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2008-11-16 Michael Kifer <kifer@cs.stonybrook.edu>
[emacs.git] / lisp / calc / calc-funcs.el
blobf5553475c38d3fbd48dfd67ce436290fe0992e23
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, 2008 Free Software Foundation, Inc.
5
6 ;; Author: David Gillespie <daveg@synaptics.com>
7 ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
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 3 of the License, or
14 ;; (at your option) 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. If not, see <http://www.gnu.org/licenses/>.
23
24 ;;; Commentary:
25
26 ;;; Code:
27
28 ;; This file is autoloaded from calc-ext.el.
29
30 (require 'calc-ext)
31 (require 'calc-macs)
32
33 (defun calc-inc-gamma (arg)
34 (interactive "P")
35 (calc-slow-wrapper
36 (if (calc-is-inverse)
37 (if (calc-is-hyperbolic)
38 (calc-binary-op "gamG" 'calcFunc-gammaG arg)
39 (calc-binary-op "gamQ" 'calcFunc-gammaQ arg))
40 (if (calc-is-hyperbolic)
41 (calc-binary-op "gamg" 'calcFunc-gammag arg)
42 (calc-binary-op "gamP" 'calcFunc-gammaP arg)))))
43
44 (defun calc-erf (arg)
45 (interactive "P")
46 (calc-slow-wrapper
47 (if (calc-is-inverse)
48 (calc-unary-op "erfc" 'calcFunc-erfc arg)
49 (calc-unary-op "erf" 'calcFunc-erf arg))))
50
51 (defun calc-erfc (arg)
52 (interactive "P")
53 (calc-invert-func)
54 (calc-erf arg))
55
56 (defun calc-beta (arg)
57 (interactive "P")
58 (calc-slow-wrapper
59 (calc-binary-op "beta" 'calcFunc-beta arg)))
60
61 (defun calc-inc-beta ()
62 (interactive)
63 (calc-slow-wrapper
64 (if (calc-is-hyperbolic)
65 (calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3)))
66 (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3))))))
67
68 (defun calc-bessel-J (arg)
69 (interactive "P")
70 (calc-slow-wrapper
71 (calc-binary-op "besJ" 'calcFunc-besJ arg)))
72
73 (defun calc-bessel-Y (arg)
74 (interactive "P")
75 (calc-slow-wrapper
76 (calc-binary-op "besY" 'calcFunc-besY arg)))
77
78 (defun calc-bernoulli-number (arg)
79 (interactive "P")
80 (calc-slow-wrapper
81 (if (calc-is-hyperbolic)
82 (calc-binary-op "bern" 'calcFunc-bern arg)
83 (calc-unary-op "bern" 'calcFunc-bern arg))))
84
85 (defun calc-euler-number (arg)
86 (interactive "P")
87 (calc-slow-wrapper
88 (if (calc-is-hyperbolic)
89 (calc-binary-op "eulr" 'calcFunc-euler arg)
90 (calc-unary-op "eulr" 'calcFunc-euler arg))))
91
92 (defun calc-stirling-number (arg)
93 (interactive "P")
94 (calc-slow-wrapper
95 (if (calc-is-hyperbolic)
96 (calc-binary-op "str2" 'calcFunc-stir2 arg)
97 (calc-binary-op "str1" 'calcFunc-stir1 arg))))
98
99 (defun calc-utpb ()
100 (interactive)
101 (calc-prob-dist "b" 3))
102
103 (defun calc-utpc ()
104 (interactive)
105 (calc-prob-dist "c" 2))
106
107 (defun calc-utpf ()
108 (interactive)
109 (calc-prob-dist "f" 3))
110
111 (defun calc-utpn ()
112 (interactive)
113 (calc-prob-dist "n" 3))
114
115 (defun calc-utpp ()
116 (interactive)
117 (calc-prob-dist "p" 2))
118
119 (defun calc-utpt ()
120 (interactive)
121 (calc-prob-dist "t" 2))
122
123 (defun calc-prob-dist (letter nargs)
124 (calc-slow-wrapper
125 (if (calc-is-inverse)
126 (calc-enter-result nargs (concat "ltp" letter)
127 (append (list (intern (concat "calcFunc-ltp" letter))
128 (calc-top-n 1))
129 (calc-top-list-n (1- nargs) 2)))
130 (calc-enter-result nargs (concat "utp" letter)
131 (append (list (intern (concat "calcFunc-utp" letter))
132 (calc-top-n 1))
133 (calc-top-list-n (1- nargs) 2))))))
134
135
136
137
138 ;;; Sources: Numerical Recipes, Press et al;
139 ;;; Handbook of Mathematical Functions, Abramowitz & Stegun.
