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[emacs.git] / lisp / calc / calc-funcs.el
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1 ;;; calc-funcs.el --- well-known functions for Calc
2
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
4 ;; 2005, 2006, 2007 Free Software Foundation, Inc.
5
6 ;; Author: David Gillespie <daveg@synaptics.com>
7 ;; Maintainer: Jay Belanger <belanger@truman.edu>
8
9 ;; This file is part of GNU Emacs.
10
11 ;; GNU Emacs is free software; you can redistribute it and/or modify
12 ;; it under the terms of the GNU General Public License as published by
13 ;; the Free Software Foundation; either version 2, or (at your option)
14 ;; any later version.
15
16 ;; GNU Emacs is distributed in the hope that it will be useful,
17 ;; but WITHOUT ANY WARRANTY; without even the implied warranty of
18 ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
19 ;; GNU General Public License for more details.
20
21 ;; You should have received a copy of the GNU General Public License
22 ;; along with GNU Emacs; see the file COPYING. If not, write to the
23 ;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
24 ;; Boston, MA 02110-1301, USA.
25
26 ;;; Commentary:
27
28 ;;; Code:
29
30 ;; This file is autoloaded from calc-ext.el.
31
32 (require 'calc-ext)
33 (require 'calc-macs)
34
35 (defun calc-inc-gamma (arg)
36 (interactive "P")
37 (calc-slow-wrapper
38 (if (calc-is-inverse)
39 (if (calc-is-hyperbolic)
40 (calc-binary-op "gamG" 'calcFunc-gammaG arg)
41 (calc-binary-op "gamQ" 'calcFunc-gammaQ arg))
42 (if (calc-is-hyperbolic)
43 (calc-binary-op "gamg" 'calcFunc-gammag arg)
44 (calc-binary-op "gamP" 'calcFunc-gammaP arg)))))
45
46 (defun calc-erf (arg)
47 (interactive "P")
48 (calc-slow-wrapper
49 (if (calc-is-inverse)
50 (calc-unary-op "erfc" 'calcFunc-erfc arg)
51 (calc-unary-op "erf" 'calcFunc-erf arg))))
52
53 (defun calc-erfc (arg)
54 (interactive "P")
55 (calc-invert-func)
56 (calc-erf arg))
57
58 (defun calc-beta (arg)
59 (interactive "P")
60 (calc-slow-wrapper
61 (calc-binary-op "beta" 'calcFunc-beta arg)))
62
63 (defun calc-inc-beta ()
64 (interactive)
65 (calc-slow-wrapper
66 (if (calc-is-hyperbolic)
67 (calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3)))
68 (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3))))))
69
70 (defun calc-bessel-J (arg)
71 (interactive "P")
72 (calc-slow-wrapper
73 (calc-binary-op "besJ" 'calcFunc-besJ arg)))
74
75 (defun calc-bessel-Y (arg)
76 (interactive "P")
77 (calc-slow-wrapper
78 (calc-binary-op "besY" 'calcFunc-besY arg)))
79
80 (defun calc-bernoulli-number (arg)
81 (interactive "P")
82 (calc-slow-wrapper
83 (if (calc-is-hyperbolic)
84 (calc-binary-op "bern" 'calcFunc-bern arg)
85 (calc-unary-op "bern" 'calcFunc-bern arg))))
86
87 (defun calc-euler-number (arg)
88 (interactive "P")
89 (calc-slow-wrapper
90 (if (calc-is-hyperbolic)
91 (calc-binary-op "eulr" 'calcFunc-euler arg)
92 (calc-unary-op "eulr" 'calcFunc-euler arg))))
93
94 (defun calc-stirling-number (arg)
95 (interactive "P")
96 (calc-slow-wrapper
97 (if (calc-is-hyperbolic)
98 (calc-binary-op "str2" 'calcFunc-stir2 arg)
99 (calc-binary-op "str1" 'calcFunc-stir1 arg))))
100
101 (defun calc-utpb ()
102 (interactive)
103 (calc-prob-dist "b" 3))
104
105 (defun calc-utpc ()
106 (interactive)
107 (calc-prob-dist "c" 2))
108
109 (defun calc-utpf ()
110 (interactive)
111 (calc-prob-dist "f" 3))
112
113 (defun calc-utpn ()
114 (interactive)
115 (calc-prob-dist "n" 3))
116
117 (defun calc-utpp ()
118 (interactive)
119 (calc-prob-dist "p" 2))
120
121 (defun calc-utpt ()
122 (interactive)
123 (calc-prob-dist "t" 2))
124
125 (defun calc-prob-dist (letter nargs)
126 (calc-slow-wrapper
127 (if (calc-is-inverse)
128 (calc-enter-result nargs (concat "ltp" letter)
129 (append (list (intern (concat "calcFunc-ltp" letter))
130 (calc-top-n 1))
131 (calc-top-list-n (1- nargs) 2)))
132 (calc-enter-result nargs (concat "utp" letter)
133 (append (list (intern (concat "calcFunc-utp" letter))
134 (calc-top-n 1))
135 (calc-top-list-n (1- nargs) 2))))))
136
137
138
139
140 ;;; Sources: Numerical Recipes, Press et al;
141 ;;; Handbook of Mathematical Functions, Abramowitz & Stegun.
