1 ;;; calc-funcs.el --- well-known functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc.
5 ;; Author: David Gillespie <daveg@synaptics.com>
6 ;; Maintainer: Jay Belanger <belanger@truman.edu>
8 ;; This file is part of GNU Emacs.
10 ;; GNU Emacs is distributed in the hope that it will be useful,
11 ;; but WITHOUT ANY WARRANTY. No author or distributor
12 ;; accepts responsibility to anyone for the consequences of using it
13 ;; or for whether it serves any particular purpose or works at all,
14 ;; unless he says so in writing. Refer to the GNU Emacs General Public
15 ;; License for full details.
17 ;; Everyone is granted permission to copy, modify and redistribute
18 ;; GNU Emacs, but only under the conditions described in the
19 ;; GNU Emacs General Public License. A copy of this license is
20 ;; supposed to have been given to you along with GNU Emacs so you
21 ;; can know your rights and responsibilities. It should be in a
22 ;; file named COPYING. Among other things, the copyright notice
23 ;; and this notice must be preserved on all copies.
29 ;; This file is autoloaded from calc-ext.el.
34 (defun calc-inc-gamma (arg)
38 (if (calc-is-hyperbolic)
39 (calc-binary-op "gamG" 'calcFunc-gammaG arg)
40 (calc-binary-op "gamQ" 'calcFunc-gammaQ arg))
41 (if (calc-is-hyperbolic)
42 (calc-binary-op "gamg" 'calcFunc-gammag arg)
43 (calc-binary-op "gamP" 'calcFunc-gammaP arg)))))
49 (calc-unary-op "erfc" 'calcFunc-erfc arg)
50 (calc-unary-op "erf" 'calcFunc-erf arg))))
52 (defun calc-erfc (arg)
57 (defun calc-beta (arg)
60 (calc-binary-op "beta" 'calcFunc-beta arg)))
62 (defun calc-inc-beta ()
65 (if (calc-is-hyperbolic)
66 (calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3)))
67 (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3))))))
69 (defun calc-bessel-J (arg)
72 (calc-binary-op "besJ" 'calcFunc-besJ arg)))
74 (defun calc-bessel-Y (arg)
77 (calc-binary-op "besY" 'calcFunc-besY arg)))
79 (defun calc-bernoulli-number (arg)
82 (if (calc-is-hyperbolic)
83 (calc-binary-op "bern" 'calcFunc-bern arg)
84 (calc-unary-op "bern" 'calcFunc-bern arg))))
86 (defun calc-euler-number (arg)
89 (if (calc-is-hyperbolic)
90 (calc-binary-op "eulr" 'calcFunc-euler arg)
91 (calc-unary-op "eulr" 'calcFunc-euler arg))))
93 (defun calc-stirling-number (arg)
96 (if (calc-is-hyperbolic)
97 (calc-binary-op "str2" 'calcFunc-stir2 arg)
98 (calc-binary-op "str1" 'calcFunc-stir1 arg))))
102 (calc-prob-dist "b" 3))
106 (calc-prob-dist "c" 2))
110 (calc-prob-dist "f" 3))
114 (calc-prob-dist "n" 3))
118 (calc-prob-dist "p" 2))
122 (calc-prob-dist "t" 2))
124 (defun calc-prob-dist (letter nargs)
126 (if (calc-is-inverse)
127 (calc-enter-result nargs (concat "ltp" letter)
128 (append (list (intern (concat "calcFunc-ltp" letter))
130 (calc-top-list-n (1- nargs) 2)))
131 (calc-enter-result nargs (concat "utp" letter)
132 (append (list (intern (concat "calcFunc-utp" letter))
134 (calc-top-list-n (1- nargs) 2))))))
139 ;;; Sources: Numerical Recipes, Press et al;
140 ;;; Handbook of Mathematical Functions, Abramowitz & Stegun.