140
141
142 ;;; Gamma function.
143
144 (defun calcFunc-gamma (x)
145 (or (math-numberp x) (math-reject-arg x 'numberp))
146 (calcFunc-fact (math-add x -1)))
147
148 (defun math-gammap1-raw (x &optional fprec nfprec)
149 "Compute gamma(1+X) to the appropriate precision."
150 (or fprec
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)
155 (math-neg (math-div
156 (math-pi)
157 (math-mul (math-gammap1-raw
158 (math-add (math-neg x)
159 '(float -1 0))
160 fprec nfprec)
161 (math-sin-raw
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))))
165 ((and (math-realp x)
166 (math-lessp-float '(float 736276 0) x))
167 (math-overflow))
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)
172 (math-exp-raw
173 (math-gamma-series
174 (math-sub (math-mul (math-add x '(float 5 -1))
175 lnx)
176 x)
177 xinv
178 (math-sqr xinv)
179 '(float 0 0)
180 2)))))))
181
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)
187 (* n (1- n)))))
188 x))
189 (next (math-add sum term)))
190 (if (math-nearly-equal sum next)
191 next
192 (if (> n (* 2 calc-internal-prec))
193 (progn
194 ;; Need this because series eventually diverges for large enough n.
195 (calc-record-why
196 "*Gamma computation stopped early, not all digits may be valid")
197 next)
198 (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2))))))
199
200
201 ;;; Incomplete gamma function.
202
203 (defvar math-current-gamma-value nil)
204 (defun calcFunc-gammaP (a x)
205 (if (equal x '(var inf var-inf))
206 '(float 1 0)
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)))
212 (> a 0) (< a 20))
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)))))
216
217 (defun calcFunc-gammaQ (a x)
218 (if (equal x '(var inf var-inf))
219 '(float 0 0)
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)))
225 (> a 0) (< a 20))
226 (let ((n 0)
227 (sum '(float 1 0))
228 (term '(float 1 0)))
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)))))
237
238 (defun calcFunc-gammag (a x)
239 (if (equal x '(var inf var-inf))
240 (calcFunc-gamma a)
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)
250 '(float 1 0))))
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))))))
254
255 (defun calcFunc-gammaG (a x)
256 (if (equal x '(var inf var-inf))
257 '(float 0 0)
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)
267 '(float 1 0))))
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)))))
271
272 (defun math-inc-gamma-series (a x)
273 (if (Math-zerop x)
274 '(float 0 0)
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))))))
279
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)
286 next
287 (math-inc-gamma-series-step next term a x))))
288
289 (defun math-inc-gamma-cfrac (a x)
290 (if (Math-zerop 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)
296 a x))))
297
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)))
306 (if (math-zerop a1)
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)
312 next
313 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x))))))
314
315
316 ;;; Error function.
317
318 (defun calcFunc-erf (x)
319 (if (equal x '(var inf var-inf))
320 '(float 1 0)
321 (if (equal x '(neg (var inf var-inf)))
322 '(float -1 0)
323 (if (Math-zerop x)
324 x
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)
330 x))))))
331
332 (defun calcFunc-erfc (x)
333 (if (equal x '(var inf var-inf))
334 '(float 0 0)
335 (if (math-posp x)
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)))))
340
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)))
345 x))
346
347 (defun math-to-same-complex-quad (x y)
348 (if (eq (car-safe y) 'cplx)
349 (if (eq (car-safe x) 'cplx)
350 (list '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))
354 (if (math-negp y)
355 (if (eq (car-safe x) 'cplx)
356 (list 'cplx (math-neg (nth 1 x)) (nth 2 x))
357 (math-neg x))
358 x)))
359
360
361 ;;; Beta function.