142
143
144 ;;; Gamma function.
145
146 (defun calcFunc-gamma (x)
147 (or (math-numberp x) (math-reject-arg x 'numberp))
148 (calcFunc-fact (math-add x -1)))
149
150 (defun math-gammap1-raw (x &optional fprec nfprec) ; compute gamma(1 + x)
151 (or fprec
152 (setq fprec (math-float calc-internal-prec)
153 nfprec (math-float (- calc-internal-prec))))
154 (cond ((math-lessp-float (calcFunc-re x) fprec)
155 (if (math-lessp-float (calcFunc-re x) nfprec)
156 (math-neg (math-div
157 (math-pi)
158 (math-mul (math-gammap1-raw
159 (math-add (math-neg x)
160 '(float -1 0))
161 fprec nfprec)
162 (math-sin-raw
163 (math-mul (math-pi) x)))))
164 (let ((xplus1 (math-add x '(float 1 0))))
165 (math-div (math-gammap1-raw xplus1 fprec nfprec) xplus1))))
166 ((and (math-realp x)
167 (math-lessp-float '(float 736276 0) x))
168 (math-overflow))
169 (t ; re(x) now >= 10.0
170 (let ((xinv (math-div 1 x))
171 (lnx (math-ln-raw x)))
172 (math-mul (math-sqrt-two-pi)
173 (math-exp-raw
174 (math-gamma-series
175 (math-sub (math-mul (math-add x '(float 5 -1))
176 lnx)
177 x)
178 xinv
179 (math-sqr xinv)
180 '(float 0 0)
181 2)))))))
182
183 (defun math-gamma-series (sum x xinvsqr oterm n)
184 (math-working "gamma" sum)
185 (let* ((bn (math-bernoulli-number n))
186 (term (math-mul (math-div-float (math-float (nth 1 bn))
187 (math-float (* (nth 2 bn)
188 (* n (1- n)))))
189 x))
190 (next (math-add sum term)))
191 (if (math-nearly-equal sum next)
192 next
193 (if (> n (* 2 calc-internal-prec))
194 (progn
195 ;; Need this because series eventually diverges for large enough n.
196 (calc-record-why
197 "*Gamma computation stopped early, not all digits may be valid")
198 next)
199 (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2))))))
200
201
202 ;;; Incomplete gamma function.
203
204 (defvar math-current-gamma-value nil)
205 (defun calcFunc-gammaP (a x)
206 (if (equal x '(var inf var-inf))
207 '(float 1 0)
208 (math-inexact-result)
209 (or (Math-numberp a) (math-reject-arg a 'numberp))
210 (or (math-numberp x) (math-reject-arg x 'numberp))
211 (if (and (math-num-integerp a)
212 (integerp (setq a (math-trunc a)))
213 (> a 0) (< a 20))
214 (math-sub 1 (calcFunc-gammaQ a x))
215 (let ((math-current-gamma-value (calcFunc-gamma a)))
216 (math-div (calcFunc-gammag a x) math-current-gamma-value)))))
217
218 (defun calcFunc-gammaQ (a x)
219 (if (equal x '(var inf var-inf))
220 '(float 0 0)
221 (math-inexact-result)
222 (or (Math-numberp a) (math-reject-arg a 'numberp))
223 (or (math-numberp x) (math-reject-arg x 'numberp))
224 (if (and (math-num-integerp a)
225 (integerp (setq a (math-trunc a)))
226 (> a 0) (< a 20))
227 (let ((n 0)
228 (sum '(float 1 0))
229 (term '(float 1 0)))
230 (math-with-extra-prec 1
231 (while (< (setq n (1+ n)) a)
232 (setq term (math-div (math-mul term x) n)
233 sum (math-add sum term))
234 (math-working "gamma" sum))
235 (math-mul sum (calcFunc-exp (math-neg x)))))
236 (let ((math-current-gamma-value (calcFunc-gamma a)))
237 (math-div (calcFunc-gammaG a x) math-current-gamma-value)))))
238
239 (defun calcFunc-gammag (a x)
240 (if (equal x '(var inf var-inf))
241 (calcFunc-gamma a)
242 (math-inexact-result)
243 (or (Math-numberp a) (math-reject-arg a 'numberp))
244 (or (Math-numberp x) (math-reject-arg x 'numberp))
245 (math-with-extra-prec 2
246 (setq a (math-float a))
247 (setq x (math-float x))
248 (if (or (math-negp (calcFunc-re a))
249 (math-lessp-float (calcFunc-re x)
250 (math-add-float (calcFunc-re a)
251 '(float 1 0))))
252 (math-inc-gamma-series a x)
253 (math-sub (or math-current-gamma-value (calcFunc-gamma a))
254 (math-inc-gamma-cfrac a x))))))
255
256 (defun calcFunc-gammaG (a x)
257 (if (equal x '(var inf var-inf))
258 '(float 0 0)
259 (math-inexact-result)
260 (or (Math-numberp a) (math-reject-arg a 'numberp))