145 (defun calcFunc-gamma (x)
146 (or (math-numberp x) (math-reject-arg x 'numberp))
147 (calcFunc-fact (math-add x -1)))
149 (defun math-gammap1-raw (x &optional fprec nfprec) ; compute gamma(1 + x)
151 (setq fprec (math-float calc-internal-prec)
152 nfprec (math-float (- calc-internal-prec))))
153 (cond ((math-lessp-float (calcFunc-re x) fprec)
154 (if (math-lessp-float (calcFunc-re x) nfprec)
157 (math-mul (math-gammap1-raw
158 (math-add (math-neg x)
162 (math-mul (math-pi) x)))))
163 (let ((xplus1 (math-add x '(float 1 0))))
164 (math-div (math-gammap1-raw xplus1 fprec nfprec) xplus1))))
166 (math-lessp-float '(float 736276 0) x))
168 (t ; re(x) now >= 10.0
169 (let ((xinv (math-div 1 x))
170 (lnx (math-ln-raw x)))
171 (math-mul (math-sqrt-two-pi)
174 (math-sub (math-mul (math-add x '(float 5 -1))
182 (defun math-gamma-series (sum x xinvsqr oterm n)
183 (math-working "gamma" sum)
184 (let* ((bn (math-bernoulli-number n))
185 (term (math-mul (math-div-float (math-float (nth 1 bn))
186 (math-float (* (nth 2 bn)
189 (next (math-add sum term)))
190 (if (math-nearly-equal sum next)
192 (if (> n (* 2 calc-internal-prec))
194 ;; Need this because series eventually diverges for large enough n.
196 "*Gamma computation stopped early, not all digits may be valid")
198 (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2))))))
201 ;;; Incomplete gamma function.
203 (defvar math-current-gamma-value nil)
204 (defun calcFunc-gammaP (a x)
205 (if (equal x '(var inf var-inf))
207 (math-inexact-result)
208 (or (Math-numberp a) (math-reject-arg a 'numberp))
209 (or (math-numberp x) (math-reject-arg x 'numberp))
210 (if (and (math-num-integerp a)
211 (integerp (setq a (math-trunc a)))
213 (math-sub 1 (calcFunc-gammaQ a x))
214 (let ((math-current-gamma-value (calcFunc-gamma a)))
215 (math-div (calcFunc-gammag a x) math-current-gamma-value)))))
217 (defun calcFunc-gammaQ (a x)
218 (if (equal x '(var inf var-inf))
220 (math-inexact-result)
221 (or (Math-numberp a) (math-reject-arg a 'numberp))
222 (or (math-numberp x) (math-reject-arg x 'numberp))
223 (if (and (math-num-integerp a)
224 (integerp (setq a (math-trunc a)))
229 (math-with-extra-prec 1
230 (while (< (setq n (1+ n)) a)
231 (setq term (math-div (math-mul term x) n)
232 sum (math-add sum term))
233 (math-working "gamma" sum))
234 (math-mul sum (calcFunc-exp (math-neg x)))))
235 (let ((math-current-gamma-value (calcFunc-gamma a)))
236 (math-div (calcFunc-gammaG a x) math-current-gamma-value)))))
238 (defun calcFunc-gammag (a x)
239 (if (equal x '(var inf var-inf))
241 (math-inexact-result)
242 (or (Math-numberp a) (math-reject-arg a 'numberp))
243 (or (Math-numberp x) (math-reject-arg x 'numberp))
244 (math-with-extra-prec 2
245 (setq a (math-float a))
246 (setq x (math-float x))
247 (if (or (math-negp (calcFunc-re a))
248 (math-lessp-float (calcFunc-re x)
249 (math-add-float (calcFunc-re a)
251 (math-inc-gamma-series a x)
252 (math-sub (or math-current-gamma-value (calcFunc-gamma a))
253 (math-inc-gamma-cfrac a x))))))
255 (defun calcFunc-gammaG (a x)
256 (if (equal x '(var inf var-inf))
258 (math-inexact-result)
259 (or (Math-numberp a) (math-reject-arg a 'numberp))
260 (or (Math-numberp x) (math-reject-arg x 'numberp))
