assignment of a variable. It stores the value on the
28538 top of the stack into the specified variable, then evaluates the
28539 second-to-top stack entry, then restores the original value (or lack of one)
28540 in the variable. Thus after @kbd{'@w{ }a+b^2 @key{RET} 3 s l b @key{RET}},
28541 the stack will contain the formula @samp{a + 9}. The subsequent command
28542 @kbd{@w{5 s l a} @key{RET}} will replace this formula with the number 14.
28543 The variables @samp{a} and @samp{b} are not permanently affected in any way
28546 The value on the top of the stack may be an equation or assignment, or
28547 a vector of equations or assignments, in which case the default will be
28548 analogous to the case of @kbd{s t @key{RET}}. @xref{Storing Variables}.
28550 Also, you can answer the variable-name prompt with an equation or
28551 assignment: @kbd{s l b=3 @key{RET}} is the same as storing 3 on the stack
28552 and typing @kbd{s l b @key{RET}}.
28554 The @kbd{a b} (@code{calc-substitute}) command is another way to substitute
28555 a variable with a value in a formula. It does an actual substitution
28556 rather than temporarily assigning the variable and evaluating. For
28557 example, letting @expr{n=2} in @samp{f(n pi)} with @kbd{a b} will
28558 produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)}
28559 since the evaluation step will also evaluate @code{pi}.
28561 @node Evaluates-To Operator, , Let Command, Store and Recall
28562 @section The Evaluates-To Operator
28567 @cindex Evaluates-to operator
28568 @cindex @samp{=>} operator
28569 The special algebraic symbol @samp{=>} is known as the @dfn{evaluates-to
28570 operator}. (It will show up as an @code{evalto} function call in
28571 other language modes like Pascal and La@TeX{}.) This is a binary
28572 operator, that is, it has a lefthand and a righthand argument,
28573 although it can be entered with the righthand argument omitted.
28575 A formula like @samp{@var{a} => @var{b}} is evaluated by Calc as
28576 follows: First, @var{a} is not simplified or modified in any
28577 way. The previous value of argument @var{b} is thrown away; the
28578 formula @var{a} is then copied and evaluated as if by the @kbd{=}
28579 command according to all current modes and stored variable values,
28580 and the result is installed as the new value of @var{b}.
28582 For example, suppose you enter the algebraic formula @samp{2 + 3 => 17}.
28583 The number 17 is ignored, and the lefthand argument is left in its
28584 unevaluated form; the result is the formula @samp{2 + 3 => 5}.
28587 @pindex calc-evalto
28588 You can enter an @samp{=>} formula either directly using algebraic
28589 entry (in which case the righthand side may be omitted since it is
28590 going to be replaced right away anyhow), or by using the @kbd{s =}
28591 (@code{calc-evalto}) command, which takes @var{a} from the stack
28592 and replaces it with @samp{@var{a} => @var{b}}.
28594 Calc keeps track of all @samp{=>} operators on the stack, and
28595 recomputes them whenever anything changes that might affect their
28596 values, i.e., a mode setting or variable value. This occurs only
28597 if the @samp{=>} operator is at the top level of the formula, or
28598 if it is part of a top-level vector. In other words, pushing
28599 @samp{2 + (a => 17)} will change the 17 to the actual value of
28600 @samp{a} when you enter the formula, but the result will not be
28601 dynamically updated when @samp{a} is changed later because the
28602 @samp{=>} operator is buried inside a sum. However, a vector
28603 of @samp{=>} operators will be recomputed, since it is convenient
28604 to push a vector like @samp{[a =>, b =>, c =>]} on the stack to
28605 make a concise display of all the variables in your problem.
28606 (Another way to do this would be to use @samp{[a, b, c] =>},
28607 which provides a slightly different format of display. You
28608 can use whichever you find easiest to read.)
28611 @pindex calc-auto-recompute
28612 The @kbd{m C} (@code{calc-auto-recompute}) command allows you to
28613 turn this automatic recomputation on or off. If you turn
28614 recomputation off, you must explicitly recompute an @samp{=>}
28615 operator on the stack in one of the usual ways, such as by
28616 pressing @kbd{=}. Turning recomputation off temporarily can save
28617 a lot of time if you will be changing several modes or variables
28618 before you look at the @samp{=>} entries again.
28620 Most commands are not especially useful with @samp{=>} operators
28621 as arguments. For example, given @samp{x + 2 => 17}, it won't
28622 work to type @kbd{1 +} to get @samp{x + 3 => 18}. If you want
28623 to operate on the lefthand side of the @samp{=>} operator on
28624 the top of the stack, type @kbd{j 1} (that's the digit ``one'')
28625 to select the lefthand side, execute your commands, then type
28626 @kbd{j u} to unselect.
28628 All current modes apply when an @samp{=>} operator is computed,
28629 including the current simplification mode. Recall that the
28630 formula @samp{x + y + x} is not handled by Calc's default
28631 simplifications, but the @kbd{a s} command will reduce it to
28632 the simpler form @samp{y + 2 x}. You can also type @kbd{m A}
28633 to enable an Algebraic Simplification mode in which the
28634 equivalent of @kbd{a s} is used on all of Calc's results.
28635 If you enter @samp{x + y + x =>} normally, the result will
28636 be @samp{x + y + x => x + y + x}. If you change to
28637 Algebraic Simplification mode, the result will be
28638 @samp{x + y + x => y + 2 x}. However, just pressing @kbd{a s}
28639 once will have no effect on @samp{x + y + x => x + y + x},
28640 because the righthand side depends only on the lefthand side
28641 and the current mode settings, and the lefthand side is not
28642 affected by commands like @kbd{a s}.
28644 The ``let'' command (@kbd{s l}) has an interesting interaction
28645 with the @samp{=>} operator. The @kbd{s l} command evaluates the
28646 second-to-top stack entry with the top stack entry supplying
28647 a temporary value for a given variable. As you might expect,
28648 if that stack entry is an @samp{=>} operator its righthand
28649 side will temporarily show this value for the variable. In
28650 fact, all @samp{=>}s on the stack will be updated if they refer
28651 to that variable. But this change is temporary in the sense
28652 that the next command that causes Calc to look at those stack
28653 entries will make them revert to the old variable value.
28657 2: a => a 2: a => 17 2: a => a
28658 1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1
28661 17 s l a @key{RET} p 8 @key{RET}
28665 Here the @kbd{p 8} command changes the current precision,
28666 thus causing the @samp{=>} forms to be recomputed after the
28667 influence of the ``let'' is gone. The @kbd{d @key{SPC}} command
28668 (@code{calc-refresh}) is a handy way to force the @samp{=>}
28669 operators on the stack to be recomputed without any other
28673 @pindex calc-assign
28676 Embedded mode also uses @samp{=>} operators. In Embedded mode,
28677 the lefthand side of an @samp{=>} operator can refer to variables
28678 assigned elsewhere in the file by @samp{:=} operators. The
28679 assignment operator @samp{a := 17} does not actually do anything
28680 by itself. But Embedded mode recognizes it and marks it as a sort
28681 of file-local definition of the variable. You can enter @samp{:=}
28682 operators in Algebraic mode, or by using the @kbd{s :}
28683 (@code{calc-assign}) [@code{assign}] command which takes a variable
28684 and value from the stack and replaces them with an assignment.
28686 @xref{TeX and LaTeX Language Modes}, for the way @samp{=>} appears in
28687 @TeX{} language output. The @dfn{eqn} mode gives similar
28688 treatment to @samp{=>}.
28690 @node Graphics, Kill and Yank, Store and Recall, Top
28694 The commands for graphing data begin with the @kbd{g} prefix key. Calc
28695 uses GNUPLOT 2.0 or 3.0 to do graphics. These commands will only work
28696 if GNUPLOT is available on your system. (While GNUPLOT sounds like
28697 a relative of GNU Emacs, it is actually completely unrelated.
28698 However, it is free software and can be obtained from the Free
28699 Software Foundation's machine @samp{prep.ai.mit.edu}.)
28701 @vindex calc-gnuplot-name
28702 If you have GNUPLOT installed on your system but Calc is unable to
28703 find it, you may need to set the @code{calc-gnuplot-name} variable
28704 in your Calc init file or @file{.emacs}. You may also need to set some Lisp
28705 variables to show Calc how to run GNUPLOT on your system; these
28706 are described under @kbd{g D} and @kbd{g O} below. If you are
28707 using the X window system, Calc will configure GNUPLOT for you
28708 automatically. If you have GNUPLOT 3.0 and you are not using X,
28709 Calc will configure GNUPLOT to display graphs using simple character
28710 graphics that will work on any terminal.
28714 * Three Dimensional Graphics::
28715 * Managing Curves::
28716 * Graphics Options::
28720 @node Basic Graphics, Three Dimensional Graphics, Graphics, Graphics
28721 @section Basic Graphics
28725 @pindex calc-graph-fast
28726 The easiest graphics command is @kbd{g f} (@code{calc-graph-fast}).
28727 This command takes two vectors of equal length from the stack.
28728 The vector at the top of the stack represents the ``y'' values of
28729 the various data points. The vector in the second-to-top position
28730 represents the corresponding ``x'' values. This command runs
28731 GNUPLOT (if it has not already been started by previous graphing
28732 commands) and displays the set of data points. The points will
28733 be connected by lines, and there will also be some kind of symbol
28734 to indicate the points themselves.
28736 The ``x'' entry may instead be an interval form, in which case suitable
28737 ``x'' values are interpolated between the minimum and maximum values of
28738 the interval (whether the interval is open or closed is ignored).
28740 The ``x'' entry may also be a number, in which case Calc uses the
28741 sequence of ``x'' values @expr{x}, @expr{x+1}, @expr{x+2}, etc.
28742 (Generally the number 0 or 1 would be used for @expr{x} in this case.)
28744 The ``y'' entry may be any formula instead of a vector. Calc effectively
28745 uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula;
28746 the result of this must be a formula in a single (unassigned) variable.
28747 The formula is plotted with this variable taking on the various ``x''
28748 values. Graphs of formulas by default use lines without symbols at the
28749 computed data points. Note that if neither ``x'' nor ``y'' is a vector,
28750 Calc guesses at a reasonable number of data points to use. See the
28751 @kbd{g N} command below. (The ``x'' values must be either a vector
28752 or an interval if ``y'' is a formula.)
28758 If ``y'' is (or evaluates to) a formula of the form
28759 @samp{xy(@var{x}, @var{y})} then the result is a
28760 parametric plot. The two arguments of the fictitious @code{xy} function
28761 are used as the ``x'' and ``y'' coordinates of the curve, respectively.
28762 In this case the ``x'' vector or interval you specified is not directly
28763 visible in the graph. For example, if ``x'' is the interval @samp{[0..360]}
28764 and ``y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph
28767 Also, ``x'' and ``y'' may each be variable names, in which case Calc
28768 looks for suitable vectors, intervals, or formulas stored in those
28771 The ``x'' and ``y'' values for the data points (as pulled from the vectors,
28772 calculated from the formulas, or interpolated from the intervals) should
28773 be real numbers (integers, fractions, or floats). If either the ``x''
28774 value or the ``y'' value of a given data point is not a real number, that
28775 data point will be omitted from the graph. The points on either side
28776 of the invalid point will @emph{not} be connected by a line.
28778 See the documentation for @kbd{g a} below for a description of the way
28779 numeric prefix arguments affect @kbd{g f}.
28781 @cindex @code{PlotRejects} variable
28782 @vindex PlotRejects
28783 If you store an empty vector in the variable @code{PlotRejects}
28784 (i.e., @kbd{[ ] s t PlotRejects}), Calc will append information to
28785 this vector for every data point which was rejected because its
28786 ``x'' or ``y'' values were not real numbers. The result will be
28787 a matrix where each row holds the curve number, data point number,
28788 ``x'' value, and ``y'' value for a rejected data point.
28789 @xref{Evaluates-To Operator}, for a handy way to keep tabs on the
28790 current value of @code{PlotRejects}. @xref{Operations on Variables},
28791 for the @kbd{s R} command which is another easy way to examine
28792 @code{PlotRejects}.
28795 @pindex calc-graph-clear
28796 To clear the graphics display, type @kbd{g c} (@code{calc-graph-clear}).
28797 If the GNUPLOT output device is an X window, the window will go away.
28798 Effects on other kinds of output devices will vary. You don't need
28799 to use @kbd{g c} if you don't want to---if you give another @kbd{g f}
28800 or @kbd{g p} command later on, it will reuse the existing graphics
28801 window if there is one.
28803 @node Three Dimensional Graphics, Managing Curves, Basic Graphics, Graphics
28804 @section Three-Dimensional Graphics
28807 @pindex calc-graph-fast-3d
28808 The @kbd{g F} (@code{calc-graph-fast-3d}) command makes a three-dimensional
28809 graph. It works only if you have GNUPLOT 3.0 or later; with GNUPLOT 2.0,
28810 you will see a GNUPLOT error message if you try this command.
28812 The @kbd{g F} command takes three values from the stack, called ``x'',
28813 ``y'', and ``z'', respectively. As was the case for 2D graphs, there
28814 are several options for these values.
28816 In the first case, ``x'' and ``y'' are each vectors (not necessarily of
28817 the same length); either or both may instead be interval forms. The
28818 ``z'' value must be a matrix with the same number of rows as elements
28819 in ``x'', and the same number of columns as elements in ``y''. The
28820 result is a surface plot where
28821 @texline @math{z_{ij}}
28822 @infoline @expr{z_ij}
28823 is the height of the point
28824 at coordinate @expr{(x_i, y_j)} on the surface. The 3D graph will
28825 be displayed from a certain default viewpoint; you can change this
28826 viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*}
28827 buffer as described later. See the GNUPLOT 3.0 documentation for a
28828 description of the @samp{set view} command.
28830 Each point in the matrix will be displayed as a dot in the graph,
28831 and these points will be connected by a grid of lines (@dfn{isolines}).
28833 In the second case, ``x'', ``y'', and ``z'' are all vectors of equal
28834 length. The resulting graph displays a 3D line instead of a surface,
28835 where the coordinates of points along the line are successive triplets
28836 of values from the input vectors.
28838 In the third case, ``x'' and ``y'' are vectors or interval forms, and
28839 ``z'' is any formula involving two variables (not counting variables
28840 with assigned values). These variables are sorted into alphabetical
28841 order; the first takes on values from ``x'' and the second takes on
28842 values from ``y'' to form a matrix of results that are graphed as a
28849 If the ``z'' formula evaluates to a call to the fictitious function
28850 @samp{xyz(@var{x}, @var{y}, @var{z})}, then the result is a
28851 ``parametric surface.'' In this case, the axes of the graph are
28852 taken from the @var{x} and @var{y} values in these calls, and the
28853 ``x'' and ``y'' values from the input vectors or intervals are used only
28854 to specify the range of inputs to the formula. For example, plotting
28855 @samp{[0..360], [0..180], xyz(sin(x)*sin(y), cos(x)*sin(y), cos(y))}
28856 will draw a sphere. (Since the default resolution for 3D plots is
28857 5 steps in each of ``x'' and ``y'', this will draw a very crude
28858 sphere. You could use the @kbd{g N} command, described below, to
28859 increase this resolution, or specify the ``x'' and ``y'' values as
28860 vectors with more than 5 elements.
28862 It is also possible to have a function in a regular @kbd{g f} plot
28863 evaluate to an @code{xyz} call. Since @kbd{g f} plots a line, not
28864 a surface, the result will be a 3D parametric line. For example,
28865 @samp{[[0..720], xyz(sin(x), cos(x), x)]} will plot two turns of a
28866 helix (a three-dimensional spiral).
28868 As for @kbd{g f}, each of ``x'', ``y'', and ``z'' may instead be
28869 variables containing the relevant data.
28871 @node Managing Curves, Graphics Options, Three Dimensional Graphics, Graphics
28872 @section Managing Curves
28875 The @kbd{g f} command is really shorthand for the following commands:
28876 @kbd{C-u g d g a g p}. Likewise, @w{@kbd{g F}} is shorthand for
28877 @kbd{C-u g d g A g p}. You can gain more control over your graph
28878 by using these commands directly.
28881 @pindex calc-graph-add
28882 The @kbd{g a} (@code{calc-graph-add}) command adds the ``curve''
28883 represented by the two values on the top of the stack to the current
28884 graph. You can have any number of curves in the same graph. When
28885 you give the @kbd{g p} command, all the curves will be drawn superimposed
28888 The @kbd{g a} command (and many others that affect the current graph)
28889 will cause a special buffer, @samp{*Gnuplot Commands*}, to be displayed
28890 in another window. This buffer is a template of the commands that will
28891 be sent to GNUPLOT when it is time to draw the graph. The first
28892 @kbd{g a} command adds a @code{plot} command to this buffer. Succeeding
28893 @kbd{g a} commands add extra curves onto that @code{plot} command.
28894 Other graph-related commands put other GNUPLOT commands into this
28895 buffer. In normal usage you never need to work with this buffer
28896 directly, but you can if you wish. The only constraint is that there
28897 must be only one @code{plot} command, and it must be the last command
28898 in the buffer. If you want to save and later restore a complete graph
28899 configuration, you can use regular Emacs commands to save and restore
28900 the contents of the @samp{*Gnuplot Commands*} buffer.
28904 If the values on the stack are not variable names, @kbd{g a} will invent
28905 variable names for them (of the form @samp{PlotData@var{n}}) and store
28906 the values in those variables. The ``x'' and ``y'' variables are what
28907 go into the @code{plot} command in the template. If you add a curve
28908 that uses a certain variable and then later change that variable, you
28909 can replot the graph without having to delete and re-add the curve.
28910 That's because the variable name, not the vector, interval or formula
28911 itself, is what was added by @kbd{g a}.
28913 A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way
28914 stack entries are interpreted as curves. With a positive prefix
28915 argument @expr{n}, the top @expr{n} stack entries are ``y'' values
28916 for @expr{n} different curves which share a common ``x'' value in
28917 the @expr{n+1}st stack entry. (Thus @kbd{g a} with no prefix
28918 argument is equivalent to @kbd{C-u 1 g a}.)
28920 A prefix of zero or plain @kbd{C-u} means to take two stack entries,
28921 ``x'' and ``y'' as usual, but to interpret ``y'' as a vector of
28922 ``y'' values for several curves that share a common ``x''.
28924 A negative prefix argument tells Calc to read @expr{n} vectors from
28925 the stack; each vector @expr{[x, y]} describes an independent curve.
28926 This is the only form of @kbd{g a} that creates several curves at once
28927 that don't have common ``x'' values. (Of course, the range of ``x''
28928 values covered by all the curves ought to be roughly the same if
28929 they are to look nice on the same graph.)
28931 For example, to plot
28932 @texline @math{\sin n x}
28933 @infoline @expr{sin(n x)}
28934 for integers @expr{n}
28935 from 1 to 5, you could use @kbd{v x} to create a vector of integers
28936 (@expr{n}), then @kbd{V M '} or @kbd{V M $} to map @samp{sin(n x)}
28937 across this vector. The resulting vector of formulas is suitable
28938 for use as the ``y'' argument to a @kbd{C-u g a} or @kbd{C-u g f}
28942 @pindex calc-graph-add-3d
28943 The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve
28944 to the graph. It is not valid to intermix 2D and 3D curves in a
28945 single graph. This command takes three arguments, ``x'', ``y'',
28946 and ``z'', from the stack. With a positive prefix @expr{n}, it
28947 takes @expr{n+2} arguments (common ``x'' and ``y'', plus @expr{n}
28948 separate ``z''s). With a zero prefix, it takes three stack entries
28949 but the ``z'' entry is a vector of curve values. With a negative
28950 prefix @expr{-n}, it takes @expr{n} vectors of the form @expr{[x, y, z]}.
28951 The @kbd{g A} command works by adding a @code{splot} (surface-plot)
28952 command to the @samp{*Gnuplot Commands*} buffer.
28954 (Although @kbd{g a} adds a 2D @code{plot} command to the
28955 @samp{*Gnuplot Commands*} buffer, Calc changes this to @code{splot}
28956 before sending it to GNUPLOT if it notices that the data points are
28957 evaluating to @code{xyz} calls. It will not work to mix 2D and 3D
28958 @kbd{g a} curves in a single graph, although Calc does not currently
28962 @pindex calc-graph-delete
28963 The @kbd{g d} (@code{calc-graph-delete}) command deletes the most
28964 recently added curve from the graph. It has no effect if there are
28965 no curves in the graph. With a numeric prefix argument of any kind,
28966 it deletes all of the curves from the graph.
28969 @pindex calc-graph-hide
28970 The @kbd{g H} (@code{calc-graph-hide}) command ``hides'' or ``unhides''
28971 the most recently added curve. A hidden curve will not appear in
28972 the actual plot, but information about it such as its name and line and
28973 point styles will be retained.
28976 @pindex calc-graph-juggle
28977 The @kbd{g j} (@code{calc-graph-juggle}) command moves the curve
28978 at the end of the list (the ``most recently added curve'') to the
28979 front of the list. The next-most-recent curve is thus exposed for
28980 @w{@kbd{g d}} or similar commands to use. With @kbd{g j} you can work
28981 with any curve in the graph even though curve-related commands only
28982 affect the last curve in the list.
28985 @pindex calc-graph-plot
28986 The @kbd{g p} (@code{calc-graph-plot}) command uses GNUPLOT to draw
28987 the graph described in the @samp{*Gnuplot Commands*} buffer. Any
28988 GNUPLOT parameters which are not defined by commands in this buffer
28989 are reset to their default values. The variables named in the @code{plot}
28990 command are written to a temporary data file and the variable names
28991 are then replaced by the file name in the template. The resulting
28992 plotting commands are fed to the GNUPLOT program. See the documentation
28993 for the GNUPLOT program for more specific information. All temporary
28994 files are removed when Emacs or GNUPLOT exits.
28996 If you give a formula for ``y'', Calc will remember all the values that
28997 it calculates for the formula so that later plots can reuse these values.
28998 Calc throws out these saved values when you change any circumstances
28999 that may affect the data, such as switching from Degrees to Radians
29000 mode, or changing the value of a parameter in the formula. You can
29001 force Calc to recompute the data from scratch by giving a negative
29002 numeric prefix argument to @kbd{g p}.
29004 Calc uses a fairly rough step size when graphing formulas over intervals.
29005 This is to ensure quick response. You can ``refine'' a plot by giving
29006 a positive numeric prefix argument to @kbd{g p}. Calc goes through
29007 the data points it has computed and saved from previous plots of the
29008 function, and computes and inserts a new data point midway between
29009 each of the existing points. You can refine a plot any number of times,
29010 but beware that the amount of calculation involved doubles each time.
29012 Calc does not remember computed values for 3D graphs. This means the
29013 numerix prefix argument, if any, to @kbd{g p} is effectively ignored if
29014 the current graph is three-dimensional.
29017 @pindex calc-graph-print
29018 The @kbd{g P} (@code{calc-graph-print}) command is like @kbd{g p},
29019 except that it sends the output to a printer instead of to the
29020 screen. More precisely, @kbd{g p} looks for @samp{set terminal}
29021 or @samp{set output} commands in the @samp{*Gnuplot Commands*} buffer;
29022 lacking these it uses the default settings. However, @kbd{g P}
29023 ignores @samp{set terminal} and @samp{set output} commands and
29024 uses a different set of default values. All of these values are
29025 controlled by the @kbd{g D} and @kbd{g O} commands discussed below.
29026 Provided everything is set up properly, @kbd{g p} will plot to
29027 the screen unless you have specified otherwise and @kbd{g P} will
29028 always plot to the printer.
29030 @node Graphics Options, Devices, Managing Curves, Graphics
29031 @section Graphics Options
29035 @pindex calc-graph-grid
29036 The @kbd{g g} (@code{calc-graph-grid}) command turns the ``grid''
29037 on and off. It is off by default; tick marks appear only at the
29038 edges of the graph. With the grid turned on, dotted lines appear
29039 across the graph at each tick mark. Note that this command only
29040 changes the setting in @samp{*Gnuplot Commands*}; to see the effects
29041 of the change you must give another @kbd{g p} command.
29044 @pindex calc-graph-border
29045 The @kbd{g b} (@code{calc-graph-border}) command turns the border
29046 (the box that surrounds the graph) on and off. It is on by default.
29047 This command will only work with GNUPLOT 3.0 and later versions.
29050 @pindex calc-graph-key
29051 The @kbd{g k} (@code{calc-graph-key}) command turns the ``key''
29052 on and off. The key is a chart in the corner of the graph that
29053 shows the correspondence between curves and line styles. It is
29054 off by default, and is only really useful if you have several
29055 curves on the same graph.
29058 @pindex calc-graph-num-points
29059 The @kbd{g N} (@code{calc-graph-num-points}) command allows you
29060 to select the number of data points in the graph. This only affects
29061 curves where neither ``x'' nor ``y'' is specified as a vector.
29062 Enter a blank line to revert to the default value (initially 15).
29063 With no prefix argument, this command affects only the current graph.
29064 With a positive prefix argument this command changes or, if you enter
29065 a blank line, displays the default number of points used for all
29066 graphs created by @kbd{g a} that don't specify the resolution explicitly.
29067 With a negative prefix argument, this command changes or displays
29068 the default value (initially 5) used for 3D graphs created by @kbd{g A}.
29069 Note that a 3D setting of 5 means that a total of @expr{5^2 = 25} points
29070 will be computed for the surface.
29072 Data values in the graph of a function are normally computed to a
29073 precision of five digits, regardless of the current precision at the
29074 time. This is usually more than adequate, but there are cases where
29075 it will not be. For example, plotting @expr{1 + x} with @expr{x} in the
29076 interval @samp{[0 ..@: 1e-6]} will round all the data points down
29077 to 1.0! Putting the command @samp{set precision @var{n}} in the
29078 @samp{*Gnuplot Commands*} buffer will cause the data to be computed
29079 at precision @var{n} instead of 5. Since this is such a rare case,
29080 there is no keystroke-based command to set the precision.
29083 @pindex calc-graph-header
29084 The @kbd{g h} (@code{calc-graph-header}) command sets the title
29085 for the graph. This will show up centered above the graph.
29086 The default title is blank (no title).
29089 @pindex calc-graph-name
29090 The @kbd{g n} (@code{calc-graph-name}) command sets the title of an
29091 individual curve. Like the other curve-manipulating commands, it
29092 affects the most recently added curve, i.e., the last curve on the
29093 list in the @samp{*Gnuplot Commands*} buffer. To set the title of
29094 the other curves you must first juggle them to the end of the list
29095 with @kbd{g j}, or edit the @samp{*Gnuplot Commands*} buffer by hand.
29096 Curve titles appear in the key; if the key is turned off they are
29101 @pindex calc-graph-title-x
29102 @pindex calc-graph-title-y
29103 The @kbd{g t} (@code{calc-graph-title-x}) and @kbd{g T}
29104 (@code{calc-graph-title-y}) commands set the titles on the ``x''
29105 and ``y'' axes, respectively. These titles appear next to the
29106 tick marks on the left and bottom edges of the graph, respectively.
29107 Calc does not have commands to control the tick marks themselves,
29108 but you can edit them into the @samp{*Gnuplot Commands*} buffer if
29109 you wish. See the GNUPLOT documentation for details.
29113 @pindex calc-graph-range-x
29114 @pindex calc-graph-range-y
29115 The @kbd{g r} (@code{calc-graph-range-x}) and @kbd{g R}
29116 (@code{calc-graph-range-y}) commands set the range of values on the
29117 ``x'' and ``y'' axes, respectively. You are prompted to enter a
29118 suitable range. This should be either a pair of numbers of the
29119 form, @samp{@var{min}:@var{max}}, or a blank line to revert to the
29120 default behavior of setting the range based on the range of values
29121 in the data, or @samp{$} to take the range from the top of the stack.
29122 Ranges on the stack can be represented as either interval forms or
29123 vectors: @samp{[@var{min} ..@: @var{max}]} or @samp{[@var{min}, @var{max}]}.
29127 @pindex calc-graph-log-x
29128 @pindex calc-graph-log-y
29129 The @kbd{g l} (@code{calc-graph-log-x}) and @kbd{g L} (@code{calc-graph-log-y})
29130 commands allow you to set either or both of the axes of the graph to
29131 be logarithmic instead of linear.
29136 @pindex calc-graph-log-z
29137 @pindex calc-graph-range-z
29138 @pindex calc-graph-title-z
29139 For 3D plots, @kbd{g C-t}, @kbd{g C-r}, and @kbd{g C-l} (those are
29140 letters with the Control key held down) are the corresponding commands
29141 for the ``z'' axis.
29145 @pindex calc-graph-zero-x
29146 @pindex calc-graph-zero-y
29147 The @kbd{g z} (@code{calc-graph-zero-x}) and @kbd{g Z}
29148 (@code{calc-graph-zero-y}) commands control whether a dotted line is
29149 drawn to indicate the ``x'' and/or ``y'' zero axes. (These are the same
29150 dotted lines that would be drawn there anyway if you used @kbd{g g} to
29151 turn the ``grid'' feature on.) Zero-axis lines are on by default, and
29152 may be turned off only in GNUPLOT 3.0 and later versions. They are
29153 not available for 3D plots.
29156 @pindex calc-graph-line-style
29157 The @kbd{g s} (@code{calc-graph-line-style}) command turns the connecting
29158 lines on or off for the most recently added curve, and optionally selects
29159 the style of lines to be used for that curve. Plain @kbd{g s} simply
29160 toggles the lines on and off. With a numeric prefix argument, @kbd{g s}
29161 turns lines on and sets a particular line style. Line style numbers
29162 start at one and their meanings vary depending on the output device.
29163 GNUPLOT guarantees that there will be at least six different line styles
29164 available for any device.
29167 @pindex calc-graph-point-style
29168 The @kbd{g S} (@code{calc-graph-point-style}) command similarly turns
29169 the symbols at the data points on or off, or sets the point style.
29170 If you turn both lines and points off, the data points will show as
29173 @cindex @code{LineStyles} variable
29174 @cindex @code{PointStyles} variable
29176 @vindex PointStyles
29177 Another way to specify curve styles is with the @code{LineStyles} and
29178 @code{PointStyles} variables. These variables initially have no stored
29179 values, but if you store a vector of integers in one of these variables,
29180 the @kbd{g a} and @kbd{g f} commands will use those style numbers
29181 instead of the defaults for new curves that are added to the graph.
29182 An entry should be a positive integer for a specific style, or 0 to let
29183 the style be chosen automatically, or @mathit{-1} to turn off lines or points
29184 altogether. If there are more curves than elements in the vector, the
29185 last few curves will continue to have the default styles. Of course,
29186 you can later use @kbd{g s} and @kbd{g S} to change any of these styles.
29188 For example, @kbd{'[2 -1 3] @key{RET} s t LineStyles} causes the first curve
29189 to have lines in style number 2, the second curve to have no connecting
29190 lines, and the third curve to have lines in style 3. Point styles will
29191 still be assigned automatically, but you could store another vector in
29192 @code{PointStyles} to define them, too.
29194 @node Devices, , Graphics Options, Graphics
29195 @section Graphical Devices
29199 @pindex calc-graph-device
29200 The @kbd{g D} (@code{calc-graph-device}) command sets the device name
29201 (or ``terminal name'' in GNUPLOT lingo) to be used by @kbd{g p} commands
29202 on this graph. It does not affect the permanent default device name.
29203 If you enter a blank name, the device name reverts to the default.
29204 Enter @samp{?} to see a list of supported devices.
29206 With a positive numeric prefix argument, @kbd{g D} instead sets
29207 the default device name, used by all plots in the future which do
29208 not override it with a plain @kbd{g D} command. If you enter a
29209 blank line this command shows you the current default. The special
29210 name @code{default} signifies that Calc should choose @code{x11} if
29211 the X window system is in use (as indicated by the presence of a
29212 @code{DISPLAY} environment variable), or otherwise @code{dumb} under
29213 GNUPLOT 3.0 and later, or @code{postscript} under GNUPLOT 2.0.
29214 This is the initial default value.
29216 The @code{dumb} device is an interface to ``dumb terminals,'' i.e.,
29217 terminals with no special graphics facilities. It writes a crude
29218 picture of the graph composed of characters like @code{-} and @code{|}
29219 to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays.
29220 The graph is made the same size as the Emacs screen, which on most
29221 dumb terminals will be
29222 @texline @math{80\times24}
29224 characters. The graph is displayed in
29225 an Emacs ``recursive edit''; type @kbd{q} or @kbd{C-c C-c} to exit
29226 the recursive edit and return to Calc. Note that the @code{dumb}
29227 device is present only in GNUPLOT 3.0 and later versions.
29229 The word @code{dumb} may be followed by two numbers separated by
29230 spaces. These are the desired width and height of the graph in
29231 characters. Also, the device name @code{big} is like @code{dumb}
29232 but creates a graph four times the width and height of the Emacs
29233 screen. You will then have to scroll around to view the entire
29234 graph. In the @samp{*Gnuplot Trail*} buffer, @key{SPC}, @key{DEL},
29235 @kbd{<}, and @kbd{>} are defined to scroll by one screenful in each
29236 of the four directions.
29238 With a negative numeric prefix argument, @kbd{g D} sets or displays
29239 the device name used by @kbd{g P} (@code{calc-graph-print}). This
29240 is initially @code{postscript}. If you don't have a PostScript
29241 printer, you may decide once again to use @code{dumb} to create a
29242 plot on any text-only printer.
29245 @pindex calc-graph-output
29246 The @kbd{g O} (@code{calc-graph-output}) command sets the name of
29247 the output file used by GNUPLOT. For some devices, notably @code{x11},
29248 there is no output file and this information is not used. Many other
29249 ``devices'' are really file formats like @code{postscript}; in these
29250 cases the output in the desired format goes into the file you name
29251 with @kbd{g O}. Type @kbd{g O stdout @key{RET}} to set GNUPLOT to write
29252 to its standard output stream, i.e., to @samp{*Gnuplot Trail*}.
29253 This is the default setting.
29255 Another special output name is @code{tty}, which means that GNUPLOT
29256 is going to write graphics commands directly to its standard output,
29257 which you wish Emacs to pass through to your terminal. Tektronix
29258 graphics terminals, among other devices, operate this way. Calc does
29259 this by telling GNUPLOT to write to a temporary file, then running a
29260 sub-shell executing the command @samp{cat tempfile >/dev/tty}. On
29261 typical Unix systems, this will copy the temporary file directly to
29262 the terminal, bypassing Emacs entirely. You will have to type @kbd{C-l}
29263 to Emacs afterwards to refresh the screen.
29265 Once again, @kbd{g O} with a positive or negative prefix argument
29266 sets the default or printer output file names, respectively. In each
29267 case you can specify @code{auto}, which causes Calc to invent a temporary
29268 file name for each @kbd{g p} (or @kbd{g P}) command. This temporary file
29269 will be deleted once it has been displayed or printed. If the output file
29270 name is not @code{auto}, the file is not automatically deleted.
29272 The default and printer devices and output files can be saved
29273 permanently by the @kbd{m m} (@code{calc-save-modes}) command. The
29274 default number of data points (see @kbd{g N}) and the X geometry
29275 (see @kbd{g X}) are also saved. Other graph information is @emph{not}
29276 saved; you can save a graph's configuration simply by saving the contents
29277 of the @samp{*Gnuplot Commands*} buffer.
29279 @vindex calc-gnuplot-plot-command
29280 @vindex calc-gnuplot-default-device
29281 @vindex calc-gnuplot-default-output
29282 @vindex calc-gnuplot-print-command
29283 @vindex calc-gnuplot-print-device
29284 @vindex calc-gnuplot-print-output
29285 You may wish to configure the default and
29286 printer devices and output files for the whole system. The relevant
29287 Lisp variables are @code{calc-gnuplot-default-device} and @code{-output},
29288 and @code{calc-gnuplot-print-device} and @code{-output}. The output
29289 file names must be either strings as described above, or Lisp
29290 expressions which are evaluated on the fly to get the output file names.
29292 Other important Lisp variables are @code{calc-gnuplot-plot-command} and
29293 @code{calc-gnuplot-print-command}, which give the system commands to
29294 display or print the output of GNUPLOT, respectively. These may be
29295 @code{nil} if no command is necessary, or strings which can include
29296 @samp{%s} to signify the name of the file to be displayed or printed.
29297 Or, these variables may contain Lisp expressions which are evaluated
29298 to display or print the output. These variables are customizable
29299 (@pxref{Customizable Variables}).
29302 @pindex calc-graph-display
29303 The @kbd{g x} (@code{calc-graph-display}) command lets you specify
29304 on which X window system display your graphs should be drawn. Enter
29305 a blank line to see the current display name. This command has no
29306 effect unless the current device is @code{x11}.
29309 @pindex calc-graph-geometry
29310 The @kbd{g X} (@code{calc-graph-geometry}) command is a similar
29311 command for specifying the position and size of the X window.
29312 The normal value is @code{default}, which generally means your
29313 window manager will let you place the window interactively.
29314 Entering @samp{800x500+0+0} would create an 800-by-500 pixel
29315 window in the upper-left corner of the screen.
29317 The buffer called @samp{*Gnuplot Trail*} holds a transcript of the
29318 session with GNUPLOT. This shows the commands Calc has ``typed'' to
29319 GNUPLOT and the responses it has received. Calc tries to notice when an
29320 error message has appeared here and display the buffer for you when
29321 this happens. You can check this buffer yourself if you suspect
29322 something has gone wrong.
29325 @pindex calc-graph-command
29326 The @kbd{g C} (@code{calc-graph-command}) command prompts you to
29327 enter any line of text, then simply sends that line to the current
29328 GNUPLOT process. The @samp{*Gnuplot Trail*} buffer looks deceptively
29329 like a Shell buffer but you can't type commands in it yourself.
29330 Instead, you must use @kbd{g C} for this purpose.
29334 @pindex calc-graph-view-commands
29335 @pindex calc-graph-view-trail
29336 The @kbd{g v} (@code{calc-graph-view-commands}) and @kbd{g V}
29337 (@code{calc-graph-view-trail}) commands display the @samp{*Gnuplot Commands*}
29338 and @samp{*Gnuplot Trail*} buffers, respectively, in another window.
29339 This happens automatically when Calc thinks there is something you
29340 will want to see in either of these buffers. If you type @kbd{g v}
29341 or @kbd{g V} when the relevant buffer is already displayed, the
29342 buffer is hidden again.
29344 One reason to use @kbd{g v} is to add your own commands to the
29345 @samp{*Gnuplot Commands*} buffer. Press @kbd{g v}, then use
29346 @kbd{C-x o} to switch into that window. For example, GNUPLOT has
29347 @samp{set label} and @samp{set arrow} commands that allow you to
29348 annotate your plots. Since Calc doesn't understand these commands,
29349 you have to add them to the @samp{*Gnuplot Commands*} buffer
29350 yourself, then use @w{@kbd{g p}} to replot using these new commands. Note
29351 that your commands must appear @emph{before} the @code{plot} command.
29352 To get help on any GNUPLOT feature, type, e.g., @kbd{g C help set label}.
29353 You may have to type @kbd{g C @key{RET}} a few times to clear the
29354 ``press return for more'' or ``subtopic of @dots{}'' requests.
29355 Note that Calc always sends commands (like @samp{set nolabel}) to
29356 reset all plotting parameters to the defaults before each plot, so
29357 to delete a label all you need to do is delete the @samp{set label}
29358 line you added (or comment it out with @samp{#}) and then replot
29362 @pindex calc-graph-quit
29363 You can use @kbd{g q} (@code{calc-graph-quit}) to kill the GNUPLOT
29364 process that is running. The next graphing command you give will
29365 start a fresh GNUPLOT process. The word @samp{Graph} appears in
29366 the Calc window's mode line whenever a GNUPLOT process is currently
29367 running. The GNUPLOT process is automatically killed when you
29368 exit Emacs if you haven't killed it manually by then.