261 (or (Math-numberp x) (math-reject-arg x 'numberp))
262 (math-with-extra-prec 2
263 (setq a (math-float a))
264 (setq x (math-float x))
265 (if (or (math-negp (calcFunc-re a))
266 (math-lessp-float (calcFunc-re x)
267 (math-add-float (math-abs-approx a)
268 '(float 1 0))))
269 (math-sub (or math-current-gamma-value (calcFunc-gamma a))
270 (math-inc-gamma-series a x))
271 (math-inc-gamma-cfrac a x)))))
272
273 (defun math-inc-gamma-series (a x)
274 (if (Math-zerop x)
275 '(float 0 0)
276 (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
277 (math-with-extra-prec 2
278 (let ((start (math-div '(float 1 0) a)))
279 (math-inc-gamma-series-step start start a x))))))
280
281 (defun math-inc-gamma-series-step (sum term a x)
282 (math-working "gamma" sum)
283 (setq a (math-add a '(float 1 0))
284 term (math-div (math-mul term x) a))
285 (let ((next (math-add sum term)))
286 (if (math-nearly-equal sum next)
287 next
288 (math-inc-gamma-series-step next term a x))))
289
290 (defun math-inc-gamma-cfrac (a x)
291 (if (Math-zerop x)
292 (or math-current-gamma-value (calcFunc-gamma a))
293 (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
294 (math-inc-gamma-cfrac-step '(float 1 0) x
295 '(float 0 0) '(float 1 0)
296 '(float 1 0) '(float 1 0) '(float 0 0)
297 a x))))
298
299 (defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x)
300 (let ((ana (math-sub n a))
301 (anf (math-mul n fac)))
302 (setq n (math-add n '(float 1 0))
303 a0 (math-mul (math-add a1 (math-mul a0 ana)) fac)
304 b0 (math-mul (math-add b1 (math-mul b0 ana)) fac)
305 a1 (math-add (math-mul x a0) (math-mul anf a1))
306 b1 (math-add (math-mul x b0) (math-mul anf b1)))
307 (if (math-zerop a1)
308 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x)
309 (setq fac (math-div '(float 1 0) a1))
310 (let ((next (math-mul b1 fac)))
311 (math-working "gamma" next)
312 (if (math-nearly-equal next g)
313 next
314 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x))))))
315
316
317 ;;; Error function.
318
319 (defun calcFunc-erf (x)
320 (if (equal x '(var inf var-inf))
321 '(float 1 0)
322 (if (equal x '(neg (var inf var-inf)))
323 '(float -1 0)
324 (if (Math-zerop x)
325 x
326 (let ((math-current-gamma-value (math-sqrt-pi)))
327 (math-to-same-complex-quad
328 (math-div (calcFunc-gammag '(float 5 -1)
329 (math-sqr (math-to-complex-quad-one x)))
330 math-current-gamma-value)
331 x))))))
332
333 (defun calcFunc-erfc (x)
334 (if (equal x '(var inf var-inf))
335 '(float 0 0)
336 (if (math-posp x)
337 (let ((math-current-gamma-value (math-sqrt-pi)))
338 (math-div (calcFunc-gammaG '(float 5 -1) (math-sqr x))
339 math-current-gamma-value))
340 (math-sub 1 (calcFunc-erf x)))))
341
342 (defun math-to-complex-quad-one (x)
343 (if (eq (car-safe x) 'polar) (setq x (math-complex x)))
344 (if (eq (car-safe x) 'cplx)
345 (list 'cplx (math-abs (nth 1 x)) (math-abs (nth 2 x)))
346 x))
347
348 (defun math-to-same-complex-quad (x y)
349 (if (eq (car-safe y) 'cplx)
350 (if (eq (car-safe x) 'cplx)
351 (list 'cplx
352 (if (math-negp (nth 1 y)) (math-neg (nth 1 x)) (nth 1 x))
353 (if (math-negp (nth 2 y)) (math-neg (nth 2 x)) (nth 2 x)))
354 (if (math-negp (nth 1 y)) (math-neg x) x))
355 (if (math-negp y)
356 (if (eq (car-safe x) 'cplx)
357 (list 'cplx (math-neg (nth 1 x)) (nth 2 x))
358 (math-neg x))
359 x)))
360
361
362 ;;; Beta function.
363
364 (defun calcFunc-beta (a b)
365 (if (math-num-integerp a)
366 (let ((am (math-add a -1)))
367 (or (math-numberp b) (math-reject-arg b 'numberp))
368 (math-div 1 (math-mul b (calcFunc-choose (math-add b am) am))))
369 (if (math-num-integerp b)
370 (calcFunc-beta b a)
371 (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b))
372 (calcFunc-gamma (math-add a b))))))
373
374
375 ;;; Incomplete beta function.