261 (math-with-extra-prec 2
262 (setq a (math-float a))
263 (setq x (math-float x))
264 (if (or (math-negp (calcFunc-re a))
265 (math-lessp-float (calcFunc-re x)
266 (math-add-float (math-abs-approx a)
268 (math-sub (or math-current-gamma-value (calcFunc-gamma a))
269 (math-inc-gamma-series a x))
270 (math-inc-gamma-cfrac a x)))))
272 (defun math-inc-gamma-series (a x)
275 (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
276 (math-with-extra-prec 2
277 (let ((start (math-div '(float 1 0) a)))
278 (math-inc-gamma-series-step start start a x))))))
280 (defun math-inc-gamma-series-step (sum term a x)
281 (math-working "gamma" sum)
282 (setq a (math-add a '(float 1 0))
283 term (math-div (math-mul term x) a))
284 (let ((next (math-add sum term)))
285 (if (math-nearly-equal sum next)
287 (math-inc-gamma-series-step next term a x))))
289 (defun math-inc-gamma-cfrac (a x)
291 (or math-current-gamma-value (calcFunc-gamma a))
292 (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
293 (math-inc-gamma-cfrac-step '(float 1 0) x
294 '(float 0 0) '(float 1 0)
295 '(float 1 0) '(float 1 0) '(float 0 0)
298 (defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x)
299 (let ((ana (math-sub n a))
300 (anf (math-mul n fac)))
301 (setq n (math-add n '(float 1 0))
302 a0 (math-mul (math-add a1 (math-mul a0 ana)) fac)
303 b0 (math-mul (math-add b1 (math-mul b0 ana)) fac)
304 a1 (math-add (math-mul x a0) (math-mul anf a1))
305 b1 (math-add (math-mul x b0) (math-mul anf b1)))
307 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x)
308 (setq fac (math-div '(float 1 0) a1))
309 (let ((next (math-mul b1 fac)))
310 (math-working "gamma" next)
311 (if (math-nearly-equal next g)
313 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x))))))
318 (defun calcFunc-erf (x)
319 (if (equal x '(var inf var-inf))
321 (if (equal x '(neg (var inf var-inf)))
325 (let ((math-current-gamma-value (math-sqrt-pi)))
326 (math-to-same-complex-quad
327 (math-div (calcFunc-gammag '(float 5 -1)
328 (math-sqr (math-to-complex-quad-one x)))
329 math-current-gamma-value)
332 (defun calcFunc-erfc (x)
333 (if (equal x '(var inf var-inf))
336 (let ((math-current-gamma-value (math-sqrt-pi)))
337 (math-div (calcFunc-gammaG '(float 5 -1) (math-sqr x))
338 math-current-gamma-value))
339 (math-sub 1 (calcFunc-erf x)))))
341 (defun math-to-complex-quad-one (x)
342 (if (eq (car-safe x) 'polar) (setq x (math-complex x)))
343 (if (eq (car-safe x) 'cplx)
344 (list 'cplx (math-abs (nth 1 x)) (math-abs (nth 2 x)))
347 (defun math-to-same-complex-quad (x y)
348 (if (eq (car-safe y) 'cplx)
349 (if (eq (car-safe x) 'cplx)
351 (if (math-negp (nth 1 y)) (math-neg (nth 1 x)) (nth 1 x))
352 (if (math-negp (nth 2 y)) (math-neg (nth 2 x)) (nth 2 x)))
353 (if (math-negp (nth 1 y)) (math-neg x) x))
355 (if (eq (car-safe x) 'cplx)
356 (list 'cplx (math-neg (nth 1 x)) (nth 2 x))
363 (defun calcFunc-beta (a b)
364 (if (math-num-integerp a)
365 (let ((am (math-add a -1)))
366 (or (math-numberp b) (math-reject-arg b 'numberp))
367 (math-div 1 (math-mul b (calcFunc-choose (math-add b am) am))))
368 (if (math-num-integerp b)
370 (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b))
371 (calcFunc-gamma (math-add a b))))))
374 ;;; Incomplete beta function.