362
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)
369 (calcFunc-beta b a)
370 (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b))
371 (calcFunc-gamma (math-add a b))))))
372
373
374 ;;; Incomplete beta function.
375
376 (defvar math-current-beta-value nil)
377 (defun calcFunc-betaI (x a b)
378 (cond ((math-zerop x)
379 '(float 0 0))
380 ((math-equal-int x 1)
381 '(float 1 0))
382 ((or (math-zerop a)
383 (and (math-num-integerp a)
384 (math-negp a)))
385 (if (or (math-zerop b)
386 (and (math-num-integerp b)
387 (math-negp b)))
388 (math-reject-arg b 'range)
389 '(float 1 0)))
390 ((or (math-zerop b)
391 (and (math-num-integerp b)
392 (math-negp b)))
393 '(float 0 0))
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)))))
399
400 (defun calcFunc-betaB (x a b)
401 (cond
402 ((math-zerop x)
403 '(float 0 0))
404 ((math-equal-int x 1)
405 (calcFunc-beta a b))
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)
412 (if (math-negp 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))
417 (math-mul
418 (math-with-extra-prec 2
419 (let* ((i 0)
420 (term 1)
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))
426 sum))
427 (math-pow x a)))
428 ((and (math-num-integerp a)
429 (if (math-negp 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)))
434 (t
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)
443 x)))))))
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)))
450 b))))))))
451
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)
460 '(float 1 0)
461 qab qap qam a b x)))
462
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)))
469 (d2 (math-neg
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)
477 next
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))))
482
483
484 ;;; Bessel functions.
485
486 ;;; Should generalize this to handle arbitrary precision!
487
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)
496 (if (math-zerop v)
497 '(float 1 0)
498 '(float 0 0)))
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
503 0
504 (math-mul '(float -25 -2)
505 (math-sqr x))
506 v)
507 (math-pow (math-div x 2) v))))
508 ((math-negp (setq v (math-trunc v)))
509 (if (math-oddp v)
510 (math-neg (calcFunc-besJ (math-neg v) x))
511 (calcFunc-besJ (math-neg v) x)))
512 ((eq v 0)
513 (math-besJ0 x))
514 ((eq v 1)
515 (math-besJ1 x))
516 ((Math-lessp v (math-abs-approx x))
517 (let ((j 0)
518 (bjm (math-besJ0 x))
519 (bj (math-besJ1 x))
520 (two-over-x (math-div 2 x))
521 bjp)
522 (while (< (setq j (1+ j)) v)
523 (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj)
524 bjm)
525 bjm bj
526 bj bjp))
527 bj))
528 (t
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))
532 (jsum nil)
533 (bjp '(float 0 0))
534 (sum '(float 0 0))
535 (bj '(float 1 0))
536 bjm ans)
537 (while (> (setq j (1- j)) 0)
538 (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj)
539 bjp)
540 bjp bj
541 bj bjm)
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)))
549 (if (= j v)
550 (setq ans bjp)))
551 (math-div ans (math-sub (math-mul 2 sum) bj))))))))
552
553 (defun math-besJ-series (sum term k zz vk)
554 (math-working "besJ" sum)
555 (setq k (1+ k)
556 vk (math-add 1 vk)
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)
560 next
561 (math-besJ-series next term k zz vk))))
562
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))
568 (y (math-sqr z))