29371 @pindex calc-graph-kill
29372 The @kbd{g K} (@code{calc-graph-kill}) command is like @kbd{g q}
29373 except that it also views the @samp{*Gnuplot Trail*} buffer so that
29374 you can see the process being killed. This is better if you are
29375 killing GNUPLOT because you think it has gotten stuck.
29377 @node Kill and Yank, Keypad Mode, Graphics, Top
29378 @chapter Kill and Yank Functions
29381 The commands in this chapter move information between the Calculator and
29382 other Emacs editing buffers.
29384 In many cases Embedded mode is an easier and more natural way to
29385 work with Calc from a regular editing buffer. @xref{Embedded Mode}.
29388 * Killing From Stack::
29389 * Yanking Into Stack::
29390 * Grabbing From Buffers::
29391 * Yanking Into Buffers::
29392 * X Cut and Paste::
29395 @node Killing From Stack, Yanking Into Stack, Kill and Yank, Kill and Yank
29396 @section Killing from the Stack
29402 @pindex calc-copy-as-kill
29404 @pindex calc-kill-region
29406 @pindex calc-copy-region-as-kill
29408 @dfn{Kill} commands are Emacs commands that insert text into the
29409 ``kill ring,'' from which it can later be ``yanked'' by a @kbd{C-y}
29410 command. Three common kill commands in normal Emacs are @kbd{C-k}, which
29411 kills one line, @kbd{C-w}, which kills the region between mark and point,
29412 and @kbd{M-w}, which puts the region into the kill ring without actually
29413 deleting it. All of these commands work in the Calculator, too. Also,
29414 @kbd{M-k} has been provided to complete the set; it puts the current line
29415 into the kill ring without deleting anything.
29417 The kill commands are unusual in that they pay attention to the location
29418 of the cursor in the Calculator buffer. If the cursor is on or below the
29419 bottom line, the kill commands operate on the top of the stack. Otherwise,
29420 they operate on whatever stack element the cursor is on. Calc's kill
29421 commands always operate on whole stack entries. (They act the same as their
29422 standard Emacs cousins except they ``round up'' the specified region to
29423 encompass full lines.) The text is copied into the kill ring exactly as
29424 it appears on the screen, including line numbers if they are enabled.
29426 A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number
29427 of lines killed. A positive argument kills the current line and @expr{n-1}
29428 lines below it. A negative argument kills the @expr{-n} lines above the
29429 current line. Again this mirrors the behavior of the standard Emacs
29430 @kbd{C-k} command. Although a whole line is always deleted, @kbd{C-k}
29431 with no argument copies only the number itself into the kill ring, whereas
29432 @kbd{C-k} with a prefix argument of 1 copies the number with its trailing
29435 @node Yanking Into Stack, Grabbing From Buffers, Killing From Stack, Kill and Yank
29436 @section Yanking into the Stack
29441 The @kbd{C-y} command yanks the most recently killed text back into the
29442 Calculator. It pushes this value onto the top of the stack regardless of
29443 the cursor position. In general it re-parses the killed text as a number
29444 or formula (or a list of these separated by commas or newlines). However if
29445 the thing being yanked is something that was just killed from the Calculator
29446 itself, its full internal structure is yanked. For example, if you have
29447 set the floating-point display mode to show only four significant digits,
29448 then killing and re-yanking 3.14159 (which displays as 3.142) will yank the
29449 full 3.14159, even though yanking it into any other buffer would yank the
29450 number in its displayed form, 3.142. (Since the default display modes
29451 show all objects to their full precision, this feature normally makes no
29454 @node Grabbing From Buffers, Yanking Into Buffers, Yanking Into Stack, Kill and Yank
29455 @section Grabbing from Other Buffers
29459 @pindex calc-grab-region
29460 The @kbd{M-# g} (@code{calc-grab-region}) command takes the text between
29461 point and mark in the current buffer and attempts to parse it as a
29462 vector of values. Basically, it wraps the text in vector brackets
29463 @samp{[ ]} unless the text already is enclosed in vector brackets,
29464 then reads the text as if it were an algebraic entry. The contents
29465 of the vector may be numbers, formulas, or any other Calc objects.
29466 If the @kbd{M-# g} command works successfully, it does an automatic
29467 @kbd{M-# c} to enter the Calculator buffer.
29469 A numeric prefix argument grabs the specified number of lines around
29470 point, ignoring the mark. A positive prefix grabs from point to the
29471 @expr{n}th following newline (so that @kbd{M-1 M-# g} grabs from point
29472 to the end of the current line); a negative prefix grabs from point
29473 back to the @expr{n+1}st preceding newline. In these cases the text
29474 that is grabbed is exactly the same as the text that @kbd{C-k} would
29475 delete given that prefix argument.
29477 A prefix of zero grabs the current line; point may be anywhere on the
29480 A plain @kbd{C-u} prefix interprets the region between point and mark
29481 as a single number or formula rather than a vector. For example,
29482 @kbd{M-# g} on the text @samp{2 a b} produces the vector of three
29483 values @samp{[2, a, b]}, but @kbd{C-u M-# g} on the same region
29484 reads a formula which is a product of three things: @samp{2 a b}.
29485 (The text @samp{a + b}, on the other hand, will be grabbed as a
29486 vector of one element by plain @kbd{M-# g} because the interpretation
29487 @samp{[a, +, b]} would be a syntax error.)
29489 If a different language has been specified (@pxref{Language Modes}),
29490 the grabbed text will be interpreted according to that language.
29493 @pindex calc-grab-rectangle
29494 The @kbd{M-# r} (@code{calc-grab-rectangle}) command takes the text between
29495 point and mark and attempts to parse it as a matrix. If point and mark
29496 are both in the leftmost column, the lines in between are parsed in their
29497 entirety. Otherwise, point and mark define the corners of a rectangle
29498 whose contents are parsed.
29500 Each line of the grabbed area becomes a row of the matrix. The result
29501 will actually be a vector of vectors, which Calc will treat as a matrix
29502 only if every row contains the same number of values.
29504 If a line contains a portion surrounded by square brackets (or curly
29505 braces), that portion is interpreted as a vector which becomes a row
29506 of the matrix. Any text surrounding the bracketed portion on the line
29509 Otherwise, the entire line is interpreted as a row vector as if it
29510 were surrounded by square brackets. Leading line numbers (in the
29511 format used in the Calc stack buffer) are ignored. If you wish to
29512 force this interpretation (even if the line contains bracketed
29513 portions), give a negative numeric prefix argument to the
29514 @kbd{M-# r} command.
29516 If you give a numeric prefix argument of zero or plain @kbd{C-u}, each
29517 line is instead interpreted as a single formula which is converted into
29518 a one-element vector. Thus the result of @kbd{C-u M-# r} will be a
29519 one-column matrix. For example, suppose one line of the data is the
29520 expression @samp{2 a}. A plain @w{@kbd{M-# r}} will interpret this as
29521 @samp{[2 a]}, which in turn is read as a two-element vector that forms
29522 one row of the matrix. But a @kbd{C-u M-# r} will interpret this row
29525 If you give a positive numeric prefix argument @var{n}, then each line
29526 will be split up into columns of width @var{n}; each column is parsed
29527 separately as a matrix element. If a line contained
29528 @w{@samp{2 +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8
29529 would correctly split the line into two error forms.
29531 @xref{Matrix Functions}, to see how to pull the matrix apart into its
29532 constituent rows and columns. (If it is a
29533 @texline @math{1\times1}
29535 matrix, just hit @kbd{v u} (@code{calc-unpack}) twice.)
29539 @pindex calc-grab-sum-across
29540 @pindex calc-grab-sum-down
29541 @cindex Summing rows and columns of data
29542 The @kbd{M-# :} (@code{calc-grab-sum-down}) command is a handy way to
29543 grab a rectangle of data and sum its columns. It is equivalent to
29544 typing @kbd{M-# r}, followed by @kbd{V R : +} (the vector reduction
29545 command that sums the columns of a matrix; @pxref{Reducing}). The
29546 result of the command will be a vector of numbers, one for each column
29547 in the input data. The @kbd{M-# _} (@code{calc-grab-sum-across}) command
29548 similarly grabs a rectangle and sums its rows by executing @w{@kbd{V R _ +}}.
29550 As well as being more convenient, @kbd{M-# :} and @kbd{M-# _} are also
29551 much faster because they don't actually place the grabbed vector on
29552 the stack. In a @kbd{M-# r V R : +} sequence, formatting the vector
29553 for display on the stack takes a large fraction of the total time
29554 (unless you have planned ahead and used @kbd{v .} and @kbd{t .} modes).
29556 For example, suppose we have a column of numbers in a file which we
29557 wish to sum. Go to one corner of the column and press @kbd{C-@@} to
29558 set the mark; go to the other corner and type @kbd{M-# :}. Since there
29559 is only one column, the result will be a vector of one number, the sum.
29560 (You can type @kbd{v u} to unpack this vector into a plain number if
29561 you want to do further arithmetic with it.)
29563 To compute the product of the column of numbers, we would have to do
29564 it ``by hand'' since there's no special grab-and-multiply command.
29565 Use @kbd{M-# r} to grab the column of numbers into the calculator in
29566 the form of a column matrix. The statistics command @kbd{u *} is a
29567 handy way to find the product of a vector or matrix of numbers.
29568 @xref{Statistical Operations}. Another approach would be to use
29569 an explicit column reduction command, @kbd{V R : *}.
29571 @node Yanking Into Buffers, X Cut and Paste, Grabbing From Buffers, Kill and Yank
29572 @section Yanking into Other Buffers
29576 @pindex calc-copy-to-buffer
29577 The plain @kbd{y} (@code{calc-copy-to-buffer}) command inserts the number
29578 at the top of the stack into the most recently used normal editing buffer.
29579 (More specifically, this is the most recently used buffer which is displayed
29580 in a window and whose name does not begin with @samp{*}. If there is no
29581 such buffer, this is the most recently used buffer except for Calculator
29582 and Calc Trail buffers.) The number is inserted exactly as it appears and
29583 without a newline. (If line-numbering is enabled, the line number is
29584 normally not included.) The number is @emph{not} removed from the stack.
29586 With a prefix argument, @kbd{y} inserts several numbers, one per line.
29587 A positive argument inserts the specified number of values from the top
29588 of the stack. A negative argument inserts the @expr{n}th value from the
29589 top of the stack. An argument of zero inserts the entire stack. Note
29590 that @kbd{y} with an argument of 1 is slightly different from @kbd{y}
29591 with no argument; the former always copies full lines, whereas the
29592 latter strips off the trailing newline.
29594 With a lone @kbd{C-u} as a prefix argument, @kbd{y} @emph{replaces} the
29595 region in the other buffer with the yanked text, then quits the
29596 Calculator, leaving you in that buffer. A typical use would be to use
29597 @kbd{M-# g} to read a region of data into the Calculator, operate on the
29598 data to produce a new matrix, then type @kbd{C-u y} to replace the
29599 original data with the new data. One might wish to alter the matrix
29600 display style (@pxref{Vector and Matrix Formats}) or change the current
29601 display language (@pxref{Language Modes}) before doing this. Also, note
29602 that this command replaces a linear region of text (as grabbed by
29603 @kbd{M-# g}), not a rectangle (as grabbed by @kbd{M-# r}).
29605 If the editing buffer is in overwrite (as opposed to insert) mode,
29606 and the @kbd{C-u} prefix was not used, then the yanked number will
29607 overwrite the characters following point rather than being inserted
29608 before those characters. The usual conventions of overwrite mode
29609 are observed; for example, characters will be inserted at the end of
29610 a line rather than overflowing onto the next line. Yanking a multi-line
29611 object such as a matrix in overwrite mode overwrites the next @var{n}
29612 lines in the buffer, lengthening or shortening each line as necessary.
29613 Finally, if the thing being yanked is a simple integer or floating-point
29614 number (like @samp{-1.2345e-3}) and the characters following point also
29615 make up such a number, then Calc will replace that number with the new
29616 number, lengthening or shortening as necessary. The concept of
29617 ``overwrite mode'' has thus been generalized from overwriting characters
29618 to overwriting one complete number with another.
29621 The @kbd{M-# y} key sequence is equivalent to @kbd{y} except that
29622 it can be typed anywhere, not just in Calc. This provides an easy
29623 way to guarantee that Calc knows which editing buffer you want to use!
29625 @node X Cut and Paste, , Yanking Into Buffers, Kill and Yank
29626 @section X Cut and Paste
29629 If you are using Emacs with the X window system, there is an easier
29630 way to move small amounts of data into and out of the calculator:
29631 Use the mouse-oriented cut and paste facilities of X.
29633 The default bindings for a three-button mouse cause the left button
29634 to move the Emacs cursor to the given place, the right button to
29635 select the text between the cursor and the clicked location, and
29636 the middle button to yank the selection into the buffer at the
29637 clicked location. So, if you have a Calc window and an editing
29638 window on your Emacs screen, you can use left-click/right-click
29639 to select a number, vector, or formula from one window, then
29640 middle-click to paste that value into the other window. When you
29641 paste text into the Calc window, Calc interprets it as an algebraic
29642 entry. It doesn't matter where you click in the Calc window; the
29643 new value is always pushed onto the top of the stack.
29645 The @code{xterm} program that is typically used for general-purpose
29646 shell windows in X interprets the mouse buttons in the same way.
29647 So you can use the mouse to move data between Calc and any other
29648 Unix program. One nice feature of @code{xterm} is that a double
29649 left-click selects one word, and a triple left-click selects a
29650 whole line. So you can usually transfer a single number into Calc
29651 just by double-clicking on it in the shell, then middle-clicking
29652 in the Calc window.
29654 @node Keypad Mode, Embedded Mode, Kill and Yank, Top
29655 @chapter Keypad Mode
29659 @pindex calc-keypad
29660 The @kbd{M-# k} (@code{calc-keypad}) command starts the Calculator
29661 and displays a picture of a calculator-style keypad. If you are using
29662 the X window system, you can click on any of the ``keys'' in the
29663 keypad using the left mouse button to operate the calculator.
29664 The original window remains the selected window; in Keypad mode
29665 you can type in your file while simultaneously performing
29666 calculations with the mouse.
29668 @pindex full-calc-keypad
29669 If you have used @kbd{M-# b} first, @kbd{M-# k} instead invokes
29670 the @code{full-calc-keypad} command, which takes over the whole
29671 Emacs screen and displays the keypad, the Calc stack, and the Calc
29672 trail all at once. This mode would normally be used when running
29673 Calc standalone (@pxref{Standalone Operation}).
29675 If you aren't using the X window system, you must switch into
29676 the @samp{*Calc Keypad*} window, place the cursor on the desired
29677 ``key,'' and type @key{SPC} or @key{RET}. If you think this
29678 is easier than using Calc normally, go right ahead.
29680 Calc commands are more or less the same in Keypad mode. Certain
29681 keypad keys differ slightly from the corresponding normal Calc
29682 keystrokes; all such deviations are described below.
29684 Keypad mode includes many more commands than will fit on the keypad
29685 at once. Click the right mouse button [@code{calc-keypad-menu}]
29686 to switch to the next menu. The bottom five rows of the keypad
29687 stay the same; the top three rows change to a new set of commands.
29688 To return to earlier menus, click the middle mouse button
29689 [@code{calc-keypad-menu-back}] or simply advance through the menus
29690 until you wrap around. Typing @key{TAB} inside the keypad window
29691 is equivalent to clicking the right mouse button there.
29693 You can always click the @key{EXEC} button and type any normal
29694 Calc key sequence. This is equivalent to switching into the
29695 Calc buffer, typing the keys, then switching back to your
29699 * Keypad Main Menu::
29700 * Keypad Functions Menu::
29701 * Keypad Binary Menu::
29702 * Keypad Vectors Menu::
29703 * Keypad Modes Menu::
29706 @node Keypad Main Menu, Keypad Functions Menu, Keypad Mode, Keypad Mode
29711 |----+-----Calc 2.1------+----1
29712 |FLR |CEIL|RND |TRNC|CLN2|FLT |
29713 |----+----+----+----+----+----|
29714 | LN |EXP | |ABS |IDIV|MOD |
29715 |----+----+----+----+----+----|
29716 |SIN |COS |TAN |SQRT|y^x |1/x |
29717 |----+----+----+----+----+----|
29718 | ENTER |+/- |EEX |UNDO| <- |
29719 |-----+---+-+--+--+-+---++----|
29720 | INV | 7 | 8 | 9 | / |
29721 |-----+-----+-----+-----+-----|
29722 | HYP | 4 | 5 | 6 | * |
29723 |-----+-----+-----+-----+-----|
29724 |EXEC | 1 | 2 | 3 | - |
29725 |-----+-----+-----+-----+-----|
29726 | OFF | 0 | . | PI | + |
29727 |-----+-----+-----+-----+-----+
29732 This is the menu that appears the first time you start Keypad mode.
29733 It will show up in a vertical window on the right side of your screen.
29734 Above this menu is the traditional Calc stack display. On a 24-line
29735 screen you will be able to see the top three stack entries.
29737 The ten digit keys, decimal point, and @key{EEX} key are used for
29738 entering numbers in the obvious way. @key{EEX} begins entry of an
29739 exponent in scientific notation. Just as with regular Calc, the
29740 number is pushed onto the stack as soon as you press @key{ENTER}
29741 or any other function key.
29743 The @key{+/-} key corresponds to normal Calc's @kbd{n} key. During
29744 numeric entry it changes the sign of the number or of the exponent.
29745 At other times it changes the sign of the number on the top of the
29748 The @key{INV} and @key{HYP} keys modify other keys. As well as
29749 having the effects described elsewhere in this manual, Keypad mode
29750 defines several other ``inverse'' operations. These are described
29751 below and in the following sections.
29753 The @key{ENTER} key finishes the current numeric entry, or otherwise
29754 duplicates the top entry on the stack.
29756 The @key{UNDO} key undoes the most recent Calc operation.
29757 @kbd{INV UNDO} is the ``redo'' command, and @kbd{HYP UNDO} is
29758 ``last arguments'' (@kbd{M-@key{RET}}).
29760 The @key{<-} key acts as a ``backspace'' during numeric entry.
29761 At other times it removes the top stack entry. @kbd{INV <-}
29762 clears the entire stack. @kbd{HYP <-} takes an integer from
29763 the stack, then removes that many additional stack elements.
29765 The @key{EXEC} key prompts you to enter any keystroke sequence
29766 that would normally work in Calc mode. This can include a
29767 numeric prefix if you wish. It is also possible simply to
29768 switch into the Calc window and type commands in it; there is
29769 nothing ``magic'' about this window when Keypad mode is active.
29771 The other keys in this display perform their obvious calculator
29772 functions. @key{CLN2} rounds the top-of-stack by temporarily
29773 reducing the precision by 2 digits. @key{FLT} converts an
29774 integer or fraction on the top of the stack to floating-point.
29776 The @key{INV} and @key{HYP} keys combined with several of these keys
29777 give you access to some common functions even if the appropriate menu
29778 is not displayed. Obviously you don't need to learn these keys
29779 unless you find yourself wasting time switching among the menus.
29783 is the same as @key{1/x}.
29785 is the same as @key{SQRT}.
29787 is the same as @key{CONJ}.
29789 is the same as @key{y^x}.
29791 is the same as @key{INV y^x} (the @expr{x}th root of @expr{y}).
29793 are the same as @key{SIN} / @kbd{INV SIN}.
29795 are the same as @key{COS} / @kbd{INV COS}.
29797 are the same as @key{TAN} / @kbd{INV TAN}.
29799 are the same as @key{LN} / @kbd{HYP LN}.
29801 are the same as @key{EXP} / @kbd{HYP EXP}.
29803 is the same as @key{ABS}.
29805 is the same as @key{RND} (@code{calc-round}).
29807 is the same as @key{CLN2}.
29809 is the same as @key{FLT} (@code{calc-float}).
29811 is the same as @key{IMAG}.
29813 is the same as @key{PREC}.
29815 is the same as @key{SWAP}.
29817 is the same as @key{RLL3}.
29818 @item INV HYP ENTER
29819 is the same as @key{OVER}.
29821 packs the top two stack entries as an error form.
29823 packs the top two stack entries as a modulo form.
29825 creates an interval form; this removes an integer which is one
29826 of 0 @samp{[]}, 1 @samp{[)}, 2 @samp{(]} or 3 @samp{()}, followed
29827 by the two limits of the interval.
29830 The @kbd{OFF} key turns Calc off; typing @kbd{M-# k} or @kbd{M-# M-#}
29831 again has the same effect. This is analogous to typing @kbd{q} or
29832 hitting @kbd{M-# c} again in the normal calculator. If Calc is
29833 running standalone (the @code{full-calc-keypad} command appeared in the
29834 command line that started Emacs), then @kbd{OFF} is replaced with
29835 @kbd{EXIT}; clicking on this actually exits Emacs itself.
29837 @node Keypad Functions Menu, Keypad Binary Menu, Keypad Main Menu, Keypad Mode
29838 @section Functions Menu
29842 |----+----+----+----+----+----2
29843 |IGAM|BETA|IBET|ERF |BESJ|BESY|
29844 |----+----+----+----+----+----|
29845 |IMAG|CONJ| RE |ATN2|RAND|RAGN|
29846 |----+----+----+----+----+----|
29847 |GCD |FACT|DFCT|BNOM|PERM|NXTP|
29848 |----+----+----+----+----+----|
29853 This menu provides various operations from the @kbd{f} and @kbd{k}
29856 @key{IMAG} multiplies the number on the stack by the imaginary
29857 number @expr{i = (0, 1)}.
29859 @key{RE} extracts the real part a complex number. @kbd{INV RE}
29860 extracts the imaginary part.
29862 @key{RAND} takes a number from the top of the stack and computes
29863 a random number greater than or equal to zero but less than that
29864 number. (@xref{Random Numbers}.) @key{RAGN} is the ``random
29865 again'' command; it computes another random number using the
29866 same limit as last time.
29868 @key{INV GCD} computes the LCM (least common multiple) function.
29870 @key{INV FACT} is the gamma function.
29871 @texline @math{\Gamma(x) = (x-1)!}.
29872 @infoline @expr{gamma(x) = (x-1)!}.
29874 @key{PERM} is the number-of-permutations function, which is on the
29875 @kbd{H k c} key in normal Calc.
29877 @key{NXTP} finds the next prime after a number. @kbd{INV NXTP}
29878 finds the previous prime.
29880 @node Keypad Binary Menu, Keypad Vectors Menu, Keypad Functions Menu, Keypad Mode
29881 @section Binary Menu
29885 |----+----+----+----+----+----3
29886 |AND | OR |XOR |NOT |LSH |RSH |
29887 |----+----+----+----+----+----|
29888 |DEC |HEX |OCT |BIN |WSIZ|ARSH|
29889 |----+----+----+----+----+----|
29890 | A | B | C | D | E | F |
29891 |----+----+----+----+----+----|
29896 The keys in this menu perform operations on binary integers.
29897 Note that both logical and arithmetic right-shifts are provided.
29898 @key{INV LSH} rotates one bit to the left.
29900 The ``difference'' function (normally on @kbd{b d}) is on @key{INV AND}.
29901 The ``clip'' function (normally on @w{@kbd{b c}}) is on @key{INV NOT}.
29903 The @key{DEC}, @key{HEX}, @key{OCT}, and @key{BIN} keys select the
29904 current radix for display and entry of numbers: Decimal, hexadecimal,
29905 octal, or binary. The six letter keys @key{A} through @key{F} are used
29906 for entering hexadecimal numbers.
29908 The @key{WSIZ} key displays the current word size for binary operations
29909 and allows you to enter a new word size. You can respond to the prompt
29910 using either the keyboard or the digits and @key{ENTER} from the keypad.
29911 The initial word size is 32 bits.
29913 @node Keypad Vectors Menu, Keypad Modes Menu, Keypad Binary Menu, Keypad Mode
29914 @section Vectors Menu
29918 |----+----+----+----+----+----4
29919 |SUM |PROD|MAX |MAP*|MAP^|MAP$|
29920 |----+----+----+----+----+----|
29921 |MINV|MDET|MTRN|IDNT|CRSS|"x" |
29922 |----+----+----+----+----+----|
29923 |PACK|UNPK|INDX|BLD |LEN |... |
29924 |----+----+----+----+----+----|
29929 The keys in this menu operate on vectors and matrices.
29931 @key{PACK} removes an integer @var{n} from the top of the stack;
29932 the next @var{n} stack elements are removed and packed into a vector,
29933 which is replaced onto the stack. Thus the sequence
29934 @kbd{1 ENTER 3 ENTER 5 ENTER 3 PACK} enters the vector
29935 @samp{[1, 3, 5]} onto the stack. To enter a matrix, build each row
29936 on the stack as a vector, then use a final @key{PACK} to collect the
29937 rows into a matrix.
29939 @key{UNPK} unpacks the vector on the stack, pushing each of its
29940 components separately.
29942 @key{INDX} removes an integer @var{n}, then builds a vector of
29943 integers from 1 to @var{n}. @kbd{INV INDX} takes three numbers
29944 from the stack: The vector size @var{n}, the starting number,
29945 and the increment. @kbd{BLD} takes an integer @var{n} and any
29946 value @var{x} and builds a vector of @var{n} copies of @var{x}.
29948 @key{IDNT} removes an integer @var{n}, then builds an @var{n}-by-@var{n}
29951 @key{LEN} replaces a vector by its length, an integer.
29953 @key{...} turns on or off ``abbreviated'' display mode for large vectors.
29955 @key{MINV}, @key{MDET}, @key{MTRN}, and @key{CROSS} are the matrix
29956 inverse, determinant, and transpose, and vector cross product.
29958 @key{SUM} replaces a vector by the sum of its elements. It is
29959 equivalent to @kbd{u +} in normal Calc (@pxref{Statistical Operations}).
29960 @key{PROD} computes the product of the elements of a vector, and
29961 @key{MAX} computes the maximum of all the elements of a vector.
29963 @key{INV SUM} computes the alternating sum of the first element
29964 minus the second, plus the third, minus the fourth, and so on.
29965 @key{INV MAX} computes the minimum of the vector elements.
29967 @key{HYP SUM} computes the mean of the vector elements.
29968 @key{HYP PROD} computes the sample standard deviation.
29969 @key{HYP MAX} computes the median.
29971 @key{MAP*} multiplies two vectors elementwise. It is equivalent
29972 to the @kbd{V M *} command. @key{MAP^} computes powers elementwise.
29973 The arguments must be vectors of equal length, or one must be a vector
29974 and the other must be a plain number. For example, @kbd{2 MAP^} squares
29975 all the elements of a vector.
29977 @key{MAP$} maps the formula on the top of the stack across the
29978 vector in the second-to-top position. If the formula contains
29979 several variables, Calc takes that many vectors starting at the
29980 second-to-top position and matches them to the variables in
29981 alphabetical order. The result is a vector of the same size as
29982 the input vectors, whose elements are the formula evaluated with
29983 the variables set to the various sets of numbers in those vectors.
29984 For example, you could simulate @key{MAP^} using @key{MAP$} with
29985 the formula @samp{x^y}.
29987 The @kbd{"x"} key pushes the variable name @expr{x} onto the
29988 stack. To build the formula @expr{x^2 + 6}, you would use the
29989 key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be
29990 suitable for use with the @key{MAP$} key described above.
29991 With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the
29992 @kbd{"x"} key pushes the variable names @expr{y}, @expr{z}, and
29993 @expr{t}, respectively.
29995 @node Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode
29996 @section Modes Menu
30000 |----+----+----+----+----+----5
30001 |FLT |FIX |SCI |ENG |GRP | |
30002 |----+----+----+----+----+----|
30003 |RAD |DEG |FRAC|POLR|SYMB|PREC|
30004 |----+----+----+----+----+----|
30005 |SWAP|RLL3|RLL4|OVER|STO |RCL |
30006 |----+----+----+----+----+----|
30011 The keys in this menu manipulate modes, variables, and the stack.
30013 The @key{FLT}, @key{FIX}, @key{SCI}, and @key{ENG} keys select
30014 floating-point, fixed-point, scientific, or engineering notation.
30015 @key{FIX} displays two digits after the decimal by default; the
30016 others display full precision. With the @key{INV} prefix, these
30017 keys pop a number-of-digits argument from the stack.
30019 The @key{GRP} key turns grouping of digits with commas on or off.
30020 @kbd{INV GRP} enables grouping to the right of the decimal point as
30021 well as to the left.
30023 The @key{RAD} and @key{DEG} keys switch between radians and degrees
30024 for trigonometric functions.
30026 The @key{FRAC} key turns Fraction mode on or off. This affects
30027 whether commands like @kbd{/} with integer arguments produce
30028 fractional or floating-point results.
30030 The @key{POLR} key turns Polar mode on or off, determining whether
30031 polar or rectangular complex numbers are used by default.
30033 The @key{SYMB} key turns Symbolic mode on or off, in which
30034 operations that would produce inexact floating-point results
30035 are left unevaluated as algebraic formulas.
30037 The @key{PREC} key selects the current precision. Answer with
30038 the keyboard or with the keypad digit and @key{ENTER} keys.
30040 The @key{SWAP} key exchanges the top two stack elements.
30041 The @key{RLL3} key rotates the top three stack elements upwards.
30042 The @key{RLL4} key rotates the top four stack elements upwards.
30043 The @key{OVER} key duplicates the second-to-top stack element.
30045 The @key{STO} and @key{RCL} keys are analogous to @kbd{s t} and
30046 @kbd{s r} in regular Calc. @xref{Store and Recall}. Click the
30047 @key{STO} or @key{RCL} key, then one of the ten digits. (Named
30048 variables are not available in Keypad mode.) You can also use,
30049 for example, @kbd{STO + 3} to add to register 3.
30051 @node Embedded Mode, Programming, Keypad Mode, Top
30052 @chapter Embedded Mode
30055 Embedded mode in Calc provides an alternative to copying numbers
30056 and formulas back and forth between editing buffers and the Calc
30057 stack. In Embedded mode, your editing buffer becomes temporarily
30058 linked to the stack and this copying is taken care of automatically.
30061 * Basic Embedded Mode::
30062 * More About Embedded Mode::
30063 * Assignments in Embedded Mode::
30064 * Mode Settings in Embedded Mode::
30065 * Customizing Embedded Mode::
30068 @node Basic Embedded Mode, More About Embedded Mode, Embedded Mode, Embedded Mode
30069 @section Basic Embedded Mode
30073 @pindex calc-embedded
30074 To enter Embedded mode, position the Emacs point (cursor) on a
30075 formula in any buffer and press @kbd{M-# e} (@code{calc-embedded}).
30076 Note that @kbd{M-# e} is not to be used in the Calc stack buffer
30077 like most Calc commands, but rather in regular editing buffers that
30078 are visiting your own files.
30080 Calc will try to guess an appropriate language based on the major mode
30081 of the editing buffer. (@xref{Language Modes}.) If the current buffer is
30082 in @code{latex-mode}, for example, Calc will set its language to La@TeX{}.
30083 Similarly, Calc will use @TeX{} language for @code{tex-mode},
30084 @code{plain-tex-mode} and @code{context-mode}, C language for
30085 @code{c-mode} and @code{c++-mode}, FORTRAN language for
30086 @code{fortran-mode} and @code{f90-mode}, Pascal for @code{pascal-mode},
30087 and eqn for @code{nroff-mode} (@pxref{Customizable Variables}).
30088 These can be overridden with Calc's mode
30089 changing commands (@pxref{Mode Settings in Embedded Mode}). If no
30090 suitable language is available, Calc will continue with its current language.
30092 Calc normally scans backward and forward in the buffer for the
30093 nearest opening and closing @dfn{formula delimiters}. The simplest
30094 delimiters are blank lines. Other delimiters that Embedded mode
30099 The @TeX{} and La@TeX{} math delimiters @samp{$ $}, @samp{$$ $$},
30100 @samp{\[ \]}, and @samp{\( \)};
30102 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
30104 Lines beginning with @samp{@@} (Texinfo delimiters).
30106 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
30108 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
30111 @xref{Customizing Embedded Mode}, to see how to make Calc recognize
30112 your own favorite delimiters. Delimiters like @samp{$ $} can appear
30113 on their own separate lines or in-line with the formula.
30115 If you give a positive or negative numeric prefix argument, Calc
30116 instead uses the current point as one end of the formula, and moves
30117 forward or backward (respectively) by that many lines to find the
30118 other end. Explicit delimiters are not necessary in this case.
30120 With a prefix argument of zero, Calc uses the current region
30121 (delimited by point and mark) instead of formula delimiters.
30124 @pindex calc-embedded-word
30125 With a prefix argument of @kbd{C-u} only, Calc scans for the first
30126 non-numeric character (i.e., the first character that is not a
30127 digit, sign, decimal point, or upper- or lower-case @samp{e})
30128 forward and backward to delimit the formula. @kbd{M-# w}
30129 (@code{calc-embedded-word}) is equivalent to @kbd{C-u M-# e}.
30131 When you enable Embedded mode for a formula, Calc reads the text
30132 between the delimiters and tries to interpret it as a Calc formula.
30133 Calc can generally identify @TeX{} formulas and
30134 Big-style formulas even if the language mode is wrong. If Calc
30135 can't make sense of the formula, it beeps and refuses to enter
30136 Embedded mode. But if the current language is wrong, Calc can
30137 sometimes parse the formula successfully (but incorrectly);
30138 for example, the C expression @samp{atan(a[1])} can be parsed
30139 in Normal language mode, but the @code{atan} won't correspond to
30140 the built-in @code{arctan} function, and the @samp{a[1]} will be
30141 interpreted as @samp{a} times the vector @samp{[1]}!
30143 If you press @kbd{M-# e} or @kbd{M-# w} to activate an embedded
30144 formula which is blank, say with the cursor on the space between
30145 the two delimiters @samp{$ $}, Calc will immediately prompt for
30146 an algebraic entry.
30148 Only one formula in one buffer can be enabled at a time. If you
30149 move to another area of the current buffer and give Calc commands,
30150 Calc turns Embedded mode off for the old formula and then tries
30151 to restart Embedded mode at the new position. Other buffers are
30152 not affected by Embedded mode.
30154 When Embedded mode begins, Calc pushes the current formula onto
30155 the stack. No Calc stack window is created; however, Calc copies
30156 the top-of-stack position into the original buffer at all times.
30157 You can create a Calc window by hand with @kbd{M-# o} if you
30158 find you need to see the entire stack.
30160 For example, typing @kbd{M-# e} while somewhere in the formula
30161 @samp{n>2} in the following line enables Embedded mode on that
30165 We define $F_n = F_(n-1)+F_(n-2)$ for all $n>2$.
30169 The formula @expr{n>2} will be pushed onto the Calc stack, and
30170 the top of stack will be copied back into the editing buffer.
30171 This means that spaces will appear around the @samp{>} symbol
30172 to match Calc's usual display style:
30175 We define $F_n = F_(n-1)+F_(n-2)$ for all $n > 2$.
30179 No spaces have appeared around the @samp{+} sign because it's
30180 in a different formula, one which we have not yet touched with
30183 Now that Embedded mode is enabled, keys you type in this buffer
30184 are interpreted as Calc commands. At this point we might use
30185 the ``commute'' command @kbd{j C} to reverse the inequality.
30186 This is a selection-based command for which we first need to
30187 move the cursor onto the operator (@samp{>} in this case) that
30188 needs to be commuted.
30191 We define $F_n = F_(n-1)+F_(n-2)$ for all $2 < n$.
30194 The @kbd{M-# o} command is a useful way to open a Calc window
30195 without actually selecting that window. Giving this command
30196 verifies that @samp{2 < n} is also on the Calc stack. Typing
30197 @kbd{17 @key{RET}} would produce:
30200 We define $F_n = F_(n-1)+F_(n-2)$ for all $17$.
30204 with @samp{2 < n} and @samp{17} on the stack; typing @key{TAB}
30205 at this point will exchange the two stack values and restore
30206 @samp{2 < n} to the embedded formula. Even though you can't
30207 normally see the stack in Embedded mode, it is still there and
30208 it still operates in the same way. But, as with old-fashioned
30209 RPN calculators, you can only see the value at the top of the
30210 stack at any given time (unless you use @kbd{M-# o}).
30212 Typing @kbd{M-# e} again turns Embedded mode off. The Calc
30213 window reveals that the formula @w{@samp{2 < n}} is automatically
30214 removed from the stack, but the @samp{17} is not. Entering
30215 Embedded mode always pushes one thing onto the stack, and
30216 leaving Embedded mode always removes one thing. Anything else
30217 that happens on the stack is entirely your business as far as
30218 Embedded mode is concerned.
30220 If you press @kbd{M-# e} in the wrong place by accident, it is
30221 possible that Calc will be able to parse the nearby text as a
30222 formula and will mangle that text in an attempt to redisplay it
30223 ``properly'' in the current language mode. If this happens,
30224 press @kbd{M-# e} again to exit Embedded mode, then give the
30225 regular Emacs ``undo'' command (@kbd{C-_} or @kbd{C-x u}) to put
30226 the text back the way it was before Calc edited it. Note that Calc's
30227 own Undo command (typed before you turn Embedded mode back off)
30228 will not do you any good, because as far as Calc is concerned
30229 you haven't done anything with this formula yet.
30231 @node More About Embedded Mode, Assignments in Embedded Mode, Basic Embedded Mode, Embedded Mode
30232 @section More About Embedded Mode
30235 When Embedded mode ``activates'' a formula, i.e., when it examines
30236 the formula for the first time since the buffer was created or
30237 loaded, Calc tries to sense the language in which the formula was
30238 written. If the formula contains any La@TeX{}-like @samp{\} sequences,
30239 it is parsed (i.e., read) in La@TeX{} mode. If the formula appears to
30240 be written in multi-line Big mode, it is parsed in Big mode. Otherwise,
30241 it is parsed according to the current language mode.
30243 Note that Calc does not change the current language mode according
30244 the formula it reads in. Even though it can read a La@TeX{} formula when
30245 not in La@TeX{} mode, it will immediately rewrite this formula using
30246 whatever language mode is in effect.
30253 @pindex calc-show-plain
30254 Calc's parser is unable to read certain kinds of formulas. For
30255 example, with @kbd{v ]} (@code{calc-matrix-brackets}) you can
30256 specify matrix display styles which the parser is unable to
30257 recognize as matrices. The @kbd{d p} (@code{calc-show-plain})
30258 command turns on a mode in which a ``plain'' version of a
30259 formula is placed in front of the fully-formatted version.
30260 When Calc reads a formula that has such a plain version in
30261 front, it reads the plain version and ignores the formatted
30264 Plain formulas are preceded and followed by @samp{%%%} signs
30265 by default. This notation has the advantage that the @samp{%}
30266 character begins a comment in @TeX{} and La@TeX{}, so if your formula is
30267 embedded in a @TeX{} or La@TeX{} document its plain version will be
30268 invisible in the final printed copy. @xref{Customizing
30269 Embedded Mode}, to see how to change the ``plain'' formula
30270 delimiters, say to something that @dfn{eqn} or some other
30271 formatter will treat as a comment.
30273 There are several notations which Calc's parser for ``big''
30274 formatted formulas can't yet recognize. In particular, it can't
30275 read the large symbols for @code{sum}, @code{prod}, and @code{integ},
30276 and it can't handle @samp{=>} with the righthand argument omitted.
30277 Also, Calc won't recognize special formats you have defined with
30278 the @kbd{Z C} command (@pxref{User-Defined Compositions}). In
30279 these cases it is important to use ``plain'' mode to make sure
30280 Calc will be able to read your formula later.
30282 Another example where ``plain'' mode is important is if you have
30283 specified a float mode with few digits of precision. Normally
30284 any digits that are computed but not displayed will simply be
30285 lost when you save and re-load your embedded buffer, but ``plain''
30286 mode allows you to make sure that the complete number is present
30287 in the file as well as the rounded-down number.