376 (defvar math-current-beta-value nil)
377 (defun calcFunc-betaI (x a b)
378 (cond ((math-zerop x)
380 ((math-equal-int x 1)
383 (and (math-num-integerp a)
385 (if (or (math-zerop b)
386 (and (math-num-integerp b)
388 (math-reject-arg b 'range)
391 (and (math-num-integerp b)
394 ((not (math-numberp a)) (math-reject-arg a 'numberp))
395 ((not (math-numberp b)) (math-reject-arg b 'numberp))
396 ((math-inexact-result))
397 (t (let ((math-current-beta-value (calcFunc-beta a b)))
398 (math-div (calcFunc-betaB x a b) math-current-beta-value)))))
400 (defun calcFunc-betaB (x a b)
404 ((math-equal-int x 1)
406 ((not (math-numberp x)) (math-reject-arg x 'numberp))
407 ((not (math-numberp a)) (math-reject-arg a 'numberp))
408 ((not (math-numberp b)) (math-reject-arg b 'numberp))
409 ((math-zerop a) (math-reject-arg a 'nonzerop))
410 ((math-zerop b) (math-reject-arg b 'nonzerop))
411 ((and (math-num-integerp b)
413 (math-reject-arg b 'range)
414 (Math-natnum-lessp (setq b (math-trunc b)) 20)))
415 (and calc-symbolic-mode (or (math-floatp a) (math-floatp b))
416 (math-inexact-result))
418 (math-with-extra-prec 2
421 (sum (math-div term a)))
422 (while (< (setq i (1+ i)) b)
423 (setq term (math-mul (math-div (math-mul term (- i b)) i) x)
424 sum (math-add sum (math-div term (math-add a i))))
425 (math-working "beta" sum))
428 ((and (math-num-integerp a)
430 (math-reject-arg a 'range)
431 (Math-natnum-lessp (setq a (math-trunc a)) 20)))
432 (math-sub (or math-current-beta-value (calcFunc-beta a b))
433 (calcFunc-betaB (math-sub 1 x) b a)))
435 (math-inexact-result)
436 (math-with-extra-prec 2
437 (setq x (math-float x))
438 (setq a (math-float a))
439 (setq b (math-float b))
440 (let ((bt (math-exp-raw (math-add (math-mul a (math-ln-raw x))
441 (math-mul b (math-ln-raw
442 (math-sub '(float 1 0)
444 (if (Math-lessp x (math-div (math-add a '(float 1 0))
445 (math-add (math-add a b) '(float 2 0))))
446 (math-div (math-mul bt (math-beta-cfrac a b x)) a)
447 (math-sub (or math-current-beta-value (calcFunc-beta a b))
448 (math-div (math-mul bt
449 (math-beta-cfrac b a (math-sub 1 x)))
452 (defun math-beta-cfrac (a b x)
453 (let ((qab (math-add a b))
454 (qap (math-add a '(float 1 0)))
455 (qam (math-add a '(float -1 0))))
456 (math-beta-cfrac-step '(float 1 0)
457 (math-sub '(float 1 0)
458 (math-div (math-mul qab x) qap))
459 '(float 1 0) '(float 1 0)
463 (defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x)
464 (let* ((two-m (math-mul m '(float 2 0)))
465 (d (math-div (math-mul (math-mul (math-sub b m) m) x)
466 (math-mul (math-add qam two-m) (math-add a two-m))))
467 (ap (math-add az (math-mul d am)))
468 (bp (math-add bz (math-mul d bm)))
470 (math-div (math-mul (math-mul (math-add a m) (math-add qab m)) x)
471 (math-mul (math-add qap two-m) (math-add a two-m)))))
472 (app (math-add ap (math-mul d2 az)))
473 (bpp (math-add bp (math-mul d2 bz)))
474 (next (math-div app bpp)))
475 (math-working "beta" next)
476 (if (math-nearly-equal next az)
478 (math-beta-cfrac-step next '(float 1 0)
479 (math-div ap bpp) (math-div bp bpp)
480 (math-add m '(float 1 0))
481 qab qap qam a b x))))
484 ;;; Bessel functions.
486 ;;; Should generalize this to handle arbitrary precision!
