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12f1951a CW |
1 | ;;; calc-funcs.el --- well-known functions for Calc |
2 | ||
ab422c4d | 3 | ;; Copyright (C) 1990-1993, 2001-2013 Free Software Foundation, Inc. |
12f1951a CW |
4 | |
5 | ;; Author: David Gillespie <daveg@synaptics.com> | |
e8fff8ed | 6 | ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com> |
136211a9 EZ |
7 | |
8 | ;; This file is part of GNU Emacs. | |
9 | ||
662c9c64 | 10 | ;; GNU Emacs is free software: you can redistribute it and/or modify |
7c671b23 | 11 | ;; it under the terms of the GNU General Public License as published by |
662c9c64 GM |
12 | ;; the Free Software Foundation, either version 3 of the License, or |
13 | ;; (at your option) any later version. | |
7c671b23 | 14 | |
136211a9 | 15 | ;; GNU Emacs is distributed in the hope that it will be useful, |
7c671b23 GM |
16 | ;; but WITHOUT ANY WARRANTY; without even the implied warranty of |
17 | ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
18 | ;; GNU General Public License for more details. | |
19 | ||
20 | ;; You should have received a copy of the GNU General Public License | |
662c9c64 | 21 | ;; along with GNU Emacs. If not, see <http://www.gnu.org/licenses/>. |
136211a9 | 22 | |
12f1951a | 23 | ;;; Commentary: |
136211a9 | 24 | |
12f1951a | 25 | ;;; Code: |
136211a9 EZ |
26 | |
27 | ;; This file is autoloaded from calc-ext.el. | |
136211a9 | 28 | |
300f28d3 | 29 | (require 'calc-ext) |
136211a9 EZ |
30 | (require 'calc-macs) |
31 | ||
136211a9 EZ |
32 | (defun calc-inc-gamma (arg) |
33 | (interactive "P") | |
34 | (calc-slow-wrapper | |
35 | (if (calc-is-inverse) | |
36 | (if (calc-is-hyperbolic) | |
37 | (calc-binary-op "gamG" 'calcFunc-gammaG arg) | |
38 | (calc-binary-op "gamQ" 'calcFunc-gammaQ arg)) | |
39 | (if (calc-is-hyperbolic) | |
40 | (calc-binary-op "gamg" 'calcFunc-gammag arg) | |
bf77c646 | 41 | (calc-binary-op "gamP" 'calcFunc-gammaP arg))))) |
136211a9 EZ |
42 | |
43 | (defun calc-erf (arg) | |
44 | (interactive "P") | |
45 | (calc-slow-wrapper | |
46 | (if (calc-is-inverse) | |
47 | (calc-unary-op "erfc" 'calcFunc-erfc arg) | |
bf77c646 | 48 | (calc-unary-op "erf" 'calcFunc-erf arg)))) |
136211a9 EZ |
49 | |
50 | (defun calc-erfc (arg) | |
51 | (interactive "P") | |
52 | (calc-invert-func) | |
bf77c646 | 53 | (calc-erf arg)) |
136211a9 EZ |
54 | |
55 | (defun calc-beta (arg) | |
56 | (interactive "P") | |
57 | (calc-slow-wrapper | |
bf77c646 | 58 | (calc-binary-op "beta" 'calcFunc-beta arg))) |
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59 | |
60 | (defun calc-inc-beta () | |
61 | (interactive) | |
62 | (calc-slow-wrapper | |
63 | (if (calc-is-hyperbolic) | |
64 | (calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3))) | |
bf77c646 | 65 | (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3)))))) |
136211a9 EZ |
66 | |
67 | (defun calc-bessel-J (arg) | |
68 | (interactive "P") | |
69 | (calc-slow-wrapper | |
bf77c646 | 70 | (calc-binary-op "besJ" 'calcFunc-besJ arg))) |
136211a9 EZ |
71 | |
72 | (defun calc-bessel-Y (arg) | |
73 | (interactive "P") | |
74 | (calc-slow-wrapper | |
bf77c646 | 75 | (calc-binary-op "besY" 'calcFunc-besY arg))) |
136211a9 EZ |
76 | |
77 | (defun calc-bernoulli-number (arg) | |
78 | (interactive "P") | |
79 | (calc-slow-wrapper | |
80 | (if (calc-is-hyperbolic) | |
81 | (calc-binary-op "bern" 'calcFunc-bern arg) | |
bf77c646 | 82 | (calc-unary-op "bern" 'calcFunc-bern arg)))) |
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83 | |
84 | (defun calc-euler-number (arg) | |