376
377 (defvar math-current-beta-value nil)
378 (defun calcFunc-betaI (x a b)
379 (cond ((math-zerop x)
380 '(float 0 0))
381 ((math-equal-int x 1)
382 '(float 1 0))
383 ((or (math-zerop a)
384 (and (math-num-integerp a)
385 (math-negp a)))
386 (if (or (math-zerop b)
387 (and (math-num-integerp b)
388 (math-negp b)))
389 (math-reject-arg b 'range)
390 '(float 1 0)))
391 ((or (math-zerop b)
392 (and (math-num-integerp b)
393 (math-negp b)))
394 '(float 0 0))
395 ((not (math-numberp a)) (math-reject-arg a 'numberp))
396 ((not (math-numberp b)) (math-reject-arg b 'numberp))
397 ((math-inexact-result))
398 (t (let ((math-current-beta-value (calcFunc-beta a b)))
399 (math-div (calcFunc-betaB x a b) math-current-beta-value)))))
400
401 (defun calcFunc-betaB (x a b)
402 (cond
403 ((math-zerop x)
404 '(float 0 0))
405 ((math-equal-int x 1)
406 (calcFunc-beta a b))
407 ((not (math-numberp x)) (math-reject-arg x 'numberp))
408 ((not (math-numberp a)) (math-reject-arg a 'numberp))
409 ((not (math-numberp b)) (math-reject-arg b 'numberp))
410 ((math-zerop a) (math-reject-arg a 'nonzerop))
411 ((math-zerop b) (math-reject-arg b 'nonzerop))
412 ((and (math-num-integerp b)
413 (if (math-negp b)
414 (math-reject-arg b 'range)
415 (Math-natnum-lessp (setq b (math-trunc b)) 20)))
416 (and calc-symbolic-mode (or (math-floatp a) (math-floatp b))
417 (math-inexact-result))
418 (math-mul
419 (math-with-extra-prec 2
420 (let* ((i 0)
421 (term 1)
422 (sum (math-div term a)))
423 (while (< (setq i (1+ i)) b)
424 (setq term (math-mul (math-div (math-mul term (- i b)) i) x)
425 sum (math-add sum (math-div term (math-add a i))))
426 (math-working "beta" sum))
427 sum))
428 (math-pow x a)))
429 ((and (math-num-integerp a)
430 (if (math-negp a)
431 (math-reject-arg a 'range)
432 (Math-natnum-lessp (setq a (math-trunc a)) 20)))
433 (math-sub (or math-current-beta-value (calcFunc-beta a b))
434 (calcFunc-betaB (math-sub 1 x) b a)))
435 (t
436 (math-inexact-result)
437 (math-with-extra-prec 2
438 (setq x (math-float x))
439 (setq a (math-float a))
440 (setq b (math-float b))
441 (let ((bt (math-exp-raw (math-add (math-mul a (math-ln-raw x))
442 (math-mul b (math-ln-raw
443 (math-sub '(float 1 0)
444 x)))))))
445 (if (Math-lessp x (math-div (math-add a '(float 1 0))
446 (math-add (math-add a b) '(float 2 0))))
447 (math-div (math-mul bt (math-beta-cfrac a b x)) a)
448 (math-sub (or math-current-beta-value (calcFunc-beta a b))
449 (math-div (math-mul bt
450 (math-beta-cfrac b a (math-sub 1 x)))
451 b))))))))
452
453 (defun math-beta-cfrac (a b x)
454 (let ((qab (math-add a b))
455 (qap (math-add a '(float 1 0)))
456 (qam (math-add a '(float -1 0))))
457 (math-beta-cfrac-step '(float 1 0)
458 (math-sub '(float 1 0)
459 (math-div (math-mul qab x) qap))
460 '(float 1 0) '(float 1 0)
461 '(float 1 0)
462 qab qap qam a b x)))
463
464 (defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x)
465 (let* ((two-m (math-mul m '(float 2 0)))
466 (d (math-div (math-mul (math-mul (math-sub b m) m) x)
467 (math-mul (math-add qam two-m) (math-add a two-m))))
468 (ap (math-add az (math-mul d am)))
469 (bp (math-add bz (math-mul d bm)))
470 (d2 (math-neg
471 (math-div (math-mul (math-mul (math-add a m) (math-add qab m)) x)
472 (math-mul (math-add qap two-m) (math-add a two-m)))))
473 (app (math-add ap (math-mul d2 az)))
474 (bpp (math-add bp (math-mul d2 bz)))
475 (next (math-div app bpp)))
476 (math-working "beta" next)
477 (if (math-nearly-equal next az)
478 next
479 (math-beta-cfrac-step next '(float 1 0)
480 (math-div ap bpp) (math-div bp bpp)
481 (math-add m '(float 1 0))
482 qab qap qam a b x))))
483
484
485 ;;; Bessel functions.
486
487 ;;; Should generalize this to handle arbitrary precision!