488 (defun calcFunc-besJ (v x)
489 (or (math-numberp v) (math-reject-arg v 'numberp))
490 (or (math-numberp x) (math-reject-arg x 'numberp))
491 (let ((calc-internal-prec (min 8 calc-internal-prec)))
492 (math-with-extra-prec 3
493 (setq x (math-float (math-normalize x)))
494 (setq v (math-float (math-normalize v)))
495 (cond ((math-zerop x)
499 ((math-inexact-result))
500 ((not (math-num-integerp v))
501 (let ((start (math-div 1 (calcFunc-fact v))))
502 (math-mul (math-besJ-series start start
504 (math-mul '(float -25 -2)
507 (math-pow (math-div x 2) v))))
508 ((math-negp (setq v (math-trunc v)))
510 (math-neg (calcFunc-besJ (math-neg v) x))
511 (calcFunc-besJ (math-neg v) x)))
516 ((Math-lessp v (math-abs-approx x))
520 (two-over-x (math-div 2 x))
522 (while (< (setq j (1+ j)) v)
523 (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj)
529 (if (Math-lessp 100 v) (math-reject-arg v 'range))
530 (let* ((j (logior (+ v (math-isqrt-small (* 40 v))) 1))
531 (two-over-x (math-div 2 x))
537 (while (> (setq j (1- j)) 0)
538 (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj)
542 (if (> (nth 2 (math-abs-approx bj)) 10)
543 (setq bj (math-mul bj '(float 1 -10))
544 bjp (math-mul bjp '(float 1 -10))
545 ans (and ans (math-mul ans '(float 1 -10)))
546 sum (math-mul sum '(float 1 -10))))
547 (or (setq jsum (not jsum))
548 (setq sum (math-add sum bj)))
551 (math-div ans (math-sub (math-mul 2 sum) bj))))))))
553 (defun math-besJ-series (sum term k zz vk)
554 (math-working "besJ" sum)
557 term (math-div (math-mul term zz) (math-mul k vk)))
558 (let ((next (math-add sum term)))
559 (if (math-nearly-equal next sum)
561 (math-besJ-series next term k zz vk))))
563 (defun math-besJ0 (x &optional yflag)
564 (cond ((and (not yflag) (math-negp (calcFunc-re x)))
565 (math-besJ0 (math-neg x)))
566 ((Math-lessp '(float 8 0) (math-abs-approx x))
567 (let* ((z (math-div '(float 8 0) x))
569 (xx (math-add x '(float (bigneg 164 398 785) -9)))
570 (a1 (math-poly-eval y
571 '((float (bigpos 211 887 093 2) -16)
572 (float (bigneg 639 370 073 2) -15)
573 (float (bigpos 407 510 734 2) -14)
574 (float (bigneg 627 628 098 1) -12)
576 (a2 (math-poly-eval y
577 '((float (bigneg 152 935 934) -16)
578 (float (bigpos 161 095 621 7) -16)
579 (float (bigneg 651 147 911 6) -15)
580 (float (bigpos 765 488 430 1) -13)
581 (float (bigneg 995 499 562 1) -11))))
582 (sc (math-sin-cos-raw xx)))
584 (setq sc (cons (math-neg (cdr sc)) (car sc))))
586 (math-div '(float (bigpos 722 619 636) -9) x))
587 (math-sub (math-mul (cdr sc) a1)
588 (math-mul (car sc) (math-mul z a2))))))
590 (let ((y (math-sqr x)))
591 (math-div (math-poly-eval y
592 '((float (bigneg 456 052 849 1) -7)
593 (float (bigpos 017 233 739 7) -5)
594 (float (bigneg 418 442 121 1) -2)
595 (float (bigpos 407 196 516 6) -1)
596 (float (bigneg 354 590 362 13) 0)
597 (float (bigpos 574 490 568 57) 0)))
600 (float (bigpos 712 532 678 2) -7)
601 (float (bigpos 853 264 927 5) -5)
602 (float (bigpos 718 680 494 9) -3)
603 (float (bigpos 985 532 029 1) 0)
604 (float (bigpos 411 490 568 57) 0))))))))
606 (defun math-besJ1 (x &optional yflag)
607 (cond ((and (math-negp (calcFunc-re x)) (not yflag))
608 (math-neg (math-besJ1 (math-neg x))))
609 ((Math-lessp '(float 8 0) (math-abs-approx x))
610 (let* ((z (math-div '(float 8 0) x))
612 (xx (math-add x '(float (bigneg 491 194 356 2) -9)))
613 (a1 (math-poly-eval y
614 '((float (bigneg 019 337 240) -15)
615 (float (bigpos 174 520 457 2) -15)
616 (float (bigneg 496 396 516 3) -14)
619 (a2 (math-poly-eval y
620 '((float (bigpos 412 787 105) -15)