85 | (interactive "P") | |
86 | (calc-slow-wrapper | |
87 | (if (calc-is-hyperbolic) | |
88 | (calc-binary-op "eulr" 'calcFunc-euler arg) | |
bf77c646 | 89 | (calc-unary-op "eulr" 'calcFunc-euler arg)))) |
136211a9 EZ |
90 | |
91 | (defun calc-stirling-number (arg) | |
92 | (interactive "P") | |
93 | (calc-slow-wrapper | |
94 | (if (calc-is-hyperbolic) | |
95 | (calc-binary-op "str2" 'calcFunc-stir2 arg) | |
bf77c646 | 96 | (calc-binary-op "str1" 'calcFunc-stir1 arg)))) |
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97 | |
98 | (defun calc-utpb () | |
99 | (interactive) | |
bf77c646 | 100 | (calc-prob-dist "b" 3)) |
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101 | |
102 | (defun calc-utpc () | |
103 | (interactive) | |
bf77c646 | 104 | (calc-prob-dist "c" 2)) |
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105 | |
106 | (defun calc-utpf () | |
107 | (interactive) | |
bf77c646 | 108 | (calc-prob-dist "f" 3)) |
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109 | |
110 | (defun calc-utpn () | |
111 | (interactive) | |
bf77c646 | 112 | (calc-prob-dist "n" 3)) |
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113 | |
114 | (defun calc-utpp () | |
115 | (interactive) | |
bf77c646 | 116 | (calc-prob-dist "p" 2)) |
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117 | |
118 | (defun calc-utpt () | |
119 | (interactive) | |
bf77c646 | 120 | (calc-prob-dist "t" 2)) |
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121 | |
122 | (defun calc-prob-dist (letter nargs) | |
123 | (calc-slow-wrapper | |
124 | (if (calc-is-inverse) | |
125 | (calc-enter-result nargs (concat "ltp" letter) | |
126 | (append (list (intern (concat "calcFunc-ltp" letter)) | |
127 | (calc-top-n 1)) | |
128 | (calc-top-list-n (1- nargs) 2))) | |
129 | (calc-enter-result nargs (concat "utp" letter) | |
130 | (append (list (intern (concat "calcFunc-utp" letter)) | |
131 | (calc-top-n 1)) | |
bf77c646 | 132 | (calc-top-list-n (1- nargs) 2)))))) |
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133 | |
134 | ||
135 | ||
136 | ||
137 | ;;; Sources: Numerical Recipes, Press et al; | |
138 | ;;; Handbook of Mathematical Functions, Abramowitz & Stegun. | |
139 | ||
140 | ||
141 | ;;; Gamma function. | |
142 | ||
143 | (defun calcFunc-gamma (x) | |
144 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
bf77c646 | 145 | (calcFunc-fact (math-add x -1))) |
136211a9 | 146 | |
3bf97114 JB |
147 | (defun math-gammap1-raw (x &optional fprec nfprec) |
148 | "Compute gamma(1+X) to the appropriate precision." | |
136211a9 EZ |
149 | (or fprec |
150 | (setq fprec (math-float calc-internal-prec) | |
151 | nfprec (math-float (- calc-internal-prec)))) | |
152 | (cond ((math-lessp-float (calcFunc-re x) fprec) | |
153 | (if (math-lessp-float (calcFunc-re x) nfprec) | |
154 | (math-neg (math-div | |
155 | (math-pi) | |
156 | (math-mul (math-gammap1-raw | |
157 | (math-add (math-neg x) | |
158 | '(float -1 0)) | |
159 | fprec nfprec) | |
160 | (math-sin-raw | |
161 | (math-mul (math-pi) x))))) | |
162 | (let ((xplus1 (math-add x '(float 1 0)))) | |
163 | (math-div (math-gammap1-raw xplus1 fprec nfprec) xplus1)))) | |
164 | ((and (math-realp x) | |
165 | (math-lessp-float '(float 736276 0) x)) | |
166 | (math-overflow)) | |
167 | (t ; re(x) now >= 10.0 | |
168 | (let ((xinv (math-div 1 x)) | |
169 | (lnx (math-ln-raw x))) | |
170 | (math-mul (math-sqrt-two-pi) | |
171 | (math-exp-raw | |
172 | (math-gamma-series | |
173 | (math-sub (math-mul (math-add x '(float 5 -1)) | |