30293 Embedded buffers remember active formulas for as long as they
30294 exist in Emacs memory. Suppose you have an embedded formula
30295 which is @cpi{} to the normal 12 decimal places, and then
30296 type @w{@kbd{C-u 5 d n}} to display only five decimal places.
30297 If you then type @kbd{d n}, all 12 places reappear because the
30298 full number is still there on the Calc stack. More surprisingly,
30299 even if you exit Embedded mode and later re-enter it for that
30300 formula, typing @kbd{d n} will restore all 12 places because
30301 each buffer remembers all its active formulas. However, if you
30302 save the buffer in a file and reload it in a new Emacs session,
30303 all non-displayed digits will have been lost unless you used
30310 In some applications of Embedded mode, you will want to have a
30311 sequence of copies of a formula that show its evolution as you
30312 work on it. For example, you might want to have a sequence
30313 like this in your file (elaborating here on the example from
30314 the ``Getting Started'' chapter):
30323 @r{(the derivative of }ln(ln(x))@r{)}
30325 whose value at x = 2 is
30335 @pindex calc-embedded-duplicate
30336 The @kbd{M-# d} (@code{calc-embedded-duplicate}) command is a
30337 handy way to make sequences like this. If you type @kbd{M-# d},
30338 the formula under the cursor (which may or may not have Embedded
30339 mode enabled for it at the time) is copied immediately below and
30340 Embedded mode is then enabled for that copy.
30342 For this example, you would start with just
30351 and press @kbd{M-# d} with the cursor on this formula. The result
30364 with the second copy of the formula enabled in Embedded mode.
30365 You can now press @kbd{a d x @key{RET}} to take the derivative, and
30366 @kbd{M-# d M-# d} to make two more copies of the derivative.
30367 To complete the computations, type @kbd{3 s l x @key{RET}} to evaluate
30368 the last formula, then move up to the second-to-last formula
30369 and type @kbd{2 s l x @key{RET}}.
30371 Finally, you would want to press @kbd{M-# e} to exit Embedded
30372 mode, then go up and insert the necessary text in between the
30373 various formulas and numbers.
30381 @pindex calc-embedded-new-formula
30382 The @kbd{M-# f} (@code{calc-embedded-new-formula}) command
30383 creates a new embedded formula at the current point. It inserts
30384 some default delimiters, which are usually just blank lines,
30385 and then does an algebraic entry to get the formula (which is
30386 then enabled for Embedded mode). This is just shorthand for
30387 typing the delimiters yourself, positioning the cursor between
30388 the new delimiters, and pressing @kbd{M-# e}. The key sequence
30389 @kbd{M-# '} is equivalent to @kbd{M-# f}.
30393 @pindex calc-embedded-next
30394 @pindex calc-embedded-previous
30395 The @kbd{M-# n} (@code{calc-embedded-next}) and @kbd{M-# p}
30396 (@code{calc-embedded-previous}) commands move the cursor to the
30397 next or previous active embedded formula in the buffer. They
30398 can take positive or negative prefix arguments to move by several
30399 formulas. Note that these commands do not actually examine the
30400 text of the buffer looking for formulas; they only see formulas
30401 which have previously been activated in Embedded mode. In fact,
30402 @kbd{M-# n} and @kbd{M-# p} are a useful way to tell which
30403 embedded formulas are currently active. Also, note that these
30404 commands do not enable Embedded mode on the next or previous
30405 formula, they just move the cursor. (By the way, @kbd{M-# n} is
30406 not as awkward to type as it may seem, because @kbd{M-#} ignores
30407 Shift and Meta on the second keystroke: @kbd{M-# M-N} can be typed
30408 by holding down Shift and Meta and alternately typing two keys.)
30411 @pindex calc-embedded-edit
30412 The @kbd{M-# `} (@code{calc-embedded-edit}) command edits the
30413 embedded formula at the current point as if by @kbd{`} (@code{calc-edit}).
30414 Embedded mode does not have to be enabled for this to work. Press
30415 @kbd{C-c C-c} to finish the edit, or @kbd{C-x k} to cancel.
30417 @node Assignments in Embedded Mode, Mode Settings in Embedded Mode, More About Embedded Mode, Embedded Mode
30418 @section Assignments in Embedded Mode
30421 The @samp{:=} (assignment) and @samp{=>} (``evaluates-to'') operators
30422 are especially useful in Embedded mode. They allow you to make
30423 a definition in one formula, then refer to that definition in
30424 other formulas embedded in the same buffer.
30426 An embedded formula which is an assignment to a variable, as in
30433 records @expr{5} as the stored value of @code{foo} for the
30434 purposes of Embedded mode operations in the current buffer. It
30435 does @emph{not} actually store @expr{5} as the ``global'' value
30436 of @code{foo}, however. Regular Calc operations, and Embedded
30437 formulas in other buffers, will not see this assignment.
30439 One way to use this assigned value is simply to create an
30440 Embedded formula elsewhere that refers to @code{foo}, and to press
30441 @kbd{=} in that formula. However, this permanently replaces the
30442 @code{foo} in the formula with its current value. More interesting
30443 is to use @samp{=>} elsewhere:
30449 @xref{Evaluates-To Operator}, for a general discussion of @samp{=>}.
30451 If you move back and change the assignment to @code{foo}, any
30452 @samp{=>} formulas which refer to it are automatically updated.
30460 The obvious question then is, @emph{how} can one easily change the
30461 assignment to @code{foo}? If you simply select the formula in
30462 Embedded mode and type 17, the assignment itself will be replaced
30463 by the 17. The effect on the other formula will be that the
30464 variable @code{foo} becomes unassigned:
30472 The right thing to do is first to use a selection command (@kbd{j 2}
30473 will do the trick) to select the righthand side of the assignment.
30474 Then, @kbd{17 @key{TAB} @key{DEL}} will swap the 17 into place (@pxref{Selecting
30475 Subformulas}, to see how this works).
30478 @pindex calc-embedded-select
30479 The @kbd{M-# j} (@code{calc-embedded-select}) command provides an
30480 easy way to operate on assignments. It is just like @kbd{M-# e},
30481 except that if the enabled formula is an assignment, it uses
30482 @kbd{j 2} to select the righthand side. If the enabled formula
30483 is an evaluates-to, it uses @kbd{j 1} to select the lefthand side.
30484 A formula can also be a combination of both:
30487 bar := foo + 3 => 20
30491 in which case @kbd{M-# j} will select the middle part (@samp{foo + 3}).
30493 The formula is automatically deselected when you leave Embedded
30498 @pindex calc-embedded-update
30499 Another way to change the assignment to @code{foo} would simply be
30500 to edit the number using regular Emacs editing rather than Embedded
30501 mode. Then, we have to find a way to get Embedded mode to notice
30502 the change. The @kbd{M-# u} or @kbd{M-# =}
30503 (@code{calc-embedded-update-formula}) command is a convenient way
30512 Pressing @kbd{M-# u} is much like pressing @kbd{M-# e = M-# e}, that
30513 is, temporarily enabling Embedded mode for the formula under the
30514 cursor and then evaluating it with @kbd{=}. But @kbd{M-# u} does
30515 not actually use @kbd{M-# e}, and in fact another formula somewhere
30516 else can be enabled in Embedded mode while you use @kbd{M-# u} and
30517 that formula will not be disturbed.
30519 With a numeric prefix argument, @kbd{M-# u} updates all active
30520 @samp{=>} formulas in the buffer. Formulas which have not yet
30521 been activated in Embedded mode, and formulas which do not have
30522 @samp{=>} as their top-level operator, are not affected by this.
30523 (This is useful only if you have used @kbd{m C}; see below.)
30525 With a plain @kbd{C-u} prefix, @kbd{C-u M-# u} updates only in the
30526 region between mark and point rather than in the whole buffer.
30528 @kbd{M-# u} is also a handy way to activate a formula, such as an
30529 @samp{=>} formula that has freshly been typed in or loaded from a
30533 @pindex calc-embedded-activate
30534 The @kbd{M-# a} (@code{calc-embedded-activate}) command scans
30535 through the current buffer and activates all embedded formulas
30536 that contain @samp{:=} or @samp{=>} symbols. This does not mean
30537 that Embedded mode is actually turned on, but only that the
30538 formulas' positions are registered with Embedded mode so that
30539 the @samp{=>} values can be properly updated as assignments are
30542 It is a good idea to type @kbd{M-# a} right after loading a file
30543 that uses embedded @samp{=>} operators. Emacs includes a nifty
30544 ``buffer-local variables'' feature that you can use to do this
30545 automatically. The idea is to place near the end of your file
30546 a few lines that look like this:
30549 --- Local Variables: ---
30550 --- eval:(calc-embedded-activate) ---
30555 where the leading and trailing @samp{---} can be replaced by
30556 any suitable strings (which must be the same on all three lines)
30557 or omitted altogether; in a @TeX{} or La@TeX{} file, @samp{%} would be a good
30558 leading string and no trailing string would be necessary. In a
30559 C program, @samp{/*} and @samp{*/} would be good leading and
30562 When Emacs loads a file into memory, it checks for a Local Variables
30563 section like this one at the end of the file. If it finds this
30564 section, it does the specified things (in this case, running
30565 @kbd{M-# a} automatically) before editing of the file begins.
30566 The Local Variables section must be within 3000 characters of the
30567 end of the file for Emacs to find it, and it must be in the last
30568 page of the file if the file has any page separators.
30569 @xref{File Variables, , Local Variables in Files, emacs, the
30572 Note that @kbd{M-# a} does not update the formulas it finds.
30573 To do this, type, say, @kbd{M-1 M-# u} after @w{@kbd{M-# a}}.
30574 Generally this should not be a problem, though, because the
30575 formulas will have been up-to-date already when the file was
30578 Normally, @kbd{M-# a} activates all the formulas it finds, but
30579 any previous active formulas remain active as well. With a
30580 positive numeric prefix argument, @kbd{M-# a} first deactivates
30581 all current active formulas, then actives the ones it finds in
30582 its scan of the buffer. With a negative prefix argument,
30583 @kbd{M-# a} simply deactivates all formulas.
30585 Embedded mode has two symbols, @samp{Active} and @samp{~Active},
30586 which it puts next to the major mode name in a buffer's mode line.
30587 It puts @samp{Active} if it has reason to believe that all
30588 formulas in the buffer are active, because you have typed @kbd{M-# a}
30589 and Calc has not since had to deactivate any formulas (which can
30590 happen if Calc goes to update an @samp{=>} formula somewhere because
30591 a variable changed, and finds that the formula is no longer there
30592 due to some kind of editing outside of Embedded mode). Calc puts
30593 @samp{~Active} in the mode line if some, but probably not all,
30594 formulas in the buffer are active. This happens if you activate
30595 a few formulas one at a time but never use @kbd{M-# a}, or if you
30596 used @kbd{M-# a} but then Calc had to deactivate a formula
30597 because it lost track of it. If neither of these symbols appears
30598 in the mode line, no embedded formulas are active in the buffer
30599 (e.g., before Embedded mode has been used, or after a @kbd{M-- M-# a}).
30601 Embedded formulas can refer to assignments both before and after them
30602 in the buffer. If there are several assignments to a variable, the
30603 nearest preceding assignment is used if there is one, otherwise the
30604 following assignment is used.
30618 As well as simple variables, you can also assign to subscript
30619 expressions of the form @samp{@var{var}_@var{number}} (as in
30620 @code{x_0}), or @samp{@var{var}_@var{var}} (as in @code{x_max}).
30621 Assignments to other kinds of objects can be represented by Calc,
30622 but the automatic linkage between assignments and references works
30623 only for plain variables and these two kinds of subscript expressions.
30625 If there are no assignments to a given variable, the global
30626 stored value for the variable is used (@pxref{Storing Variables}),
30627 or, if no value is stored, the variable is left in symbolic form.
30628 Note that global stored values will be lost when the file is saved
30629 and loaded in a later Emacs session, unless you have used the
30630 @kbd{s p} (@code{calc-permanent-variable}) command to save them;
30631 @pxref{Operations on Variables}.
30633 The @kbd{m C} (@code{calc-auto-recompute}) command turns automatic
30634 recomputation of @samp{=>} forms on and off. If you turn automatic
30635 recomputation off, you will have to use @kbd{M-# u} to update these
30636 formulas manually after an assignment has been changed. If you
30637 plan to change several assignments at once, it may be more efficient
30638 to type @kbd{m C}, change all the assignments, then use @kbd{M-1 M-# u}
30639 to update the entire buffer afterwards. The @kbd{m C} command also
30640 controls @samp{=>} formulas on the stack; @pxref{Evaluates-To
30641 Operator}. When you turn automatic recomputation back on, the
30642 stack will be updated but the Embedded buffer will not; you must
30643 use @kbd{M-# u} to update the buffer by hand.
30645 @node Mode Settings in Embedded Mode, Customizing Embedded Mode, Assignments in Embedded Mode, Embedded Mode
30646 @section Mode Settings in Embedded Mode
30649 @pindex calc-embedded-preserve-modes
30651 The mode settings can be changed while Calc is in embedded mode, but
30652 by default they will revert to their original values when embedded mode
30653 is ended. However, the modes saved when the mode-recording mode is
30654 @code{Save} (see below) and the modes in effect when the @kbd{m e}
30655 (@code{calc-embedded-preserve-modes}) command is given
30656 will be preserved when embedded mode is ended.
30658 Embedded mode has a rather complicated mechanism for handling mode
30659 settings in Embedded formulas. It is possible to put annotations
30660 in the file that specify mode settings either global to the entire
30661 file or local to a particular formula or formulas. In the latter
30662 case, different modes can be specified for use when a formula
30663 is the enabled Embedded mode formula.
30665 When you give any mode-setting command, like @kbd{m f} (for Fraction
30666 mode) or @kbd{d s} (for scientific notation), Embedded mode adds
30667 a line like the following one to the file just before the opening
30668 delimiter of the formula.
30671 % [calc-mode: fractions: t]
30672 % [calc-mode: float-format: (sci 0)]
30675 When Calc interprets an embedded formula, it scans the text before
30676 the formula for mode-setting annotations like these and sets the
30677 Calc buffer to match these modes. Modes not explicitly described
30678 in the file are not changed. Calc scans all the way to the top of
30679 the file, or up to a line of the form
30686 which you can insert at strategic places in the file if this backward
30687 scan is getting too slow, or just to provide a barrier between one
30688 ``zone'' of mode settings and another.
30690 If the file contains several annotations for the same mode, the
30691 closest one before the formula is used. Annotations after the
30692 formula are never used (except for global annotations, described
30695 The scan does not look for the leading @samp{% }, only for the
30696 square brackets and the text they enclose. You can edit the mode
30697 annotations to a style that works better in context if you wish.
30698 @xref{Customizing Embedded Mode}, to see how to change the style
30699 that Calc uses when it generates the annotations. You can write
30700 mode annotations into the file yourself if you know the syntax;
30701 the easiest way to find the syntax for a given mode is to let
30702 Calc write the annotation for it once and see what it does.
30704 If you give a mode-changing command for a mode that already has
30705 a suitable annotation just above the current formula, Calc will
30706 modify that annotation rather than generating a new, conflicting
30709 Mode annotations have three parts, separated by colons. (Spaces
30710 after the colons are optional.) The first identifies the kind
30711 of mode setting, the second is a name for the mode itself, and
30712 the third is the value in the form of a Lisp symbol, number,
30713 or list. Annotations with unrecognizable text in the first or
30714 second parts are ignored. The third part is not checked to make
30715 sure the value is of a valid type or range; if you write an
30716 annotation by hand, be sure to give a proper value or results
30717 will be unpredictable. Mode-setting annotations are case-sensitive.
30719 While Embedded mode is enabled, the word @code{Local} appears in
30720 the mode line. This is to show that mode setting commands generate
30721 annotations that are ``local'' to the current formula or set of
30722 formulas. The @kbd{m R} (@code{calc-mode-record-mode}) command
30723 causes Calc to generate different kinds of annotations. Pressing
30724 @kbd{m R} repeatedly cycles through the possible modes.
30726 @code{LocEdit} and @code{LocPerm} modes generate annotations
30727 that look like this, respectively:
30730 % [calc-edit-mode: float-format: (sci 0)]
30731 % [calc-perm-mode: float-format: (sci 5)]
30734 The first kind of annotation will be used only while a formula
30735 is enabled in Embedded mode. The second kind will be used only
30736 when the formula is @emph{not} enabled. (Whether the formula
30737 is ``active'' or not, i.e., whether Calc has seen this formula
30738 yet, is not relevant here.)
30740 @code{Global} mode generates an annotation like this at the end
30744 % [calc-global-mode: fractions t]
30747 Global mode annotations affect all formulas throughout the file,
30748 and may appear anywhere in the file. This allows you to tuck your
30749 mode annotations somewhere out of the way, say, on a new page of
30750 the file, as long as those mode settings are suitable for all
30751 formulas in the file.
30753 Enabling a formula with @kbd{M-# e} causes a fresh scan for local
30754 mode annotations; you will have to use this after adding annotations
30755 above a formula by hand to get the formula to notice them. Updating
30756 a formula with @kbd{M-# u} will also re-scan the local modes, but
30757 global modes are only re-scanned by @kbd{M-# a}.
30759 Another way that modes can get out of date is if you add a local
30760 mode annotation to a formula that has another formula after it.
30761 In this example, we have used the @kbd{d s} command while the
30762 first of the two embedded formulas is active. But the second
30763 formula has not changed its style to match, even though by the
30764 rules of reading annotations the @samp{(sci 0)} applies to it, too.
30767 % [calc-mode: float-format: (sci 0)]
30773 We would have to go down to the other formula and press @kbd{M-# u}
30774 on it in order to get it to notice the new annotation.
30776 Two more mode-recording modes selectable by @kbd{m R} are available
30777 which are also available outside of Embedded mode.
30778 (@pxref{General Mode Commands}.) They are @code{Save}, in which mode
30779 settings are recorded permanently in your Calc init file (the file given
30780 by the variable @code{calc-settings-file}, typically @file{~/.calc.el})
30781 rather than by annotating the current document, and no-recording
30782 mode (where there is no symbol like @code{Save} or @code{Local} in
30783 the mode line), in which mode-changing commands do not leave any
30784 annotations at all.
30786 When Embedded mode is not enabled, mode-recording modes except
30787 for @code{Save} have no effect.
30789 @node Customizing Embedded Mode, , Mode Settings in Embedded Mode, Embedded Mode
30790 @section Customizing Embedded Mode
30793 You can modify Embedded mode's behavior by setting various Lisp
30794 variables described here. These variables are customizable
30795 (@pxref{Customizable Variables}), or you can use @kbd{M-x set-variable}
30796 or @kbd{M-x edit-options} to adjust a variable on the fly.
30797 (Another possibility would
30798 be to use a file-local variable annotation at the end of the
30799 file; @pxref{File Variables, , Local Variables in Files, emacs, the
30802 While none of these variables will be buffer-local by default, you
30803 can make any of them local to any Embedded mode buffer. (Their
30804 values in the @samp{*Calculator*} buffer are never used.)
30806 @vindex calc-embedded-open-formula
30807 The @code{calc-embedded-open-formula} variable holds a regular
30808 expression for the opening delimiter of a formula. @xref{Regexp Search,
30809 , Regular Expression Search, emacs, the Emacs manual}, to see
30810 how regular expressions work. Basically, a regular expression is a
30811 pattern that Calc can search for. A regular expression that considers
30812 blank lines, @samp{$}, and @samp{$$} to be opening delimiters is
30813 @code{"\\`\\|^\n\\|\\$\\$?"}. Just in case the meaning of this
30814 regular expression is not completely plain, let's go through it
30817 The surrounding @samp{" "} marks quote the text between them as a
30818 Lisp string. If you left them off, @code{set-variable} or
30819 @code{edit-options} would try to read the regular expression as a
30822 The most obvious property of this regular expression is that it
30823 contains indecently many backslashes. There are actually two levels
30824 of backslash usage going on here. First, when Lisp reads a quoted
30825 string, all pairs of characters beginning with a backslash are
30826 interpreted as special characters. Here, @code{\n} changes to a
30827 new-line character, and @code{\\} changes to a single backslash.
30828 So the actual regular expression seen by Calc is
30829 @samp{\`\|^ @r{(newline)} \|\$\$?}.
30831 Regular expressions also consider pairs beginning with backslash
30832 to have special meanings. Sometimes the backslash is used to quote
30833 a character that otherwise would have a special meaning in a regular
30834 expression, like @samp{$}, which normally means ``end-of-line,''
30835 or @samp{?}, which means that the preceding item is optional. So
30836 @samp{\$\$?} matches either one or two dollar signs.
30838 The other codes in this regular expression are @samp{^}, which matches
30839 ``beginning-of-line,'' @samp{\|}, which means ``or,'' and @samp{\`},
30840 which matches ``beginning-of-buffer.'' So the whole pattern means
30841 that a formula begins at the beginning of the buffer, or on a newline
30842 that occurs at the beginning of a line (i.e., a blank line), or at
30843 one or two dollar signs.
30845 The default value of @code{calc-embedded-open-formula} looks just
30846 like this example, with several more alternatives added on to
30847 recognize various other common kinds of delimiters.
30849 By the way, the reason to use @samp{^\n} rather than @samp{^$}
30850 or @samp{\n\n}, which also would appear to match blank lines,
30851 is that the former expression actually ``consumes'' only one
30852 newline character as @emph{part of} the delimiter, whereas the
30853 latter expressions consume zero or two newlines, respectively.
30854 The former choice gives the most natural behavior when Calc
30855 must operate on a whole formula including its delimiters.
30857 See the Emacs manual for complete details on regular expressions.
30858 But just for your convenience, here is a list of all characters
30859 which must be quoted with backslash (like @samp{\$}) to avoid
30860 some special interpretation: @samp{. * + ? [ ] ^ $ \}. (Note
30861 the backslash in this list; for example, to match @samp{\[} you
30862 must use @code{"\\\\\\["}. An exercise for the reader is to
30863 account for each of these six backslashes!)
30865 @vindex calc-embedded-close-formula
30866 The @code{calc-embedded-close-formula} variable holds a regular
30867 expression for the closing delimiter of a formula. A closing
30868 regular expression to match the above example would be
30869 @code{"\\'\\|\n$\\|\\$\\$?"}. This is almost the same as the
30870 other one, except it now uses @samp{\'} (``end-of-buffer'') and
30871 @samp{\n$} (newline occurring at end of line, yet another way
30872 of describing a blank line that is more appropriate for this
30875 @vindex calc-embedded-open-word
30876 @vindex calc-embedded-close-word
30877 The @code{calc-embedded-open-word} and @code{calc-embedded-close-word}
30878 variables are similar expressions used when you type @kbd{M-# w}
30879 instead of @kbd{M-# e} to enable Embedded mode.
30881 @vindex calc-embedded-open-plain
30882 The @code{calc-embedded-open-plain} variable is a string which
30883 begins a ``plain'' formula written in front of the formatted
30884 formula when @kbd{d p} mode is turned on. Note that this is an
30885 actual string, not a regular expression, because Calc must be able
30886 to write this string into a buffer as well as to recognize it.
30887 The default string is @code{"%%% "} (note the trailing space).
30889 @vindex calc-embedded-close-plain
30890 The @code{calc-embedded-close-plain} variable is a string which
30891 ends a ``plain'' formula. The default is @code{" %%%\n"}. Without
30892 the trailing newline here, the first line of a Big mode formula
30893 that followed might be shifted over with respect to the other lines.
30895 @vindex calc-embedded-open-new-formula
30896 The @code{calc-embedded-open-new-formula} variable is a string
30897 which is inserted at the front of a new formula when you type
30898 @kbd{M-# f}. Its default value is @code{"\n\n"}. If this
30899 string begins with a newline character and the @kbd{M-# f} is
30900 typed at the beginning of a line, @kbd{M-# f} will skip this
30901 first newline to avoid introducing unnecessary blank lines in
30904 @vindex calc-embedded-close-new-formula
30905 The @code{calc-embedded-close-new-formula} variable is the corresponding
30906 string which is inserted at the end of a new formula. Its default
30907 value is also @code{"\n\n"}. The final newline is omitted by
30908 @w{@kbd{M-# f}} if typed at the end of a line. (It follows that if
30909 @kbd{M-# f} is typed on a blank line, both a leading opening
30910 newline and a trailing closing newline are omitted.)
30912 @vindex calc-embedded-announce-formula
30913 The @code{calc-embedded-announce-formula} variable is a regular
30914 expression which is sure to be followed by an embedded formula.
30915 The @kbd{M-# a} command searches for this pattern as well as for
30916 @samp{=>} and @samp{:=} operators. Note that @kbd{M-# a} will
30917 not activate just anything surrounded by formula delimiters; after
30918 all, blank lines are considered formula delimiters by default!
30919 But if your language includes a delimiter which can only occur
30920 actually in front of a formula, you can take advantage of it here.
30921 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which
30922 checks for @samp{%Embed} followed by any number of lines beginning
30923 with @samp{%} and a space. This last is important to make Calc
30924 consider mode annotations part of the pattern, so that the formula's
30925 opening delimiter really is sure to follow the pattern.
30927 @vindex calc-embedded-open-mode
30928 The @code{calc-embedded-open-mode} variable is a string (not a
30929 regular expression) which should precede a mode annotation.
30930 Calc never scans for this string; Calc always looks for the
30931 annotation itself. But this is the string that is inserted before
30932 the opening bracket when Calc adds an annotation on its own.
30933 The default is @code{"% "}.
30935 @vindex calc-embedded-close-mode
30936 The @code{calc-embedded-close-mode} variable is a string which
30937 follows a mode annotation written by Calc. Its default value
30938 is simply a newline, @code{"\n"}. If you change this, it is a
30939 good idea still to end with a newline so that mode annotations
30940 will appear on lines by themselves.
30942 @node Programming, Customizable Variables, Embedded Mode, Top
30943 @chapter Programming
30946 There are several ways to ``program'' the Emacs Calculator, depending
30947 on the nature of the problem you need to solve.
30951 @dfn{Keyboard macros} allow you to record a sequence of keystrokes
30952 and play them back at a later time. This is just the standard Emacs
30953 keyboard macro mechanism, dressed up with a few more features such
30954 as loops and conditionals.
30957 @dfn{Algebraic definitions} allow you to use any formula to define a
30958 new function. This function can then be used in algebraic formulas or
30959 as an interactive command.
30962 @dfn{Rewrite rules} are discussed in the section on algebra commands.
30963 @xref{Rewrite Rules}. If you put your rewrite rules in the variable
30964 @code{EvalRules}, they will be applied automatically to all Calc
30965 results in just the same way as an internal ``rule'' is applied to
30966 evaluate @samp{sqrt(9)} to 3 and so on. @xref{Automatic Rewrites}.
30969 @dfn{Lisp} is the programming language that Calc (and most of Emacs)
30970 is written in. If the above techniques aren't powerful enough, you
30971 can write Lisp functions to do anything that built-in Calc commands
30972 can do. Lisp code is also somewhat faster than keyboard macros or
30977 Programming features are available through the @kbd{z} and @kbd{Z}
30978 prefix keys. New commands that you define are two-key sequences
30979 beginning with @kbd{z}. Commands for managing these definitions
30980 use the shift-@kbd{Z} prefix. (The @kbd{Z T} (@code{calc-timing})
30981 command is described elsewhere; @pxref{Troubleshooting Commands}.
30982 The @kbd{Z C} (@code{calc-user-define-composition}) command is also
30983 described elsewhere; @pxref{User-Defined Compositions}.)
30986 * Creating User Keys::
30987 * Keyboard Macros::
30988 * Invocation Macros::
30989 * Algebraic Definitions::
30990 * Lisp Definitions::
30993 @node Creating User Keys, Keyboard Macros, Programming, Programming
30994 @section Creating User Keys
30998 @pindex calc-user-define
30999 Any Calculator command may be bound to a key using the @kbd{Z D}
31000 (@code{calc-user-define}) command. Actually, it is bound to a two-key
31001 sequence beginning with the lower-case @kbd{z} prefix.
31003 The @kbd{Z D} command first prompts for the key to define. For example,
31004 press @kbd{Z D a} to define the new key sequence @kbd{z a}. You are then
31005 prompted for the name of the Calculator command that this key should
31006 run. For example, the @code{calc-sincos} command is not normally
31007 available on a key. Typing @kbd{Z D s sincos @key{RET}} programs the
31008 @kbd{z s} key sequence to run @code{calc-sincos}. This definition will remain
31009 in effect for the rest of this Emacs session, or until you redefine
31010 @kbd{z s} to be something else.
31012 You can actually bind any Emacs command to a @kbd{z} key sequence by
31013 backspacing over the @samp{calc-} when you are prompted for the command name.
31015 As with any other prefix key, you can type @kbd{z ?} to see a list of
31016 all the two-key sequences you have defined that start with @kbd{z}.
31017 Initially, no @kbd{z} sequences (except @kbd{z ?} itself) are defined.
31019 User keys are typically letters, but may in fact be any key.
31020 (@key{META}-keys are not permitted, nor are a terminal's special
31021 function keys which generate multi-character sequences when pressed.)
31022 You can define different commands on the shifted and unshifted versions
31023 of a letter if you wish.
31026 @pindex calc-user-undefine
31027 The @kbd{Z U} (@code{calc-user-undefine}) command unbinds a user key.
31028 For example, the key sequence @kbd{Z U s} will undefine the @code{sincos}
31029 key we defined above.
31032 @pindex calc-user-define-permanent
31033 @cindex Storing user definitions
31034 @cindex Permanent user definitions
31035 @cindex Calc init file, user-defined commands
31036 The @kbd{Z P} (@code{calc-user-define-permanent}) command makes a key
31037 binding permanent so that it will remain in effect even in future Emacs
31038 sessions. (It does this by adding a suitable bit of Lisp code into
31039 your Calc init file; that is, the file given by the variable
31040 @code{calc-settings-file}, typically @file{~/.calc.el}.) For example,
31041 @kbd{Z P s} would register our @code{sincos} command permanently. If
31042 you later wish to unregister this command you must edit your Calc init
31043 file by hand. (@xref{General Mode Commands}, for a way to tell Calc to
31044 use a different file for the Calc init file.)
31046 The @kbd{Z P} command also saves the user definition, if any, for the
31047 command bound to the key. After @kbd{Z F} and @kbd{Z C}, a given user
31048 key could invoke a command, which in turn calls an algebraic function,
31049 which might have one or more special display formats. A single @kbd{Z P}
31050 command will save all of these definitions.
31051 To save an algebraic function, type @kbd{'} (the apostrophe)
31052 when prompted for a key, and type the function name. To save a command
31053 without its key binding, type @kbd{M-x} and enter a function name. (The
31054 @samp{calc-} prefix will automatically be inserted for you.)
31055 (If the command you give implies a function, the function will be saved,
31056 and if the function has any display formats, those will be saved, but
31057 not the other way around: Saving a function will not save any commands
31058 or key bindings associated with the function.)
31061 @pindex calc-user-define-edit
31062 @cindex Editing user definitions
31063 The @kbd{Z E} (@code{calc-user-define-edit}) command edits the definition
31064 of a user key. This works for keys that have been defined by either
31065 keyboard macros or formulas; further details are contained in the relevant
31066 following sections.
31068 @node Keyboard Macros, Invocation Macros, Creating User Keys, Programming
31069 @section Programming with Keyboard Macros
31073 @cindex Programming with keyboard macros
31074 @cindex Keyboard macros
31075 The easiest way to ``program'' the Emacs Calculator is to use standard
31076 keyboard macros. Press @w{@kbd{C-x (}} to begin recording a macro. From
31077 this point on, keystrokes you type will be saved away as well as
31078 performing their usual functions. Press @kbd{C-x )} to end recording.
31079 Press shift-@kbd{X} (or the standard Emacs key sequence @kbd{C-x e}) to
31080 execute your keyboard macro by replaying the recorded keystrokes.
31081 @xref{Keyboard Macros, , , emacs, the Emacs Manual}, for further
31084 When you use @kbd{X} to invoke a keyboard macro, the entire macro is
31085 treated as a single command by the undo and trail features. The stack
31086 display buffer is not updated during macro execution, but is instead
31087 fixed up once the macro completes. Thus, commands defined with keyboard
31088 macros are convenient and efficient. The @kbd{C-x e} command, on the
31089 other hand, invokes the keyboard macro with no special treatment: Each
31090 command in the macro will record its own undo information and trail entry,
31091 and update the stack buffer accordingly. If your macro uses features
31092 outside of Calc's control to operate on the contents of the Calc stack
31093 buffer, or if it includes Undo, Redo, or last-arguments commands, you
31094 must use @kbd{C-x e} to make sure the buffer and undo list are up-to-date
31095 at all times. You could also consider using @kbd{K} (@code{calc-keep-args})
31096 instead of @kbd{M-@key{RET}} (@code{calc-last-args}).
31098 Calc extends the standard Emacs keyboard macros in several ways.
31099 Keyboard macros can be used to create user-defined commands. Keyboard
31100 macros can include conditional and iteration structures, somewhat
31101 analogous to those provided by a traditional programmable calculator.
31104 * Naming Keyboard Macros::
31105 * Conditionals in Macros::
31106 * Loops in Macros::
31107 * Local Values in Macros::
31108 * Queries in Macros::
31111 @node Naming Keyboard Macros, Conditionals in Macros, Keyboard Macros, Keyboard Macros
31112 @subsection Naming Keyboard Macros
31116 @pindex calc-user-define-kbd-macro
31117 Once you have defined a keyboard macro, you can bind it to a @kbd{z}
31118 key sequence with the @kbd{Z K} (@code{calc-user-define-kbd-macro}) command.
31119 This command prompts first for a key, then for a command name. For
31120 example, if you type @kbd{C-x ( n @key{TAB} n @key{TAB} C-x )} you will
31121 define a keyboard macro which negates the top two numbers on the stack
31122 (@key{TAB} swaps the top two stack elements). Now you can type
31123 @kbd{Z K n @key{RET}} to define this keyboard macro onto the @kbd{z n} key
31124 sequence. The default command name (if you answer the second prompt with
31125 just the @key{RET} key as in this example) will be something like
31126 @samp{calc-User-n}. The keyboard macro will now be available as both
31127 @kbd{z n} and @kbd{M-x calc-User-n}. You can backspace and enter a more
31128 descriptive command name if you wish.
31130 Macros defined by @kbd{Z K} act like single commands; they are executed
31131 in the same way as by the @kbd{X} key. If you wish to define the macro
31132 as a standard no-frills Emacs macro (to be executed as if by @kbd{C-x e}),
31133 give a negative prefix argument to @kbd{Z K}.
31135 Once you have bound your keyboard macro to a key, you can use
31136 @kbd{Z P} to register it permanently with Emacs. @xref{Creating User Keys}.
31138 @cindex Keyboard macros, editing
31139 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
31140 been defined by a keyboard macro tries to use the @code{edmacro} package
31141 edit the macro. Type @kbd{C-c C-c} to finish editing and update
31142 the definition stored on the key, or, to cancel the edit, kill the
31143 buffer with @kbd{C-x k}.
31144 The special characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC},
31145 @code{DEL}, and @code{NUL} must be entered as these three character
31146 sequences, written in all uppercase, as must the prefixes @code{C-} and
31147 @code{M-}. Spaces and line breaks are ignored. Other characters are
31148 copied verbatim into the keyboard macro. Basically, the notation is the
31149 same as is used in all of this manual's examples, except that the manual
31150 takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}},
31151 we take it for granted that it is clear we really mean
31152 @kbd{' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}.
31155 @pindex read-kbd-macro
31156 The @kbd{M-# m} (@code{read-kbd-macro}) command reads an Emacs ``region''
31157 of spelled-out keystrokes and defines it as the current keyboard macro.
31158 It is a convenient way to define a keyboard macro that has been stored
31159 in a file, or to define a macro without executing it at the same time.
31161 @node Conditionals in Macros, Loops in Macros, Naming Keyboard Macros, Keyboard Macros
31162 @subsection Conditionals in Keyboard Macros
31167 @pindex calc-kbd-if
31168 @pindex calc-kbd-else
31169 @pindex calc-kbd-else-if
31170 @pindex calc-kbd-end-if
31171 @cindex Conditional structures
31172 The @kbd{Z [} (@code{calc-kbd-if}) and @kbd{Z ]} (@code{calc-kbd-end-if})
31173 commands allow you to put simple tests in a keyboard macro. When Calc
31174 sees the @kbd{Z [}, it pops an object from the stack and, if the object is
31175 a non-zero value, continues executing keystrokes. But if the object is
31176 zero, or if it is not provably nonzero, Calc skips ahead to the matching
31177 @kbd{Z ]} keystroke. @xref{Logical Operations}, for a set of commands for
31178 performing tests which conveniently produce 1 for true and 0 for false.
31180 For example, @kbd{@key{RET} 0 a < Z [ n Z ]} implements an absolute-value
31181 function in the form of a keyboard macro. This macro duplicates the
31182 number on the top of the stack, pushes zero and compares using @kbd{a <}
31183 (@code{calc-less-than}), then, if the number was less than zero,
31184 executes @kbd{n} (@code{calc-change-sign}). Otherwise, the change-sign
31185 command is skipped.
31187 To program this macro, type @kbd{C-x (}, type the above sequence of
31188 keystrokes, then type @kbd{C-x )}. Note that the keystrokes will be
31189 executed while you are making the definition as well as when you later
31190 re-execute the macro by typing @kbd{X}. Thus you should make sure a
31191 suitable number is on the stack before defining the macro so that you
31192 don't get a stack-underflow error during the definition process.
31194 Conditionals can be nested arbitrarily. However, there should be exactly
31195 one @kbd{Z ]} for each @kbd{Z [} in a keyboard macro.
31198 The @kbd{Z :} (@code{calc-kbd-else}) command allows you to choose between
31199 two keystroke sequences. The general format is @kbd{@var{cond} Z [
31200 @var{then-part} Z : @var{else-part} Z ]}. If @var{cond} is true
31201 (i.e., if the top of stack contains a non-zero number after @var{cond}
31202 has been executed), the @var{then-part} will be executed and the
31203 @var{else-part} will be skipped. Otherwise, the @var{then-part} will
31204 be skipped and the @var{else-part} will be executed.
31207 The @kbd{Z |} (@code{calc-kbd-else-if}) command allows you to choose
31208 between any number of alternatives. For example,
31209 @kbd{@var{cond1} Z [ @var{part1} Z : @var{cond2} Z | @var{part2} Z :
31210 @var{part3} Z ]} will execute @var{part1} if @var{cond1} is true,
31211 otherwise it will execute @var{part2} if @var{cond2} is true, otherwise
31212 it will execute @var{part3}.
31214 More precisely, @kbd{Z [} pops a number and conditionally skips to the
31215 next matching @kbd{Z :} or @kbd{Z ]} key. @w{@kbd{Z ]}} has no effect when
31216 actually executed. @kbd{Z :} skips to the next matching @kbd{Z ]}.
31217 @kbd{Z |} pops a number and conditionally skips to the next matching
31218 @kbd{Z :} or @kbd{Z ]}; thus, @kbd{Z [} and @kbd{Z |} are functionally
31219 equivalent except that @kbd{Z [} participates in nesting but @kbd{Z |}
31222 Calc's conditional and looping constructs work by scanning the
31223 keyboard macro for occurrences of character sequences like @samp{Z:}
31224 and @samp{Z]}. One side-effect of this is that if you use these
31225 constructs you must be careful that these character pairs do not
31226 occur by accident in other parts of the macros. Since Calc rarely
31227 uses shift-@kbd{Z} for any purpose except as a prefix character, this
31228 is not likely to be a problem. Another side-effect is that it will
31229 not work to define your own custom key bindings for these commands.