621 (float (bigneg 987 228 88) -14)
622 (float (bigpos 096 199 449 8) -15)
623 (float (bigneg 873 690 002 2) -13)
624 (float (bigpos 995 499 687 4) -11))))
625 (sc (math-sin-cos-raw xx)))
627 (setq sc (cons (math-neg (cdr sc)) (car sc)))
629 (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc))))))
630 (math-mul (math-sqrt (math-div '(float (bigpos 722 619 636) -9) x))
631 (math-sub (math-mul (cdr sc) a1)
632 (math-mul (car sc) (math-mul z a2))))))
634 (let ((y (math-sqr x)))
637 (math-div (math-poly-eval y
638 '((float (bigneg 606 036 016 3) -8)
639 (float (bigpos 826 044 157) -4)
640 (float (bigneg 439 611 972 2) -3)
641 (float (bigpos 531 968 423 2) -1)
642 (float (bigneg 235 059 895 7) 0)
643 (float (bigpos 232 614 362 72) 0)))
646 (float (bigpos 397 991 769 3) -7)
647 (float (bigpos 394 743 944 9) -5)
648 (float (bigpos 474 330 858 1) -2)
649 (float (bigpos 178 535 300 2) 0)
650 (float (bigpos 442 228 725 144)
653 (defun calcFunc-besY (v x)
654 (math-inexact-result)
655 (or (math-numberp v) (math-reject-arg v 'numberp))
656 (or (math-numberp x) (math-reject-arg x 'numberp))
657 (let ((calc-internal-prec (min 8 calc-internal-prec)))
658 (math-with-extra-prec 3
659 (setq x (math-float (math-normalize x)))
660 (setq v (math-float (math-normalize v)))
661 (cond ((not (math-num-integerp v))
662 (let ((sc (math-sin-cos-raw (math-mul v (math-pi)))))
663 (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc))
664 (calcFunc-besJ (math-neg v) x))
666 ((math-negp (setq v (math-trunc v)))
668 (math-neg (calcFunc-besY (math-neg v) x))
669 (calcFunc-besY (math-neg v) x)))
678 (two-over-x (math-div 2 x))
680 (while (< (setq j (1+ j)) v)
681 (setq byp (math-sub (math-mul (math-mul j two-over-x) by)
687 (defun math-besY0 (x)
688 (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
689 (let ((y (math-sqr x)))
691 (math-div (math-poly-eval y
692 '((float (bigpos 733 622 284 2) -7)
693 (float (bigneg 757 792 632 8) -5)
694 (float (bigpos 129 988 087 1) -2)
695 (float (bigneg 036 598 123 5) -1)
696 (float (bigpos 065 834 062 7) 0)
697 (float (bigneg 389 821 957 2) 0)))
700 (float (bigpos 244 030 261 2) -7)
701 (float (bigpos 647 472 474) -4)
702 (float (bigpos 438 466 189 7) -3)
703 (float (bigpos 648 499 452 7) -1)
704 (float (bigpos 269 544 076 40) 0))))
705 (math-mul '(float (bigpos 772 619 636) -9)
706 (math-mul (math-besJ0 x) (math-ln-raw x))))))
707 ((math-negp (calcFunc-re x))
708 (math-add (math-besJ0 (math-neg x) t)
709 (math-mul '(cplx 0 2)
710 (math-besJ0 (math-neg x)))))
714 (defun math-besY1 (x)
715 (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
716 (let ((y (math-sqr x)))
720 (math-div (math-poly-eval y
721 '((float (bigpos 935 937 511 8) -6)
722 (float (bigneg 726 922 237 4) -3)
723 (float (bigpos 551 264 349 7) -1)
724 (float (bigneg 139 438 153 5) 1)
725 (float (bigpos 439 527 127) 4)
726 (float (bigneg 943 604 900 4) 3)))
729 (float (bigpos 885 632 549 3) -7)
730 (float (bigpos 605 042 102) -3)
731 (float (bigpos 002 904 245 2) -2)
732 (float (bigpos 367 650 733 3) 0)
733 (float (bigpos 664 419 244 4) 2)
734 (float (bigpos 057 958 249) 5)))))
735 (math-mul '(float (bigpos 772 619 636) -9)
736 (math-sub (math-mul (math-besJ1 x) (math-ln-raw x))
738 ((math-negp (calcFunc-re x))
740 (math-add (math-besJ1 (math-neg x) t)
741 (math-mul '(cplx 0 2)
742 (math-besJ1 (math-neg x))))))
746 (defun math-poly-eval (x coefs)
747 (let ((accum (car coefs)))
748 (while (setq coefs (cdr coefs))
749 (setq accum (math-add (car coefs) (math-mul accum x))))
753 ;;;; Bernoulli and Euler polynomials and numbers.