488
489 (defun calcFunc-besJ (v x)
490 (or (math-numberp v) (math-reject-arg v 'numberp))
491 (or (math-numberp x) (math-reject-arg x 'numberp))
492 (let ((calc-internal-prec (min 8 calc-internal-prec)))
493 (math-with-extra-prec 3
494 (setq x (math-float (math-normalize x)))
495 (setq v (math-float (math-normalize v)))
496 (cond ((math-zerop x)
497 (if (math-zerop v)
498 '(float 1 0)
499 '(float 0 0)))
500 ((math-inexact-result))
501 ((not (math-num-integerp v))
502 (let ((start (math-div 1 (calcFunc-fact v))))
503 (math-mul (math-besJ-series start start
504 0
505 (math-mul '(float -25 -2)
506 (math-sqr x))
507 v)
508 (math-pow (math-div x 2) v))))
509 ((math-negp (setq v (math-trunc v)))
510 (if (math-oddp v)
511 (math-neg (calcFunc-besJ (math-neg v) x))
512 (calcFunc-besJ (math-neg v) x)))
513 ((eq v 0)
514 (math-besJ0 x))
515 ((eq v 1)
516 (math-besJ1 x))
517 ((Math-lessp v (math-abs-approx x))
518 (let ((j 0)
519 (bjm (math-besJ0 x))
520 (bj (math-besJ1 x))
521 (two-over-x (math-div 2 x))
522 bjp)
523 (while (< (setq j (1+ j)) v)
524 (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj)
525 bjm)
526 bjm bj
527 bj bjp))
528 bj))
529 (t
530 (if (Math-lessp 100 v) (math-reject-arg v 'range))
531 (let* ((j (logior (+ v (math-isqrt-small (* 40 v))) 1))
532 (two-over-x (math-div 2 x))
533 (jsum nil)
534 (bjp '(float 0 0))
535 (sum '(float 0 0))
536 (bj '(float 1 0))
537 bjm ans)
538 (while (> (setq j (1- j)) 0)
539 (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj)
540 bjp)
541 bjp bj
542 bj bjm)
543 (if (> (nth 2 (math-abs-approx bj)) 10)
544 (setq bj (math-mul bj '(float 1 -10))
545 bjp (math-mul bjp '(float 1 -10))
546 ans (and ans (math-mul ans '(float 1 -10)))
547 sum (math-mul sum '(float 1 -10))))
548 (or (setq jsum (not jsum))
549 (setq sum (math-add sum bj)))
550 (if (= j v)
551 (setq ans bjp)))
552 (math-div ans (math-sub (math-mul 2 sum) bj))))))))
553
554 (defun math-besJ-series (sum term k zz vk)
555 (math-working "besJ" sum)
556 (setq k (1+ k)
557 vk (math-add 1 vk)
558 term (math-div (math-mul term zz) (math-mul k vk)))
559 (let ((next (math-add sum term)))
560 (if (math-nearly-equal next sum)
561 next
562 (math-besJ-series next term k zz vk))))
563
564 (defun math-besJ0 (x &optional yflag)
565 (cond ((and (not yflag) (math-negp (calcFunc-re x)))
566 (math-besJ0 (math-neg x)))
567 ((Math-lessp '(float 8 0) (math-abs-approx x))
568 (let* ((z (math-div '(float 8 0) x))
569 (y (math-sqr z))
570 (xx (math-add x '(float (bigneg 164 398 785) -9)))
571 (a1 (math-poly-eval y
572 '((float (bigpos 211 887 093 2) -16)
573 (float (bigneg 639 370 073 2) -15)
574 (float (bigpos 407 510 734 2) -14)
575 (float (bigneg 627 628 098 1) -12)
576 (float 1 0))))
577 (a2 (math-poly-eval y
578 '((float (bigneg 152 935 934) -16)
579 (float (bigpos 161 095 621 7) -16)
580 (float (bigneg 651 147 911 6) -15)
581 (float (bigpos 765 488 430 1) -13)
582 (float (bigneg 995 499 562 1) -11))))
583 (sc (math-sin-cos-raw xx)))
584 (if yflag
585 (setq sc (cons (math-neg (cdr sc)) (car sc))))
586 (math-mul (math-sqrt
587 (math-div '(float (bigpos 722 619 636) -9) x))
588 (math-sub (math-mul (cdr sc) a1)
589 (math-mul (car sc) (math-mul z a2))))))
590 (t
591 (let ((y (math-sqr x)))
592 (math-div (math-poly-eval y
593 '((float (bigneg 456 052 849 1) -7)
594 (float (bigpos 017 233 739 7) -5)
595 (float (bigneg 418 442 121 1) -2)
596 (float (bigpos 407 196 516 6) -1)
597 (float (bigneg 354 590 362 13) 0)
598 (float (bigpos 574 490 568 57) 0)))
599 (math-poly-eval y
600 '((float 1 0)
601 (float (bigpos 712 532 678 2) -7)
602 (float (bigpos 853 264 927 5) -5)
603 (float (bigpos 718 680 494 9) -3)
604 (float (bigpos 985 532 029 1) 0)
605 (float (bigpos 411 490 568 57) 0))))))))
606
607 (defun math-besJ1 (x &optional yflag)
608 (cond ((and (math-negp (calcFunc-re x)) (not yflag))
609 (math-neg (math-besJ1 (math-neg x))))
610 ((Math-lessp '(float 8 0) (math-abs-approx x))
611 (let* ((z (math-div '(float 8 0) x))
612 (y (math-sqr z))
613 (xx (math-add x '(float (bigneg 491 194 356 2) -9)))
614 (a1 (math-poly-eval y
615 '((float (bigneg 019 337 240) -15)
616 (float (bigpos 174 520 457 2) -15)
617 (float (bigneg 496 396 516 3) -14)
618 (float 183105 -8)
619 (float 1 0))))
620 (a2 (math-poly-eval y