569 (xx (math-add x
570 (math-read-number-simple "-0.785398164")))
571 (a1 (math-poly-eval y
572 (list
573 (math-read-number-simple "0.0000002093887211")
574 (math-read-number-simple "-0.000002073370639")
575 (math-read-number-simple "0.00002734510407")
576 (math-read-number-simple "-0.001098628627")
577 '(float 1 0))))
578 (a2 (math-poly-eval y
579 (list
580 (math-read-number-simple "-0.0000000934935152")
581 (math-read-number-simple "0.0000007621095161")
582 (math-read-number-simple "-0.000006911147651")
583 (math-read-number-simple "0.0001430488765")
584 (math-read-number-simple "-0.01562499995"))))
585 (sc (math-sin-cos-raw xx)))
586 (if yflag
587 (setq sc (cons (math-neg (cdr sc)) (car sc))))
588 (math-mul (math-sqrt
589 (math-div (math-read-number-simple "0.636619722")
590 x))
591 (math-sub (math-mul (cdr sc) a1)
592 (math-mul (car sc) (math-mul z a2))))))
593 (t
594 (let ((y (math-sqr x)))
595 (math-div (math-poly-eval y
596 (list
597 (math-read-number-simple "-184.9052456")
598 (math-read-number-simple "77392.33017")
599 (math-read-number-simple "-11214424.18")
600 (math-read-number-simple "651619640.7")
601 (math-read-number-simple "-13362590354.0")
602 (math-read-number-simple "57568490574.0")))
603 (math-poly-eval y
604 (list
605 '(float 1 0)
606 (math-read-number-simple "267.8532712")
607 (math-read-number-simple "59272.64853")
608 (math-read-number-simple "9494680.718")
609 (math-read-number-simple "1029532985.0")
610 (math-read-number-simple "57568490411.0"))))))))
611
612 (defun math-besJ1 (x &optional yflag)
613 (cond ((and (math-negp (calcFunc-re x)) (not yflag))
614 (math-neg (math-besJ1 (math-neg x))))
615 ((Math-lessp '(float 8 0) (math-abs-approx x))
616 (let* ((z (math-div '(float 8 0) x))
617 (y (math-sqr z))
618 (xx (math-add x (math-read-number-simple "-2.356194491")))
619 (a1 (math-poly-eval y
620 (list
621 (math-read-number-simple "-0.000000240337019")
622 (math-read-number-simple "0.000002457520174")
623 (math-read-number-simple "-0.00003516396496")
624 '(float 183105 -8)
625 '(float 1 0))))
626 (a2 (math-poly-eval y
627 (list
628 (math-read-number-simple "0.000000105787412")
629 (math-read-number-simple "-0.00000088228987")
630 (math-read-number-simple "0.000008449199096")
631 (math-read-number-simple "-0.0002002690873")
632 (math-read-number-simple "0.04687499995"))))
633 (sc (math-sin-cos-raw xx)))
634 (if yflag
635 (setq sc (cons (math-neg (cdr sc)) (car sc)))
636 (if (math-negp x)
637 (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc))))))
638 (math-mul (math-sqrt (math-div
639 (math-read-number-simple "0.636619722")
640 x))
641 (math-sub (math-mul (cdr sc) a1)
642 (math-mul (car sc) (math-mul z a2))))))
643 (t
644 (let ((y (math-sqr x)))
645 (math-mul
646 x
647 (math-div (math-poly-eval y
648 (list
649 (math-read-number-simple "-30.16036606")
650 (math-read-number-simple "15704.4826")
651 (math-read-number-simple "-2972611.439")
652 (math-read-number-simple "242396853.1")
653 (math-read-number-simple "-7895059235.0")
654 (math-read-number-simple "72362614232.0")))
655 (math-poly-eval y
656 (list
657 '(float 1 0)
658 (math-read-number-simple "376.9991397")
659 (math-read-number-simple "99447.43394")
660 (math-read-number-simple "18583304.74")
661 (math-read-number-simple "2300535178.0")
662 (math-read-number-simple "144725228442.0")))))))))
663