755 (defun calcFunc-bern (n &optional x)
756 (if (and x (not (math-zerop x)))
757 (if (and calc-symbolic-mode (math-floatp x))
758 (math-inexact-result)
759 (math-build-polynomial-expr (math-bernoulli-coefs n) x))
760 (or (math-num-natnump n) (math-reject-arg n 'natnump))
763 (math-inexact-result)
764 (math-float (math-bernoulli-number (math-trunc n))))
765 (math-bernoulli-number n))))
767 (defun calcFunc-euler (n &optional x)
768 (or (math-num-natnump n) (math-reject-arg n 'natnump))
770 (let* ((n1 (math-add n 1))
771 (coefs (math-bernoulli-coefs n1))
772 (fac (math-div (math-pow 2 n1) n1))
774 (x1 (math-div (math-add x 1) 2))
777 (if (and calc-symbolic-mode (math-floatp x))
778 (math-inexact-result)
780 (math-sub (math-build-polynomial-expr coefs x1)
781 (math-build-polynomial-expr coefs x2))))
789 (math-mul (math-mul fac c)
790 (math-sub (math-pow x1 k)
794 (math-mul (math-pow 2 n)
797 (math-inexact-result)
798 (calcFunc-euler n '(float 5 -1)))
799 (calcFunc-euler n '(frac 1 2))))))
801 (defvar math-bernoulli-b-cache '((frac -174611
802 (bigpos 0 200 291 698 662 857 802))
803 (frac 43867 (bigpos 0 944 170 217 94 109 5))
804 (frac -3617 (bigpos 0 880 842 622 670 10))
805 (frac 1 (bigpos 600 249 724 74))
806 (frac -691 (bigpos 0 368 674 307 1))
807 (frac 1 (bigpos 160 900 47))
808 (frac -1 (bigpos 600 209 1))
809 (frac 1 30240) (frac -1 720)
812 (defvar math-bernoulli-B-cache '((frac -174611 330) (frac 43867 798)
813 (frac -3617 510) (frac 7 6) (frac -691 2730)
814 (frac 5 66) (frac -1 30) (frac 1 42)
815 (frac -1 30) (frac 1 6) 1 ))
817 (defvar math-bernoulli-cache-size 11)
818 (defun math-bernoulli-coefs (n)
819 (let* ((coefs (list (calcFunc-bern n)))
824 (calc-prefer-frac (or (integerp n) calc-prefer-frac)))
825 (while (>= (setq k (1- k)) 0)
826 (setq term (math-div term (- nn k))
827 coef (math-mul term (math-bernoulli-number k))
828 coefs (cons (if (consp n) (math-float coef) coef) coefs)
829 term (math-mul term k)))
832 (defun math-bernoulli-number (n)
838 (while (>= n math-bernoulli-cache-size)
841 (fact 1) ; fact = (n-k+1)!