621 '((float (bigpos 412 787 105) -15)
622 (float (bigneg 987 228 88) -14)
623 (float (bigpos 096 199 449 8) -15)
624 (float (bigneg 873 690 002 2) -13)
625 (float (bigpos 995 499 687 4) -11))))
626 (sc (math-sin-cos-raw xx)))
627 (if yflag
628 (setq sc (cons (math-neg (cdr sc)) (car sc)))
629 (if (math-negp x)
630 (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc))))))
631 (math-mul (math-sqrt (math-div '(float (bigpos 722 619 636) -9) x))
632 (math-sub (math-mul (cdr sc) a1)
633 (math-mul (car sc) (math-mul z a2))))))
634 (t
635 (let ((y (math-sqr x)))
636 (math-mul
637 x
638 (math-div (math-poly-eval y
639 '((float (bigneg 606 036 016 3) -8)
640 (float (bigpos 826 044 157) -4)
641 (float (bigneg 439 611 972 2) -3)
642 (float (bigpos 531 968 423 2) -1)
643 (float (bigneg 235 059 895 7) 0)
644 (float (bigpos 232 614 362 72) 0)))
645 (math-poly-eval y
646 '((float 1 0)
647 (float (bigpos 397 991 769 3) -7)
648 (float (bigpos 394 743 944 9) -5)
649 (float (bigpos 474 330 858 1) -2)
650 (float (bigpos 178 535 300 2) 0)
651 (float (bigpos 442 228 725 144)
652 0)))))))))
653
654 (defun calcFunc-besY (v x)
655 (math-inexact-result)
656 (or (math-numberp v) (math-reject-arg v 'numberp))
657 (or (math-numberp x) (math-reject-arg x 'numberp))
658 (let ((calc-internal-prec (min 8 calc-internal-prec)))
659 (math-with-extra-prec 3
660 (setq x (math-float (math-normalize x)))
661 (setq v (math-float (math-normalize v)))
662 (cond ((not (math-num-integerp v))
663 (let ((sc (math-sin-cos-raw (math-mul v (math-pi)))))
664 (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc))
665 (calcFunc-besJ (math-neg v) x))
666 (car sc))))
667 ((math-negp (setq v (math-trunc v)))
668 (if (math-oddp v)
669 (math-neg (calcFunc-besY (math-neg v) x))
670 (calcFunc-besY (math-neg v) x)))
671 ((eq v 0)
672 (math-besY0 x))
673 ((eq v 1)
674 (math-besY1 x))
675 (t
676 (let ((j 0)
677 (bym (math-besY0 x))
678 (by (math-besY1 x))
679 (two-over-x (math-div 2 x))
680 byp)
681 (while (< (setq j (1+ j)) v)
682 (setq byp (math-sub (math-mul (math-mul j two-over-x) by)
683 bym)
684 bym by
685 by byp))
686 by))))))
687
688 (defun math-besY0 (x)
689 (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
690 (let ((y (math-sqr x)))
691 (math-add
692 (math-div (math-poly-eval y
693 '((float (bigpos 733 622 284 2) -7)
694 (float (bigneg 757 792 632 8) -5)
695 (float (bigpos 129 988 087 1) -2)
696 (float (bigneg 036 598 123 5) -1)
697 (float (bigpos 065 834 062 7) 0)
698 (float (bigneg 389 821 957 2) 0)))
699 (math-poly-eval y
700 '((float 1 0)
701 (float (bigpos 244 030 261 2) -7)
702 (float (bigpos 647 472 474) -4)
703 (float (bigpos 438 466 189 7) -3)
704 (float (bigpos 648 499 452 7) -1)
705 (float (bigpos 269 544 076 40) 0))))
706 (math-mul '(float (bigpos 772 619 636) -9)
707 (math-mul (math-besJ0 x) (math-ln-raw x))))))
708 ((math-negp (calcFunc-re x))
709 (math-add (math-besJ0 (math-neg x) t)
710 (math-mul '(cplx 0 2)
711 (math-besJ0 (math-neg x)))))
712 (t
713 (math-besJ0 x t))))
714
715 (defun math-besY1 (x)
716 (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
717 (let ((y (math-sqr x)))
718 (math-add
719 (math-mul
720 x
721 (math-div (math-poly-eval y
722 '((float (bigpos 935 937 511 8) -6)
723 (float (bigneg 726 922 237 4) -3)
724 (float (bigpos 551 264 349 7) -1)
725 (float (bigneg 139 438 153 5) 1)
726 (float (bigpos 439 527 127) 4)
727 (float (bigneg 943 604 900 4) 3)))
728 (math-poly-eval y
729 '((float 1 0)
730 (float (bigpos 885 632 549 3) -7)
731 (float (bigpos 605 042 102) -3)
732 (float (bigpos 002 904 245 2) -2)
733 (float (bigpos 367 650 733 3) 0)
734 (float (bigpos 664 419 244 4) 2)
735 (float (bigpos 057 958 249) 5)))))
736 (math-mul '(float (bigpos 772 619 636) -9)
737 (math-sub (math-mul (math-besJ1 x) (math-ln-raw x))
738 (math-div 1 x))))))
739 ((math-negp (calcFunc-re x))
740 (math-neg
741 (math-add (math-besJ1 (math-neg x) t)
742 (math-mul '(cplx 0 2)
743 (math-besJ1 (math-neg x))))))
744 (t
745 (math-besJ1 x t))))
746
747 (defun math-poly-eval (x coefs)
748 (let ((accum (car coefs)))
749 (while (setq coefs (cdr coefs))
750 (setq accum (math-add (car coefs) (math-mul accum x))))
751 accum))
752
753
754 ;;;; Bernoulli and Euler polynomials and numbers.