664 (defun calcFunc-besY (v x)
665 (math-inexact-result)
666 (or (math-numberp v) (math-reject-arg v 'numberp))
667 (or (math-numberp x) (math-reject-arg x 'numberp))
668 (let ((calc-internal-prec (min 8 calc-internal-prec)))
669 (math-with-extra-prec 3
670 (setq x (math-float (math-normalize x)))
671 (setq v (math-float (math-normalize v)))
672 (cond ((not (math-num-integerp v))
673 (let ((sc (math-sin-cos-raw (math-mul v (math-pi)))))
674 (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc))
675 (calcFunc-besJ (math-neg v) x))
676 (car sc))))
677 ((math-negp (setq v (math-trunc v)))
678 (if (math-oddp v)
679 (math-neg (calcFunc-besY (math-neg v) x))
680 (calcFunc-besY (math-neg v) x)))
681 ((eq v 0)
682 (math-besY0 x))
683 ((eq v 1)
684 (math-besY1 x))
685 (t
686 (let ((j 0)
687 (bym (math-besY0 x))
688 (by (math-besY1 x))
689 (two-over-x (math-div 2 x))
690 byp)
691 (while (< (setq j (1+ j)) v)
692 (setq byp (math-sub (math-mul (math-mul j two-over-x) by)
693 bym)
694 bym by
695 by byp))
696 by))))))
697
698 (defun math-besY0 (x)
699 (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
700 (let ((y (math-sqr x)))
701 (math-add
702 (math-div (math-poly-eval y
703 (list
704 (math-read-number-simple "228.4622733")
705 (math-read-number-simple "-86327.92757")
706 (math-read-number-simple "10879881.29")
707 (math-read-number-simple "-512359803.6")
708 (math-read-number-simple "7062834065.0")
709 (math-read-number-simple "-2957821389.0")))
710 (math-poly-eval y
711 (list
712 '(float 1 0)
713 (math-read-number-simple "226.1030244")
714 (math-read-number-simple "47447.2647")
715 (math-read-number-simple "7189466.438")
716 (math-read-number-simple "745249964.8")
717 (math-read-number-simple "40076544269.0"))))
718 (math-mul (math-read-number-simple "0.636619772")
719 (math-mul (math-besJ0 x) (math-ln-raw x))))))
720 ((math-negp (calcFunc-re x))
721 (math-add (math-besJ0 (math-neg x) t)
722 (math-mul '(cplx 0 2)
723 (math-besJ0 (math-neg x)))))
724 (t
725 (math-besJ0 x t))))
726
727 (defun math-besY1 (x)
728 (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
729 (let ((y (math-sqr x)))
730 (math-add
731 (math-mul
732 x
733 (math-div (math-poly-eval y
734 (list
735 (math-read-number-simple "8511.937935")
736 (math-read-number-simple "-4237922.726")
737 (math-read-number-simple "734926455.1")
738 (math-read-number-simple "-51534381390.0")
739 (math-read-number-simple "1275274390000.0")
740 (math-read-number-simple "-4900604943000.0")))
741 (math-poly-eval y
742 (list
743 '(float 1 0)
744 (math-read-number-simple "354.9632885")
745 (math-read-number-simple "102042.605")
746 (math-read-number-simple "22459040.02")
747 (math-read-number-simple "3733650367.0")
748 (math-read-number-simple "424441966400.0")
749 (math-read-number-simple "24995805700000.0")))))
750 (math-mul (math-read-number-simple "0.636619772")
751 (math-sub (math-mul (math-besJ1 x) (math-ln-raw x))
752 (math-div 1 x))))))
753 ((math-negp (calcFunc-re x))
754 (math-neg
755 (math-add (math-besJ1 (math-neg x) t)
756 (math-mul '(cplx 0 2)
757 (math-besJ1 (math-neg x))))))
758 (t
759 (math-besJ1 x t))))
760
761 (defun math-poly-eval (x coefs)
762 (let ((accum (car coefs)))
763 (while (setq coefs (cdr coefs))
764 (setq accum (math-add (car coefs) (math-mul accum x))))
765 accum))
766
767
768 ;;;; Bernoulli and Euler polynomials and numbers.