31230 Only the standard shift-@kbd{Z} bindings will work correctly.
31233 If Calc gets stuck while skipping characters during the definition of a
31234 macro, type @kbd{Z C-g} to cancel the definition. (Typing plain @kbd{C-g}
31235 actually adds a @kbd{C-g} keystroke to the macro.)
31237 @node Loops in Macros, Local Values in Macros, Conditionals in Macros, Keyboard Macros
31238 @subsection Loops in Keyboard Macros
31243 @pindex calc-kbd-repeat
31244 @pindex calc-kbd-end-repeat
31245 @cindex Looping structures
31246 @cindex Iterative structures
31247 The @kbd{Z <} (@code{calc-kbd-repeat}) and @kbd{Z >}
31248 (@code{calc-kbd-end-repeat}) commands pop a number from the stack,
31249 which must be an integer, then repeat the keystrokes between the brackets
31250 the specified number of times. If the integer is zero or negative, the
31251 body is skipped altogether. For example, @kbd{1 @key{TAB} Z < 2 * Z >}
31252 computes two to a nonnegative integer power. First, we push 1 on the
31253 stack and then swap the integer argument back to the top. The @kbd{Z <}
31254 pops that argument leaving the 1 back on top of the stack. Then, we
31255 repeat a multiply-by-two step however many times.
31257 Once again, the keyboard macro is executed as it is being entered.
31258 In this case it is especially important to set up reasonable initial
31259 conditions before making the definition: Suppose the integer 1000 just
31260 happened to be sitting on the stack before we typed the above definition!
31261 Another approach is to enter a harmless dummy definition for the macro,
31262 then go back and edit in the real one with a @kbd{Z E} command. Yet
31263 another approach is to type the macro as written-out keystroke names
31264 in a buffer, then use @kbd{M-# m} (@code{read-kbd-macro}) to read the
31269 The @kbd{Z /} (@code{calc-kbd-break}) command allows you to break out
31270 of a keyboard macro loop prematurely. It pops an object from the stack;
31271 if that object is true (a non-zero number), control jumps out of the
31272 innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues
31273 after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no
31274 effect. Thus @kbd{@var{cond} Z /} is similar to @samp{if (@var{cond}) break;}
31279 @pindex calc-kbd-for
31280 @pindex calc-kbd-end-for
31281 The @kbd{Z (} (@code{calc-kbd-for}) and @kbd{Z )} (@code{calc-kbd-end-for})
31282 commands are similar to @kbd{Z <} and @kbd{Z >}, except that they make the
31283 value of the counter available inside the loop. The general layout is
31284 @kbd{@var{init} @var{final} Z ( @var{body} @var{step} Z )}. The @kbd{Z (}
31285 command pops initial and final values from the stack. It then creates
31286 a temporary internal counter and initializes it with the value @var{init}.
31287 The @kbd{Z (} command then repeatedly pushes the counter value onto the
31288 stack and executes @var{body} and @var{step}, adding @var{step} to the
31289 counter each time until the loop finishes.
31291 @cindex Summations (by keyboard macros)
31292 By default, the loop finishes when the counter becomes greater than (or
31293 less than) @var{final}, assuming @var{initial} is less than (greater
31294 than) @var{final}. If @var{initial} is equal to @var{final}, the body
31295 executes exactly once. The body of the loop always executes at least
31296 once. For example, @kbd{0 1 10 Z ( 2 ^ + 1 Z )} computes the sum of the
31297 squares of the integers from 1 to 10, in steps of 1.
31299 If you give a numeric prefix argument of 1 to @kbd{Z (}, the loop is
31300 forced to use upward-counting conventions. In this case, if @var{initial}
31301 is greater than @var{final} the body will not be executed at all.
31302 Note that @var{step} may still be negative in this loop; the prefix
31303 argument merely constrains the loop-finished test. Likewise, a prefix
31304 argument of @mathit{-1} forces downward-counting conventions.
31308 @pindex calc-kbd-loop
31309 @pindex calc-kbd-end-loop
31310 The @kbd{Z @{} (@code{calc-kbd-loop}) and @kbd{Z @}}
31311 (@code{calc-kbd-end-loop}) commands are similar to @kbd{Z <} and
31312 @kbd{Z >}, except that they do not pop a count from the stack---they
31313 effectively create an infinite loop. Every @kbd{Z @{} @dots{} @kbd{Z @}}
31314 loop ought to include at least one @kbd{Z /} to make sure the loop
31315 doesn't run forever. (If any error message occurs which causes Emacs
31316 to beep, the keyboard macro will also be halted; this is a standard
31317 feature of Emacs. You can also generally press @kbd{C-g} to halt a
31318 running keyboard macro, although not all versions of Unix support
31321 The conditional and looping constructs are not actually tied to
31322 keyboard macros, but they are most often used in that context.
31323 For example, the keystrokes @kbd{10 Z < 23 @key{RET} Z >} push
31324 ten copies of 23 onto the stack. This can be typed ``live'' just
31325 as easily as in a macro definition.
31327 @xref{Conditionals in Macros}, for some additional notes about
31328 conditional and looping commands.
31330 @node Local Values in Macros, Queries in Macros, Loops in Macros, Keyboard Macros
31331 @subsection Local Values in Macros
31334 @cindex Local variables
31335 @cindex Restoring saved modes
31336 Keyboard macros sometimes want to operate under known conditions
31337 without affecting surrounding conditions. For example, a keyboard
31338 macro may wish to turn on Fraction mode, or set a particular
31339 precision, independent of the user's normal setting for those
31344 @pindex calc-kbd-push
31345 @pindex calc-kbd-pop
31346 Macros also sometimes need to use local variables. Assignments to
31347 local variables inside the macro should not affect any variables
31348 outside the macro. The @kbd{Z `} (@code{calc-kbd-push}) and @kbd{Z '}
31349 (@code{calc-kbd-pop}) commands give you both of these capabilities.
31351 When you type @kbd{Z `} (with a backquote or accent grave character),
31352 the values of various mode settings are saved away. The ten ``quick''
31353 variables @code{q0} through @code{q9} are also saved. When
31354 you type @w{@kbd{Z '}} (with an apostrophe), these values are restored.
31355 Pairs of @kbd{Z `} and @kbd{Z '} commands may be nested.
31357 If a keyboard macro halts due to an error in between a @kbd{Z `} and
31358 a @kbd{Z '}, the saved values will be restored correctly even though
31359 the macro never reaches the @kbd{Z '} command. Thus you can use
31360 @kbd{Z `} and @kbd{Z '} without having to worry about what happens
31361 in exceptional conditions.
31363 If you type @kbd{Z `} ``live'' (not in a keyboard macro), Calc puts
31364 you into a ``recursive edit.'' You can tell you are in a recursive
31365 edit because there will be extra square brackets in the mode line,
31366 as in @samp{[(Calculator)]}. These brackets will go away when you
31367 type the matching @kbd{Z '} command. The modes and quick variables
31368 will be saved and restored in just the same way as if actual keyboard
31369 macros were involved.
31371 The modes saved by @kbd{Z `} and @kbd{Z '} are the current precision
31372 and binary word size, the angular mode (Deg, Rad, or HMS), the
31373 simplification mode, Algebraic mode, Symbolic mode, Infinite mode,
31374 Matrix or Scalar mode, Fraction mode, and the current complex mode
31375 (Polar or Rectangular). The ten ``quick'' variables' values (or lack
31376 thereof) are also saved.
31378 Most mode-setting commands act as toggles, but with a numeric prefix
31379 they force the mode either on (positive prefix) or off (negative
31380 or zero prefix). Since you don't know what the environment might
31381 be when you invoke your macro, it's best to use prefix arguments
31382 for all mode-setting commands inside the macro.
31384 In fact, @kbd{C-u Z `} is like @kbd{Z `} except that it sets the modes
31385 listed above to their default values. As usual, the matching @kbd{Z '}
31386 will restore the modes to their settings from before the @kbd{C-u Z `}.
31387 Also, @w{@kbd{Z `}} with a negative prefix argument resets the algebraic mode
31388 to its default (off) but leaves the other modes the same as they were
31389 outside the construct.
31391 The contents of the stack and trail, values of non-quick variables, and
31392 other settings such as the language mode and the various display modes,
31393 are @emph{not} affected by @kbd{Z `} and @kbd{Z '}.
31395 @node Queries in Macros, , Local Values in Macros, Keyboard Macros
31396 @subsection Queries in Keyboard Macros
31400 @c @pindex calc-kbd-report
31401 @c The @kbd{Z =} (@code{calc-kbd-report}) command displays an informative
31402 @c message including the value on the top of the stack. You are prompted
31403 @c to enter a string. That string, along with the top-of-stack value,
31404 @c is displayed unless @kbd{m w} (@code{calc-working}) has been used
31405 @c to turn such messages off.
31409 @pindex calc-kbd-query
31410 The @kbd{Z #} (@code{calc-kbd-query}) command prompts for an algebraic
31411 entry which takes its input from the keyboard, even during macro
31412 execution. All the normal conventions of algebraic input, including the
31413 use of @kbd{$} characters, are supported. The prompt message itself is
31414 taken from the top of the stack, and so must be entered (as a string)
31415 before the @kbd{Z #} command. (Recall, as a string it can be entered by
31416 pressing the @kbd{"} key and will appear as a vector when it is put on
31417 the stack. The prompt message is only put on the stack to provide a
31418 prompt for the @kbd{Z #} command; it will not play any role in any
31419 subsequent calculations.) This command allows your keyboard macros to
31420 accept numbers or formulas as interactive input.
31423 @kbd{2 @key{RET} "Power: " @key{RET} Z # 3 @key{RET}} will prompt for
31424 input with ``Power: '' in the minibuffer, then return 2 to the provided
31425 power. (The response to the prompt that's given, 3 in this example,
31426 will not be part of the macro.)
31428 @xref{Keyboard Macro Query, , , emacs, the Emacs Manual}, for a description of
31429 @kbd{C-x q} (@code{kbd-macro-query}), the standard Emacs way to accept
31430 keyboard input during a keyboard macro. In particular, you can use
31431 @kbd{C-x q} to enter a recursive edit, which allows the user to perform
31432 any Calculator operations interactively before pressing @kbd{C-M-c} to
31433 return control to the keyboard macro.
31435 @node Invocation Macros, Algebraic Definitions, Keyboard Macros, Programming
31436 @section Invocation Macros
31440 @pindex calc-user-invocation
31441 @pindex calc-user-define-invocation
31442 Calc provides one special keyboard macro, called up by @kbd{M-# z}
31443 (@code{calc-user-invocation}), that is intended to allow you to define
31444 your own special way of starting Calc. To define this ``invocation
31445 macro,'' create the macro in the usual way with @kbd{C-x (} and
31446 @kbd{C-x )}, then type @kbd{Z I} (@code{calc-user-define-invocation}).
31447 There is only one invocation macro, so you don't need to type any
31448 additional letters after @kbd{Z I}. From now on, you can type
31449 @kbd{M-# z} at any time to execute your invocation macro.
31451 For example, suppose you find yourself often grabbing rectangles of
31452 numbers into Calc and multiplying their columns. You can do this
31453 by typing @kbd{M-# r} to grab, and @kbd{V R : *} to multiply columns.
31454 To make this into an invocation macro, just type @kbd{C-x ( M-# r
31455 V R : * C-x )}, then @kbd{Z I}. Then, to multiply a rectangle of data,
31456 just mark the data in its buffer in the usual way and type @kbd{M-# z}.
31458 Invocation macros are treated like regular Emacs keyboard macros;
31459 all the special features described above for @kbd{Z K}-style macros
31460 do not apply. @kbd{M-# z} is just like @kbd{C-x e}, except that it
31461 uses the macro that was last stored by @kbd{Z I}. (In fact, the
31462 macro does not even have to have anything to do with Calc!)
31464 The @kbd{m m} command saves the last invocation macro defined by
31465 @kbd{Z I} along with all the other Calc mode settings.
31466 @xref{General Mode Commands}.
31468 @node Algebraic Definitions, Lisp Definitions, Invocation Macros, Programming
31469 @section Programming with Formulas
31473 @pindex calc-user-define-formula
31474 @cindex Programming with algebraic formulas
31475 Another way to create a new Calculator command uses algebraic formulas.
31476 The @kbd{Z F} (@code{calc-user-define-formula}) command stores the
31477 formula at the top of the stack as the definition for a key. This
31478 command prompts for five things: The key, the command name, the function
31479 name, the argument list, and the behavior of the command when given
31480 non-numeric arguments.
31482 For example, suppose we type @kbd{' a+2b @key{RET}} to push the formula
31483 @samp{a + 2*b} onto the stack. We now type @kbd{Z F m} to define this
31484 formula on the @kbd{z m} key sequence. The next prompt is for a command
31485 name, beginning with @samp{calc-}, which should be the long (@kbd{M-x}) form
31486 for the new command. If you simply press @key{RET}, a default name like
31487 @code{calc-User-m} will be constructed. In our example, suppose we enter
31488 @kbd{spam @key{RET}} to define the new command as @code{calc-spam}.
31490 If you want to give the formula a long-style name only, you can press
31491 @key{SPC} or @key{RET} when asked which single key to use. For example
31492 @kbd{Z F @key{RET} spam @key{RET}} defines the new command as
31493 @kbd{M-x calc-spam}, with no keyboard equivalent.
31495 The third prompt is for an algebraic function name. The default is to
31496 use the same name as the command name but without the @samp{calc-}
31497 prefix. (If this is of the form @samp{User-m}, the hyphen is removed so
31498 it won't be taken for a minus sign in algebraic formulas.)
31499 This is the name you will use if you want to enter your
31500 new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}.
31501 Then the new function can be invoked by pushing two numbers on the
31502 stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic
31503 formula @samp{yow(x,y)}.
31505 The fourth prompt is for the function's argument list. This is used to
31506 associate values on the stack with the variables that appear in the formula.
31507 The default is a list of all variables which appear in the formula, sorted
31508 into alphabetical order. In our case, the default would be @samp{(a b)}.
31509 This means that, when the user types @kbd{z m}, the Calculator will remove
31510 two numbers from the stack, substitute these numbers for @samp{a} and
31511 @samp{b} (respectively) in the formula, then simplify the formula and
31512 push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m}
31513 would replace the 10 and 100 on the stack with the number 210, which is
31514 @expr{a + 2 b} with @expr{a=10} and @expr{b=100}. Likewise, the formula
31515 @samp{yow(10, 100)} will be evaluated by substituting @expr{a=10} and
31516 @expr{b=100} in the definition.
31518 You can rearrange the order of the names before pressing @key{RET} to
31519 control which stack positions go to which variables in the formula. If
31520 you remove a variable from the argument list, that variable will be left
31521 in symbolic form by the command. Thus using an argument list of @samp{(b)}
31522 for our function would cause @kbd{10 z m} to replace the 10 on the stack
31523 with the formula @samp{a + 20}. If we had used an argument list of
31524 @samp{(b a)}, the result with inputs 10 and 100 would have been 120.
31526 You can also put a nameless function on the stack instead of just a
31527 formula, as in @samp{<a, b : a + 2 b>}. @xref{Specifying Operators}.
31528 In this example, the command will be defined by the formula @samp{a + 2 b}
31529 using the argument list @samp{(a b)}.
31531 The final prompt is a y-or-n question concerning what to do if symbolic
31532 arguments are given to your function. If you answer @kbd{y}, then
31533 executing @kbd{z m} (using the original argument list @samp{(a b)}) with
31534 arguments @expr{10} and @expr{x} will leave the function in symbolic
31535 form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n},
31536 then the formula will always be expanded, even for non-constant
31537 arguments: @samp{10 + 2 x}. If you never plan to feed algebraic
31538 formulas to your new function, it doesn't matter how you answer this
31541 If you answered @kbd{y} to this question you can still cause a function
31542 call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}).
31543 Also, Calc will expand the function if necessary when you take a
31544 derivative or integral or solve an equation involving the function.
31547 @pindex calc-get-user-defn
31548 Once you have defined a formula on a key, you can retrieve this formula
31549 with the @kbd{Z G} (@code{calc-user-define-get-defn}) command. Press a
31550 key, and this command pushes the formula that was used to define that
31551 key onto the stack. Actually, it pushes a nameless function that
31552 specifies both the argument list and the defining formula. You will get
31553 an error message if the key is undefined, or if the key was not defined
31554 by a @kbd{Z F} command.
31556 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
31557 been defined by a formula uses a variant of the @code{calc-edit} command
31558 to edit the defining formula. Press @kbd{C-c C-c} to finish editing and
31559 store the new formula back in the definition, or kill the buffer with
31561 cancel the edit. (The argument list and other properties of the
31562 definition are unchanged; to adjust the argument list, you can use
31563 @kbd{Z G} to grab the function onto the stack, edit with @kbd{`}, and
31564 then re-execute the @kbd{Z F} command.)
31566 As usual, the @kbd{Z P} command records your definition permanently.
31567 In this case it will permanently record all three of the relevant
31568 definitions: the key, the command, and the function.
31570 You may find it useful to turn off the default simplifications with
31571 @kbd{m O} (@code{calc-no-simplify-mode}) when entering a formula to be
31572 used as a function definition. For example, the formula @samp{deriv(a^2,v)}
31573 which might be used to define a new function @samp{dsqr(a,v)} will be
31574 ``simplified'' to 0 immediately upon entry since @code{deriv} considers
31575 @expr{a} to be constant with respect to @expr{v}. Turning off
31576 default simplifications cures this problem: The definition will be stored
31577 in symbolic form without ever activating the @code{deriv} function. Press
31578 @kbd{m D} to turn the default simplifications back on afterwards.
31580 @node Lisp Definitions, , Algebraic Definitions, Programming
31581 @section Programming with Lisp
31584 The Calculator can be programmed quite extensively in Lisp. All you
31585 do is write a normal Lisp function definition, but with @code{defmath}
31586 in place of @code{defun}. This has the same form as @code{defun}, but it
31587 automagically replaces calls to standard Lisp functions like @code{+} and
31588 @code{zerop} with calls to the corresponding functions in Calc's own library.
31589 Thus you can write natural-looking Lisp code which operates on all of the
31590 standard Calculator data types. You can then use @kbd{Z D} if you wish to
31591 bind your new command to a @kbd{z}-prefix key sequence. The @kbd{Z E} command
31592 will not edit a Lisp-based definition.
31594 Emacs Lisp is described in the GNU Emacs Lisp Reference Manual. This section
31595 assumes a familiarity with Lisp programming concepts; if you do not know
31596 Lisp, you may find keyboard macros or rewrite rules to be an easier way
31597 to program the Calculator.
31599 This section first discusses ways to write commands, functions, or
31600 small programs to be executed inside of Calc. Then it discusses how
31601 your own separate programs are able to call Calc from the outside.
31602 Finally, there is a list of internal Calc functions and data structures
31603 for the true Lisp enthusiast.
31606 * Defining Functions::
31607 * Defining Simple Commands::
31608 * Defining Stack Commands::
31609 * Argument Qualifiers::
31610 * Example Definitions::
31612 * Calling Calc from Your Programs::
31616 @node Defining Functions, Defining Simple Commands, Lisp Definitions, Lisp Definitions
31617 @subsection Defining New Functions
31621 The @code{defmath} function (actually a Lisp macro) is like @code{defun}
31622 except that code in the body of the definition can make use of the full
31623 range of Calculator data types. The prefix @samp{calcFunc-} is added
31624 to the specified name to get the actual Lisp function name. As a simple
31628 (defmath myfact (n)
31630 (* n (myfact (1- n)))
31635 This actually expands to the code,
31638 (defun calcFunc-myfact (n)
31640 (math-mul n (calcFunc-myfact (math-add n -1)))
31645 This function can be used in algebraic expressions, e.g., @samp{myfact(5)}.
31647 The @samp{myfact} function as it is defined above has the bug that an
31648 expression @samp{myfact(a+b)} will be simplified to 1 because the
31649 formula @samp{a+b} is not considered to be @code{posp}. A robust
31650 factorial function would be written along the following lines:
31653 (defmath myfact (n)
31655 (* n (myfact (1- n)))
31658 nil))) ; this could be simplified as: (and (= n 0) 1)
31661 If a function returns @code{nil}, it is left unsimplified by the Calculator
31662 (except that its arguments will be simplified). Thus, @samp{myfact(a+1+2)}
31663 will be simplified to @samp{myfact(a+3)} but no further. Beware that every
31664 time the Calculator reexamines this formula it will attempt to resimplify
31665 it, so your function ought to detect the returning-@code{nil} case as
31666 efficiently as possible.
31668 The following standard Lisp functions are treated by @code{defmath}:
31669 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^} or
31670 @code{expt}, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=},
31671 @code{/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior}, @code{logxor},
31672 @code{logandc2}, @code{lognot}. Also, @code{~=} is an abbreviation for
31673 @code{math-nearly-equal}, which is useful in implementing Taylor series.
31675 For other functions @var{func}, if a function by the name
31676 @samp{calcFunc-@var{func}} exists it is used, otherwise if a function by the
31677 name @samp{math-@var{func}} exists it is used, otherwise if @var{func} itself
31678 is defined as a function it is used, otherwise @samp{calcFunc-@var{func}} is
31679 used on the assumption that this is a to-be-defined math function. Also, if
31680 the function name is quoted as in @samp{('integerp a)} the function name is
31681 always used exactly as written (but not quoted).
31683 Variable names have @samp{var-} prepended to them unless they appear in
31684 the function's argument list or in an enclosing @code{let}, @code{let*},
31685 @code{for}, or @code{foreach} form,
31686 or their names already contain a @samp{-} character. Thus a reference to
31687 @samp{foo} is the same as a reference to @samp{var-foo}.
31689 A few other Lisp extensions are available in @code{defmath} definitions:
31693 The @code{elt} function accepts any number of index variables.
31694 Note that Calc vectors are stored as Lisp lists whose first
31695 element is the symbol @code{vec}; thus, @samp{(elt v 2)} yields
31696 the second element of vector @code{v}, and @samp{(elt m i j)}
31697 yields one element of a Calc matrix.
31700 The @code{setq} function has been extended to act like the Common
31701 Lisp @code{setf} function. (The name @code{setf} is recognized as
31702 a synonym of @code{setq}.) Specifically, the first argument of
31703 @code{setq} can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form,
31704 in which case the effect is to store into the specified
31705 element of a list. Thus, @samp{(setq (elt m i j) x)} stores @expr{x}
31706 into one element of a matrix.
31709 A @code{for} looping construct is available. For example,
31710 @samp{(for ((i 0 10)) body)} executes @code{body} once for each
31711 binding of @expr{i} from zero to 10. This is like a @code{let}
31712 form in that @expr{i} is temporarily bound to the loop count
31713 without disturbing its value outside the @code{for} construct.
31714 Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)},
31715 are also available. For each value of @expr{i} from zero to 10,
31716 @expr{j} counts from 0 to @expr{i-1} in steps of two. Note that
31717 @code{for} has the same general outline as @code{let*}, except
31718 that each element of the header is a list of three or four
31719 things, not just two.
31722 The @code{foreach} construct loops over elements of a list.
31723 For example, @samp{(foreach ((x (cdr v))) body)} executes
31724 @code{body} with @expr{x} bound to each element of Calc vector
31725 @expr{v} in turn. The purpose of @code{cdr} here is to skip over
31726 the initial @code{vec} symbol in the vector.
31729 The @code{break} function breaks out of the innermost enclosing
31730 @code{while}, @code{for}, or @code{foreach} loop. If given a
31731 value, as in @samp{(break x)}, this value is returned by the
31732 loop. (Lisp loops otherwise always return @code{nil}.)
31735 The @code{return} function prematurely returns from the enclosing
31736 function. For example, @samp{(return (+ x y))} returns @expr{x+y}
31737 as the value of a function. You can use @code{return} anywhere
31738 inside the body of the function.
31741 Non-integer numbers (and extremely large integers) cannot be included
31742 directly into a @code{defmath} definition. This is because the Lisp
31743 reader will fail to parse them long before @code{defmath} ever gets control.
31744 Instead, use the notation, @samp{:"3.1415"}. In fact, any algebraic
31745 formula can go between the quotes. For example,
31748 (defmath sqexp (x) ; sqexp(x) == sqrt(exp(x)) == exp(x*0.5)
31756 (defun calcFunc-sqexp (x)
31757 (and (math-numberp x)
31758 (calcFunc-exp (math-mul x '(float 5 -1)))))
31761 Note the use of @code{numberp} as a guard to ensure that the argument is
31762 a number first, returning @code{nil} if not. The exponential function
31763 could itself have been included in the expression, if we had preferred:
31764 @samp{:"exp(x * 0.5)"}. As another example, the multiplication-and-recursion
31765 step of @code{myfact} could have been written
31771 A good place to put your @code{defmath} commands is your Calc init file
31772 (the file given by @code{calc-settings-file}, typically
31773 @file{~/.calc.el}), which will not be loaded until Calc starts.
31774 If a file named @file{.emacs} exists in your home directory, Emacs reads
31775 and executes the Lisp forms in this file as it starts up. While it may
31776 seem reasonable to put your favorite @code{defmath} commands there,
31777 this has the unfortunate side-effect that parts of the Calculator must be
31778 loaded in to process the @code{defmath} commands whether or not you will
31779 actually use the Calculator! If you want to put the @code{defmath}
31780 commands there (for example, if you redefine @code{calc-settings-file}
31781 to be @file{.emacs}), a better effect can be had by writing
31784 (put 'calc-define 'thing '(progn
31791 @vindex calc-define
31792 The @code{put} function adds a @dfn{property} to a symbol. Each Lisp
31793 symbol has a list of properties associated with it. Here we add a
31794 property with a name of @code{thing} and a @samp{(progn ...)} form as
31795 its value. When Calc starts up, and at the start of every Calc command,
31796 the property list for the symbol @code{calc-define} is checked and the
31797 values of any properties found are evaluated as Lisp forms. The
31798 properties are removed as they are evaluated. The property names
31799 (like @code{thing}) are not used; you should choose something like the
31800 name of your project so as not to conflict with other properties.
31802 The net effect is that you can put the above code in your @file{.emacs}
31803 file and it will not be executed until Calc is loaded. Or, you can put
31804 that same code in another file which you load by hand either before or
31805 after Calc itself is loaded.
31807 The properties of @code{calc-define} are evaluated in the same order
31808 that they were added. They can assume that the Calc modules @file{calc.el},
31809 @file{calc-ext.el}, and @file{calc-macs.el} have been fully loaded, and
31810 that the @samp{*Calculator*} buffer will be the current buffer.
31812 If your @code{calc-define} property only defines algebraic functions,
31813 you can be sure that it will have been evaluated before Calc tries to
31814 call your function, even if the file defining the property is loaded
31815 after Calc is loaded. But if the property defines commands or key
31816 sequences, it may not be evaluated soon enough. (Suppose it defines the
31817 new command @code{tweak-calc}; the user can load your file, then type
31818 @kbd{M-x tweak-calc} before Calc has had chance to do anything.) To
31819 protect against this situation, you can put
31822 (run-hooks 'calc-check-defines)
31825 @findex calc-check-defines
31827 at the end of your file. The @code{calc-check-defines} function is what
31828 looks for and evaluates properties on @code{calc-define}; @code{run-hooks}
31829 has the advantage that it is quietly ignored if @code{calc-check-defines}
31830 is not yet defined because Calc has not yet been loaded.
31832 Examples of things that ought to be enclosed in a @code{calc-define}
31833 property are @code{defmath} calls, @code{define-key} calls that modify
31834 the Calc key map, and any calls that redefine things defined inside Calc.
31835 Ordinary @code{defun}s need not be enclosed with @code{calc-define}.
31837 @node Defining Simple Commands, Defining Stack Commands, Defining Functions, Lisp Definitions
31838 @subsection Defining New Simple Commands
31841 @findex interactive
31842 If a @code{defmath} form contains an @code{interactive} clause, it defines
31843 a Calculator command. Actually such a @code{defmath} results in @emph{two}
31844 function definitions: One, a @samp{calcFunc-} function as was just described,
31845 with the @code{interactive} clause removed. Two, a @samp{calc-} function
31846 with a suitable @code{interactive} clause and some sort of wrapper to make
31847 the command work in the Calc environment.
31849 In the simple case, the @code{interactive} clause has the same form as
31850 for normal Emacs Lisp commands:
31853 (defmath increase-precision (delta)
31854 "Increase precision by DELTA." ; This is the "documentation string"
31855 (interactive "p") ; Register this as a M-x-able command
31856 (setq calc-internal-prec (+ calc-internal-prec delta)))
31859 This expands to the pair of definitions,
31862 (defun calc-increase-precision (delta)
31863 "Increase precision by DELTA."
31866 (setq calc-internal-prec (math-add calc-internal-prec delta))))
31868 (defun calcFunc-increase-precision (delta)
31869 "Increase precision by DELTA."
31870 (setq calc-internal-prec (math-add calc-internal-prec delta)))
31874 where in this case the latter function would never really be used! Note
31875 that since the Calculator stores small integers as plain Lisp integers,
31876 the @code{math-add} function will work just as well as the native
31877 @code{+} even when the intent is to operate on native Lisp integers.
31879 @findex calc-wrapper
31880 The @samp{calc-wrapper} call invokes a macro which surrounds the body of
31881 the function with code that looks roughly like this:
31884 (let ((calc-command-flags nil))
31887 (calc-select-buffer)
31888 @emph{body of function}
31889 @emph{renumber stack}
31890 @emph{clear} Working @emph{message})
31891 @emph{realign cursor and window}
31892 @emph{clear Inverse, Hyperbolic, and Keep Args flags}
31893 @emph{update Emacs mode line}))
31896 @findex calc-select-buffer
31897 The @code{calc-select-buffer} function selects the @samp{*Calculator*}
31898 buffer if necessary, say, because the command was invoked from inside
31899 the @samp{*Calc Trail*} window.
31901 @findex calc-set-command-flag
31902 You can call, for example, @code{(calc-set-command-flag 'no-align)} to
31903 set the above-mentioned command flags. Calc routines recognize the
31904 following command flags:
31908 Stack line numbers @samp{1:}, @samp{2:}, and so on must be renumbered
31909 after this command completes. This is set by routines like
31912 @item clear-message
31913 Calc should call @samp{(message "")} if this command completes normally
31914 (to clear a ``Working@dots{}'' message out of the echo area).
31917 Do not move the cursor back to the @samp{.} top-of-stack marker.
31919 @item position-point
31920 Use the variables @code{calc-position-point-line} and
31921 @code{calc-position-point-column} to position the cursor after
31922 this command finishes.
31925 Do not clear @code{calc-inverse-flag}, @code{calc-hyperbolic-flag},
31926 and @code{calc-keep-args-flag} at the end of this command.
31929 Switch to buffer @samp{*Calc Edit*} after this command.
31932 Do not move trail pointer to end of trail when something is recorded
31938 @vindex calc-Y-help-msgs
31939 Calc reserves a special prefix key, shift-@kbd{Y}, for user-written
31940 extensions to Calc. There are no built-in commands that work with
31941 this prefix key; you must call @code{define-key} from Lisp (probably
31942 from inside a @code{calc-define} property) to add to it. Initially only
31943 @kbd{Y ?} is defined; it takes help messages from a list of strings
31944 (initially @code{nil}) in the variable @code{calc-Y-help-msgs}. All
31945 other undefined keys except for @kbd{Y} are reserved for use by
31946 future versions of Calc.
31948 If you are writing a Calc enhancement which you expect to give to
31949 others, it is best to minimize the number of @kbd{Y}-key sequences
31950 you use. In fact, if you have more than one key sequence you should
31951 consider defining three-key sequences with a @kbd{Y}, then a key that
31952 stands for your package, then a third key for the particular command
31953 within your package.
31955 Users may wish to install several Calc enhancements, and it is possible
31956 that several enhancements will choose to use the same key. In the
31957 example below, a variable @code{inc-prec-base-key} has been defined
31958 to contain the key that identifies the @code{inc-prec} package. Its
31959 value is initially @code{"P"}, but a user can change this variable
31960 if necessary without having to modify the file.
31962 Here is a complete file, @file{inc-prec.el}, which makes a @kbd{Y P I}
31963 command that increases the precision, and a @kbd{Y P D} command that
31964 decreases the precision.
31967 ;;; Increase and decrease Calc precision. Dave Gillespie, 5/31/91.
31968 ;;; (Include copyright or copyleft stuff here.)
31970 (defvar inc-prec-base-key "P"
31971 "Base key for inc-prec.el commands.")
31973 (put 'calc-define 'inc-prec '(progn
31975 (define-key calc-mode-map (format "Y%sI" inc-prec-base-key)
31976 'increase-precision)
31977 (define-key calc-mode-map (format "Y%sD" inc-prec-base-key)
31978 'decrease-precision)
31980 (setq calc-Y-help-msgs
31981 (cons (format "%s + Inc-prec, Dec-prec" inc-prec-base-key)
31984 (defmath increase-precision (delta)
31985 "Increase precision by DELTA."
31987 (setq calc-internal-prec (+ calc-internal-prec delta)))
31989 (defmath decrease-precision (delta)
31990 "Decrease precision by DELTA."
31992 (setq calc-internal-prec (- calc-internal-prec delta)))
31994 )) ; end of calc-define property
31996 (run-hooks 'calc-check-defines)
31999 @node Defining Stack Commands, Argument Qualifiers, Defining Simple Commands, Lisp Definitions
32000 @subsection Defining New Stack-Based Commands
32003 To define a new computational command which takes and/or leaves arguments
32004 on the stack, a special form of @code{interactive} clause is used.
32007 (interactive @var{num} @var{tag})
32011 where @var{num} is an integer, and @var{tag} is a string. The effect is
32012 to pop @var{num} values off the stack, resimplify them by calling
32013 @code{calc-normalize}, and hand them to your function according to the
32014 function's argument list. Your function may include @code{&optional} and
32015 @code{&rest} parameters, so long as calling the function with @var{num}
32016 parameters is valid.
32018 Your function must return either a number or a formula in a form
32019 acceptable to Calc, or a list of such numbers or formulas. These value(s)
32020 are pushed onto the stack when the function completes. They are also
32021 recorded in the Calc Trail buffer on a line beginning with @var{tag},
32022 a string of (normally) four characters or less. If you omit @var{tag}
32023 or use @code{nil} as a tag, the result is not recorded in the trail.
32025 As an example, the definition
32028 (defmath myfact (n)
32029 "Compute the factorial of the integer at the top of the stack."
32030 (interactive 1 "fact")
32032 (* n (myfact (1- n)))
32037 is a version of the factorial function shown previously which can be used
32038 as a command as well as an algebraic function. It expands to
32041 (defun calc-myfact ()
32042 "Compute the factorial of the integer at the top of the stack."
32045 (calc-enter-result 1 "fact"
32046 (cons 'calcFunc-myfact (calc-top-list-n 1)))))
32048 (defun calcFunc-myfact (n)
32049 "Compute the factorial of the integer at the top of the stack."
32051 (math-mul n (calcFunc-myfact (math-add n -1)))
32052 (and (math-zerop n) 1)))
32055 @findex calc-slow-wrapper
32056 The @code{calc-slow-wrapper} function is a version of @code{calc-wrapper}
32057 that automatically puts up a @samp{Working...} message before the
32058 computation begins. (This message can be turned off by the user
32059 with an @kbd{m w} (@code{calc-working}) command.)
32061 @findex calc-top-list-n
32062 The @code{calc-top-list-n} function returns a list of the specified number
32063 of values from the top of the stack. It resimplifies each value by
32064 calling @code{calc-normalize}. If its argument is zero it returns an
32065 empty list. It does not actually remove these values from the stack.
32067 @findex calc-enter-result
32068 The @code{calc-enter-result} function takes an integer @var{num} and string
32069 @var{tag} as described above, plus a third argument which is either a
32070 Calculator data object or a list of such objects. These objects are
32071 resimplified and pushed onto the stack after popping the specified number
32072 of values from the stack. If @var{tag} is non-@code{nil}, the values
32073 being pushed are also recorded in the trail.
32075 Note that if @code{calcFunc-myfact} returns @code{nil} this represents
32076 ``leave the function in symbolic form.'' To return an actual empty list,
32077 in the sense that @code{calc-enter-result} will push zero elements back
32078 onto the stack, you should return the special value @samp{'(nil)}, a list
32079 containing the single symbol @code{nil}.
32081 The @code{interactive} declaration can actually contain a limited
32082 Emacs-style code string as well which comes just before @var{num} and
32083 @var{tag}. Currently the only Emacs code supported is @samp{"p"}, as in
32086 (defmath foo (a b &optional c)
32087 (interactive "p" 2 "foo")
32091 In this example, the command @code{calc-foo} will evaluate the expression
32092 @samp{foo(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if
32093 executed with a numeric prefix argument of @expr{n}.
32095 The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"}
32096 code as used with @code{defun}). It uses the numeric prefix argument as the
32097 number of objects to remove from the stack and pass to the function.
32098 In this case, the integer @var{num} serves as a default number of
32099 arguments to be used when no prefix is supplied.
32101 @node Argument Qualifiers, Example Definitions, Defining Stack Commands, Lisp Definitions
32102 @subsection Argument Qualifiers
32105 Anywhere a parameter name can appear in the parameter list you can also use
32106 an @dfn{argument qualifier}. Thus the general form of a definition is:
32109 (defmath @var{name} (@var{param} @var{param...}
32110 &optional @var{param} @var{param...}
32116 where each @var{param} is either a symbol or a list of the form
32119 (@var{qual} @var{param})
32122 The following qualifiers are recognized:
32127 The argument must not be an incomplete vector, interval, or complex number.
32128 (This is rarely needed since the Calculator itself will never call your
32129 function with an incomplete argument. But there is nothing stopping your
32130 own Lisp code from calling your function with an incomplete argument.)
32134 The argument must be an integer. If it is an integer-valued float
32135 it will be accepted but converted to integer form. Non-integers and
32136 formulas are rejected.
32140 Like @samp{integer}, but the argument must be non-negative.
32144 Like @samp{integer}, but the argument must fit into a native Lisp integer,
32145 which on most systems means less than 2^23 in absolute value. The
32146 argument is converted into Lisp-integer form if necessary.
32150 The argument is converted to floating-point format if it is a number or
32151 vector. If it is a formula it is left alone. (The argument is never
32152 actually rejected by this qualifier.)
32155 The argument must satisfy predicate @var{pred}, which is one of the
32156 standard Calculator predicates. @xref{Predicates}.
32158 @item not-@var{pred}
32159 The argument must @emph{not} satisfy predicate @var{pred}.
32165 (defmath foo (a (constp (not-matrixp b)) &optional (float c)
32174 (defun calcFunc-foo (a b &optional c &rest d)
32175 (and (math-matrixp b)
32176 (math-reject-arg b 'not-matrixp))
32177 (or (math-constp b)
32178 (math-reject-arg b 'constp))
32179 (and c (setq c (math-check-float c)))
32180 (setq d (mapcar 'math-check-integer d))
32185 which performs the necessary checks and conversions before executing the
32186 body of the function.
32188 @node Example Definitions, Calling Calc from Your Programs, Argument Qualifiers, Lisp Definitions
32189 @subsection Example Definitions
32192 This section includes some Lisp programming examples on a larger scale.
32193 These programs make use of some of the Calculator's internal functions;
32197 * Bit Counting Example::
32201 @node Bit Counting Example, Sine Example, Example Definitions, Example Definitions
32202 @subsubsection Bit-Counting
32209 Calc does not include a built-in function for counting the number of
32210 ``one'' bits in a binary integer. It's easy to invent one using @kbd{b u}
32211 to convert the integer to a set, and @kbd{V #} to count the elements of
32212 that set; let's write a function that counts the bits without having to
32213 create an intermediate set.