843 (p math-bernoulli-b-cache)
844 (calc-prefer-frac t))
845 (math-working "bernoulli B" (* 2 math-bernoulli-cache-size))
849 fact (math-mul fact (* nk (1- nk)))
850 sum (math-add sum (math-div (car p) fact))
852 (setq ofact (math-mul ofact (1- nk))
853 sum (math-sub (math-div '(frac 1 2) ofact) sum)
854 math-bernoulli-b-cache (cons sum math-bernoulli-b-cache)
855 math-bernoulli-B-cache (cons (math-mul sum ofact)
856 math-bernoulli-B-cache)
857 math-bernoulli-cache-size (1+ math-bernoulli-cache-size))))
858 (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache)))
861 ;;; bn = - sum_k=0^n-1 bk / (n-k+1)!
863 ;;; A faster method would be to use "tangent numbers", c.f., Concrete
864 ;;; Mathematics pg. 273.
867 ;;; Probability distributions.
870 (defun calcFunc-utpb (x n p)
871 (if math-expand-formulas
872 (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1)))
873 (calcFunc-betaI p x (math-add (math-sub n x) 1))))
874 (put 'calcFunc-utpb 'math-expandable t)
876 (defun calcFunc-ltpb (x n p)
877 (math-sub 1 (calcFunc-utpb x n p)))
878 (put 'calcFunc-ltpb 'math-expandable t)
881 (defun calcFunc-utpc (chisq v)
882 (if math-expand-formulas
883 (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2)))
884 (calcFunc-gammaQ (math-div v 2) (math-div chisq 2))))
885 (put 'calcFunc-utpc 'math-expandable t)
887 (defun calcFunc-ltpc (chisq v)
888 (if math-expand-formulas
889 (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2)))
890 (calcFunc-gammaP (math-div v 2) (math-div chisq 2))))
891 (put 'calcFunc-ltpc 'math-expandable t)
894 (defun calcFunc-utpf (f v1 v2)
895 (if math-expand-formulas
896 (math-normalize (list 'calcFunc-betaI
897 (list '/ v2 (list '+ v2 (list '* v1 f)))
900 (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f)))
903 (put 'calcFunc-utpf 'math-expandable t)
905 (defun calcFunc-ltpf (f v1 v2)
906 (math-sub 1 (calcFunc-utpf f v1 v2)))
907 (put 'calcFunc-ltpf 'math-expandable t)
910 (defun calcFunc-utpn (x mean sdev)
911 (if math-expand-formulas
916 (list '/ (list '- mean x)
917 (list '* sdev (list 'calcFunc-sqrt 2)))))
919 (math-mul (math-add '(float 1 0)
921 (math-div (math-sub mean x)
922 (math-mul sdev (math-sqrt-2)))))
924 (put 'calcFunc-utpn 'math-expandable t)
926 (defun calcFunc-ltpn (x mean sdev)
927 (if math-expand-formulas
932 (list '/ (list '- x mean)
933 (list '* sdev (list 'calcFunc-sqrt 2)))))
935 (math-mul (math-add '(float 1 0)
937 (math-div (math-sub x mean)
938 (math-mul sdev (math-sqrt-2)))))
940 (put 'calcFunc-ltpn 'math-expandable t)
943 (defun calcFunc-utpp (n x)
944 (if math-expand-formulas
945 (math-normalize (list 'calcFunc-gammaP x n))
946 (calcFunc-gammaP x n)))
947 (put 'calcFunc-utpp 'math-expandable t)
949 (defun calcFunc-ltpp (n x)
950 (if math-expand-formulas
951 (math-normalize (list 'calcFunc-gammaQ x n))
952 (calcFunc-gammaQ x n)))
953 (put 'calcFunc-ltpp 'math-expandable t)
955 ;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.)
956 (defun calcFunc-utpt (tt v)
957 (if math-expand-formulas
958 (math-normalize (list 'calcFunc-betaI
959 (list '/ v (list '+ v (list '^ tt 2)))
962 (calcFunc-betaI (math-div v (math-add v (math-sqr tt)))
965 (put 'calcFunc-utpt 'math-expandable t)
967 (defun calcFunc-ltpt (tt v)
968 (math-sub 1 (calcFunc-utpt tt v)))
969 (put 'calcFunc-ltpt 'math-expandable t)
971 (provide 'calc-funcs)
973 ;;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a
974 ;;; calc-funcs.el ends here