755
756 (defun calcFunc-bern (n &optional x)
757 (if (and x (not (math-zerop x)))
758 (if (and calc-symbolic-mode (math-floatp x))
759 (math-inexact-result)
760 (math-build-polynomial-expr (math-bernoulli-coefs n) x))
761 (or (math-num-natnump n) (math-reject-arg n 'natnump))
762 (if (consp n)
763 (progn
764 (math-inexact-result)
765 (math-float (math-bernoulli-number (math-trunc n))))
766 (math-bernoulli-number n))))
767
768 (defun calcFunc-euler (n &optional x)
769 (or (math-num-natnump n) (math-reject-arg n 'natnump))
770 (if x
771 (let* ((n1 (math-add n 1))
772 (coefs (math-bernoulli-coefs n1))
773 (fac (math-div (math-pow 2 n1) n1))
774 (k -1)
775 (x1 (math-div (math-add x 1) 2))
776 (x2 (math-div x 2)))
777 (if (math-numberp x)
778 (if (and calc-symbolic-mode (math-floatp x))
779 (math-inexact-result)
780 (math-mul fac
781 (math-sub (math-build-polynomial-expr coefs x1)
782 (math-build-polynomial-expr coefs x2))))
783 (calcFunc-collect
784 (math-reduce-vec
785 'math-add
786 (cons 'vec
787 (mapcar (function
788 (lambda (c)
789 (setq k (1+ k))
790 (math-mul (math-mul fac c)
791 (math-sub (math-pow x1 k)
792 (math-pow x2 k)))))
793 coefs)))
794 x)))
795 (math-mul (math-pow 2 n)
796 (if (consp n)
797 (progn
798 (math-inexact-result)
799 (calcFunc-euler n '(float 5 -1)))
800 (calcFunc-euler n '(frac 1 2))))))
801
802 (defvar math-bernoulli-b-cache '((frac -174611
803 (bigpos 0 200 291 698 662 857 802))
804 (frac 43867 (bigpos 0 944 170 217 94 109 5))
805 (frac -3617 (bigpos 0 880 842 622 670 10))
806 (frac 1 (bigpos 600 249 724 74))
807 (frac -691 (bigpos 0 368 674 307 1))
808 (frac 1 (bigpos 160 900 47))
809 (frac -1 (bigpos 600 209 1))
810 (frac 1 30240) (frac -1 720)
811 (frac 1 12) 1 ))
812
813 (defvar math-bernoulli-B-cache '((frac -174611 330) (frac 43867 798)
814 (frac -3617 510) (frac 7 6) (frac -691 2730)
815 (frac 5 66) (frac -1 30) (frac 1 42)
816 (frac -1 30) (frac 1 6) 1 ))
817
818 (defvar math-bernoulli-cache-size 11)
819 (defun math-bernoulli-coefs (n)
820 (let* ((coefs (list (calcFunc-bern n)))
821 (nn (math-trunc n))
822 (k nn)
823 (term nn)
824 coef
825 (calc-prefer-frac (or (integerp n) calc-prefer-frac)))
826 (while (>= (setq k (1- k)) 0)
827 (setq term (math-div term (- nn k))
828 coef (math-mul term (math-bernoulli-number k))
829 coefs (cons (if (consp n) (math-float coef) coef) coefs)
830 term (math-mul term k)))
831 (nreverse coefs)))
832
833 (defun math-bernoulli-number (n)
834 (if (= (% n 2) 1)
835 (if (= n 1)
836 '(frac -1 2)
837 0)
838 (setq n (/ n 2))
839 (while (>= n math-bernoulli-cache-size)
840 (let* ((sum 0)
841 (nk 1) ; nk = n-k+1
842 (fact 1) ; fact = (n-k+1)!