32216 (defmath bcount ((natnum n))
32217 (interactive 1 "bcnt")
32221 (setq count (1+ count)))
32222 (setq n (lsh n -1)))
32227 When this is expanded by @code{defmath}, it will become the following
32228 Emacs Lisp function:
32231 (defun calcFunc-bcount (n)
32232 (setq n (math-check-natnum n))
32234 (while (math-posp n)
32236 (setq count (math-add count 1)))
32237 (setq n (calcFunc-lsh n -1)))
32241 If the input numbers are large, this function involves a fair amount
32242 of arithmetic. A binary right shift is essentially a division by two;
32243 recall that Calc stores integers in decimal form so bit shifts must
32244 involve actual division.
32246 To gain a bit more efficiency, we could divide the integer into
32247 @var{n}-bit chunks, each of which can be handled quickly because
32248 they fit into Lisp integers. It turns out that Calc's arithmetic
32249 routines are especially fast when dividing by an integer less than
32250 1000, so we can set @var{n = 9} bits and use repeated division by 512:
32253 (defmath bcount ((natnum n))
32254 (interactive 1 "bcnt")
32256 (while (not (fixnump n))
32257 (let ((qr (idivmod n 512)))
32258 (setq count (+ count (bcount-fixnum (cdr qr)))
32260 (+ count (bcount-fixnum n))))
32262 (defun bcount-fixnum (n)
32265 (setq count (+ count (logand n 1))
32271 Note that the second function uses @code{defun}, not @code{defmath}.
32272 Because this function deals only with native Lisp integers (``fixnums''),
32273 it can use the actual Emacs @code{+} and related functions rather
32274 than the slower but more general Calc equivalents which @code{defmath}
32277 The @code{idivmod} function does an integer division, returning both
32278 the quotient and the remainder at once. Again, note that while it
32279 might seem that @samp{(logand n 511)} and @samp{(lsh n -9)} are
32280 more efficient ways to split off the bottom nine bits of @code{n},
32281 actually they are less efficient because each operation is really
32282 a division by 512 in disguise; @code{idivmod} allows us to do the
32283 same thing with a single division by 512.
32285 @node Sine Example, , Bit Counting Example, Example Definitions
32286 @subsubsection The Sine Function
32293 A somewhat limited sine function could be defined as follows, using the
32294 well-known Taylor series expansion for
32295 @texline @math{\sin x}:
32296 @infoline @samp{sin(x)}:
32299 (defmath mysin ((float (anglep x)))
32300 (interactive 1 "mysn")
32301 (setq x (to-radians x)) ; Convert from current angular mode.
32302 (let ((sum x) ; Initial term of Taylor expansion of sin.
32304 (nfact 1) ; "nfact" equals "n" factorial at all times.
32305 (xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2.
32306 (for ((n 3 100 2)) ; Upper limit of 100 is a good precaution.
32307 (working "mysin" sum) ; Display "Working" message, if enabled.
32308 (setq nfact (* nfact (1- n) n)
32310 newsum (+ sum (/ x nfact)))
32311 (if (~= newsum sum) ; If newsum is "nearly equal to" sum,
32312 (break)) ; then we are done.
32317 The actual @code{sin} function in Calc works by first reducing the problem
32318 to a sine or cosine of a nonnegative number less than @cpiover{4}. This
32319 ensures that the Taylor series will converge quickly. Also, the calculation
32320 is carried out with two extra digits of precision to guard against cumulative
32321 round-off in @samp{sum}. Finally, complex arguments are allowed and handled
32322 by a separate algorithm.
32325 (defmath mysin ((float (scalarp x)))
32326 (interactive 1 "mysn")
32327 (setq x (to-radians x)) ; Convert from current angular mode.
32328 (with-extra-prec 2 ; Evaluate with extra precision.
32329 (cond ((complexp x)
32332 (- (mysin-raw (- x))) ; Always call mysin-raw with x >= 0.
32333 (t (mysin-raw x))))))
32335 (defmath mysin-raw (x)
32337 (mysin-raw (% x (two-pi)))) ; Now x < 7.
32339 (- (mysin-raw (- x (pi))))) ; Now -pi/2 <= x <= pi/2.
32341 (mycos-raw (- x (pi-over-2)))) ; Now -pi/2 <= x <= pi/4.
32342 ((< x (- (pi-over-4)))
32343 (- (mycos-raw (+ x (pi-over-2))))) ; Now -pi/4 <= x <= pi/4,
32344 (t (mysin-series x)))) ; so the series will be efficient.
32348 where @code{mysin-complex} is an appropriate function to handle complex
32349 numbers, @code{mysin-series} is the routine to compute the sine Taylor
32350 series as before, and @code{mycos-raw} is a function analogous to
32351 @code{mysin-raw} for cosines.
32353 The strategy is to ensure that @expr{x} is nonnegative before calling
32354 @code{mysin-raw}. This function then recursively reduces its argument
32355 to a suitable range, namely, plus-or-minus @cpiover{4}. Note that each
32356 test, and particularly the first comparison against 7, is designed so
32357 that small roundoff errors cannot produce an infinite loop. (Suppose
32358 we compared with @samp{(two-pi)} instead; if due to roundoff problems
32359 the modulo operator ever returned @samp{(two-pi)} exactly, an infinite
32360 recursion could result!) We use modulo only for arguments that will
32361 clearly get reduced, knowing that the next rule will catch any reductions
32362 that this rule misses.
32364 If a program is being written for general use, it is important to code
32365 it carefully as shown in this second example. For quick-and-dirty programs,
32366 when you know that your own use of the sine function will never encounter
32367 a large argument, a simpler program like the first one shown is fine.
32369 @node Calling Calc from Your Programs, Internals, Example Definitions, Lisp Definitions
32370 @subsection Calling Calc from Your Lisp Programs
32373 A later section (@pxref{Internals}) gives a full description of
32374 Calc's internal Lisp functions. It's not hard to call Calc from
32375 inside your programs, but the number of these functions can be daunting.
32376 So Calc provides one special ``programmer-friendly'' function called
32377 @code{calc-eval} that can be made to do just about everything you
32378 need. It's not as fast as the low-level Calc functions, but it's
32379 much simpler to use!
32381 It may seem that @code{calc-eval} itself has a daunting number of
32382 options, but they all stem from one simple operation.
32384 In its simplest manifestation, @samp{(calc-eval "1+2")} parses the
32385 string @code{"1+2"} as if it were a Calc algebraic entry and returns
32386 the result formatted as a string: @code{"3"}.
32388 Since @code{calc-eval} is on the list of recommended @code{autoload}
32389 functions, you don't need to make any special preparations to load
32390 Calc before calling @code{calc-eval} the first time. Calc will be
32391 loaded and initialized for you.
32393 All the Calc modes that are currently in effect will be used when
32394 evaluating the expression and formatting the result.
32401 @subsubsection Additional Arguments to @code{calc-eval}
32404 If the input string parses to a list of expressions, Calc returns
32405 the results separated by @code{", "}. You can specify a different
32406 separator by giving a second string argument to @code{calc-eval}:
32407 @samp{(calc-eval "1+2,3+4" ";")} returns @code{"3;7"}.
32409 The ``separator'' can also be any of several Lisp symbols which
32410 request other behaviors from @code{calc-eval}. These are discussed
32413 You can give additional arguments to be substituted for
32414 @samp{$}, @samp{$$}, and so on in the main expression. For
32415 example, @samp{(calc-eval "$/$$" nil "7" "1+1")} evaluates the
32416 expression @code{"7/(1+1)"} to yield the result @code{"3.5"}
32417 (assuming Fraction mode is not in effect). Note the @code{nil}
32418 used as a placeholder for the item-separator argument.
32425 @subsubsection Error Handling
32428 If @code{calc-eval} encounters an error, it returns a list containing
32429 the character position of the error, plus a suitable message as a
32430 string. Note that @samp{1 / 0} is @emph{not} an error by Calc's
32431 standards; it simply returns the string @code{"1 / 0"} which is the
32432 division left in symbolic form. But @samp{(calc-eval "1/")} will
32433 return the list @samp{(2 "Expected a number")}.
32435 If you bind the variable @code{calc-eval-error} to @code{t}
32436 using a @code{let} form surrounding the call to @code{calc-eval},
32437 errors instead call the Emacs @code{error} function which aborts
32438 to the Emacs command loop with a beep and an error message.
32440 If you bind this variable to the symbol @code{string}, error messages
32441 are returned as strings instead of lists. The character position is
32444 As a courtesy to other Lisp code which may be using Calc, be sure
32445 to bind @code{calc-eval-error} using @code{let} rather than changing
32446 it permanently with @code{setq}.
32453 @subsubsection Numbers Only
32456 Sometimes it is preferable to treat @samp{1 / 0} as an error
32457 rather than returning a symbolic result. If you pass the symbol
32458 @code{num} as the second argument to @code{calc-eval}, results
32459 that are not constants are treated as errors. The error message
32460 reported is the first @code{calc-why} message if there is one,
32461 or otherwise ``Number expected.''
32463 A result is ``constant'' if it is a number, vector, or other
32464 object that does not include variables or function calls. If it
32465 is a vector, the components must themselves be constants.
32472 @subsubsection Default Modes
32475 If the first argument to @code{calc-eval} is a list whose first
32476 element is a formula string, then @code{calc-eval} sets all the
32477 various Calc modes to their default values while the formula is
32478 evaluated and formatted. For example, the precision is set to 12
32479 digits, digit grouping is turned off, and the Normal language
32482 This same principle applies to the other options discussed below.
32483 If the first argument would normally be @var{x}, then it can also
32484 be the list @samp{(@var{x})} to use the default mode settings.
32486 If there are other elements in the list, they are taken as
32487 variable-name/value pairs which override the default mode
32488 settings. Look at the documentation at the front of the
32489 @file{calc.el} file to find the names of the Lisp variables for
32490 the various modes. The mode settings are restored to their
32491 original values when @code{calc-eval} is done.
32493 For example, @samp{(calc-eval '("$+$$" calc-internal-prec 8) 'num a b)}
32494 computes the sum of two numbers, requiring a numeric result, and
32495 using default mode settings except that the precision is 8 instead
32496 of the default of 12.
32498 It's usually best to use this form of @code{calc-eval} unless your
32499 program actually considers the interaction with Calc's mode settings
32500 to be a feature. This will avoid all sorts of potential ``gotchas'';
32501 consider what happens with @samp{(calc-eval "sqrt(2)" 'num)}
32502 when the user has left Calc in Symbolic mode or No-Simplify mode.
32504 As another example, @samp{(equal (calc-eval '("$<$$") nil a b) "1")}
32505 checks if the number in string @expr{a} is less than the one in
32506 string @expr{b}. Without using a list, the integer 1 might
32507 come out in a variety of formats which would be hard to test for
32508 conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But
32509 see ``Predicates'' mode, below.)
32516 @subsubsection Raw Numbers
32519 Normally all input and output for @code{calc-eval} is done with strings.
32520 You can do arithmetic with, say, @samp{(calc-eval "$+$$" nil a b)}
32521 in place of @samp{(+ a b)}, but this is very inefficient since the
32522 numbers must be converted to and from string format as they are passed
32523 from one @code{calc-eval} to the next.
32525 If the separator is the symbol @code{raw}, the result will be returned
32526 as a raw Calc data structure rather than a string. You can read about
32527 how these objects look in the following sections, but usually you can
32528 treat them as ``black box'' objects with no important internal
32531 There is also a @code{rawnum} symbol, which is a combination of
32532 @code{raw} (returning a raw Calc object) and @code{num} (signaling
32533 an error if that object is not a constant).
32535 You can pass a raw Calc object to @code{calc-eval} in place of a
32536 string, either as the formula itself or as one of the @samp{$}
32537 arguments. Thus @samp{(calc-eval "$+$$" 'raw a b)} is an
32538 addition function that operates on raw Calc objects. Of course
32539 in this case it would be easier to call the low-level @code{math-add}
32540 function in Calc, if you can remember its name.
32542 In particular, note that a plain Lisp integer is acceptable to Calc
32543 as a raw object. (All Lisp integers are accepted on input, but
32544 integers of more than six decimal digits are converted to ``big-integer''
32545 form for output. @xref{Data Type Formats}.)
32547 When it comes time to display the object, just use @samp{(calc-eval a)}
32548 to format it as a string.
32550 It is an error if the input expression evaluates to a list of
32551 values. The separator symbol @code{list} is like @code{raw}
32552 except that it returns a list of one or more raw Calc objects.
32554 Note that a Lisp string is not a valid Calc object, nor is a list
32555 containing a string. Thus you can still safely distinguish all the
32556 various kinds of error returns discussed above.
32563 @subsubsection Predicates
32566 If the separator symbol is @code{pred}, the result of the formula is
32567 treated as a true/false value; @code{calc-eval} returns @code{t} or
32568 @code{nil}, respectively. A value is considered ``true'' if it is a
32569 non-zero number, or false if it is zero or if it is not a number.
32571 For example, @samp{(calc-eval "$<$$" 'pred a b)} tests whether
32572 one value is less than another.
32574 As usual, it is also possible for @code{calc-eval} to return one of
32575 the error indicators described above. Lisp will interpret such an
32576 indicator as ``true'' if you don't check for it explicitly. If you
32577 wish to have an error register as ``false'', use something like
32578 @samp{(eq (calc-eval ...) t)}.
32585 @subsubsection Variable Values
32588 Variables in the formula passed to @code{calc-eval} are not normally
32589 replaced by their values. If you wish this, you can use the
32590 @code{evalv} function (@pxref{Algebraic Manipulation}). For example,
32591 if 4 is stored in Calc variable @code{a} (i.e., in Lisp variable
32592 @code{var-a}), then @samp{(calc-eval "a+pi")} will return the
32593 formula @code{"a + pi"}, but @samp{(calc-eval "evalv(a+pi)")}
32594 will return @code{"7.14159265359"}.
32596 To store in a Calc variable, just use @code{setq} to store in the
32597 corresponding Lisp variable. (This is obtained by prepending
32598 @samp{var-} to the Calc variable name.) Calc routines will
32599 understand either string or raw form values stored in variables,
32600 although raw data objects are much more efficient. For example,
32601 to increment the Calc variable @code{a}:
32604 (setq var-a (calc-eval "evalv(a+1)" 'raw))
32612 @subsubsection Stack Access
32615 If the separator symbol is @code{push}, the formula argument is
32616 evaluated (with possible @samp{$} expansions, as usual). The
32617 result is pushed onto the Calc stack. The return value is @code{nil}
32618 (unless there is an error from evaluating the formula, in which
32619 case the return value depends on @code{calc-eval-error} in the
32622 If the separator symbol is @code{pop}, the first argument to
32623 @code{calc-eval} must be an integer instead of a string. That
32624 many values are popped from the stack and thrown away. A negative
32625 argument deletes the entry at that stack level. The return value
32626 is the number of elements remaining in the stack after popping;
32627 @samp{(calc-eval 0 'pop)} is a good way to measure the size of
32630 If the separator symbol is @code{top}, the first argument to
32631 @code{calc-eval} must again be an integer. The value at that
32632 stack level is formatted as a string and returned. Thus
32633 @samp{(calc-eval 1 'top)} returns the top-of-stack value. If the
32634 integer is out of range, @code{nil} is returned.
32636 The separator symbol @code{rawtop} is just like @code{top} except
32637 that the stack entry is returned as a raw Calc object instead of
32640 In all of these cases the first argument can be made a list in
32641 order to force the default mode settings, as described above.
32642 Thus @samp{(calc-eval '(2 calc-number-radix 16) 'top)} returns the
32643 second-to-top stack entry, formatted as a string using the default
32644 instead of current display modes, except that the radix is
32645 hexadecimal instead of decimal.
32647 It is, of course, polite to put the Calc stack back the way you
32648 found it when you are done, unless the user of your program is
32649 actually expecting it to affect the stack.
32651 Note that you do not actually have to switch into the @samp{*Calculator*}
32652 buffer in order to use @code{calc-eval}; it temporarily switches into
32653 the stack buffer if necessary.
32660 @subsubsection Keyboard Macros
32663 If the separator symbol is @code{macro}, the first argument must be a
32664 string of characters which Calc can execute as a sequence of keystrokes.
32665 This switches into the Calc buffer for the duration of the macro.
32666 For example, @samp{(calc-eval "vx5\rVR+" 'macro)} pushes the
32667 vector @samp{[1,2,3,4,5]} on the stack and then replaces it
32668 with the sum of those numbers. Note that @samp{\r} is the Lisp
32669 notation for the carriage-return, @key{RET}, character.
32671 If your keyboard macro wishes to pop the stack, @samp{\C-d} is
32672 safer than @samp{\177} (the @key{DEL} character) because some
32673 installations may have switched the meanings of @key{DEL} and
32674 @kbd{C-h}. Calc always interprets @kbd{C-d} as a synonym for
32675 ``pop-stack'' regardless of key mapping.
32677 If you provide a third argument to @code{calc-eval}, evaluation
32678 of the keyboard macro will leave a record in the Trail using
32679 that argument as a tag string. Normally the Trail is unaffected.
32681 The return value in this case is always @code{nil}.
32688 @subsubsection Lisp Evaluation
32691 Finally, if the separator symbol is @code{eval}, then the Lisp
32692 @code{eval} function is called on the first argument, which must
32693 be a Lisp expression rather than a Calc formula. Remember to
32694 quote the expression so that it is not evaluated until inside
32697 The difference from plain @code{eval} is that @code{calc-eval}
32698 switches to the Calc buffer before evaluating the expression.
32699 For example, @samp{(calc-eval '(setq calc-internal-prec 17) 'eval)}
32700 will correctly affect the buffer-local Calc precision variable.
32702 An alternative would be @samp{(calc-eval '(calc-precision 17) 'eval)}.
32703 This is evaluating a call to the function that is normally invoked
32704 by the @kbd{p} key, giving it 17 as its ``numeric prefix argument.''
32705 Note that this function will leave a message in the echo area as
32706 a side effect. Also, all Calc functions switch to the Calc buffer
32707 automatically if not invoked from there, so the above call is
32708 also equivalent to @samp{(calc-precision 17)} by itself.
32709 In all cases, Calc uses @code{save-excursion} to switch back to
32710 your original buffer when it is done.
32712 As usual the first argument can be a list that begins with a Lisp
32713 expression to use default instead of current mode settings.
32715 The result of @code{calc-eval} in this usage is just the result
32716 returned by the evaluated Lisp expression.
32723 @subsubsection Example
32726 @findex convert-temp
32727 Here is a sample Emacs command that uses @code{calc-eval}. Suppose
32728 you have a document with lots of references to temperatures on the
32729 Fahrenheit scale, say ``98.6 F'', and you wish to convert these
32730 references to Centigrade. The following command does this conversion.
32731 Place the Emacs cursor right after the letter ``F'' and invoke the
32732 command to change ``98.6 F'' to ``37 C''. Or, if the temperature is
32733 already in Centigrade form, the command changes it back to Fahrenheit.
32736 (defun convert-temp ()
32739 (re-search-backward "[^-.0-9]\\([-.0-9]+\\) *\\([FC]\\)")
32740 (let* ((top1 (match-beginning 1))
32741 (bot1 (match-end 1))
32742 (number (buffer-substring top1 bot1))
32743 (top2 (match-beginning 2))
32744 (bot2 (match-end 2))
32745 (type (buffer-substring top2 bot2)))
32746 (if (equal type "F")
32748 number (calc-eval "($ - 32)*5/9" nil number))
32750 number (calc-eval "$*9/5 + 32" nil number)))
32752 (delete-region top2 bot2)
32753 (insert-before-markers type)
32755 (delete-region top1 bot1)
32756 (if (string-match "\\.$" number) ; change "37." to "37"
32757 (setq number (substring number 0 -1)))
32761 Note the use of @code{insert-before-markers} when changing between
32762 ``F'' and ``C'', so that the character winds up before the cursor
32763 instead of after it.
32765 @node Internals, , Calling Calc from Your Programs, Lisp Definitions
32766 @subsection Calculator Internals
32769 This section describes the Lisp functions defined by the Calculator that
32770 may be of use to user-written Calculator programs (as described in the
32771 rest of this chapter). These functions are shown by their names as they
32772 conventionally appear in @code{defmath}. Their full Lisp names are
32773 generally gotten by prepending @samp{calcFunc-} or @samp{math-} to their
32774 apparent names. (Names that begin with @samp{calc-} are already in
32775 their full Lisp form.) You can use the actual full names instead if you
32776 prefer them, or if you are calling these functions from regular Lisp.
32778 The functions described here are scattered throughout the various
32779 Calc component files. Note that @file{calc.el} includes @code{autoload}s
32780 for only a few component files; when Calc wants to call an advanced
32781 function it calls @samp{(calc-extensions)} first; this function
32782 autoloads @file{calc-ext.el}, which in turn autoloads all the functions
32783 in the remaining component files.
32785 Because @code{defmath} itself uses the extensions, user-written code
32786 generally always executes with the extensions already loaded, so
32787 normally you can use any Calc function and be confident that it will
32788 be autoloaded for you when necessary. If you are doing something
32789 special, check carefully to make sure each function you are using is
32790 from @file{calc.el} or its components, and call @samp{(calc-extensions)}
32791 before using any function based in @file{calc-ext.el} if you can't
32792 prove this file will already be loaded.
32795 * Data Type Formats::
32796 * Interactive Lisp Functions::
32797 * Stack Lisp Functions::
32799 * Computational Lisp Functions::
32800 * Vector Lisp Functions::
32801 * Symbolic Lisp Functions::
32802 * Formatting Lisp Functions::
32806 @node Data Type Formats, Interactive Lisp Functions, Internals, Internals
32807 @subsubsection Data Type Formats
32810 Integers are stored in either of two ways, depending on their magnitude.
32811 Integers less than one million in absolute value are stored as standard
32812 Lisp integers. This is the only storage format for Calc data objects
32813 which is not a Lisp list.
32815 Large integers are stored as lists of the form @samp{(bigpos @var{d0}
32816 @var{d1} @var{d2} @dots{})} for positive integers 1000000 or more, or
32817 @samp{(bigneg @var{d0} @var{d1} @var{d2} @dots{})} for negative integers
32818 @mathit{-1000000} or less. Each @var{d} is a base-1000 ``digit,'' a Lisp integer
32819 from 0 to 999. The least significant digit is @var{d0}; the last digit,
32820 @var{dn}, which is always nonzero, is the most significant digit. For
32821 example, the integer @mathit{-12345678} is stored as @samp{(bigneg 678 345 12)}.
32823 The distinction between small and large integers is entirely hidden from
32824 the user. In @code{defmath} definitions, the Lisp predicate @code{integerp}
32825 returns true for either kind of integer, and in general both big and small
32826 integers are accepted anywhere the word ``integer'' is used in this manual.
32827 If the distinction must be made, native Lisp integers are called @dfn{fixnums}
32828 and large integers are called @dfn{bignums}.
32830 Fractions are stored as a list of the form, @samp{(frac @var{n} @var{d})}
32831 where @var{n} is an integer (big or small) numerator, @var{d} is an
32832 integer denominator greater than one, and @var{n} and @var{d} are relatively
32833 prime. Note that fractions where @var{d} is one are automatically converted
32834 to plain integers by all math routines; fractions where @var{d} is negative
32835 are normalized by negating the numerator and denominator.
32837 Floating-point numbers are stored in the form, @samp{(float @var{mant}
32838 @var{exp})}, where @var{mant} (the ``mantissa'') is an integer less than
32839 @samp{10^@var{p}} in absolute value (@var{p} represents the current
32840 precision), and @var{exp} (the ``exponent'') is a fixnum. The value of
32841 the float is @samp{@var{mant} * 10^@var{exp}}. For example, the number
32842 @mathit{-3.14} is stored as @samp{(float -314 -2) = -314*10^-2}. Other constraints
32843 are that the number 0.0 is always stored as @samp{(float 0 0)}, and,
32844 except for the 0.0 case, the rightmost base-10 digit of @var{mant} is
32845 always nonzero. (If the rightmost digit is zero, the number is
32846 rearranged by dividing @var{mant} by ten and incrementing @var{exp}.)
32848 Rectangular complex numbers are stored in the form @samp{(cplx @var{re}
32849 @var{im})}, where @var{re} and @var{im} are each real numbers, either
32850 integers, fractions, or floats. The value is @samp{@var{re} + @var{im}i}.
32851 The @var{im} part is nonzero; complex numbers with zero imaginary
32852 components are converted to real numbers automatically.
32854 Polar complex numbers are stored in the form @samp{(polar @var{r}
32855 @var{theta})}, where @var{r} is a positive real value and @var{theta}
32856 is a real value or HMS form representing an angle. This angle is
32857 usually normalized to lie in the interval @samp{(-180 ..@: 180)} degrees,
32858 or @samp{(-pi ..@: pi)} radians, according to the current angular mode.
32859 If the angle is 0 the value is converted to a real number automatically.
32860 (If the angle is 180 degrees, the value is usually also converted to a
32861 negative real number.)
32863 Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m}
32864 @var{s})}, where @var{h} is an integer or an integer-valued float (i.e.,
32865 a float with @samp{@var{exp} >= 0}), @var{m} is an integer or integer-valued
32866 float in the range @w{@samp{[0 ..@: 60)}}, and @var{s} is any real number
32867 in the range @samp{[0 ..@: 60)}.
32869 Date forms are stored as @samp{(date @var{n})}, where @var{n} is
32870 a real number that counts days since midnight on the morning of
32871 January 1, 1 AD. If @var{n} is an integer, this is a pure date
32872 form. If @var{n} is a fraction or float, this is a date/time form.
32874 Modulo forms are stored as @samp{(mod @var{n} @var{m})}, where @var{m} is a
32875 positive real number or HMS form, and @var{n} is a real number or HMS
32876 form in the range @samp{[0 ..@: @var{m})}.
32878 Error forms are stored as @samp{(sdev @var{x} @var{sigma})}, where @var{x}
32879 is the mean value and @var{sigma} is the standard deviation. Each
32880 component is either a number, an HMS form, or a symbolic object
32881 (a variable or function call). If @var{sigma} is zero, the value is
32882 converted to a plain real number. If @var{sigma} is negative or
32883 complex, it is automatically normalized to be a positive real.
32885 Interval forms are stored as @samp{(intv @var{mask} @var{lo} @var{hi})},
32886 where @var{mask} is one of the integers 0, 1, 2, or 3, and @var{lo} and
32887 @var{hi} are real numbers, HMS forms, or symbolic objects. The @var{mask}
32888 is a binary integer where 1 represents the fact that the interval is
32889 closed on the high end, and 2 represents the fact that it is closed on
32890 the low end. (Thus 3 represents a fully closed interval.) The interval
32891 @w{@samp{(intv 3 @var{x} @var{x})}} is converted to the plain number @var{x};
32892 intervals @samp{(intv @var{mask} @var{x} @var{x})} for any other @var{mask}
32893 represent empty intervals. If @var{hi} is less than @var{lo}, the interval
32894 is converted to a standard empty interval by replacing @var{hi} with @var{lo}.
32896 Vectors are stored as @samp{(vec @var{v1} @var{v2} @dots{})}, where @var{v1}
32897 is the first element of the vector, @var{v2} is the second, and so on.
32898 An empty vector is stored as @samp{(vec)}. A matrix is simply a vector
32899 where all @var{v}'s are themselves vectors of equal lengths. Note that
32900 Calc vectors are unrelated to the Emacs Lisp ``vector'' type, which is
32901 generally unused by Calc data structures.
32903 Variables are stored as @samp{(var @var{name} @var{sym})}, where
32904 @var{name} is a Lisp symbol whose print name is used as the visible name
32905 of the variable, and @var{sym} is a Lisp symbol in which the variable's
32906 value is actually stored. Thus, @samp{(var pi var-pi)} represents the
32907 special constant @samp{pi}. Almost always, the form is @samp{(var
32908 @var{v} var-@var{v})}. If the variable name was entered with @code{#}
32909 signs (which are converted to hyphens internally), the form is
32910 @samp{(var @var{u} @var{v})}, where @var{u} is a symbol whose name
32911 contains @code{#} characters, and @var{v} is a symbol that contains
32912 @code{-} characters instead. The value of a variable is the Calc
32913 object stored in its @var{sym} symbol's value cell. If the symbol's
32914 value cell is void or if it contains @code{nil}, the variable has no
32915 value. Special constants have the form @samp{(special-const
32916 @var{value})} stored in their value cell, where @var{value} is a formula
32917 which is evaluated when the constant's value is requested. Variables
32918 which represent units are not stored in any special way; they are units
32919 only because their names appear in the units table. If the value
32920 cell contains a string, it is parsed to get the variable's value when
32921 the variable is used.
32923 A Lisp list with any other symbol as the first element is a function call.
32924 The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^},
32925 and @code{|} represent special binary operators; these lists are always
32926 of the form @samp{(@var{op} @var{lhs} @var{rhs})} where @var{lhs} is the
32927 sub-formula on the lefthand side and @var{rhs} is the sub-formula on the
32928 right. The symbol @code{neg} represents unary negation; this list is always
32929 of the form @samp{(neg @var{arg})}. Any other symbol @var{func} represents a
32930 function that would be displayed in function-call notation; the symbol
32931 @var{func} is in general always of the form @samp{calcFunc-@var{name}}.
32932 The function cell of the symbol @var{func} should contain a Lisp function
32933 for evaluating a call to @var{func}. This function is passed the remaining
32934 elements of the list (themselves already evaluated) as arguments; such
32935 functions should return @code{nil} or call @code{reject-arg} to signify
32936 that they should be left in symbolic form, or they should return a Calc
32937 object which represents their value, or a list of such objects if they
32938 wish to return multiple values. (The latter case is allowed only for
32939 functions which are the outer-level call in an expression whose value is
32940 about to be pushed on the stack; this feature is considered obsolete
32941 and is not used by any built-in Calc functions.)
32943 @node Interactive Lisp Functions, Stack Lisp Functions, Data Type Formats, Internals
32944 @subsubsection Interactive Functions
32947 The functions described here are used in implementing interactive Calc
32948 commands. Note that this list is not exhaustive! If there is an
32949 existing command that behaves similarly to the one you want to define,
32950 you may find helpful tricks by checking the source code for that command.
32952 @defun calc-set-command-flag flag
32953 Set the command flag @var{flag}. This is generally a Lisp symbol, but
32954 may in fact be anything. The effect is to add @var{flag} to the list
32955 stored in the variable @code{calc-command-flags}, unless it is already
32956 there. @xref{Defining Simple Commands}.
32959 @defun calc-clear-command-flag flag
32960 If @var{flag} appears among the list of currently-set command flags,
32961 remove it from that list.
32964 @defun calc-record-undo rec
32965 Add the ``undo record'' @var{rec} to the list of steps to take if the
32966 current operation should need to be undone. Stack push and pop functions
32967 automatically call @code{calc-record-undo}, so the kinds of undo records
32968 you might need to create take the form @samp{(set @var{sym} @var{value})},
32969 which says that the Lisp variable @var{sym} was changed and had previously
32970 contained @var{value}; @samp{(store @var{var} @var{value})} which says that
32971 the Calc variable @var{var} (a string which is the name of the symbol that
32972 contains the variable's value) was stored and its previous value was
32973 @var{value} (either a Calc data object, or @code{nil} if the variable was
32974 previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})},
32975 which means that to undo requires calling the function @samp{(@var{undo}
32976 @var{args} @dots{})} and, if the undo is later redone, calling
32977 @samp{(@var{redo} @var{args} @dots{})}.
32980 @defun calc-record-why msg args
32981 Record the error or warning message @var{msg}, which is normally a string.
32982 This message will be replayed if the user types @kbd{w} (@code{calc-why});
32983 if the message string begins with a @samp{*}, it is considered important
32984 enough to display even if the user doesn't type @kbd{w}. If one or more
32985 @var{args} are present, the displayed message will be of the form,
32986 @samp{@var{msg}: @var{arg1}, @var{arg2}, @dots{}}, where the arguments are
32987 formatted on the assumption that they are either strings or Calc objects of
32988 some sort. If @var{msg} is a symbol, it is the name of a Calc predicate
32989 (such as @code{integerp} or @code{numvecp}) which the arguments did not
32990 satisfy; it is expanded to a suitable string such as ``Expected an
32991 integer.'' The @code{reject-arg} function calls @code{calc-record-why}
32992 automatically; @pxref{Predicates}.
32995 @defun calc-is-inverse
32996 This predicate returns true if the current command is inverse,
32997 i.e., if the Inverse (@kbd{I} key) flag was set.
33000 @defun calc-is-hyperbolic
33001 This predicate is the analogous function for the @kbd{H} key.
33004 @node Stack Lisp Functions, Predicates, Interactive Lisp Functions, Internals
33005 @subsubsection Stack-Oriented Functions
33008 The functions described here perform various operations on the Calc
33009 stack and trail. They are to be used in interactive Calc commands.
33011 @defun calc-push-list vals n
33012 Push the Calc objects in list @var{vals} onto the stack at stack level
33013 @var{n}. If @var{n} is omitted it defaults to 1, so that the elements
33014 are pushed at the top of the stack. If @var{n} is greater than 1, the
33015 elements will be inserted into the stack so that the last element will
33016 end up at level @var{n}, the next-to-last at level @var{n}+1, etc.
33017 The elements of @var{vals} are assumed to be valid Calc objects, and
33018 are not evaluated, rounded, or renormalized in any way. If @var{vals}
33019 is an empty list, nothing happens.
33021 The stack elements are pushed without any sub-formula selections.
33022 You can give an optional third argument to this function, which must
33023 be a list the same size as @var{vals} of selections. Each selection
33024 must be @code{eq} to some sub-formula of the corresponding formula
33025 in @var{vals}, or @code{nil} if that formula should have no selection.
33028 @defun calc-top-list n m
33029 Return a list of the @var{n} objects starting at level @var{m} of the
33030 stack. If @var{m} is omitted it defaults to 1, so that the elements are
33031 taken from the top of the stack. If @var{n} is omitted, it also
33032 defaults to 1, so that the top stack element (in the form of a
33033 one-element list) is returned. If @var{m} is greater than 1, the
33034 @var{m}th stack element will be at the end of the list, the @var{m}+1st
33035 element will be next-to-last, etc. If @var{n} or @var{m} are out of
33036 range, the command is aborted with a suitable error message. If @var{n}
33037 is zero, the function returns an empty list. The stack elements are not
33038 evaluated, rounded, or renormalized.
33040 If any stack elements contain selections, and selections have not
33041 been disabled by the @kbd{j e} (@code{calc-enable-selections}) command,
33042 this function returns the selected portions rather than the entire
33043 stack elements. It can be given a third ``selection-mode'' argument
33044 which selects other behaviors. If it is the symbol @code{t}, then
33045 a selection in any of the requested stack elements produces an
33046 ``invalid operation on selections'' error. If it is the symbol @code{full},
33047 the whole stack entry is always returned regardless of selections.
33048 If it is the symbol @code{sel}, the selected portion is always returned,
33049 or @code{nil} if there is no selection. (This mode ignores the @kbd{j e}
33050 command.) If the symbol is @code{entry}, the complete stack entry in
33051 list form is returned; the first element of this list will be the whole
33052 formula, and the third element will be the selection (or @code{nil}).
33055 @defun calc-pop-stack n m
33056 Remove the specified elements from the stack. The parameters @var{n}
33057 and @var{m} are defined the same as for @code{calc-top-list}. The return
33058 value of @code{calc-pop-stack} is uninteresting.
33060 If there are any selected sub-formulas among the popped elements, and
33061 @kbd{j e} has not been used to disable selections, this produces an
33062 error without changing the stack. If you supply an optional third
33063 argument of @code{t}, the stack elements are popped even if they
33064 contain selections.
33067 @defun calc-record-list vals tag
33068 This function records one or more results in the trail. The @var{vals}
33069 are a list of strings or Calc objects. The @var{tag} is the four-character
33070 tag string to identify the values. If @var{tag} is omitted, a blank tag
33074 @defun calc-normalize n
33075 This function takes a Calc object and ``normalizes'' it. At the very
33076 least this involves re-rounding floating-point values according to the
33077 current precision and other similar jobs. Also, unless the user has
33078 selected No-Simplify mode (@pxref{Simplification Modes}), this involves
33079 actually evaluating a formula object by executing the function calls
33080 it contains, and possibly also doing algebraic simplification, etc.
33083 @defun calc-top-list-n n m
33084 This function is identical to @code{calc-top-list}, except that it calls
33085 @code{calc-normalize} on the values that it takes from the stack. They
33086 are also passed through @code{check-complete}, so that incomplete
33087 objects will be rejected with an error message. All computational
33088 commands should use this in preference to @code{calc-top-list}; the only
33089 standard Calc commands that operate on the stack without normalizing
33090 are stack management commands like @code{calc-enter} and @code{calc-roll-up}.
33091 This function accepts the same optional selection-mode argument as
33092 @code{calc-top-list}.
33095 @defun calc-top-n m
33096 This function is a convenient form of @code{calc-top-list-n} in which only
33097 a single element of the stack is taken and returned, rather than a list
33098 of elements. This also accepts an optional selection-mode argument.
33101 @defun calc-enter-result n tag vals
33102 This function is a convenient interface to most of the above functions.
33103 The @var{vals} argument should be either a single Calc object, or a list
33104 of Calc objects; the object or objects are normalized, and the top @var{n}
33105 stack entries are replaced by the normalized objects. If @var{tag} is
33106 non-@code{nil}, the normalized objects are also recorded in the trail.
33107 A typical stack-based computational command would take the form,
33110 (calc-enter-result @var{n} @var{tag} (cons 'calcFunc-@var{func}
33111 (calc-top-list-n @var{n})))
33114 If any of the @var{n} stack elements replaced contain sub-formula
33115 selections, and selections have not been disabled by @kbd{j e},
33116 this function takes one of two courses of action. If @var{n} is
33117 equal to the number of elements in @var{vals}, then each element of
33118 @var{vals} is spliced into the corresponding selection; this is what
33119 happens when you use the @key{TAB} key, or when you use a unary
33120 arithmetic operation like @code{sqrt}. If @var{vals} has only one
33121 element but @var{n} is greater than one, there must be only one
33122 selection among the top @var{n} stack elements; the element from
33123 @var{vals} is spliced into that selection. This is what happens when
33124 you use a binary arithmetic operation like @kbd{+}. Any other
33125 combination of @var{n} and @var{vals} is an error when selections
33129 @defun calc-unary-op tag func arg
33130 This function implements a unary operator that allows a numeric prefix
33131 argument to apply the operator over many stack entries. If the prefix
33132 argument @var{arg} is @code{nil}, this uses @code{calc-enter-result}
33133 as outlined above. Otherwise, it maps the function over several stack
33134 elements; @pxref{Prefix Arguments}. For example,
33137 (defun calc-zeta (arg)
33139 (calc-unary-op "zeta" 'calcFunc-zeta arg))
33143 @defun calc-binary-op tag func arg ident unary
33144 This function implements a binary operator, analogously to
33145 @code{calc-unary-op}. The optional @var{ident} and @var{unary}
33146 arguments specify the behavior when the prefix argument is zero or
33147 one, respectively. If the prefix is zero, the value @var{ident}
33148 is pushed onto the stack, if specified, otherwise an error message
33149 is displayed. If the prefix is one, the unary function @var{unary}
33150 is applied to the top stack element, or, if @var{unary} is not
33151 specified, nothing happens. When the argument is two or more,
33152 the binary function @var{func} is reduced across the top @var{arg}
33153 stack elements; when the argument is negative, the function is
33154 mapped between the next-to-top @mathit{-@var{arg}} stack elements and the
33158 @defun calc-stack-size
33159 Return the number of elements on the stack as an integer. This count
33160 does not include elements that have been temporarily hidden by stack
33161 truncation; @pxref{Truncating the Stack}.
33164 @defun calc-cursor-stack-index n
33165 Move the point to the @var{n}th stack entry. If @var{n} is zero, this
33166 will be the @samp{.} line. If @var{n} is from 1 to the current stack size,
33167 this will be the beginning of the first line of that stack entry's display.