769
770 (defun calcFunc-bern (n &optional x)
771 (if (and x (not (math-zerop x)))
772 (if (and calc-symbolic-mode (math-floatp x))
773 (math-inexact-result)
774 (math-build-polynomial-expr (math-bernoulli-coefs n) x))
775 (or (math-num-natnump n) (math-reject-arg n 'natnump))
776 (if (consp n)
777 (progn
778 (math-inexact-result)
779 (math-float (math-bernoulli-number (math-trunc n))))
780 (math-bernoulli-number n))))
781
782 (defun calcFunc-euler (n &optional x)
783 (or (math-num-natnump n) (math-reject-arg n 'natnump))
784 (if x
785 (let* ((n1 (math-add n 1))
786 (coefs (math-bernoulli-coefs n1))
787 (fac (math-div (math-pow 2 n1) n1))
788 (k -1)
789 (x1 (math-div (math-add x 1) 2))
790 (x2 (math-div x 2)))
791 (if (math-numberp x)
792 (if (and calc-symbolic-mode (math-floatp x))
793 (math-inexact-result)
794 (math-mul fac
795 (math-sub (math-build-polynomial-expr coefs x1)
796 (math-build-polynomial-expr coefs x2))))
797 (calcFunc-collect
798 (math-reduce-vec
799 'math-add
800 (cons 'vec
801 (mapcar (function
802 (lambda (c)
803 (setq k (1+ k))
804 (math-mul (math-mul fac c)
805 (math-sub (math-pow x1 k)
806 (math-pow x2 k)))))
807 coefs)))
808 x)))
809 (math-mul (math-pow 2 n)
810 (if (consp n)
811 (progn
812 (math-inexact-result)
813 (calcFunc-euler n '(float 5 -1)))
814 (calcFunc-euler n '(frac 1 2))))))
815
816 (defvar math-bernoulli-b-cache
817 (list
818 (list 'frac
819 -174611
820 (math-read-number-simple "802857662698291200000"))
821 (list 'frac
822 43867
823 (math-read-number-simple "5109094217170944000"))
824 (list 'frac
825 -3617
826 (math-read-number-simple "10670622842880000"))
827 (list 'frac
828 1
829 (math-read-number-simple "74724249600"))
830 (list 'frac
831 -691
832 (math-read-number-simple "1307674368000"))
833 (list 'frac
834 1
835 (math-read-number-simple "47900160"))
836 (list 'frac
837 -1
838 (math-read-number-simple "1209600"))
839 (list 'frac
840 1
841 30240)
842 (list 'frac
843 -1
844 720)
845 (list 'frac
846 1
847 12)
848 1 ))
849
850 (defvar math-bernoulli-B-cache
851 '((frac -174611 330) (frac 43867 798)
852 (frac -3617 510) (frac 7 6) (frac -691 2730)
853 (frac 5 66) (frac -1 30) (frac 1 42)
854 (frac -1 30) (frac 1 6) 1 ))
855
856 (defvar math-bernoulli-cache-size 11)
857 (defun math-bernoulli-coefs (n)
858 (let* ((coefs (list (calcFunc-bern n)))
859 (nn (math-trunc n))
860 (k nn)
861 (term nn)
862 coef
863 (calc-prefer-frac (or (integerp n) calc-prefer-frac)))
864 (while (>= (setq k (1- k)) 0)
865 (setq term (math-div term (- nn k))
866 coef (math-mul term (math-bernoulli-number k))
867 coefs (cons (if (consp n) (math-float coef) coef) coefs)
868 term (math-mul term k)))
869 (nreverse coefs)))
870
871 (defun math-bernoulli-number (n)
872 (if (= (% n 2) 1)
873 (if (= n 1)
874 '(frac -1 2)
875 0)
876 (setq n (/ n 2))
877 (while (>= n math-bernoulli-cache-size)
878 (let* ((sum 0)
879 (nk 1) ; nk = n-k+1
880 (fact 1) ; fact = (n-k+1)!
881 ofact
882 (p math-bernoulli-b-cache)
883 (calc-prefer-frac t))
884 (math-working "bernoulli B" (* 2 math-bernoulli-cache-size))
885 (while p
886 (setq nk (+ nk 2)
887 ofact fact
888 fact (math-mul fact (* nk (1- nk)))
889 sum (math-add sum (math-div (car p) fact))
890 p (cdr p)))
891 (setq ofact (math-mul ofact (1- nk))
892 sum (math-sub (math-div '(frac 1 2) ofact) sum)
893 math-bernoulli-b-cache (cons sum math-bernoulli-b-cache)
894 math-bernoulli-B-cache (cons (math-mul sum ofact)
895 math-bernoulli-B-cache)
896 math-bernoulli-cache-size (1+ math-bernoulli-cache-size))))
897 (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache)))
898
899 ;;; Bn = n! bn
900 ;;; bn = - sum_k=0^n-1 bk / (n-k+1)!
901
902 ;;; A faster method would be to use "tangent numbers", c.f., Concrete
903 ;;; Mathematics pg. 273.
904
905
906 ;;; Probability distributions.
907
908 ;;; Binomial.