843 ofact
844 (p math-bernoulli-b-cache)
845 (calc-prefer-frac t))
846 (math-working "bernoulli B" (* 2 math-bernoulli-cache-size))
847 (while p
848 (setq nk (+ nk 2)
849 ofact fact
850 fact (math-mul fact (* nk (1- nk)))
851 sum (math-add sum (math-div (car p) fact))
852 p (cdr p)))
853 (setq ofact (math-mul ofact (1- nk))
854 sum (math-sub (math-div '(frac 1 2) ofact) sum)
855 math-bernoulli-b-cache (cons sum math-bernoulli-b-cache)
856 math-bernoulli-B-cache (cons (math-mul sum ofact)
857 math-bernoulli-B-cache)
858 math-bernoulli-cache-size (1+ math-bernoulli-cache-size))))
859 (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache)))
860
861 ;;; Bn = n! bn
862 ;;; bn = - sum_k=0^n-1 bk / (n-k+1)!
863
864 ;;; A faster method would be to use "tangent numbers", c.f., Concrete
865 ;;; Mathematics pg. 273.
866
867
868 ;;; Probability distributions.
869
870 ;;; Binomial.
871 (defun calcFunc-utpb (x n p)
872 (if math-expand-formulas
873 (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1)))
874 (calcFunc-betaI p x (math-add (math-sub n x) 1))))
875 (put 'calcFunc-utpb 'math-expandable t)
876
877 (defun calcFunc-ltpb (x n p)
878 (math-sub 1 (calcFunc-utpb x n p)))
879 (put 'calcFunc-ltpb 'math-expandable t)
880
881 ;;; Chi-square.
882 (defun calcFunc-utpc (chisq v)
883 (if math-expand-formulas
884 (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2)))
885 (calcFunc-gammaQ (math-div v 2) (math-div chisq 2))))
886 (put 'calcFunc-utpc 'math-expandable t)
887
888 (defun calcFunc-ltpc (chisq v)
889 (if math-expand-formulas
890 (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2)))
891 (calcFunc-gammaP (math-div v 2) (math-div chisq 2))))
892 (put 'calcFunc-ltpc 'math-expandable t)
893
894 ;;; F-distribution.
895 (defun calcFunc-utpf (f v1 v2)
896 (if math-expand-formulas
897 (math-normalize (list 'calcFunc-betaI
898 (list '/ v2 (list '+ v2 (list '* v1 f)))
899 (list '/ v2 2)
900 (list '/ v1 2)))
901 (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f)))
902 (math-div v2 2)
903 (math-div v1 2))))
904 (put 'calcFunc-utpf 'math-expandable t)
905
906 (defun calcFunc-ltpf (f v1 v2)
907 (math-sub 1 (calcFunc-utpf f v1 v2)))
908 (put 'calcFunc-ltpf 'math-expandable t)
909
910 ;;; Normal.
911 (defun calcFunc-utpn (x mean sdev)
912 (if math-expand-formulas
913 (math-normalize
914 (list '/
915 (list '+ 1
916 (list 'calcFunc-erf
917 (list '/ (list '- mean x)
918 (list '* sdev (list 'calcFunc-sqrt 2)))))
919 2))
920 (math-mul (math-add '(float 1 0)
921 (calcFunc-erf
922 (math-div (math-sub mean x)
923 (math-mul sdev (math-sqrt-2)))))
924 '(float 5 -1))))
925 (put 'calcFunc-utpn 'math-expandable t)
926
927 (defun calcFunc-ltpn (x mean sdev)
928 (if math-expand-formulas
929 (math-normalize
930 (list '/
931 (list '+ 1
932 (list 'calcFunc-erf
933 (list '/ (list '- x mean)
934 (list '* sdev (list 'calcFunc-sqrt 2)))))
935 2))
936 (math-mul (math-add '(float 1 0)
937 (calcFunc-erf
938 (math-div (math-sub x mean)
939 (math-mul sdev (math-sqrt-2)))))
940 '(float 5 -1))))
941 (put 'calcFunc-ltpn 'math-expandable t)
942
943 ;;; Poisson.
944 (defun calcFunc-utpp (n x)
945 (if math-expand-formulas
946 (math-normalize (list 'calcFunc-gammaP x n))
947 (calcFunc-gammaP x n)))
948 (put 'calcFunc-utpp 'math-expandable t)
949
950 (defun calcFunc-ltpp (n x)
951 (if math-expand-formulas
952 (math-normalize (list 'calcFunc-gammaQ x n))
953 (calcFunc-gammaQ x n)))
954 (put 'calcFunc-ltpp 'math-expandable t)
955
956 ;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.)
957 (defun calcFunc-utpt (tt v)
958 (if math-expand-formulas
959 (math-normalize (list 'calcFunc-betaI
960 (list '/ v (list '+ v (list '^ tt 2)))
961 (list '/ v 2)
962 '(float 5 -1)))
963 (calcFunc-betaI (math-div v (math-add v (math-sqr tt)))
964 (math-div v 2)
965 '(float 5 -1))))
966 (put 'calcFunc-utpt 'math-expandable t)
967
968 (defun calcFunc-ltpt (tt v)
969 (math-sub 1 (calcFunc-utpt tt v)))
970 (put 'calcFunc-ltpt 'math-expandable t)
971
972 (provide 'calc-funcs)
973
974 ;;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a
975 ;;; calc-funcs.el ends here
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