33168 If line numbers are enabled, this will move to the first character of the
33169 line number, not the stack entry itself.
33172 @defun calc-substack-height n
33173 Return the number of lines between the beginning of the @var{n}th stack
33174 entry and the bottom of the buffer. If @var{n} is zero, this
33175 will be one (assuming no stack truncation). If all stack entries are
33176 one line long (i.e., no matrices are displayed), the return value will
33177 be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big
33178 mode, the return value includes the blank lines that separate stack
33182 @defun calc-refresh
33183 Erase the @code{*Calculator*} buffer and reformat its contents from memory.
33184 This must be called after changing any parameter, such as the current
33185 display radix, which might change the appearance of existing stack
33186 entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing
33187 is suppressed, but a flag is set so that the entire stack will be refreshed
33188 rather than just the top few elements when the macro finishes.)
33191 @node Predicates, Computational Lisp Functions, Stack Lisp Functions, Internals
33192 @subsubsection Predicates
33195 The functions described here are predicates, that is, they return a
33196 true/false value where @code{nil} means false and anything else means
33197 true. These predicates are expanded by @code{defmath}, for example,
33198 from @code{zerop} to @code{math-zerop}. In many cases they correspond
33199 to native Lisp functions by the same name, but are extended to cover
33200 the full range of Calc data types.
33203 Returns true if @var{x} is numerically zero, in any of the Calc data
33204 types. (Note that for some types, such as error forms and intervals,
33205 it never makes sense to return true.) In @code{defmath}, the expression
33206 @samp{(= x 0)} will automatically be converted to @samp{(math-zerop x)},
33207 and @samp{(/= x 0)} will be converted to @samp{(not (math-zerop x))}.
33211 Returns true if @var{x} is negative. This accepts negative real numbers
33212 of various types, negative HMS and date forms, and intervals in which
33213 all included values are negative. In @code{defmath}, the expression
33214 @samp{(< x 0)} will automatically be converted to @samp{(math-negp x)},
33215 and @samp{(>= x 0)} will be converted to @samp{(not (math-negp x))}.
33219 Returns true if @var{x} is positive (and non-zero). For complex
33220 numbers, none of these three predicates will return true.
33223 @defun looks-negp x
33224 Returns true if @var{x} is ``negative-looking.'' This returns true if
33225 @var{x} is a negative number, or a formula with a leading minus sign
33226 such as @samp{-a/b}. In other words, this is an object which can be
33227 made simpler by calling @code{(- @var{x})}.
33231 Returns true if @var{x} is an integer of any size.
33235 Returns true if @var{x} is a native Lisp integer.
33239 Returns true if @var{x} is a nonnegative integer of any size.
33242 @defun fixnatnump x
33243 Returns true if @var{x} is a nonnegative Lisp integer.
33246 @defun num-integerp x
33247 Returns true if @var{x} is numerically an integer, i.e., either a
33248 true integer or a float with no significant digits to the right of
33252 @defun messy-integerp x
33253 Returns true if @var{x} is numerically, but not literally, an integer.
33254 A value is @code{num-integerp} if it is @code{integerp} or
33255 @code{messy-integerp} (but it is never both at once).
33258 @defun num-natnump x
33259 Returns true if @var{x} is numerically a nonnegative integer.
33263 Returns true if @var{x} is an even integer.
33266 @defun looks-evenp x
33267 Returns true if @var{x} is an even integer, or a formula with a leading
33268 multiplicative coefficient which is an even integer.
33272 Returns true if @var{x} is an odd integer.
33276 Returns true if @var{x} is a rational number, i.e., an integer or a
33281 Returns true if @var{x} is a real number, i.e., an integer, fraction,
33282 or floating-point number.
33286 Returns true if @var{x} is a real number or HMS form.
33290 Returns true if @var{x} is a float, or a complex number, error form,
33291 interval, date form, or modulo form in which at least one component
33296 Returns true if @var{x} is a rectangular or polar complex number
33297 (but not a real number).
33300 @defun rect-complexp x
33301 Returns true if @var{x} is a rectangular complex number.
33304 @defun polar-complexp x
33305 Returns true if @var{x} is a polar complex number.
33309 Returns true if @var{x} is a real number or a complex number.
33313 Returns true if @var{x} is a real or complex number or an HMS form.
33317 Returns true if @var{x} is a vector (this simply checks if its argument
33318 is a list whose first element is the symbol @code{vec}).
33322 Returns true if @var{x} is a number or vector.
33326 Returns true if @var{x} is a matrix, i.e., a vector of one or more vectors,
33327 all of the same size.
33330 @defun square-matrixp x
33331 Returns true if @var{x} is a square matrix.
33335 Returns true if @var{x} is any numeric Calc object, including real and
33336 complex numbers, HMS forms, date forms, error forms, intervals, and
33337 modulo forms. (Note that error forms and intervals may include formulas
33338 as their components; see @code{constp} below.)
33342 Returns true if @var{x} is an object or a vector. This also accepts
33343 incomplete objects, but it rejects variables and formulas (except as
33344 mentioned above for @code{objectp}).
33348 Returns true if @var{x} is a ``primitive'' or ``atomic'' Calc object,
33349 i.e., one whose components cannot be regarded as sub-formulas. This
33350 includes variables, and all @code{objectp} types except error forms
33355 Returns true if @var{x} is constant, i.e., a real or complex number,
33356 HMS form, date form, or error form, interval, or vector all of whose
33357 components are @code{constp}.
33361 Returns true if @var{x} is numerically less than @var{y}. Returns false
33362 if @var{x} is greater than or equal to @var{y}, or if the order is
33363 undefined or cannot be determined. Generally speaking, this works
33364 by checking whether @samp{@var{x} - @var{y}} is @code{negp}. In
33365 @code{defmath}, the expression @samp{(< x y)} will automatically be
33366 converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=},
33367 and @code{>=} are similarly converted in terms of @code{lessp}.
33371 Returns true if @var{x} comes before @var{y} in a canonical ordering
33372 of Calc objects. If @var{x} and @var{y} are both real numbers, this
33373 will be the same as @code{lessp}. But whereas @code{lessp} considers
33374 other types of objects to be unordered, @code{beforep} puts any two
33375 objects into a definite, consistent order. The @code{beforep}
33376 function is used by the @kbd{V S} vector-sorting command, and also
33377 by @kbd{a s} to put the terms of a product into canonical order:
33378 This allows @samp{x y + y x} to be simplified easily to @samp{2 x y}.
33382 This is the standard Lisp @code{equal} predicate; it returns true if
33383 @var{x} and @var{y} are structurally identical. This is the usual way
33384 to compare numbers for equality, but note that @code{equal} will treat
33385 0 and 0.0 as different.
33388 @defun math-equal x y
33389 Returns true if @var{x} and @var{y} are numerically equal, either because
33390 they are @code{equal}, or because their difference is @code{zerop}. In
33391 @code{defmath}, the expression @samp{(= x y)} will automatically be
33392 converted to @samp{(math-equal x y)}.
33395 @defun equal-int x n
33396 Returns true if @var{x} and @var{n} are numerically equal, where @var{n}
33397 is a fixnum which is not a multiple of 10. This will automatically be
33398 used by @code{defmath} in place of the more general @code{math-equal}
33402 @defun nearly-equal x y
33403 Returns true if @var{x} and @var{y}, as floating-point numbers, are
33404 equal except possibly in the last decimal place. For example,
33405 314.159 and 314.166 are considered nearly equal if the current
33406 precision is 6 (since they differ by 7 units), but not if the current
33407 precision is 7 (since they differ by 70 units). Most functions which
33408 use series expansions use @code{with-extra-prec} to evaluate the
33409 series with 2 extra digits of precision, then use @code{nearly-equal}
33410 to decide when the series has converged; this guards against cumulative
33411 error in the series evaluation without doing extra work which would be
33412 lost when the result is rounded back down to the current precision.
33413 In @code{defmath}, this can be written @samp{(~= @var{x} @var{y})}.
33414 The @var{x} and @var{y} can be numbers of any kind, including complex.
33417 @defun nearly-zerop x y
33418 Returns true if @var{x} is nearly zero, compared to @var{y}. This
33419 checks whether @var{x} plus @var{y} would by be @code{nearly-equal}
33420 to @var{y} itself, to within the current precision, in other words,
33421 if adding @var{x} to @var{y} would have a negligible effect on @var{y}
33422 due to roundoff error. @var{X} may be a real or complex number, but
33423 @var{y} must be real.
33427 Return true if the formula @var{x} represents a true value in
33428 Calc, not Lisp, terms. It tests if @var{x} is a non-zero number
33429 or a provably non-zero formula.
33432 @defun reject-arg val pred
33433 Abort the current function evaluation due to unacceptable argument values.
33434 This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a
33435 Lisp error which @code{normalize} will trap. The net effect is that the
33436 function call which led here will be left in symbolic form.
33439 @defun inexact-value
33440 If Symbolic mode is enabled, this will signal an error that causes
33441 @code{normalize} to leave the formula in symbolic form, with the message
33442 ``Inexact result.'' (This function has no effect when not in Symbolic mode.)
33443 Note that if your function calls @samp{(sin 5)} in Symbolic mode, the
33444 @code{sin} function will call @code{inexact-value}, which will cause your
33445 function to be left unsimplified. You may instead wish to call
33446 @samp{(normalize (list 'calcFunc-sin 5))}, which in Symbolic mode will
33447 return the formula @samp{sin(5)} to your function.
33451 This signals an error that will be reported as a floating-point overflow.
33455 This signals a floating-point underflow.
33458 @node Computational Lisp Functions, Vector Lisp Functions, Predicates, Internals
33459 @subsubsection Computational Functions
33462 The functions described here do the actual computational work of the
33463 Calculator. In addition to these, note that any function described in
33464 the main body of this manual may be called from Lisp; for example, if
33465 the documentation refers to the @code{calc-sqrt} [@code{sqrt}] command,
33466 this means @code{calc-sqrt} is an interactive stack-based square-root
33467 command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt})
33468 is the actual Lisp function for taking square roots.
33470 The functions @code{math-add}, @code{math-sub}, @code{math-mul},
33471 @code{math-div}, @code{math-mod}, and @code{math-neg} are not included
33472 in this list, since @code{defmath} allows you to write native Lisp
33473 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-},
33474 respectively, instead.
33476 @defun normalize val
33477 (Full form: @code{math-normalize}.)
33478 Reduce the value @var{val} to standard form. For example, if @var{val}
33479 is a fixnum, it will be converted to a bignum if it is too large, and
33480 if @var{val} is a bignum it will be normalized by clipping off trailing
33481 (i.e., most-significant) zero digits and converting to a fixnum if it is
33482 small. All the various data types are similarly converted to their standard
33483 forms. Variables are left alone, but function calls are actually evaluated
33484 in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will
33487 If a function call fails, because the function is void or has the wrong
33488 number of parameters, or because it returns @code{nil} or calls
33489 @code{reject-arg} or @code{inexact-result}, @code{normalize} returns
33490 the formula still in symbolic form.
33492 If the current simplification mode is ``none'' or ``numeric arguments
33493 only,'' @code{normalize} will act appropriately. However, the more
33494 powerful simplification modes (like Algebraic Simplification) are
33495 not handled by @code{normalize}. They are handled by @code{calc-normalize},
33496 which calls @code{normalize} and possibly some other routines, such
33497 as @code{simplify} or @code{simplify-units}. Programs generally will
33498 never call @code{calc-normalize} except when popping or pushing values
33502 @defun evaluate-expr expr
33503 Replace all variables in @var{expr} that have values with their values,
33504 then use @code{normalize} to simplify the result. This is what happens
33505 when you press the @kbd{=} key interactively.
33508 @defmac with-extra-prec n body
33509 Evaluate the Lisp forms in @var{body} with precision increased by @var{n}
33510 digits. This is a macro which expands to
33514 (let ((calc-internal-prec (+ calc-internal-prec @var{n})))
33518 The surrounding call to @code{math-normalize} causes a floating-point
33519 result to be rounded down to the original precision afterwards. This
33520 is important because some arithmetic operations assume a number's
33521 mantissa contains no more digits than the current precision allows.
33524 @defun make-frac n d
33525 Build a fraction @samp{@var{n}:@var{d}}. This is equivalent to calling
33526 @samp{(normalize (list 'frac @var{n} @var{d}))}, but more efficient.
33529 @defun make-float mant exp
33530 Build a floating-point value out of @var{mant} and @var{exp}, both
33531 of which are arbitrary integers. This function will return a
33532 properly normalized float value, or signal an overflow or underflow
33533 if @var{exp} is out of range.
33536 @defun make-sdev x sigma
33537 Build an error form out of @var{x} and the absolute value of @var{sigma}.
33538 If @var{sigma} is zero, the result is the number @var{x} directly.
33539 If @var{sigma} is negative or complex, its absolute value is used.
33540 If @var{x} or @var{sigma} is not a valid type of object for use in
33541 error forms, this calls @code{reject-arg}.
33544 @defun make-intv mask lo hi
33545 Build an interval form out of @var{mask} (which is assumed to be an
33546 integer from 0 to 3), and the limits @var{lo} and @var{hi}. If
33547 @var{lo} is greater than @var{hi}, an empty interval form is returned.
33548 This calls @code{reject-arg} if @var{lo} or @var{hi} is unsuitable.
33551 @defun sort-intv mask lo hi
33552 Build an interval form, similar to @code{make-intv}, except that if
33553 @var{lo} is less than @var{hi} they are simply exchanged, and the
33554 bits of @var{mask} are swapped accordingly.
33557 @defun make-mod n m
33558 Build a modulo form out of @var{n} and the modulus @var{m}. Since modulo
33559 forms do not allow formulas as their components, if @var{n} or @var{m}
33560 is not a real number or HMS form the result will be a formula which
33561 is a call to @code{makemod}, the algebraic version of this function.
33565 Convert @var{x} to floating-point form. Integers and fractions are
33566 converted to numerically equivalent floats; components of complex
33567 numbers, vectors, HMS forms, date forms, error forms, intervals, and
33568 modulo forms are recursively floated. If the argument is a variable
33569 or formula, this calls @code{reject-arg}.
33573 Compare the numbers @var{x} and @var{y}, and return @mathit{-1} if
33574 @samp{(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})},
33575 0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is
33576 undefined or cannot be determined.
33580 Return the number of digits of integer @var{n}, effectively
33581 @samp{ceil(log10(@var{n}))}, but much more efficient. Zero is
33582 considered to have zero digits.
33585 @defun scale-int x n
33586 Shift integer @var{x} left @var{n} decimal digits, or right @mathit{-@var{n}}
33587 digits with truncation toward zero.
33590 @defun scale-rounding x n
33591 Like @code{scale-int}, except that a right shift rounds to the nearest
33592 integer rather than truncating.
33596 Return the integer @var{n} as a fixnum, i.e., a native Lisp integer.
33597 If @var{n} is outside the permissible range for Lisp integers (usually
33598 24 binary bits) the result is undefined.
33602 Compute the square of @var{x}; short for @samp{(* @var{x} @var{x})}.
33605 @defun quotient x y
33606 Divide integer @var{x} by integer @var{y}; return an integer quotient
33607 and discard the remainder. If @var{x} or @var{y} is negative, the
33608 direction of rounding is undefined.
33612 Perform an integer division; if @var{x} and @var{y} are both nonnegative
33613 integers, this uses the @code{quotient} function, otherwise it computes
33614 @samp{floor(@var{x}/@var{y})}. Thus the result is well-defined but
33615 slower than for @code{quotient}.
33619 Divide integer @var{x} by integer @var{y}; return the integer remainder
33620 and discard the quotient. Like @code{quotient}, this works only for
33621 integer arguments and is not well-defined for negative arguments.
33622 For a more well-defined result, use @samp{(% @var{x} @var{y})}.
33626 Divide integer @var{x} by integer @var{y}; return a cons cell whose
33627 @code{car} is @samp{(quotient @var{x} @var{y})} and whose @code{cdr}
33628 is @samp{(imod @var{x} @var{y})}.
33632 Compute @var{x} to the power @var{y}. In @code{defmath} code, this can
33633 also be written @samp{(^ @var{x} @var{y})} or
33634 @w{@samp{(expt @var{x} @var{y})}}.
33637 @defun abs-approx x
33638 Compute a fast approximation to the absolute value of @var{x}. For
33639 example, for a rectangular complex number the result is the sum of
33640 the absolute values of the components.
33646 @findex pi-over-180
33647 @findex sqrt-two-pi
33653 The function @samp{(pi)} computes @samp{pi} to the current precision.
33654 Other related constant-generating functions are @code{two-pi},
33655 @code{pi-over-2}, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi},
33656 @code{e}, @code{sqrt-e}, @code{ln-2}, and @code{ln-10}. Each function
33657 returns a floating-point value in the current precision, and each uses
33658 caching so that all calls after the first are essentially free.
33661 @defmac math-defcache @var{func} @var{initial} @var{form}
33662 This macro, usually used as a top-level call like @code{defun} or
33663 @code{defvar}, defines a new cached constant analogous to @code{pi}, etc.
33664 It defines a function @code{func} which returns the requested value;
33665 if @var{initial} is non-@code{nil} it must be a @samp{(float @dots{})}
33666 form which serves as an initial value for the cache. If @var{func}
33667 is called when the cache is empty or does not have enough digits to
33668 satisfy the current precision, the Lisp expression @var{form} is evaluated
33669 with the current precision increased by four, and the result minus its
33670 two least significant digits is stored in the cache. For example,
33671 calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34
33672 digits, rounds it down to 32 digits for future use, then rounds it
33673 again to 30 digits for use in the present request.
33676 @findex half-circle
33677 @findex quarter-circle
33678 @defun full-circle symb
33679 If the current angular mode is Degrees or HMS, this function returns the
33680 integer 360. In Radians mode, this function returns either the
33681 corresponding value in radians to the current precision, or the formula
33682 @samp{2*pi}, depending on the Symbolic mode. There are also similar
33683 function @code{half-circle} and @code{quarter-circle}.
33686 @defun power-of-2 n
33687 Compute two to the integer power @var{n}, as a (potentially very large)
33688 integer. Powers of two are cached, so only the first call for a
33689 particular @var{n} is expensive.
33692 @defun integer-log2 n
33693 Compute the base-2 logarithm of @var{n}, which must be an integer which
33694 is a power of two. If @var{n} is not a power of two, this function will
33698 @defun div-mod a b m
33699 Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if
33700 there is no solution, or if any of the arguments are not integers.
33703 @defun pow-mod a b m
33704 Compute @var{a} to the power @var{b}, modulo @var{m}. If @var{a},
33705 @var{b}, and @var{m} are integers, this uses an especially efficient
33706 algorithm. Otherwise, it simply computes @samp{(% (^ a b) m)}.
33710 Compute the integer square root of @var{n}. This is the square root
33711 of @var{n} rounded down toward zero, i.e., @samp{floor(sqrt(@var{n}))}.
33712 If @var{n} is itself an integer, the computation is especially efficient.
33715 @defun to-hms a ang
33716 Convert the argument @var{a} into an HMS form. If @var{ang} is specified,
33717 it is the angular mode in which to interpret @var{a}, either @code{deg}
33718 or @code{rad}. Otherwise, the current angular mode is used. If @var{a}
33719 is already an HMS form it is returned as-is.
33722 @defun from-hms a ang
33723 Convert the HMS form @var{a} into a real number. If @var{ang} is specified,
33724 it is the angular mode in which to express the result, otherwise the
33725 current angular mode is used. If @var{a} is already a real number, it
33729 @defun to-radians a
33730 Convert the number or HMS form @var{a} to radians from the current
33734 @defun from-radians a
33735 Convert the number @var{a} from radians to the current angular mode.
33736 If @var{a} is a formula, this returns the formula @samp{deg(@var{a})}.
33739 @defun to-radians-2 a
33740 Like @code{to-radians}, except that in Symbolic mode a degrees to
33741 radians conversion yields a formula like @samp{@var{a}*pi/180}.
33744 @defun from-radians-2 a
33745 Like @code{from-radians}, except that in Symbolic mode a radians to
33746 degrees conversion yields a formula like @samp{@var{a}*180/pi}.
33749 @defun random-digit
33750 Produce a random base-1000 digit in the range 0 to 999.
33753 @defun random-digits n
33754 Produce a random @var{n}-digit integer; this will be an integer
33755 in the interval @samp{[0, 10^@var{n})}.
33758 @defun random-float
33759 Produce a random float in the interval @samp{[0, 1)}.
33762 @defun prime-test n iters
33763 Determine whether the integer @var{n} is prime. Return a list which has
33764 one of these forms: @samp{(nil @var{f})} means the number is non-prime
33765 because it was found to be divisible by @var{f}; @samp{(nil)} means it
33766 was found to be non-prime by table look-up (so no factors are known);
33767 @samp{(nil unknown)} means it is definitely non-prime but no factors
33768 are known because @var{n} was large enough that Fermat's probabilistic
33769 test had to be used; @samp{(t)} means the number is definitely prime;
33770 and @samp{(maybe @var{i} @var{p})} means that Fermat's test, after @var{i}
33771 iterations, is @var{p} percent sure that the number is prime. The
33772 @var{iters} parameter is the number of Fermat iterations to use, in the
33773 case that this is necessary. If @code{prime-test} returns ``maybe,''
33774 you can call it again with the same @var{n} to get a greater certainty;
33775 @code{prime-test} remembers where it left off.
33778 @defun to-simple-fraction f
33779 If @var{f} is a floating-point number which can be represented exactly
33780 as a small rational number. return that number, else return @var{f}.
33781 For example, 0.75 would be converted to 3:4. This function is very
33785 @defun to-fraction f tol
33786 Find a rational approximation to floating-point number @var{f} to within
33787 a specified tolerance @var{tol}; this corresponds to the algebraic
33788 function @code{frac}, and can be rather slow.
33791 @defun quarter-integer n
33792 If @var{n} is an integer or integer-valued float, this function
33793 returns zero. If @var{n} is a half-integer (i.e., an integer plus
33794 @mathit{1:2} or 0.5), it returns 2. If @var{n} is a quarter-integer,
33795 it returns 1 or 3. If @var{n} is anything else, this function
33796 returns @code{nil}.
33799 @node Vector Lisp Functions, Symbolic Lisp Functions, Computational Lisp Functions, Internals
33800 @subsubsection Vector Functions
33803 The functions described here perform various operations on vectors and
33806 @defun math-concat x y
33807 Do a vector concatenation; this operation is written @samp{@var{x} | @var{y}}
33808 in a symbolic formula. @xref{Building Vectors}.
33811 @defun vec-length v
33812 Return the length of vector @var{v}. If @var{v} is not a vector, the
33813 result is zero. If @var{v} is a matrix, this returns the number of
33814 rows in the matrix.
33817 @defun mat-dimens m
33818 Determine the dimensions of vector or matrix @var{m}. If @var{m} is not
33819 a vector, the result is an empty list. If @var{m} is a plain vector
33820 but not a matrix, the result is a one-element list containing the length
33821 of the vector. If @var{m} is a matrix with @var{r} rows and @var{c} columns,
33822 the result is the list @samp{(@var{r} @var{c})}. Higher-order tensors
33823 produce lists of more than two dimensions. Note that the object
33824 @samp{[[1, 2, 3], [4, 5]]} is a vector of vectors not all the same size,
33825 and is treated by this and other Calc routines as a plain vector of two
33829 @defun dimension-error
33830 Abort the current function with a message of ``Dimension error.''
33831 The Calculator will leave the function being evaluated in symbolic
33832 form; this is really just a special case of @code{reject-arg}.
33835 @defun build-vector args
33836 Return a Calc vector with @var{args} as elements.
33837 For example, @samp{(build-vector 1 2 3)} returns the Calc vector
33838 @samp{[1, 2, 3]}, stored internally as the list @samp{(vec 1 2 3)}.
33841 @defun make-vec obj dims
33842 Return a Calc vector or matrix all of whose elements are equal to
33843 @var{obj}. For example, @samp{(make-vec 27 3 4)} returns a 3x4 matrix
33847 @defun row-matrix v
33848 If @var{v} is a plain vector, convert it into a row matrix, i.e.,
33849 a matrix whose single row is @var{v}. If @var{v} is already a matrix,
33853 @defun col-matrix v
33854 If @var{v} is a plain vector, convert it into a column matrix, i.e., a
33855 matrix with each element of @var{v} as a separate row. If @var{v} is
33856 already a matrix, leave it alone.
33860 Map the Lisp function @var{f} over the Calc vector @var{v}. For example,
33861 @samp{(map-vec 'math-floor v)} returns a vector of the floored components
33865 @defun map-vec-2 f a b
33866 Map the Lisp function @var{f} over the two vectors @var{a} and @var{b}.
33867 If @var{a} and @var{b} are vectors of equal length, the result is a
33868 vector of the results of calling @samp{(@var{f} @var{ai} @var{bi})}
33869 for each pair of elements @var{ai} and @var{bi}. If either @var{a} or
33870 @var{b} is a scalar, it is matched with each value of the other vector.
33871 For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v}
33872 with each element increased by one. Note that using @samp{'+} would not
33873 work here, since @code{defmath} does not expand function names everywhere,
33874 just where they are in the function position of a Lisp expression.
33877 @defun reduce-vec f v
33878 Reduce the function @var{f} over the vector @var{v}. For example, if
33879 @var{v} is @samp{[10, 20, 30, 40]}, this calls @samp{(f (f (f 10 20) 30) 40)}.
33880 If @var{v} is a matrix, this reduces over the rows of @var{v}.
33883 @defun reduce-cols f m
33884 Reduce the function @var{f} over the columns of matrix @var{m}. For
33885 example, if @var{m} is @samp{[[1, 2], [3, 4], [5, 6]]}, the result
33886 is a vector of the two elements @samp{(f (f 1 3) 5)} and @samp{(f (f 2 4) 6)}.
33890 Return the @var{n}th row of matrix @var{m}. This is equivalent to
33891 @samp{(elt m n)}. For a slower but safer version, use @code{mrow}.
33892 (@xref{Extracting Elements}.)
33896 Return the @var{n}th column of matrix @var{m}, in the form of a vector.
33897 The arguments are not checked for correctness.
33900 @defun mat-less-row m n
33901 Return a copy of matrix @var{m} with its @var{n}th row deleted. The
33902 number @var{n} must be in range from 1 to the number of rows in @var{m}.
33905 @defun mat-less-col m n
33906 Return a copy of matrix @var{m} with its @var{n}th column deleted.
33910 Return the transpose of matrix @var{m}.
33913 @defun flatten-vector v
33914 Flatten nested vector @var{v} into a vector of scalars. For example,
33915 if @var{v} is @samp{[[1, 2, 3], [4, 5]]} the result is @samp{[1, 2, 3, 4, 5]}.
33918 @defun copy-matrix m
33919 If @var{m} is a matrix, return a copy of @var{m}. This maps
33920 @code{copy-sequence} over the rows of @var{m}; in Lisp terms, each
33921 element of the result matrix will be @code{eq} to the corresponding
33922 element of @var{m}, but none of the @code{cons} cells that make up
33923 the structure of the matrix will be @code{eq}. If @var{m} is a plain
33924 vector, this is the same as @code{copy-sequence}.
33927 @defun swap-rows m r1 r2
33928 Exchange rows @var{r1} and @var{r2} of matrix @var{m} in-place. In
33929 other words, unlike most of the other functions described here, this
33930 function changes @var{m} itself rather than building up a new result
33931 matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)}
33932 is true, with the side effect of exchanging the first two rows of
33936 @node Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp Functions, Internals
33937 @subsubsection Symbolic Functions
33940 The functions described here operate on symbolic formulas in the
33943 @defun calc-prepare-selection num
33944 Prepare a stack entry for selection operations. If @var{num} is
33945 omitted, the stack entry containing the cursor is used; otherwise,
33946 it is the number of the stack entry to use. This function stores
33947 useful information about the current stack entry into a set of
33948 variables. @code{calc-selection-cache-num} contains the number of
33949 the stack entry involved (equal to @var{num} if you specified it);
33950 @code{calc-selection-cache-entry} contains the stack entry as a
33951 list (such as @code{calc-top-list} would return with @code{entry}
33952 as the selection mode); and @code{calc-selection-cache-comp} contains
33953 a special ``tagged'' composition (@pxref{Formatting Lisp Functions})
33954 which allows Calc to relate cursor positions in the buffer with
33955 their corresponding sub-formulas.
33957 A slight complication arises in the selection mechanism because
33958 formulas may contain small integers. For example, in the vector
33959 @samp{[1, 2, 1]} the first and last elements are @code{eq} to each
33960 other; selections are recorded as the actual Lisp object that
33961 appears somewhere in the tree of the whole formula, but storing
33962 @code{1} would falsely select both @code{1}'s in the vector. So
33963 @code{calc-prepare-selection} also checks the stack entry and
33964 replaces any plain integers with ``complex number'' lists of the form
33965 @samp{(cplx @var{n} 0)}. This list will be displayed the same as a
33966 plain @var{n} and the change will be completely invisible to the
33967 user, but it will guarantee that no two sub-formulas of the stack
33968 entry will be @code{eq} to each other. Next time the stack entry
33969 is involved in a computation, @code{calc-normalize} will replace
33970 these lists with plain numbers again, again invisibly to the user.
33973 @defun calc-encase-atoms x
33974 This modifies the formula @var{x} to ensure that each part of the
33975 formula is a unique atom, using the @samp{(cplx @var{n} 0)} trick
33976 described above. This function may use @code{setcar} to modify
33977 the formula in-place.
33980 @defun calc-find-selected-part
33981 Find the smallest sub-formula of the current formula that contains
33982 the cursor. This assumes @code{calc-prepare-selection} has been
33983 called already. If the cursor is not actually on any part of the
33984 formula, this returns @code{nil}.
33987 @defun calc-change-current-selection selection
33988 Change the currently prepared stack element's selection to
33989 @var{selection}, which should be @code{eq} to some sub-formula
33990 of the stack element, or @code{nil} to unselect the formula.
33991 The stack element's appearance in the Calc buffer is adjusted
33992 to reflect the new selection.
33995 @defun calc-find-nth-part expr n
33996 Return the @var{n}th sub-formula of @var{expr}. This function is used
33997 by the selection commands, and (unless @kbd{j b} has been used) treats
33998 sums and products as flat many-element formulas. Thus if @var{expr}
33999 is @samp{((a + b) - c) + d}, calling @code{calc-find-nth-part} with
34000 @var{n} equal to four will return @samp{d}.
34003 @defun calc-find-parent-formula expr part
34004 Return the sub-formula of @var{expr} which immediately contains
34005 @var{part}. If @var{expr} is @samp{a*b + (c+1)*d} and @var{part}
34006 is @code{eq} to the @samp{c+1} term of @var{expr}, then this function
34007 will return @samp{(c+1)*d}. If @var{part} turns out not to be a
34008 sub-formula of @var{expr}, the function returns @code{nil}. If
34009 @var{part} is @code{eq} to @var{expr}, the function returns @code{t}.
34010 This function does not take associativity into account.
34013 @defun calc-find-assoc-parent-formula expr part
34014 This is the same as @code{calc-find-parent-formula}, except that
34015 (unless @kbd{j b} has been used) it continues widening the selection
34016 to contain a complete level of the formula. Given @samp{a} from
34017 @samp{((a + b) - c) + d}, @code{calc-find-parent-formula} will
34018 return @samp{a + b} but @code{calc-find-assoc-parent-formula} will
34019 return the whole expression.
34022 @defun calc-grow-assoc-formula expr part
34023 This expands sub-formula @var{part} of @var{expr} to encompass a
34024 complete level of the formula. If @var{part} and its immediate
34025 parent are not compatible associative operators, or if @kbd{j b}
34026 has been used, this simply returns @var{part}.
34029 @defun calc-find-sub-formula expr part
34030 This finds the immediate sub-formula of @var{expr} which contains
34031 @var{part}. It returns an index @var{n} such that
34032 @samp{(calc-find-nth-part @var{expr} @var{n})} would return @var{part}.
34033 If @var{part} is not a sub-formula of @var{expr}, it returns @code{nil}.
34034 If @var{part} is @code{eq} to @var{expr}, it returns @code{t}. This
34035 function does not take associativity into account.
34038 @defun calc-replace-sub-formula expr old new
34039 This function returns a copy of formula @var{expr}, with the
34040 sub-formula that is @code{eq} to @var{old} replaced by @var{new}.
34043 @defun simplify expr
34044 Simplify the expression @var{expr} by applying various algebraic rules.
34045 This is what the @w{@kbd{a s}} (@code{calc-simplify}) command uses. This
34046 always returns a copy of the expression; the structure @var{expr} points
34047 to remains unchanged in memory.
34049 More precisely, here is what @code{simplify} does: The expression is
34050 first normalized and evaluated by calling @code{normalize}. If any
34051 @code{AlgSimpRules} have been defined, they are then applied. Then
34052 the expression is traversed in a depth-first, bottom-up fashion; at
34053 each level, any simplifications that can be made are made until no
34054 further changes are possible. Once the entire formula has been
34055 traversed in this way, it is compared with the original formula (from
34056 before the call to @code{normalize}) and, if it has changed,
34057 the entire procedure is repeated (starting with @code{normalize})
34058 until no further changes occur. Usually only two iterations are
34059 needed:@: one to simplify the formula, and another to verify that no
34060 further simplifications were possible.
34063 @defun simplify-extended expr
34064 Simplify the expression @var{expr}, with additional rules enabled that
34065 help do a more thorough job, while not being entirely ``safe'' in all
34066 circumstances. (For example, this mode will simplify @samp{sqrt(x^2)}
34067 to @samp{x}, which is only valid when @var{x} is positive.) This is
34068 implemented by temporarily binding the variable @code{math-living-dangerously}
34069 to @code{t} (using a @code{let} form) and calling @code{simplify}.
34070 Dangerous simplification rules are written to check this variable
34071 before taking any action.
34074 @defun simplify-units expr
34075 Simplify the expression @var{expr}, treating variable names as units
34076 whenever possible. This works by binding the variable
34077 @code{math-simplifying-units} to @code{t} while calling @code{simplify}.
34080 @defmac math-defsimplify funcs body
34081 Register a new simplification rule; this is normally called as a top-level
34082 form, like @code{defun} or @code{defmath}. If @var{funcs} is a symbol
34083 (like @code{+} or @code{calcFunc-sqrt}), this simplification rule is
34084 applied to the formulas which are calls to the specified function. Or,
34085 @var{funcs} can be a list of such symbols; the rule applies to all
34086 functions on the list. The @var{body} is written like the body of a
34087 function with a single argument called @code{expr}. The body will be
34088 executed with @code{expr} bound to a formula which is a call to one of
34089 the functions @var{funcs}. If the function body returns @code{nil}, or
34090 if it returns a result @code{equal} to the original @code{expr}, it is
34091 ignored and Calc goes on to try the next simplification rule that applies.
34092 If the function body returns something different, that new formula is
34093 substituted for @var{expr} in the original formula.
34095 At each point in the formula, rules are tried in the order of the
34096 original calls to @code{math-defsimplify}; the search stops after the
34097 first rule that makes a change. Thus later rules for that same
34098 function will not have a chance to trigger until the next iteration
34099 of the main @code{simplify} loop.
34101 Note that, since @code{defmath} is not being used here, @var{body} must
34102 be written in true Lisp code without the conveniences that @code{defmath}
34103 provides. If you prefer, you can have @var{body} simply call another
34104 function (defined with @code{defmath}) which does the real work.
34106 The arguments of a function call will already have been simplified
34107 before any rules for the call itself are invoked. Since a new argument
34108 list is consed up when this happens, this means that the rule's body is
34109 allowed to rearrange the function's arguments destructively if that is
34110 convenient. Here is a typical example of a simplification rule:
34113 (math-defsimplify calcFunc-arcsinh
34114 (or (and (math-looks-negp (nth 1 expr))
34115 (math-neg (list 'calcFunc-arcsinh
34116 (math-neg (nth 1 expr)))))
34117 (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh)
34118 (or math-living-dangerously
34119 (math-known-realp (nth 1 (nth 1 expr))))
34120 (nth 1 (nth 1 expr)))))
34123 This is really a pair of rules written with one @code{math-defsimplify}
34124 for convenience; the first replaces @samp{arcsinh(-x)} with
34125 @samp{-arcsinh(x)}, and the second, which is safe only for real @samp{x},
34126 replaces @samp{arcsinh(sinh(x))} with @samp{x}.
34129 @defun common-constant-factor expr
34130 Check @var{expr} to see if it is a sum of terms all multiplied by the
34131 same rational value. If so, return this value. If not, return @code{nil}.
34132 For example, if called on @samp{6x + 9y + 12z}, it would return 3, since
34133 3 is a common factor of all the terms.
34136 @defun cancel-common-factor expr factor
34137 Assuming @var{expr} is a sum with @var{factor} as a common factor,
34138 divide each term of the sum by @var{factor}. This is done by
34139 destructively modifying parts of @var{expr}, on the assumption that
34140 it is being used by a simplification rule (where such things are
34141 allowed; see above). For example, consider this built-in rule for
34145 (math-defsimplify calcFunc-sqrt
34146 (let ((fac (math-common-constant-factor (nth 1 expr))))
34147 (and fac (not (eq fac 1))
34148 (math-mul (math-normalize (list 'calcFunc-sqrt fac))
34150 (list 'calcFunc-sqrt
34151 (math-cancel-common-factor
34152 (nth 1 expr) fac)))))))
34156 @defun frac-gcd a b
34157 Compute a ``rational GCD'' of @var{a} and @var{b}, which must both be
34158 rational numbers. This is the fraction composed of the GCD of the
34159 numerators of @var{a} and @var{b}, over the GCD of the denominators.
34160 It is used by @code{common-constant-factor}. Note that the standard
34161 @code{gcd} function uses the LCM to combine the denominators.
34164 @defun map-tree func expr many
34165 Try applying Lisp function @var{func} to various sub-expressions of
34166 @var{expr}. Initially, call @var{func} with @var{expr} itself as an
34167 argument. If this returns an expression which is not @code{equal} to
34168 @var{expr}, apply @var{func} again until eventually it does return
34169 @var{expr} with no changes. Then, if @var{expr} is a function call,
34170 recursively apply @var{func} to each of the arguments. This keeps going
34171 until no changes occur anywhere in the expression; this final expression
34172 is returned by @code{map-tree}. Note that, unlike simplification rules,
34173 @var{func} functions may @emph{not} make destructive changes to
34174 @var{expr}. If a third argument @var{many} is provided, it is an
34175 integer which says how many times @var{func} may be applied; the
34176 default, as described above, is infinitely many times.
34179 @defun compile-rewrites rules
34180 Compile the rewrite rule set specified by @var{rules}, which should
34181 be a formula that is either a vector or a variable name. If the latter,
34182 the compiled rules are saved so that later @code{compile-rules} calls
34183 for that same variable can return immediately. If there are problems
34184 with the rules, this function calls @code{error} with a suitable
34188 @defun apply-rewrites expr crules heads
34189 Apply the compiled rewrite rule set @var{crules} to the expression
34190 @var{expr}. This will make only one rewrite and only checks at the
34191 top level of the expression. The result @code{nil} if no rules
34192 matched, or if the only rules that matched did not actually change
34193 the expression. The @var{heads} argument is optional; if is given,
34194 it should be a list of all function names that (may) appear in
34195 @var{expr}. The rewrite compiler tags each rule with the
34196 rarest-looking function name in the rule; if you specify @var{heads},
34197 @code{apply-rewrites} can use this information to narrow its search
34198 down to just a few rules in the rule set.
34201 @defun rewrite-heads expr
34202 Compute a @var{heads} list for @var{expr} suitable for use with
34203 @code{apply-rewrites}, as discussed above.