909 (defun calcFunc-utpb (x n p)
910 (if math-expand-formulas
911 (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1)))
912 (calcFunc-betaI p x (math-add (math-sub n x) 1))))
913 (put 'calcFunc-utpb 'math-expandable t)
914
915 (defun calcFunc-ltpb (x n p)
916 (math-sub 1 (calcFunc-utpb x n p)))
917 (put 'calcFunc-ltpb 'math-expandable t)
918
919 ;;; Chi-square.
920 (defun calcFunc-utpc (chisq v)
921 (if math-expand-formulas
922 (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2)))
923 (calcFunc-gammaQ (math-div v 2) (math-div chisq 2))))
924 (put 'calcFunc-utpc 'math-expandable t)
925
926 (defun calcFunc-ltpc (chisq v)
927 (if math-expand-formulas
928 (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2)))
929 (calcFunc-gammaP (math-div v 2) (math-div chisq 2))))
930 (put 'calcFunc-ltpc 'math-expandable t)
931
932 ;;; F-distribution.
933 (defun calcFunc-utpf (f v1 v2)
934 (if math-expand-formulas
935 (math-normalize (list 'calcFunc-betaI
936 (list '/ v2 (list '+ v2 (list '* v1 f)))
937 (list '/ v2 2)
938 (list '/ v1 2)))
939 (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f)))
940 (math-div v2 2)
941 (math-div v1 2))))
942 (put 'calcFunc-utpf 'math-expandable t)
943
944 (defun calcFunc-ltpf (f v1 v2)
945 (math-sub 1 (calcFunc-utpf f v1 v2)))
946 (put 'calcFunc-ltpf 'math-expandable t)
947
948 ;;; Normal.
949 (defun calcFunc-utpn (x mean sdev)
950 (if math-expand-formulas
951 (math-normalize
952 (list '/
953 (list '+ 1
954 (list 'calcFunc-erf
955 (list '/ (list '- mean x)
956 (list '* sdev (list 'calcFunc-sqrt 2)))))
957 2))
958 (math-mul (math-add '(float 1 0)
959 (calcFunc-erf
960 (math-div (math-sub mean x)
961 (math-mul sdev (math-sqrt-2)))))
962 '(float 5 -1))))
963 (put 'calcFunc-utpn 'math-expandable t)
964
965 (defun calcFunc-ltpn (x mean sdev)
966 (if math-expand-formulas
967 (math-normalize
968 (list '/
969 (list '+ 1
970 (list 'calcFunc-erf
971 (list '/ (list '- x mean)
972 (list '* sdev (list 'calcFunc-sqrt 2)))))
973 2))
974 (math-mul (math-add '(float 1 0)
975 (calcFunc-erf
976 (math-div (math-sub x mean)
977 (math-mul sdev (math-sqrt-2)))))
978 '(float 5 -1))))
979 (put 'calcFunc-ltpn 'math-expandable t)
980
981 ;;; Poisson.
982 (defun calcFunc-utpp (n x)
983 (if math-expand-formulas
984 (math-normalize (list 'calcFunc-gammaP x n))
985 (calcFunc-gammaP x n)))
986 (put 'calcFunc-utpp 'math-expandable t)
987
988 (defun calcFunc-ltpp (n x)
989 (if math-expand-formulas
990 (math-normalize (list 'calcFunc-gammaQ x n))
991 (calcFunc-gammaQ x n)))
992 (put 'calcFunc-ltpp 'math-expandable t)
993
994 ;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.)
995 (defun calcFunc-utpt (tt v)
996 (if math-expand-formulas
997 (math-normalize (list 'calcFunc-betaI
998 (list '/ v (list '+ v (list '^ tt 2)))
999 (list '/ v 2)
1000 '(float 5 -1)))
1001 (calcFunc-betaI (math-div v (math-add v (math-sqr tt)))
1002 (math-div v 2)
1003 '(float 5 -1))))
1004 (put 'calcFunc-utpt 'math-expandable t)
1005
1006 (defun calcFunc-ltpt (tt v)
1007 (math-sub 1 (calcFunc-utpt tt v)))
1008 (put 'calcFunc-ltpt 'math-expandable t)
1009
1010 (provide 'calc-funcs)
1011
1012 ;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a
1013 ;;; calc-funcs.el ends here
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