34206 @defun rewrite expr rules many
34207 This is an all-in-one rewrite function. It compiles the rule set
34208 specified by @var{rules}, then uses @code{map-tree} to apply the
34209 rules throughout @var{expr} up to @var{many} (default infinity)
34213 @defun match-patterns pat vec not-flag
34214 Given a Calc vector @var{vec} and an uncompiled pattern set or
34215 pattern set variable @var{pat}, this function returns a new vector
34216 of all elements of @var{vec} which do (or don't, if @var{not-flag} is
34217 non-@code{nil}) match any of the patterns in @var{pat}.
34220 @defun deriv expr var value symb
34221 Compute the derivative of @var{expr} with respect to variable @var{var}
34222 (which may actually be any sub-expression). If @var{value} is specified,
34223 the derivative is evaluated at the value of @var{var}; otherwise, the
34224 derivative is left in terms of @var{var}. If the expression contains
34225 functions for which no derivative formula is known, new derivative
34226 functions are invented by adding primes to the names; @pxref{Calculus}.
34227 However, if @var{symb} is non-@code{nil}, the presence of undifferentiable
34228 functions in @var{expr} instead cancels the whole differentiation, and
34229 @code{deriv} returns @code{nil} instead.
34231 Derivatives of an @var{n}-argument function can be defined by
34232 adding a @code{math-derivative-@var{n}} property to the property list
34233 of the symbol for the function's derivative, which will be the
34234 function name followed by an apostrophe. The value of the property
34235 should be a Lisp function; it is called with the same arguments as the
34236 original function call that is being differentiated. It should return
34237 a formula for the derivative. For example, the derivative of @code{ln}
34241 (put 'calcFunc-ln\' 'math-derivative-1
34242 (function (lambda (u) (math-div 1 u))))
34245 The two-argument @code{log} function has two derivatives,
34247 (put 'calcFunc-log\' 'math-derivative-2 ; d(log(x,b)) / dx
34248 (function (lambda (x b) ... )))
34249 (put 'calcFunc-log\'2 'math-derivative-2 ; d(log(x,b)) / db
34250 (function (lambda (x b) ... )))
34254 @defun tderiv expr var value symb
34255 Compute the total derivative of @var{expr}. This is the same as
34256 @code{deriv}, except that variables other than @var{var} are not
34257 assumed to be constant with respect to @var{var}.
34260 @defun integ expr var low high
34261 Compute the integral of @var{expr} with respect to @var{var}.
34262 @xref{Calculus}, for further details.
34265 @defmac math-defintegral funcs body
34266 Define a rule for integrating a function or functions of one argument;
34267 this macro is very similar in format to @code{math-defsimplify}.
34268 The main difference is that here @var{body} is the body of a function
34269 with a single argument @code{u} which is bound to the argument to the
34270 function being integrated, not the function call itself. Also, the
34271 variable of integration is available as @code{math-integ-var}. If
34272 evaluation of the integral requires doing further integrals, the body
34273 should call @samp{(math-integral @var{x})} to find the integral of
34274 @var{x} with respect to @code{math-integ-var}; this function returns
34275 @code{nil} if the integral could not be done. Some examples:
34278 (math-defintegral calcFunc-conj
34279 (let ((int (math-integral u)))
34281 (list 'calcFunc-conj int))))
34283 (math-defintegral calcFunc-cos
34284 (and (equal u math-integ-var)
34285 (math-from-radians-2 (list 'calcFunc-sin u))))
34288 In the @code{cos} example, we define only the integral of @samp{cos(x) dx},
34289 relying on the general integration-by-substitution facility to handle
34290 cosines of more complicated arguments. An integration rule should return
34291 @code{nil} if it can't do the integral; if several rules are defined for
34292 the same function, they are tried in order until one returns a non-@code{nil}
34296 @defmac math-defintegral-2 funcs body
34297 Define a rule for integrating a function or functions of two arguments.
34298 This is exactly analogous to @code{math-defintegral}, except that @var{body}
34299 is written as the body of a function with two arguments, @var{u} and
34303 @defun solve-for lhs rhs var full
34304 Attempt to solve the equation @samp{@var{lhs} = @var{rhs}} by isolating
34305 the variable @var{var} on the lefthand side; return the resulting righthand
34306 side, or @code{nil} if the equation cannot be solved. The variable
34307 @var{var} must appear at least once in @var{lhs} or @var{rhs}. Note that
34308 the return value is a formula which does not contain @var{var}; this is
34309 different from the user-level @code{solve} and @code{finv} functions,
34310 which return a rearranged equation or a functional inverse, respectively.
34311 If @var{full} is non-@code{nil}, a full solution including dummy signs
34312 and dummy integers will be produced. User-defined inverses are provided
34313 as properties in a manner similar to derivatives:
34316 (put 'calcFunc-ln 'math-inverse
34317 (function (lambda (x) (list 'calcFunc-exp x))))
34320 This function can call @samp{(math-solve-get-sign @var{x})} to create
34321 a new arbitrary sign variable, returning @var{x} times that sign, and
34322 @samp{(math-solve-get-int @var{x})} to create a new arbitrary integer
34323 variable multiplied by @var{x}. These functions simply return @var{x}
34324 if the caller requested a non-``full'' solution.
34327 @defun solve-eqn expr var full
34328 This version of @code{solve-for} takes an expression which will
34329 typically be an equation or inequality. (If it is not, it will be
34330 interpreted as the equation @samp{@var{expr} = 0}.) It returns an
34331 equation or inequality, or @code{nil} if no solution could be found.
34334 @defun solve-system exprs vars full
34335 This function solves a system of equations. Generally, @var{exprs}
34336 and @var{vars} will be vectors of equal length.
34337 @xref{Solving Systems of Equations}, for other options.
34340 @defun expr-contains expr var
34341 Returns a non-@code{nil} value if @var{var} occurs as a subexpression
34344 This function might seem at first to be identical to
34345 @code{calc-find-sub-formula}. The key difference is that
34346 @code{expr-contains} uses @code{equal} to test for matches, whereas
34347 @code{calc-find-sub-formula} uses @code{eq}. In the formula
34348 @samp{f(a, a)}, the two @samp{a}s will be @code{equal} but not
34349 @code{eq} to each other.
34352 @defun expr-contains-count expr var
34353 Returns the number of occurrences of @var{var} as a subexpression
34354 of @var{expr}, or @code{nil} if there are no occurrences.
34357 @defun expr-depends expr var
34358 Returns true if @var{expr} refers to any variable the occurs in @var{var}.
34359 In other words, it checks if @var{expr} and @var{var} have any variables
34363 @defun expr-contains-vars expr
34364 Return true if @var{expr} contains any variables, or @code{nil} if @var{expr}
34365 contains only constants and functions with constant arguments.
34368 @defun expr-subst expr old new
34369 Returns a copy of @var{expr}, with all occurrences of @var{old} replaced
34370 by @var{new}. This treats @code{lambda} forms specially with respect
34371 to the dummy argument variables, so that the effect is always to return
34372 @var{expr} evaluated at @var{old} = @var{new}.
34375 @defun multi-subst expr old new
34376 This is like @code{expr-subst}, except that @var{old} and @var{new}
34377 are lists of expressions to be substituted simultaneously. If one
34378 list is shorter than the other, trailing elements of the longer list
34382 @defun expr-weight expr
34383 Returns the ``weight'' of @var{expr}, basically a count of the total
34384 number of objects and function calls that appear in @var{expr}. For
34385 ``primitive'' objects, this will be one.
34388 @defun expr-height expr
34389 Returns the ``height'' of @var{expr}, which is the deepest level to
34390 which function calls are nested. (Note that @samp{@var{a} + @var{b}}
34391 counts as a function call.) For primitive objects, this returns zero.
34394 @defun polynomial-p expr var
34395 Check if @var{expr} is a polynomial in variable (or sub-expression)
34396 @var{var}. If so, return the degree of the polynomial, that is, the
34397 highest power of @var{var} that appears in @var{expr}. For example,
34398 for @samp{(x^2 + 3)^3 + 4} this would return 6. This function returns
34399 @code{nil} unless @var{expr}, when expanded out by @kbd{a x}
34400 (@code{calc-expand}), would consist of a sum of terms in which @var{var}
34401 appears only raised to nonnegative integer powers. Note that if
34402 @var{var} does not occur in @var{expr}, then @var{expr} is considered
34403 a polynomial of degree 0.
34406 @defun is-polynomial expr var degree loose
34407 Check if @var{expr} is a polynomial in variable or sub-expression
34408 @var{var}, and, if so, return a list representation of the polynomial
34409 where the elements of the list are coefficients of successive powers of
34410 @var{var}: @samp{@var{a} + @var{b} x + @var{c} x^3} would produce the
34411 list @samp{(@var{a} @var{b} 0 @var{c})}, and @samp{(x + 1)^2} would
34412 produce the list @samp{(1 2 1)}. The highest element of the list will
34413 be non-zero, with the special exception that if @var{expr} is the
34414 constant zero, the returned value will be @samp{(0)}. Return @code{nil}
34415 if @var{expr} is not a polynomial in @var{var}. If @var{degree} is
34416 specified, this will not consider polynomials of degree higher than that
34417 value. This is a good precaution because otherwise an input of
34418 @samp{(x+1)^1000} will cause a huge coefficient list to be built. If
34419 @var{loose} is non-@code{nil}, then a looser definition of a polynomial
34420 is used in which coefficients are no longer required not to depend on
34421 @var{var}, but are only required not to take the form of polynomials
34422 themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose
34423 polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin
34424 x))}. The result will never be @code{nil} in loose mode, since any
34425 expression can be interpreted as a ``constant'' loose polynomial.
34428 @defun polynomial-base expr pred
34429 Check if @var{expr} is a polynomial in any variable that occurs in it;
34430 if so, return that variable. (If @var{expr} is a multivariate polynomial,
34431 this chooses one variable arbitrarily.) If @var{pred} is specified, it should
34432 be a Lisp function which is called as @samp{(@var{pred} @var{subexpr})},
34433 and which should return true if @code{mpb-top-expr} (a global name for
34434 the original @var{expr}) is a suitable polynomial in @var{subexpr}.
34435 The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})};
34436 you can use @var{pred} to specify additional conditions. Or, you could
34437 have @var{pred} build up a list of every suitable @var{subexpr} that
34441 @defun poly-simplify poly
34442 Simplify polynomial coefficient list @var{poly} by (destructively)
34443 clipping off trailing zeros.
34446 @defun poly-mix a ac b bc
34447 Mix two polynomial lists @var{a} and @var{b} (in the form returned by
34448 @code{is-polynomial}) in a linear combination with coefficient expressions
34449 @var{ac} and @var{bc}. The result is a (not necessarily simplified)
34450 polynomial list representing @samp{@var{ac} @var{a} + @var{bc} @var{b}}.
34453 @defun poly-mul a b
34454 Multiply two polynomial coefficient lists @var{a} and @var{b}. The
34455 result will be in simplified form if the inputs were simplified.
34458 @defun build-polynomial-expr poly var
34459 Construct a Calc formula which represents the polynomial coefficient
34460 list @var{poly} applied to variable @var{var}. The @kbd{a c}
34461 (@code{calc-collect}) command uses @code{is-polynomial} to turn an
34462 expression into a coefficient list, then @code{build-polynomial-expr}
34463 to turn the list back into an expression in regular form.
34466 @defun check-unit-name var
34467 Check if @var{var} is a variable which can be interpreted as a unit
34468 name. If so, return the units table entry for that unit. This
34469 will be a list whose first element is the unit name (not counting
34470 prefix characters) as a symbol and whose second element is the
34471 Calc expression which defines the unit. (Refer to the Calc sources
34472 for details on the remaining elements of this list.) If @var{var}
34473 is not a variable or is not a unit name, return @code{nil}.
34476 @defun units-in-expr-p expr sub-exprs
34477 Return true if @var{expr} contains any variables which can be
34478 interpreted as units. If @var{sub-exprs} is @code{t}, the entire
34479 expression is searched. If @var{sub-exprs} is @code{nil}, this
34480 checks whether @var{expr} is directly a units expression.
34483 @defun single-units-in-expr-p expr
34484 Check whether @var{expr} contains exactly one units variable. If so,
34485 return the units table entry for the variable. If @var{expr} does
34486 not contain any units, return @code{nil}. If @var{expr} contains
34487 two or more units, return the symbol @code{wrong}.
34490 @defun to-standard-units expr which
34491 Convert units expression @var{expr} to base units. If @var{which}
34492 is @code{nil}, use Calc's native base units. Otherwise, @var{which}
34493 can specify a units system, which is a list of two-element lists,
34494 where the first element is a Calc base symbol name and the second
34495 is an expression to substitute for it.
34498 @defun remove-units expr
34499 Return a copy of @var{expr} with all units variables replaced by ones.
34500 This expression is generally normalized before use.
34503 @defun extract-units expr
34504 Return a copy of @var{expr} with everything but units variables replaced
34508 @node Formatting Lisp Functions, Hooks, Symbolic Lisp Functions, Internals
34509 @subsubsection I/O and Formatting Functions
34512 The functions described here are responsible for parsing and formatting
34513 Calc numbers and formulas.
34515 @defun calc-eval str sep arg1 arg2 @dots{}
34516 This is the simplest interface to the Calculator from another Lisp program.
34517 @xref{Calling Calc from Your Programs}.
34520 @defun read-number str
34521 If string @var{str} contains a valid Calc number, either integer,
34522 fraction, float, or HMS form, this function parses and returns that
34523 number. Otherwise, it returns @code{nil}.
34526 @defun read-expr str
34527 Read an algebraic expression from string @var{str}. If @var{str} does
34528 not have the form of a valid expression, return a list of the form
34529 @samp{(error @var{pos} @var{msg})} where @var{pos} is an integer index
34530 into @var{str} of the general location of the error, and @var{msg} is
34531 a string describing the problem.
34534 @defun read-exprs str
34535 Read a list of expressions separated by commas, and return it as a
34536 Lisp list. If an error occurs in any expressions, an error list as
34537 shown above is returned instead.
34540 @defun calc-do-alg-entry initial prompt no-norm
34541 Read an algebraic formula or formulas using the minibuffer. All
34542 conventions of regular algebraic entry are observed. The return value
34543 is a list of Calc formulas; there will be more than one if the user
34544 entered a list of values separated by commas. The result is @code{nil}
34545 if the user presses Return with a blank line. If @var{initial} is
34546 given, it is a string which the minibuffer will initially contain.
34547 If @var{prompt} is given, it is the prompt string to use; the default
34548 is ``Algebraic:''. If @var{no-norm} is @code{t}, the formulas will
34549 be returned exactly as parsed; otherwise, they will be passed through
34550 @code{calc-normalize} first.
34552 To support the use of @kbd{$} characters in the algebraic entry, use
34553 @code{let} to bind @code{calc-dollar-values} to a list of the values
34554 to be substituted for @kbd{$}, @kbd{$$}, and so on, and bind
34555 @code{calc-dollar-used} to 0. Upon return, @code{calc-dollar-used}
34556 will have been changed to the highest number of consecutive @kbd{$}s
34557 that actually appeared in the input.
34560 @defun format-number a
34561 Convert the real or complex number or HMS form @var{a} to string form.
34564 @defun format-flat-expr a prec
34565 Convert the arbitrary Calc number or formula @var{a} to string form,
34566 in the style used by the trail buffer and the @code{calc-edit} command.
34567 This is a simple format designed
34568 mostly to guarantee the string is of a form that can be re-parsed by
34569 @code{read-expr}. Most formatting modes, such as digit grouping,
34570 complex number format, and point character, are ignored to ensure the
34571 result will be re-readable. The @var{prec} parameter is normally 0; if
34572 you pass a large integer like 1000 instead, the expression will be
34573 surrounded by parentheses unless it is a plain number or variable name.
34576 @defun format-nice-expr a width
34577 This is like @code{format-flat-expr} (with @var{prec} equal to 0),
34578 except that newlines will be inserted to keep lines down to the
34579 specified @var{width}, and vectors that look like matrices or rewrite
34580 rules are written in a pseudo-matrix format. The @code{calc-edit}
34581 command uses this when only one stack entry is being edited.
34584 @defun format-value a width
34585 Convert the Calc number or formula @var{a} to string form, using the
34586 format seen in the stack buffer. Beware the string returned may
34587 not be re-readable by @code{read-expr}, for example, because of digit
34588 grouping. Multi-line objects like matrices produce strings that
34589 contain newline characters to separate the lines. The @var{w}
34590 parameter, if given, is the target window size for which to format
34591 the expressions. If @var{w} is omitted, the width of the Calculator
34595 @defun compose-expr a prec
34596 Format the Calc number or formula @var{a} according to the current
34597 language mode, returning a ``composition.'' To learn about the
34598 structure of compositions, see the comments in the Calc source code.
34599 You can specify the format of a given type of function call by putting
34600 a @code{math-compose-@var{lang}} property on the function's symbol,
34601 whose value is a Lisp function that takes @var{a} and @var{prec} as
34602 arguments and returns a composition. Here @var{lang} is a language
34603 mode name, one of @code{normal}, @code{big}, @code{c}, @code{pascal},
34604 @code{fortran}, @code{tex}, @code{eqn}, @code{math}, or @code{maple}.
34605 In Big mode, Calc actually tries @code{math-compose-big} first, then
34606 tries @code{math-compose-normal}. If this property does not exist,
34607 or if the function returns @code{nil}, the function is written in the
34608 normal function-call notation for that language.
34611 @defun composition-to-string c w
34612 Convert a composition structure returned by @code{compose-expr} into
34613 a string. Multi-line compositions convert to strings containing
34614 newline characters. The target window size is given by @var{w}.
34615 The @code{format-value} function basically calls @code{compose-expr}
34616 followed by @code{composition-to-string}.
34619 @defun comp-width c
34620 Compute the width in characters of composition @var{c}.
34623 @defun comp-height c
34624 Compute the height in lines of composition @var{c}.
34627 @defun comp-ascent c
34628 Compute the portion of the height of composition @var{c} which is on or
34629 above the baseline. For a one-line composition, this will be one.
34632 @defun comp-descent c
34633 Compute the portion of the height of composition @var{c} which is below
34634 the baseline. For a one-line composition, this will be zero.
34637 @defun comp-first-char c
34638 If composition @var{c} is a ``flat'' composition, return the first
34639 (leftmost) character of the composition as an integer. Otherwise,
34643 @defun comp-last-char c
34644 If composition @var{c} is a ``flat'' composition, return the last
34645 (rightmost) character, otherwise return @code{nil}.
34648 @comment @node Lisp Variables, Hooks, Formatting Lisp Functions, Internals
34649 @comment @subsubsection Lisp Variables
34652 @comment (This section is currently unfinished.)
34654 @node Hooks, , Formatting Lisp Functions, Internals
34655 @subsubsection Hooks
34658 Hooks are variables which contain Lisp functions (or lists of functions)
34659 which are called at various times. Calc defines a number of hooks
34660 that help you to customize it in various ways. Calc uses the Lisp
34661 function @code{run-hooks} to invoke the hooks shown below. Several
34662 other customization-related variables are also described here.
34664 @defvar calc-load-hook
34665 This hook is called at the end of @file{calc.el}, after the file has
34666 been loaded, before any functions in it have been called, but after
34667 @code{calc-mode-map} and similar variables have been set up.
34670 @defvar calc-ext-load-hook
34671 This hook is called at the end of @file{calc-ext.el}.
34674 @defvar calc-start-hook
34675 This hook is called as the last step in a @kbd{M-x calc} command.
34676 At this point, the Calc buffer has been created and initialized if
34677 necessary, the Calc window and trail window have been created,
34678 and the ``Welcome to Calc'' message has been displayed.
34681 @defvar calc-mode-hook
34682 This hook is called when the Calc buffer is being created. Usually
34683 this will only happen once per Emacs session. The hook is called
34684 after Emacs has switched to the new buffer, the mode-settings file
34685 has been read if necessary, and all other buffer-local variables
34686 have been set up. After this hook returns, Calc will perform a
34687 @code{calc-refresh} operation, set up the mode line display, then
34688 evaluate any deferred @code{calc-define} properties that have not
34689 been evaluated yet.
34692 @defvar calc-trail-mode-hook
34693 This hook is called when the Calc Trail buffer is being created.
34694 It is called as the very last step of setting up the Trail buffer.
34695 Like @code{calc-mode-hook}, this will normally happen only once
34699 @defvar calc-end-hook
34700 This hook is called by @code{calc-quit}, generally because the user
34701 presses @kbd{q} or @kbd{M-# c} while in Calc. The Calc buffer will
34702 be the current buffer. The hook is called as the very first
34703 step, before the Calc window is destroyed.
34706 @defvar calc-window-hook
34707 If this hook exists, it is called to create the Calc window.
34708 Upon return, this new Calc window should be the current window.
34709 (The Calc buffer will already be the current buffer when the
34710 hook is called.) If the hook is not defined, Calc will
34711 generally use @code{split-window}, @code{set-window-buffer},
34712 and @code{select-window} to create the Calc window.
34715 @defvar calc-trail-window-hook
34716 If this hook exists, it is called to create the Calc Trail window.
34717 The variable @code{calc-trail-buffer} will contain the buffer
34718 which the window should use. Unlike @code{calc-window-hook},
34719 this hook must @emph{not} switch into the new window.
34722 @defvar calc-edit-mode-hook
34723 This hook is called by @code{calc-edit} (and the other ``edit''
34724 commands) when the temporary editing buffer is being created.
34725 The buffer will have been selected and set up to be in
34726 @code{calc-edit-mode}, but will not yet have been filled with
34727 text. (In fact it may still have leftover text from a previous
34728 @code{calc-edit} command.)
34731 @defvar calc-mode-save-hook
34732 This hook is called by the @code{calc-save-modes} command,
34733 after Calc's own mode features have been inserted into the
34734 Calc init file and just before the ``End of mode settings''
34735 message is inserted.
34738 @defvar calc-reset-hook
34739 This hook is called after @kbd{M-# 0} (@code{calc-reset}) has
34740 reset all modes. The Calc buffer will be the current buffer.
34743 @defvar calc-other-modes
34744 This variable contains a list of strings. The strings are
34745 concatenated at the end of the modes portion of the Calc
34746 mode line (after standard modes such as ``Deg'', ``Inv'' and
34747 ``Hyp''). Each string should be a short, single word followed
34748 by a space. The variable is @code{nil} by default.
34751 @defvar calc-mode-map
34752 This is the keymap that is used by Calc mode. The best time
34753 to adjust it is probably in a @code{calc-mode-hook}. If the
34754 Calc extensions package (@file{calc-ext.el}) has not yet been
34755 loaded, many of these keys will be bound to @code{calc-missing-key},
34756 which is a command that loads the extensions package and
34757 ``retypes'' the key. If your @code{calc-mode-hook} rebinds
34758 one of these keys, it will probably be overridden when the
34759 extensions are loaded.
34762 @defvar calc-digit-map
34763 This is the keymap that is used during numeric entry. Numeric
34764 entry uses the minibuffer, but this map binds every non-numeric
34765 key to @code{calcDigit-nondigit} which generally calls
34766 @code{exit-minibuffer} and ``retypes'' the key.
34769 @defvar calc-alg-ent-map
34770 This is the keymap that is used during algebraic entry. This is
34771 mostly a copy of @code{minibuffer-local-map}.
34774 @defvar calc-store-var-map
34775 This is the keymap that is used during entry of variable names for
34776 commands like @code{calc-store} and @code{calc-recall}. This is
34777 mostly a copy of @code{minibuffer-local-completion-map}.
34780 @defvar calc-edit-mode-map
34781 This is the (sparse) keymap used by @code{calc-edit} and other
34782 temporary editing commands. It binds @key{RET}, @key{LFD},
34783 and @kbd{C-c C-c} to @code{calc-edit-finish}.
34786 @defvar calc-mode-var-list
34787 This is a list of variables which are saved by @code{calc-save-modes}.
34788 Each entry is a list of two items, the variable (as a Lisp symbol)
34789 and its default value. When modes are being saved, each variable
34790 is compared with its default value (using @code{equal}) and any
34791 non-default variables are written out.
34794 @defvar calc-local-var-list
34795 This is a list of variables which should be buffer-local to the
34796 Calc buffer. Each entry is a variable name (as a Lisp symbol).
34797 These variables also have their default values manipulated by
34798 the @code{calc} and @code{calc-quit} commands; @pxref{Multiple Calculators}.
34799 Since @code{calc-mode-hook} is called after this list has been
34800 used the first time, your hook should add a variable to the
34801 list and also call @code{make-local-variable} itself.
34804 @node Customizable Variables, Reporting Bugs, Programming, Top
34805 @appendix Customizable Variables
34807 GNU Calc is controlled by many variables, most of which can be reset
34808 from within Calc. Some variables are less involved with actual
34809 calculation, and can be set outside of Calc using Emacs's
34810 customization facilities. These variables are listed below.
34811 Typing @kbd{M-x customize-variable RET @var{variable-name} RET}
34812 will bring up a buffer in which the variable's value can be redefined.
34813 Typing @kbd{M-x customize-group RET calc RET} will bring up a buffer which
34814 contains all of Calc's customizable variables. (These variables can
34815 also be reset by putting the appropriate lines in your .emacs file;
34816 @xref{Init File, ,Init File, emacs, The GNU Emacs Manual}.)
34818 Some of the customizable variables are regular expressions. A regular
34819 expression is basically a pattern that Calc can search for.
34820 See @ref{Regexp Search,, Regular Expression Search, emacs, The GNU Emacs Manual}
34821 to see how regular expressions work.
34825 @item calc-settings-file
34827 @vindex calc-settings-file
34828 The variable @code{calc-settings-file} holds the file name in
34829 which commands like @kbd{m m} and @kbd{Z P} store ``permanent''
34831 If @code{calc-settings-file} is not your user init file (typically
34832 @file{~/.emacs}) and if the variable @code{calc-loaded-settings-file} is
34833 @code{nil}, then Calc will automatically load your settings file (if it
34834 exists) the first time Calc is invoked.
34836 The default value for this variable is @code{"~/.calc.el"}.
34838 @item calc-gnuplot-name
34840 See @ref{Graphics}.@*
34841 The variable @code{calc-gnuplot-name} should be the name of the
34842 GNUPLOT program (a string). If you have GNUPLOT installed on your
34843 system but Calc is unable to find it, you may need to set this
34844 variable. (@pxref{Customizable Variables})
34845 You may also need to set some Lisp variables to show Calc how to run
34846 GNUPLOT on your system, see @ref{Devices, ,Graphical Devices} . The default value
34847 of @code{calc-gnuplot-name} is @code{"gnuplot"}.
34849 @item calc-gnuplot-plot-command
34850 @itemx calc-gnuplot-print-command
34852 See @ref{Devices, ,Graphical Devices}.@*
34853 The variables @code{calc-gnuplot-plot-command} and
34854 @code{calc-gnuplot-print-command} represent system commands to
34855 display and print the output of GNUPLOT, respectively. These may be
34856 @code{nil} if no command is necessary, or strings which can include
34857 @samp{%s} to signify the name of the file to be displayed or printed.
34858 Or, these variables may contain Lisp expressions which are evaluated
34859 to display or print the output.
34861 The default value of @code{calc-gnuplot-plot-command} is @code{nil},
34862 and the default value of @code{calc-gnuplot-print-command} is
34865 @item calc-language-alist
34867 See @ref{Basic Embedded Mode}.@*
34868 The variable @code{calc-language-alist} controls the languages that
34869 Calc will associate with major modes. When Calc embedded mode is
34870 enabled, it will try to use the current major mode to
34871 determine what language should be used. (This can be overridden using
34872 Calc's mode changing commands, @xref{Mode Settings in Embedded Mode}.)
34873 The variable @code{calc-language-alist} consists of a list of pairs of
34874 the form @code{(@var{MAJOR-MODE} . @var{LANGUAGE})}; for example,
34875 @code{(latex-mode . latex)} is one such pair. If Calc embedded is
34876 activated in a buffer whose major mode is @var{MAJOR-MODE}, it will set itself
34877 to use the language @var{LANGUAGE}.
34879 The default value of @code{calc-language-alist} is
34881 ((latex-mode . latex)
34883 (plain-tex-mode . tex)
34884 (context-mode . tex)
34886 (pascal-mode . pascal)
34889 (fortran-mode . fortran)
34890 (f90-mode . fortran))
34893 @item calc-embedded-announce-formula
34895 See @ref{Customizing Embedded Mode}.@*
34896 The variable @code{calc-embedded-announce-formula} helps determine
34897 what formulas @kbd{M-# a} will activate in a buffer. It is a
34898 regular expression, and when activating embedded formulas with
34899 @kbd{M-# a}, it will tell Calc that what follows is a formula to be
34900 activated. (Calc also uses other patterns to find formulas, such as
34901 @samp{=>} and @samp{:=}.)
34903 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which checks
34904 for @samp{%Embed} followed by any number of lines beginning with
34905 @samp{%} and a space.
34907 @item calc-embedded-open-formula
34908 @itemx calc-embedded-close-formula
34910 See @ref{Customizing Embedded Mode}.@*
34911 The variables @code{calc-embedded-open-formula} and
34912 @code{calc-embedded-open-formula} control the region that Calc will
34913 activate as a formula when Embedded mode is entered with @kbd{M-# e}.
34914 They are regular expressions;
34915 Calc normally scans backward and forward in the buffer for the
34916 nearest text matching these regular expressions to be the ``formula
34919 The simplest delimiters are blank lines. Other delimiters that
34920 Embedded mode understands by default are:
34923 The @TeX{} and La@TeX{} math delimiters @samp{$ $}, @samp{$$ $$},
34924 @samp{\[ \]}, and @samp{\( \)};
34926 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
34928 Lines beginning with @samp{@@} (Texinfo delimiters).
34930 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
34932 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
34935 @item calc-embedded-open-word
34936 @itemx calc-embedded-close-word
34938 See @ref{Customizing Embedded Mode}.@*
34939 The variables @code{calc-embedded-open-word} and
34940 @code{calc-embedded-close-word} control the region that Calc will
34941 activate when Embedded mode is entered with @kbd{M-# w}. They are
34942 regular expressions.
34944 The default values of @code{calc-embedded-open-word} and
34945 @code{calc-embedded-close-word} are @code{"^\\|[^-+0-9.eE]"} and
34946 @code{"$\\|[^-+0-9.eE]"} respectively.
34948 @item calc-embedded-open-plain
34949 @itemx calc-embedded-close-plain
34951 See @ref{Customizing Embedded Mode}.@*
34952 The variables @code{calc-embedded-open-plain} and
34953 @code{calc-embedded-open-plain} are used to delimit ``plain''
34954 formulas. Note that these are actual strings, not regular
34955 expressions, because Calc must be able to write these string into a
34956 buffer as well as to recognize them.
34958 The default string for @code{calc-embedded-open-plain} is
34959 @code{"%%% "}, note the trailing space. The default string for
34960 @code{calc-embedded-close-plain} is @code{" %%%\n"}, without
34961 the trailing newline here, the first line of a Big mode formula
34962 that followed might be shifted over with respect to the other lines.
34964 @item calc-embedded-open-new-formula
34965 @itemx calc-embedded-close-new-formula
34967 See @ref{Customizing Embedded Mode}.@*
34968 The variables @code{calc-embedded-open-new-formula} and
34969 @code{calc-embedded-close-new-formula} are strings which are
34970 inserted before and after a new formula when you type @kbd{M-# f}.
34972 The default value of @code{calc-embedded-open-new-formula} is
34973 @code{"\n\n"}. If this string begins with a newline character and the
34974 @kbd{M-# f} is typed at the beginning of a line, @kbd{M-# f} will skip
34975 this first newline to avoid introducing unnecessary blank lines in the
34976 file. The default value of @code{calc-embedded-close-new-formula} is
34977 also @code{"\n\n"}. The final newline is omitted by @w{@kbd{M-# f}}
34978 if typed at the end of a line. (It follows that if @kbd{M-# f} is
34979 typed on a blank line, both a leading opening newline and a trailing
34980 closing newline are omitted.)
34982 @item calc-embedded-open-mode
34983 @itemx calc-embedded-close-mode
34985 See @ref{Customizing Embedded Mode}.@*
34986 The variables @code{calc-embedded-open-mode} and
34987 @code{calc-embedded-close-mode} are strings which Calc will place before
34988 and after any mode annotations that it inserts. Calc never scans for
34989 these strings; Calc always looks for the annotation itself, so it is not
34990 necessary to add them to user-written annotations.
34992 The default value of @code{calc-embedded-open-mode} is @code{"% "}
34993 and the default value of @code{calc-embedded-close-mode} is
34995 If you change the value of @code{calc-embedded-close-mode}, it is a good
34996 idea still to end with a newline so that mode annotations will appear on
34997 lines by themselves.
35001 @node Reporting Bugs, Summary, Customizable Variables, Top
35002 @appendix Reporting Bugs
35005 If you find a bug in Calc, send e-mail to Jay Belanger,
35008 belanger@@truman.edu
35012 There is an automatic command @kbd{M-x report-calc-bug} which helps
35013 you to report bugs. This command prompts you for a brief subject
35014 line, then leaves you in a mail editing buffer. Type @kbd{C-c C-c} to
35015 send your mail. Make sure your subject line indicates that you are
35016 reporting a Calc bug; this command sends mail to the maintainer's
35019 If you have suggestions for additional features for Calc, please send
35020 them. Some have dared to suggest that Calc is already top-heavy with
35021 features; this obviously cannot be the case, so if you have ideas, send
35024 At the front of the source file, @file{calc.el}, is a list of ideas for
35025 future work. If any enthusiastic souls wish to take it upon themselves
35026 to work on these, please send a message (using @kbd{M-x report-calc-bug})
35027 so any efforts can be coordinated.
35029 The latest version of Calc is available from Savannah, in the Emacs
35030 CVS tree. See @uref{http://savannah.gnu.org/projects/emacs}.
35033 @node Summary, Key Index, Reporting Bugs, Top
35034 @appendix Calc Summary
35037 This section includes a complete list of Calc 2.1 keystroke commands.
35038 Each line lists the stack entries used by the command (top-of-stack
35039 last), the keystrokes themselves, the prompts asked by the command,
35040 and the result of the command (also with top-of-stack last).
35041 The result is expressed using the equivalent algebraic function.
35042 Commands which put no results on the stack show the full @kbd{M-x}
35043 command name in that position. Numbers preceding the result or
35044 command name refer to notes at the end.
35046 Algebraic functions and @kbd{M-x} commands that don't have corresponding
35047 keystrokes are not listed in this summary.
35048 @xref{Command Index}. @xref{Function Index}.
35053 \vskip-2\baselineskip \null
35054 \gdef\sumrow#1{\sumrowx#1\relax}%
35055 \gdef\sumrowx#1\:#2\:#3\:#4\:#5\:#6\relax{%
35058 \hbox to5em{\sl\hss#1}%
35059 \hbox to5em{\tt#2\hss}%
35060 \hbox to4em{\sl#3\hss}%
35061 \hbox to5em{\rm\hss#4}%
35066 \gdef\sumlpar{{\rm(}}%
35067 \gdef\sumrpar{{\rm)}}%
35068 \gdef\sumcomma{{\rm,\thinspace}}%
35069 \gdef\sumexcl{{\rm!}}%
35070 \gdef\sumbreak{\vskip-2.5\baselineskip\goodbreak}%
35071 \gdef\minus#1{{\tt-}}%
35075 @catcode`@(=@active @let(=@sumlpar
35076 @catcode`@)=@active @let)=@sumrpar
35077 @catcode`@,=@active @let,=@sumcomma
35078 @catcode`@!=@active @let!=@sumexcl
35082 @advance@baselineskip-2.5pt
35085 @r{ @: M-# a @: @: 33 @:calc-embedded-activate@:}
35086 @r{ @: M-# b @: @: @:calc-big-or-small@:}
35087 @r{ @: M-# c @: @: @:calc@:}
35088 @r{ @: M-# d @: @: @:calc-embedded-duplicate@:}
35089 @r{ @: M-# e @: @: 34 @:calc-embedded@:}
35090 @r{ @: M-# f @:formula @: @:calc-embedded-new-formula@:}
35091 @r{ @: M-# g @: @: 35 @:calc-grab-region@:}
35092 @r{ @: M-# i @: @: @:calc-info@:}
35093 @r{ @: M-# j @: @: @:calc-embedded-select@:}
35094 @r{ @: M-# k @: @: @:calc-keypad@:}
35095 @r{ @: M-# l @: @: @:calc-load-everything@:}
35096 @r{ @: M-# m @: @: @:read-kbd-macro@:}
35097 @r{ @: M-# n @: @: 4 @:calc-embedded-next@:}
35098 @r{ @: M-# o @: @: @:calc-other-window@:}
35099 @r{ @: M-# p @: @: 4 @:calc-embedded-previous@:}
35100 @r{ @: M-# q @:formula @: @:quick-calc@:}
35101 @r{ @: M-# r @: @: 36 @:calc-grab-rectangle@:}
35102 @r{ @: M-# s @: @: @:calc-info-summary@:}
35103 @r{ @: M-# t @: @: @:calc-tutorial@:}
35104 @r{ @: M-# u @: @: @:calc-embedded-update@:}
35105 @r{ @: M-# w @: @: @:calc-embedded-word@:}
35106 @r{ @: M-# x @: @: @:calc-quit@:}
35107 @r{ @: M-# y @: @:1,28,49 @:calc-copy-to-buffer@:}
35108 @r{ @: M-# z @: @: @:calc-user-invocation@:}
35109 @r{ @: M-# : @: @: 36 @:calc-grab-sum-down@:}
35110 @r{ @: M-# _ @: @: 36 @:calc-grab-sum-across@:}
35111 @r{ @: M-# ` @:editing @: 30 @:calc-embedded-edit@:}
35112 @r{ @: M-# 0 @:(zero) @: @:calc-reset@:}
35115 @r{ @: 0-9 @:number @: @:@:number}
35116 @r{ @: . @:number @: @:@:0.number}
35117 @r{ @: _ @:number @: @:-@:number}
35118 @r{ @: e @:number @: @:@:1e number}
35119 @r{ @: # @:number @: @:@:current-radix@tfn{#}number}
35120 @r{ @: P @:(in number) @: @:+/-@:}
35121 @r{ @: M @:(in number) @: @:mod@:}
35122 @r{ @: @@ ' " @: (in number)@: @:@:HMS form}
35123 @r{ @: h m s @: (in number)@: @:@:HMS form}
35126 @r{ @: ' @:formula @: 37,46 @:@:formula}
35127 @r{ @: $ @:formula @: 37,46 @:$@:formula}
35128 @r{ @: " @:string @: 37,46 @:@:string}
35131 @r{ a b@: + @: @: 2 @:add@:(a,b) a+b}
35132 @r{ a b@: - @: @: 2 @:sub@:(a,b) a@minus{}b}
35133 @r{ a b@: * @: @: 2 @:mul@:(a,b) a b, a*b}
35134 @r{ a b@: / @: @: 2 @:div@:(a,b) a/b}
35135 @r{ a b@: ^ @: @: 2 @:pow@:(a,b) a^b}
35136 @r{ a b@: I ^ @: @: 2 @:nroot@:(a,b) a^(1/b)}
35137 @r{ a b@: % @: @: 2 @:mod@:(a,b) a%b}
35138 @r{ a b@: \ @: @: 2 @:idiv@:(a,b) a\b}
35139 @r{ a b@: : @: @: 2 @:fdiv@:(a,b)}
35140 @r{ a b@: | @: @: 2 @:vconcat@:(a,b) a|b}
35141 @r{ a b@: I | @: @: @:vconcat@:(b,a) b|a}
35142 @r{ a b@: H | @: @: 2 @:append@:(a,b)}
35143 @r{ a b@: I H | @: @: @:append@:(b,a)}
35144 @r{ a@: & @: @: 1 @:inv@:(a) 1/a}
35145 @r{ a@: ! @: @: 1 @:fact@:(a) a!}
35146 @r{ a@: = @: @: 1 @:evalv@:(a)}
35147 @r{ a@: M-% @: @: @:percent@:(a) a%}
35150 @r{ ... a@: @key{RET} @: @: 1 @:@:... a a}
35151 @r{ ... a@: @key{SPC} @: @: 1 @:@:... a a}
35152 @r{... a b@: @key{TAB} @: @: 3 @:@:... b a}
35153 @r{. a b c@: M-@key{TAB} @: @: 3 @:@:... b c a}
35154 @r{... a b@: @key{LFD} @: @: 1 @:@:... a b a}
35155 @r{ ... a@: @key{DEL} @: @: 1 @:@:...}
35156 @r{... a b@: M-@key{DEL} @: @: 1 @:@:... b}
35157 @r{ @: M-@key{RET} @: @: 4 @:calc-last-args@:}
35158 @r{ a@: ` @:editing @: 1,30 @:calc-edit@:}
35161 @r{ ... a@: C-d @: @: 1 @:@:...}
35162 @r{ @: C-k @: @: 27 @:calc-kill@:}
35163 @r{ @: C-w @: @: 27 @:calc-kill-region@:}
35164 @r{ @: C-y @: @: @:calc-yank@:}
35165 @r{ @: C-_ @: @: 4 @:calc-undo@:}
35166 @r{ @: M-k @: @: 27 @:calc-copy-as-kill@:}
35167 @r{ @: M-w @: @: 27 @:calc-copy-region-as-kill@:}
35170 @r{ @: [ @: @: @:@:[...}
35171 @r{[.. a b@: ] @: @: @:@:[a,b]}
35172 @r{ @: ( @: @: @:@:(...}
35173 @r{(.. a b@: ) @: @: @:@:(a,b)}
35174 @r{ @: , @: @: @:@:vector or rect complex}
35175 @r{ @: ; @: @: @:@:matrix or polar complex}
35176 @r{ @: .. @: @: @:@:interval}
35179 @r{ @: ~ @: @: @:calc-num-prefix@:}
35180 @r{ @: < @: @: 4 @:calc-scroll-left@:}
35181 @r{ @: > @: @: 4 @:calc-scroll-right@:}
35182 @r{ @: @{ @: @: 4 @:calc-scroll-down@:}
35183 @r{ @: @} @: @: 4 @:calc-scroll-up@:}
35184 @r{ @: ? @: @: @:calc-help@:}
35187 @r{ a@: n @: @: 1 @:neg@:(a) @minus{}a}
35188 @r{ @: o @: @: 4 @:calc-realign@:}
35189 @r{ @: p @:precision @: 31 @:calc-precision@:}
35190 @r{ @: q @: @: @:calc-quit@:}
35191 @r{ @: w @: @: @:calc-why@:}
35192 @r{ @: x @:command @: @:M-x calc-@:command}
35193 @r{ a@: y @: @:1,28,49 @:calc-copy-to-buffer@:}
35196 @r{ a@: A @: @: 1 @:abs@:(a)}
35197 @r{ a b@: B @: @: 2 @:log@:(a,b)}
35198 @r{ a b@: I B @: @: 2 @:alog@:(a,b) b^a}
35199 @r{ a@: C @: @: 1 @:cos@:(a)}
35200 @r{ a@: I C @: @: 1 @:arccos@:(a)}
35201 @r{ a@: H C @: @: 1 @:cosh@:(a)}
35202 @r{ a@: I H C @: @: 1 @:arccosh@:(a)}
35203 @r{ @: D @: @: 4 @:calc-redo@:}
35204 @r{ a@: E @: @: 1 @:exp@:(a)}
35205 @r{ a@: H E @: @: 1 @:exp10@:(a) 10.^a}
35206 @r{ a@: F @: @: 1,11 @:floor@:(a,d)}
35207 @r{ a@: I F @: @: 1,11 @:ceil@:(a,d)}
35208 @r{ a@: H F @: @: 1,11 @:ffloor@:(a,d)}
35209 @r{ a@: I H F @: @: 1,11 @:fceil@:(a,d)}
35210 @r{ a@: G @: @: 1 @:arg@:(a)}
35211 @r{ @: H @:command @: 32 @:@:Hyperbolic}
35212 @r{ @: I @:command @: 32 @:@:Inverse}
35213 @r{ a@: J @: @: 1 @:conj@:(a)}
35214 @r{ @: K @:command @: 32 @:@:Keep-args}
35215 @r{ a@: L @: @: 1 @:ln@:(a)}
35216 @r{ a@: H L @: @: 1 @:log10@:(a)}
35217 @r{ @: M @: @: @:calc-more-recursion-depth@:}
35218 @r{ @: I M @: @: @:calc-less-recursion-depth@:}
35219 @r{ a@: N @: @: 5 @:evalvn@:(a)}
35220 @r{ @: P @: @: @:@:pi}
35221 @r{ @: I P @: @: @:@:gamma}
35222 @r{ @: H P @: @: @:@:e}
35223 @r{ @: I H P @: @: @:@:phi}
35224 @r{ a@: Q @: @: 1 @:sqrt@:(a)}
35225 @r{ a@: I Q @: @: 1 @:sqr@:(a) a^2}
35226 @r{ a@: R @: @: 1,11 @:round@:(a,d)}
35227 @r{ a@: I R @: @: 1,11 @:trunc@:(a,d)}
35228 @r{ a@: H R @: @: 1,11 @:fround@:(a,d)}
35229 @r{ a@: I H R @: @: 1,11 @:ftrunc@:(a,d)}
35230 @r{ a@: S @: @: 1 @:sin@:(a)}
35231 @r{ a@: I S @: @: 1 @:arcsin@:(a)}
35232 @r{ a@: H S @: @: 1 @:sinh@:(a)}
35233 @r{ a@: I H S @: @: 1 @:arcsinh@:(a)}
35234 @r{ a@: T @: @: 1 @:tan@:(a)}
35235 @r{ a@: I T @: @: 1 @:arctan@:(a)}
35236 @r{ a@: H T @: @: 1 @:tanh@:(a)}
35237 @r{ a@: I H T @: @: 1 @:arctanh@:(a)}
35238 @r{ @: U @: @: 4 @:calc-undo@:}
35239 @r{ @: X @: @: 4 @:calc-call-last-kbd-macro@:}
35242 @r{ a b@: a = @: @: 2 @:eq@:(a,b) a=b}
35243 @r{ a b@: a # @: @: 2 @:neq@:(a,b) a!=b}
35244 @r{ a b@: a < @: @: 2 @:lt@:(a,b) a<b}
35245 @r{ a b@: a > @: @: 2 @:gt@:(a,b) a>b}
35246 @r{ a b@: a [ @: @: 2 @:leq@:(a,b) a<=b}
35247 @r{ a b@: a ] @: @: 2 @:geq@:(a,b) a>=b}
35248 @r{ a b@: a @{ @: @: 2 @:in@:(a,b)}
35249 @r{ a b@: a & @: @: 2,45 @:land@:(a,b) a&&b}
35250 @r{ a b@: a | @: @: 2,45 @:lor@:(a,b) a||b}
35251 @r{ a@: a ! @: @: 1,45 @:lnot@:(a) !a}
35252 @r{ a b c@: a : @: @: 45 @:if@:(a,b,c) a?b:c}
35253 @r{ a@: a . @: @: 1 @:rmeq@:(a)}
35254 @r{ a@: a " @: @: 7,8 @:calc-expand-formula@:}
35257 @r{ a@: a + @:i, l, h @: 6,38 @:sum@:(a,i,l,h)}
35258 @r{ a@: a - @:i, l, h @: 6,38 @:asum@:(a,i,l,h)}
35259 @r{ a@: a * @:i, l, h @: 6,38 @:prod@:(a,i,l,h)}
35260 @r{ a b@: a _ @: @: 2 @:subscr@:(a,b) a_b}
35263 @r{ a b@: a \ @: @: 2 @:pdiv@:(a,b)}
35264 @r{ a b@: a % @: @: 2 @:prem@:(a,b)}
35265 @r{ a b@: a / @: @: 2 @:pdivrem@:(a,b) [q,r]}
35266 @r{ a b@: H a / @: @: 2 @:pdivide@:(a,b) q+r/b}
35269 @r{ a@: a a @: @: 1 @:apart@:(a)}
35270 @r{ a@: a b @:old, new @: 38 @:subst@:(a,old,new)}
35271 @r{ a@: a c @:v @: 38 @:collect@:(a,v)}
35272 @r{ a@: a d @:v @: 4,38 @:deriv@:(a,v)}
35273 @r{ a@: H a d @:v @: 4,38 @:tderiv@:(a,v)}
35274 @r{ a@: a e @: @: @:esimplify@:(a)}
35275 @r{ a@: a f @: @: 1 @:factor@:(a)}
35276 @r{ a@: H a f @: @: 1 @:factors@:(a)}
35277 @r{ a b@: a g @: @: 2 @:pgcd@:(a,b)}
35278 @r{ a@: a i @:v @: 38 @:integ@:(a,v)}
35279 @r{ a@: a m @:pats @: 38 @:match@:(a,pats)}
35280 @r{ a@: I a m @:pats @: 38 @:matchnot@:(a,pats)}
35281 @r{ data x@: a p @: @: 28 @:polint@:(data,x)}
35282 @r{ data x@: H a p @: @: 28 @:ratint@:(data,x)}
35283 @r{ a@: a n @: @: 1 @:nrat@:(a)}
35284 @r{ a@: a r @:rules @:4,8,38 @:rewrite@:(a,rules,n)}
35285 @r{ a@: a s @: @: @:simplify@:(a)}
35286 @r{ a@: a t @:v, n @: 31,39 @:taylor@:(a,v,n)}
35287 @r{ a@: a v @: @: 7,8 @:calc-alg-evaluate@:}
35288 @r{ a@: a x @: @: 4,8 @:expand@:(a)}
35291 @r{ data@: a F @:model, vars @: 48 @:fit@:(m,iv,pv,data)}
35292 @r{ data@: I a F @:model, vars @: 48 @:xfit@:(m,iv,pv,data)}
35293 @r{ data@: H a F @:model, vars @: 48 @:efit@:(m,iv,pv,data)}
35294 @r{ a@: a I @:v, l, h @: 38 @:ninteg@:(a,v,l,h)}
35295 @r{ a b@: a M @:op @: 22 @:mapeq@:(op,a,b)}
35296 @r{ a b@: I a M @:op @: 22 @:mapeqr@:(op,a,b)}
35297 @r{ a b@: H a M @:op @: 22 @:mapeqp@:(op,a,b)}
35298 @r{ a g@: a N @:v @: 38 @:minimize@:(a,v,g)}
35299 @r{ a g@: H a N @:v @: 38 @:wminimize@:(a,v,g)}
35300 @r{ a@: a P @:v @: 38 @:roots@:(a,v)}
35301 @r{ a g@: a R @:v @: 38 @:root@:(a,v,g)}
35302 @r{ a g@: H a R @:v @: 38 @:wroot@:(a,v,g)}
35303 @r{ a@: a S @:v @: 38 @:solve@:(a,v)}
35304 @r{ a@: I a S @:v @: 38 @:finv@:(a,v)}
35305 @r{ a@: H a S @:v @: 38 @:fsolve@:(a,v)}
35306 @r{ a@: I H a S @:v @: 38 @:ffinv@:(a,v)}
35307 @r{ a@: a T @:i, l, h @: 6,38 @:table@:(a,i,l,h)}
35308 @r{ a g@: a X @:v @: 38 @:maximize@:(a,v,g)}
35309 @r{ a g@: H a X @:v @: 38 @:wmaximize@:(a,v,g)}
35312 @r{ a b@: b a @: @: 9 @:and@:(a,b,w)}
35313 @r{ a@: b c @: @: 9 @:clip@:(a,w)}
35314 @r{ a b@: b d @: @: 9 @:diff@:(a,b,w)}
35315 @r{ a@: b l @: @: 10 @:lsh@:(a,n,w)}
35316 @r{ a n@: H b l @: @: 9 @:lsh@:(a,n,w)}
35317 @r{ a@: b n @: @: 9 @:not@:(a,w)}
35318 @r{ a b@: b o @: @: 9 @:or@:(a,b,w)}
35319 @r{ v@: b p @: @: 1 @:vpack@:(v)}
35320 @r{ a@: b r @: @: 10 @:rsh@:(a,n,w)}
35321 @r{ a n@: H b r @: @: 9 @:rsh@:(a,n,w)}
35322 @r{ a@: b t @: @: 10 @:rot@:(a,n,w)}
35323 @r{ a n@: H b t @: @: 9 @:rot@:(a,n,w)}
35324 @r{ a@: b u @: @: 1 @:vunpack@:(a)}
35325 @r{ @: b w @:w @: 9,50 @:calc-word-size@:}
35326 @r{ a b@: b x @: @: 9 @:xor@:(a,b,w)}
35329 @r{c s l p@: b D @: @: @:ddb@:(c,s,l,p)}
35330 @r{ r n p@: b F @: @: @:fv@:(r,n,p)}
35331 @r{ r n p@: I b F @: @: @:fvb@:(r,n,p)}
35332 @r{ r n p@: H b F @: @: @:fvl@:(r,n,p)}
35333 @r{ v@: b I @: @: 19 @:irr@:(v)}
35334 @r{ v@: I b I @: @: 19 @:irrb@:(v)}
35335 @r{ a@: b L @: @: 10 @:ash@:(a,n,w)}
35336 @r{ a n@: H b L @: @: 9 @:ash@:(a,n,w)}
35337 @r{ r n a@: b M @: @: @:pmt@:(r,n,a)}
35338 @r{ r n a@: I b M @: @: @:pmtb@:(r,n,a)}
35339 @r{ r n a@: H b M @: @: @:pmtl@:(r,n,a)}
35340 @r{ r v@: b N @: @: 19 @:npv@:(r,v)}
35341 @r{ r v@: I b N @: @: 19 @:npvb@:(r,v)}
35342 @r{ r n p@: b P @: @: @:pv@:(r,n,p)}
35343 @r{ r n p@: I b P @: @: @:pvb@:(r,n,p)}
35344 @r{ r n p@: H b P @: @: @:pvl@:(r,n,p)}
35345 @r{ a@: b R @: @: 10 @:rash@:(a,n,w)}
35346 @r{ a n@: H b R @: @: 9 @:rash@:(a,n,w)}
35347 @r{ c s l@: b S @: @: @:sln@:(c,s,l)}
35348 @r{ n p a@: b T @: @: @:rate@:(n,p,a)}
35349 @r{ n p a@: I b T @: @: @:rateb@:(n,p,a)}
35350 @r{ n p a@: H b T @: @: @:ratel@:(n,p,a)}
35351 @r{c s l p@: b Y @: @: @:syd@:(c,s,l,p)}
35353 @r{ r p a@: b # @: @: @:nper@:(r,p,a)}
35354 @r{ r p a@: I b # @: @: @:nperb@:(r,p,a)}
35355 @r{ r p a@: H b # @: @: @:nperl@:(r,p,a)}
35356 @r{ a b@: b % @: @: @:relch@:(a,b)}
35359 @r{ a@: c c @: @: 5 @:pclean@:(a,p)}
35360 @r{ a@: c 0-9 @: @: @:pclean@:(a,p)}
35361 @r{ a@: H c c @: @: 5 @:clean@:(a,p)}
35362 @r{ a@: H c 0-9 @: @: @:clean@:(a,p)}
35363 @r{ a@: c d @: @: 1 @:deg@:(a)}
35364 @r{ a@: c f @: @: 1 @:pfloat@:(a)}
35365 @r{ a@: H c f @: @: 1 @:float@:(a)}
35366 @r{ a@: c h @: @: 1 @:hms@:(a)}
35367 @r{ a@: c p @: @: @:polar@:(a)}
35368 @r{ a@: I c p @: @: @:rect@:(a)}
35369 @r{ a@: c r @: @: 1 @:rad@:(a)}
35372 @r{ a@: c F @: @: 5 @:pfrac@:(a,p)}
35373 @r{ a@: H c F @: @: 5 @:frac@:(a,p)}
35376 @r{ a@: c % @: @: @:percent@:(a*100)}
35379 @r{ @: d . @:char @: 50 @:calc-point-char@:}
35380 @r{ @: d , @:char @: 50 @:calc-group-char@:}
35381 @r{ @: d < @: @: 13,50 @:calc-left-justify@:}
35382 @r{ @: d = @: @: 13,50 @:calc-center-justify@:}
35383 @r{ @: d > @: @: 13,50 @:calc-right-justify@:}
35384 @r{ @: d @{ @:label @: 50 @:calc-left-label@:}
35385 @r{ @: d @} @:label @: 50 @:calc-right-label@:}
35386 @r{ @: d [ @: @: 4 @:calc-truncate-up@:}
35387 @r{ @: d ] @: @: 4 @:calc-truncate-down@:}
35388 @r{ @: d " @: @: 12,50 @:calc-display-strings@:}
35389 @r{ @: d @key{SPC} @: @: @:calc-refresh@:}
35390 @r{ @: d @key{RET} @: @: 1 @:calc-refresh-top@:}
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35729 @r{ a b@: t - @: @: 2 @:bsub@:(a,b)}
35732 @r{ @: u a @: @: 12 @:calc-autorange-units@:}
35733 @r{ a@: u b @: @: @:calc-base-units@:}
35734 @r{ a@: u c @:units @: 18 @:calc-convert-units@:}
35735 @r{ defn@: u d @:unit, descr @: @:calc-define-unit@:}
35736 @r{ @: u e @: @: @:calc-explain-units@:}
35737 @r{ @: u g @:unit @: @:calc-get-unit-definition@:}
35738 @r{ @: u p @: @: @:calc-permanent-units@:}
35739 @r{ a@: u r @: @: @:calc-remove-units@:}
35740 @r{ a@: u s @: @: @:usimplify@:(a)}
35741 @r{ a@: u t @:units @: 18 @:calc-convert-temperature@:}
35742 @r{ @: u u @:unit @: @:calc-undefine-unit@:}
35743 @r{ @: u v @: @: @:calc-enter-units-table@:}
35744 @r{ a@: u x @: @: @:calc-extract-units@:}
35745 @r{ a@: u 0-9 @: @: @:calc-quick-units@:}
35748 @r{ v1 v2@: u C @: @: 20 @:vcov@:(v1,v2)}
35749 @r{ v1 v2@: I u C @: @: 20 @:vpcov@:(v1,v2)}
35750 @r{ v1 v2@: H u C @: @: 20 @:vcorr@:(v1,v2)}
35751 @r{ v@: u G @: @: 19 @:vgmean@:(v)}
35752 @r{ a b@: H u G @: @: 2 @:agmean@:(a,b)}
35753 @r{ v@: u M @: @: 19 @:vmean@:(v)}
35754 @r{ v@: I u M @: @: 19 @:vmeane@:(v)}
35755 @r{ v@: H u M @: @: 19 @:vmedian@:(v)}
35756 @r{ v@: I H u M @: @: 19 @:vhmean@:(v)}
35757 @r{ v@: u N @: @: 19 @:vmin@:(v)}
35758 @r{ v@: u S @: @: 19 @:vsdev@:(v)}
35759 @r{ v@: I u S @: @: 19 @:vpsdev@:(v)}
35760 @r{ v@: H u S @: @: 19 @:vvar@:(v)}
35761 @r{ v@: I H u S @: @: 19 @:vpvar@:(v)}
35762 @r{ @: u V @: @: @:calc-view-units-table@:}
35763 @r{ v@: u X @: @: 19 @:vmax@:(v)}
35766 @r{ v@: u + @: @: 19 @:vsum@:(v)}
35767 @r{ v@: u * @: @: 19 @:vprod@:(v)}
35768 @r{ v@: u # @: @: 19 @:vcount@:(v)}
35771 @r{ @: V ( @: @: 50 @:calc-vector-parens@:}
35772 @r{ @: V @{ @: @: 50 @:calc-vector-braces@:}
35773 @r{ @: V [ @: @: 50 @:calc-vector-brackets@:}
35774 @r{ @: V ] @:ROCP @: 50 @:calc-matrix-brackets@:}
35775 @r{ @: V , @: @: 50 @:calc-vector-commas@:}
35776 @r{ @: V < @: @: 50 @:calc-matrix-left-justify@:}
35777 @r{ @: V = @: @: 50 @:calc-matrix-center-justify@:}
35778 @r{ @: V > @: @: 50 @:calc-matrix-right-justify@:}
35779 @r{ @: V / @: @: 12,50 @:calc-break-vectors@:}
35780 @r{ @: V . @: @: 12,50 @:calc-full-vectors@:}
35783 @r{ s t@: V ^ @: @: 2 @:vint@:(s,t)}
35784 @r{ s t@: V - @: @: 2 @:vdiff@:(s,t)}
35785 @r{ s@: V ~ @: @: 1 @:vcompl@:(s)}
35786 @r{ s@: V # @: @: 1 @:vcard@:(s)}
35787 @r{ s@: V : @: @: 1 @:vspan@:(s)}
35788 @r{ s@: V + @: @: 1 @:rdup@:(s)}
35791 @r{ m@: V & @: @: 1 @:inv@:(m) 1/m}
35794 @r{ v@: v a @:n @: @:arrange@:(v,n)}
35795 @r{ a@: v b @:n @: @:cvec@:(a,n)}
35796 @r{ v@: v c @:n >0 @: 21,31 @:mcol@:(v,n)}
35797 @r{ v@: v c @:n <0 @: 31 @:mrcol@:(v,-n)}
35798 @r{ m@: v c @:0 @: 31 @:getdiag@:(m)}
35799 @r{ v@: v d @: @: 25 @:diag@:(v,n)}
35800 @r{ v m@: v e @: @: 2 @:vexp@:(v,m)}
35801 @r{ v m f@: H v e @: @: 2 @:vexp@:(v,m,f)}
35802 @r{ v a@: v f @: @: 26 @:find@:(v,a,n)}
35803 @r{ v@: v h @: @: 1 @:head@:(v)}
35804 @r{ v@: I v h @: @: 1 @:tail@:(v)}
35805 @r{ v@: H v h @: @: 1 @:rhead@:(v)}
35806 @r{ v@: I H v h @: @: 1 @:rtail@:(v)}
35807 @r{ @: v i @:n @: 31 @:idn@:(1,n)}
35808 @r{ @: v i @:0 @: 31 @:idn@:(1)}
35809 @r{ h t@: v k @: @: 2 @:cons@:(h,t)}
35810 @r{ h t@: H v k @: @: 2 @:rcons@:(h,t)}
35811 @r{ v@: v l @: @: 1 @:vlen@:(v)}
35812 @r{ v@: H v l @: @: 1 @:mdims@:(v)}
35813 @r{ v m@: v m @: @: 2 @:vmask@:(v,m)}
35814 @r{ v@: v n @: @: 1 @:rnorm@:(v)}
35815 @r{ a b c@: v p @: @: 24 @:calc-pack@:}
35816 @r{ v@: v r @:n >0 @: 21,31 @:mrow@:(v,n)}
35817 @r{ v@: v r @:n <0 @: 31 @:mrrow@:(v,-n)}
35818 @r{ m@: v r @:0 @: 31 @:getdiag@:(m)}
35819 @r{ v i j@: v s @: @: @:subvec@:(v,i,j)}
35820 @r{ v i j@: I v s @: @: @:rsubvec@:(v,i,j)}
35821 @r{ m@: v t @: @: 1 @:trn@:(m)}
35822 @r{ v@: v u @: @: 24 @:calc-unpack@:}
35823 @r{ v@: v v @: @: 1 @:rev@:(v)}
35824 @r{ @: v x @:n @: 31 @:index@:(n)}
35825 @r{ n s i@: C-u v x @: @: @:index@:(n,s,i)}
35828 @r{ v@: V A @:op @: 22 @:apply@:(op,v)}
35829 @r{ v1 v2@: V C @: @: 2 @:cross@:(v1,v2)}
35830 @r{ m@: V D @: @: 1 @:det@:(m)}
35831 @r{ s@: V E @: @: 1 @:venum@:(s)}
35832 @r{ s@: V F @: @: 1 @:vfloor@:(s)}
35833 @r{ v@: V G @: @: @:grade@:(v)}
35834 @r{ v@: I V G @: @: @:rgrade@:(v)}
35835 @r{ v@: V H @:n @: 31 @:histogram@:(v,n)}
35836 @r{ v w@: H V H @:n @: 31 @:histogram@:(v,w,n)}
35837 @r{ v1 v2@: V I @:mop aop @: 22 @:inner@:(mop,aop,v1,v2)}
35838 @r{ m@: V J @: @: 1 @:ctrn@:(m)}
35839 @r{ m@: V L @: @: 1 @:lud@:(m)}
35840 @r{ v@: V M @:op @: 22,23 @:map@:(op,v)}
35841 @r{ v@: V N @: @: 1 @:cnorm@:(v)}
35842 @r{ v1 v2@: V O @:op @: 22 @:outer@:(op,v1,v2)}
35843 @r{ v@: V R @:op @: 22,23 @:reduce@:(op,v)}
35844 @r{ v@: I V R @:op @: 22,23 @:rreduce@:(op,v)}
35845 @r{ a n@: H V R @:op @: 22 @:nest@:(op,a,n)}
35846 @r{ a@: I H V R @:op @: 22 @:fixp@:(op,a)}
35847 @r{ v@: V S @: @: @:sort@:(v)}
35848 @r{ v@: I V S @: @: @:rsort@:(v)}
35849 @r{ m@: V T @: @: 1 @:tr@:(m)}
35850 @r{ v@: V U @:op @: 22 @:accum@:(op,v)}
35851 @r{ v@: I V U @:op @: 22 @:raccum@:(op,v)}
35852 @r{ a n@: H V U @:op @: 22 @:anest@:(op,a,n)}
35853 @r{ a@: I H V U @:op @: 22 @:afixp@:(op,a)}
35854 @r{ s t@: V V @: @: 2 @:vunion@:(s,t)}
35855 @r{ s t@: V X @: @: 2 @:vxor@:(s,t)}
35858 @r{ @: Y @: @: @:@:user commands}
35861 @r{ @: z @: @: @:@:user commands}
35864 @r{ c@: Z [ @: @: 45 @:calc-kbd-if@:}
35865 @r{ c@: Z | @: @: 45 @:calc-kbd-else-if@:}
35866 @r{ @: Z : @: @: @:calc-kbd-else@:}
35867 @r{ @: Z ] @: @: @:calc-kbd-end-if@:}
35870 @r{ @: Z @{ @: @: 4 @:calc-kbd-loop@:}
35871 @r{ c@: Z / @: @: 45 @:calc-kbd-break@:}
35872 @r{ @: Z @} @: @: @:calc-kbd-end-loop@:}
35873 @r{ n@: Z < @: @: @:calc-kbd-repeat@:}
35874 @r{ @: Z > @: @: @:calc-kbd-end-repeat@:}
35875 @r{ n m@: Z ( @: @: @:calc-kbd-for@:}
35876 @r{ s@: Z ) @: @: @:calc-kbd-end-for@:}
35879 @r{ @: Z C-g @: @: @:@:cancel if/loop command}
35882 @r{ @: Z ` @: @: @:calc-kbd-push@:}
35883 @r{ @: Z ' @: @: @:calc-kbd-pop@:}
35884 @r{ @: Z # @: @: @:calc-kbd-query@:}
35887 @r{ comp@: Z C @:func, args @: 50 @:calc-user-define-composition@:}
35888 @r{ @: Z D @:key, command @: @:calc-user-define@:}
35889 @r{ @: Z E @:key, editing @: 30 @:calc-user-define-edit@:}
35890 @r{ defn@: Z F @:k, c, f, a, n@: 28 @:calc-user-define-formula@:}
35891 @r{ @: Z G @:key @: @:calc-get-user-defn@:}
35892 @r{ @: Z I @: @: @:calc-user-define-invocation@:}
35893 @r{ @: Z K @:key, command @: @:calc-user-define-kbd-macro@:}
35894 @r{ @: Z P @:key @: @:calc-user-define-permanent@:}
35895 @r{ @: Z S @: @: 30 @:calc-edit-user-syntax@:}
35896 @r{ @: Z T @: @: 12 @:calc-timing@:}
35897 @r{ @: Z U @:key @: @:calc-user-undefine@:}
35907 Positive prefix arguments apply to @expr{n} stack entries.
35908 Negative prefix arguments apply to the @expr{-n}th stack entry.
35909 A prefix of zero applies to the entire stack. (For @key{LFD} and
35910 @kbd{M-@key{DEL}}, the meaning of the sign is reversed.)
35914 Positive prefix arguments apply to @expr{n} stack entries.
35915 Negative prefix arguments apply to the top stack entry
35916 and the next @expr{-n} stack entries.
35920 Positive prefix arguments rotate top @expr{n} stack entries by one.
35921 Negative prefix arguments rotate the entire stack by @expr{-n}.
35922 A prefix of zero reverses the entire stack.
35926 Prefix argument specifies a repeat count or distance.
35930 Positive prefix arguments specify a precision @expr{p}.
35931 Negative prefix arguments reduce the current precision by @expr{-p}.
35935 A prefix argument is interpreted as an additional step-size parameter.
35936 A plain @kbd{C-u} prefix means to prompt for the step size.
35940 A prefix argument specifies simplification level and depth.
35941 1=Default, 2=like @kbd{a s}, 3=like @kbd{a e}.
35945 A negative prefix operates only on the top level of the input formula.
35949 Positive prefix arguments specify a word size of @expr{w} bits, unsigned.
35950 Negative prefix arguments specify a word size of @expr{w} bits, signed.
35954 Prefix arguments specify the shift amount @expr{n}. The @expr{w} argument
35955 cannot be specified in the keyboard version of this command.
35959 From the keyboard, @expr{d} is omitted and defaults to zero.
35963 Mode is toggled; a positive prefix always sets the mode, and a negative
35964 prefix always clears the mode.
35968 Some prefix argument values provide special variations of the mode.
35972 A prefix argument, if any, is used for @expr{m} instead of taking
35973 @expr{m} from the stack. @expr{M} may take any of these values:
35975 {@advance@tableindent10pt
35979 Random integer in the interval @expr{[0 .. m)}.
35981 Random floating-point number in the interval @expr{[0 .. m)}.
35983 Gaussian with mean 1 and standard deviation 0.
35985 Gaussian with specified mean and standard deviation.
35987 Random integer or floating-point number in that interval.
35989 Random element from the vector.
35997 A prefix argument from 1 to 6 specifies number of date components
35998 to remove from the stack. @xref{Date Conversions}.
36002 A prefix argument specifies a time zone; @kbd{C-u} says to take the
36003 time zone number or name from the top of the stack. @xref{Time Zones}.
36007 A prefix argument specifies a day number (0-6, 0-31, or 0-366).
36011 If the input has no units, you will be prompted for both the old and
36016 With a prefix argument, collect that many stack entries to form the
36017 input data set. Each entry may be a single value or a vector of values.
36021 With a prefix argument of 1, take a single
36022 @texline @var{n}@math{\times2}
36023 @infoline @mathit{@var{N}x2}
36024 matrix from the stack instead of two separate data vectors.
36028 The row or column number @expr{n} may be given as a numeric prefix
36029 argument instead. A plain @kbd{C-u} prefix says to take @expr{n}
36030 from the top of the stack. If @expr{n} is a vector or interval,
36031 a subvector/submatrix of the input is created.
36035 The @expr{op} prompt can be answered with the key sequence for the
36036 desired function, or with @kbd{x} or @kbd{z} followed by a function name,
36037 or with @kbd{$} to take a formula from the top of the stack, or with
36038 @kbd{'} and a typed formula. In the last two cases, the formula may
36039 be a nameless function like @samp{<#1+#2>} or @samp{<x, y : x+y>}, or it
36040 may include @kbd{$}, @kbd{$$}, etc. (where @kbd{$} will correspond to the
36041 last argument of the created function), or otherwise you will be
36042 prompted for an argument list. The number of vectors popped from the
36043 stack by @kbd{V M} depends on the number of arguments of the function.
36047 One of the mapping direction keys @kbd{_} (horizontal, i.e., map
36048 by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or
36049 reduce down), or @kbd{=} (map or reduce by rows) may be used before
36050 entering @expr{op}; these modify the function name by adding the letter
36051 @code{r} for ``rows,'' @code{c} for ``columns,'' @code{a} for ``across,''
36052 or @code{d} for ``down.''
36056 The prefix argument specifies a packing mode. A nonnegative mode
36057 is the number of items (for @kbd{v p}) or the number of levels
36058 (for @kbd{v u}). A negative mode is as described below. With no
36059 prefix argument, the mode is taken from the top of the stack and
36060 may be an integer or a vector of integers.
36062 {@advance@tableindent-20pt
36066 (@var{2}) Rectangular complex number.
36068 (@var{2}) Polar complex number.
36070 (@var{3}) HMS form.
36072 (@var{2}) Error form.
36074 (@var{2}) Modulo form.
36076 (@var{2}) Closed interval.
36078 (@var{2}) Closed .. open interval.
36080 (@var{2}) Open .. closed interval.
36082 (@var{2}) Open interval.
36084 (@var{2}) Fraction.
36086 (@var{2}) Float with integer mantissa.
36088 (@var{2}) Float with mantissa in @expr{[1 .. 10)}.
36090 (@var{1}) Date form (using date numbers).
36092 (@var{3}) Date form (using year, month, day).
36094 (@var{6}) Date form (using year, month, day, hour, minute, second).
36102 A prefix argument specifies the size @expr{n} of the matrix. With no
36103 prefix argument, @expr{n} is omitted and the size is inferred from
36108 The prefix argument specifies the starting position @expr{n} (default 1).
36112 Cursor position within stack buffer affects this command.
36116 Arguments are not actually removed from the stack by this command.
36120 Variable name may be a single digit or a full name.
36124 Editing occurs in a separate buffer. Press @kbd{C-c C-c} (or
36125 @key{LFD}, or in some cases @key{RET}) to finish the edit, or kill the
36126 buffer with @kbd{C-x k} to cancel the edit. The @key{LFD} key prevents evaluation
36127 of the result of the edit.
36131 The number prompted for can also be provided as a prefix argument.
36135 Press this key a second time to cancel the prefix.
36139 With a negative prefix, deactivate all formulas. With a positive
36140 prefix, deactivate and then reactivate from scratch.
36144 Default is to scan for nearest formula delimiter symbols. With a
36145 prefix of zero, formula is delimited by mark and point. With a
36146 non-zero prefix, formula is delimited by scanning forward or
36147 backward by that many lines.
36151 Parse the region between point and mark as a vector. A nonzero prefix
36152 parses @var{n} lines before or after point as a vector. A zero prefix
36153 parses the current line as a vector. A @kbd{C-u} prefix parses the
36154 region between point and mark as a single formula.
36158 Parse the rectangle defined by point and mark as a matrix. A positive
36159 prefix @var{n} divides the rectangle into columns of width @var{n}.
36160 A zero or @kbd{C-u} prefix parses each line as one formula. A negative
36161 prefix suppresses special treatment of bracketed portions of a line.
36165 A numeric prefix causes the current language mode to be ignored.
36169 Responding to a prompt with a blank line answers that and all
36170 later prompts by popping additional stack entries.
36174 Answer for @expr{v} may also be of the form @expr{v = v_0} or
36179 With a positive prefix argument, stack contains many @expr{y}'s and one
36180 common @expr{x}. With a zero prefix, stack contains a vector of
36181 @expr{y}s and a common @expr{x}. With a negative prefix, stack
36182 contains many @expr{[x,y]} vectors. (For 3D plots, substitute
36183 @expr{z} for @expr{y} and @expr{x,y} for @expr{x}.)
36187 With any prefix argument, all curves in the graph are deleted.
36191 With a positive prefix, refines an existing plot with more data points.
36192 With a negative prefix, forces recomputation of the plot data.
36196 With any prefix argument, set the default value instead of the
36197 value for this graph.
36201 With a negative prefix argument, set the value for the printer.
36205 Condition is considered ``true'' if it is a nonzero real or complex
36206 number, or a formula whose value is known to be nonzero; it is ``false''
36211 Several formulas separated by commas are pushed as multiple stack
36212 entries. Trailing @kbd{)}, @kbd{]}, @kbd{@}}, @kbd{>}, and @kbd{"}
36213 delimiters may be omitted. The notation @kbd{$$$} refers to the value
36214 in stack level three, and causes the formula to replace the top three
36215 stack levels. The notation @kbd{$3} refers to stack level three without
36216 causing that value to be removed from the stack. Use @key{LFD} in place
36217 of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET}
36218 to evaluate variables.
36222 The variable is replaced by the formula shown on the right. The
36223 Inverse flag reverses the order of the operands, e.g., @kbd{I s - x}
36225 @texline @math{x \coloneq a-x}.
36226 @infoline @expr{x := a-x}.
36230 Press @kbd{?} repeatedly to see how to choose a model. Answer the
36231 variables prompt with @expr{iv} or @expr{iv;pv} to specify
36232 independent and parameter variables. A positive prefix argument
36233 takes @mathit{@var{n}+1} vectors from the stack; a zero prefix takes a matrix
36234 and a vector from the stack.
36238 With a plain @kbd{C-u} prefix, replace the current region of the
36239 destination buffer with the yanked text instead of inserting.
36243 All stack entries are reformatted; the @kbd{H} prefix inhibits this.
36244 The @kbd{I} prefix sets the mode temporarily, redraws the top stack
36245 entry, then restores the original setting of the mode.
36249 A negative prefix sets the default 3D resolution instead of the
36250 default 2D resolution.
36254 This grabs a vector of the form [@var{prec}, @var{wsize}, @var{ssize},
36255 @var{radix}, @var{flfmt}, @var{ang}, @var{frac}, @var{symb}, @var{polar},
36256 @var{matrix}, @var{simp}, @var{inf}]. A prefix argument from 1 to 12
36257 grabs the @var{n}th mode value only.
36261 (Space is provided below for you to keep your own written notes.)
36269 @node Key Index, Command Index, Summary, Top
36270 @unnumbered Index of Key Sequences
36274 @node Command Index, Function Index, Key Index, Top
36275 @unnumbered Index of Calculator Commands
36277 Since all Calculator commands begin with the prefix @samp{calc-}, the
36278 @kbd{x} key has been provided as a variant of @kbd{M-x} which automatically
36279 types @samp{calc-} for you. Thus, @kbd{x last-args} is short for
36280 @kbd{M-x calc-last-args}.
36284 @node Function Index, Concept Index, Command Index, Top
36285 @unnumbered Index of Algebraic Functions
36287 This is a list of built-in functions and operators usable in algebraic
36288 expressions. Their full Lisp names are derived by adding the prefix
36289 @samp{calcFunc-}, as in @code{calcFunc-sqrt}.
36291 All functions except those noted with ``*'' have corresponding
36292 Calc keystrokes and can also be found in the Calc Summary.
36297 @node Concept Index, Variable Index, Function Index, Top
36298 @unnumbered Concept Index
36302 @node Variable Index, Lisp Function Index, Concept Index, Top
36303 @unnumbered Index of Variables
36305 The variables in this list that do not contain dashes are accessible
36306 as Calc variables. Add a @samp{var-} prefix to get the name of the
36307 corresponding Lisp variable.
36309 The remaining variables are Lisp variables suitable for @code{setq}ing
36310 in your Calc init file or @file{.emacs} file.
36314 @node Lisp Function Index, , Variable Index, Top
36315 @unnumbered Index of Lisp Math Functions
36317 The following functions are meant to be used with @code{defmath}, not
36318 @code{defun} definitions. For names that do not start with @samp{calc-},
36319 the corresponding full Lisp name is derived by adding a prefix of
36333 arch-tag: 77a71809-fa4d-40be-b2cc-da3e8fb137c0