174 | lnx) | |
175 | x) | |
176 | xinv | |
177 | (math-sqr xinv) | |
178 | '(float 0 0) | |
bf77c646 | 179 | 2))))))) |
136211a9 EZ |
180 | |
181 | (defun math-gamma-series (sum x xinvsqr oterm n) | |
182 | (math-working "gamma" sum) | |
183 | (let* ((bn (math-bernoulli-number n)) | |
184 | (term (math-mul (math-div-float (math-float (nth 1 bn)) | |
185 | (math-float (* (nth 2 bn) | |
186 | (* n (1- n))))) | |
187 | x)) | |
188 | (next (math-add sum term))) | |
189 | (if (math-nearly-equal sum next) | |
190 | next | |
191 | (if (> n (* 2 calc-internal-prec)) | |
192 | (progn | |
193 | ;; Need this because series eventually diverges for large enough n. | |
194 | (calc-record-why | |
195 | "*Gamma computation stopped early, not all digits may be valid") | |
196 | next) | |
bf77c646 | 197 | (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2)))))) |
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198 | |
199 | ||
200 | ;;; Incomplete gamma function. | |
201 | ||
12f1951a | 202 | (defvar math-current-gamma-value nil) |
136211a9 EZ |
203 | (defun calcFunc-gammaP (a x) |
204 | (if (equal x '(var inf var-inf)) | |
205 | '(float 1 0) | |
206 | (math-inexact-result) | |
207 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
208 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
209 | (if (and (math-num-integerp a) | |
210 | (integerp (setq a (math-trunc a))) | |
211 | (> a 0) (< a 20)) | |
212 | (math-sub 1 (calcFunc-gammaQ a x)) | |
213 | (let ((math-current-gamma-value (calcFunc-gamma a))) | |
bf77c646 | 214 | (math-div (calcFunc-gammag a x) math-current-gamma-value))))) |
136211a9 EZ |
215 | |
216 | (defun calcFunc-gammaQ (a x) | |
217 | (if (equal x '(var inf var-inf)) | |
218 | '(float 0 0) | |
219 | (math-inexact-result) | |
220 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
221 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
222 | (if (and (math-num-integerp a) | |
223 | (integerp (setq a (math-trunc a))) | |
224 | (> a 0) (< a 20)) | |
225 | (let ((n 0) | |
226 | (sum '(float 1 0)) | |
227 | (term '(float 1 0))) | |
228 | (math-with-extra-prec 1 | |
229 | (while (< (setq n (1+ n)) a) | |
230 | (setq term (math-div (math-mul term x) n) | |
231 | sum (math-add sum term)) | |
232 | (math-working "gamma" sum)) | |
233 | (math-mul sum (calcFunc-exp (math-neg x))))) | |
234 | (let ((math-current-gamma-value (calcFunc-gamma a))) | |
bf77c646 | 235 | (math-div (calcFunc-gammaG a x) math-current-gamma-value))))) |
136211a9 EZ |
236 | |
237 | (defun calcFunc-gammag (a x) | |
238 | (if (equal x '(var inf var-inf)) | |
239 | (calcFunc-gamma a) | |
240 | (math-inexact-result) | |
241 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
242 | (or (Math-numberp x) (math-reject-arg x 'numberp)) | |
243 | (math-with-extra-prec 2 | |
244 | (setq a (math-float a)) | |
245 | (setq x (math-float x)) | |
246 | (if (or (math-negp (calcFunc-re a)) | |
247 | (math-lessp-float (calcFunc-re x) | |
248 | (math-add-float (calcFunc-re a) | |
249 | '(float 1 0)))) | |
250 | (math-inc-gamma-series a x) | |
251 | (math-sub (or math-current-gamma-value (calcFunc-gamma a)) | |
bf77c646 | 252 | (math-inc-gamma-cfrac a x)))))) |
136211a9 EZ |
253 | |
254 | (defun calcFunc-gammaG (a x) | |
255 | (if (equal x '(var inf var-inf)) | |
256 | '(float 0 0) | |
257 | (math-inexact-result) | |
258 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
259 | (or (Math-numberp x) (math-reject-arg x 'numberp)) | |
260 | (math-with-extra-prec 2 | |
261 | (setq a (math-float a)) | |
262 | (setq x (math-float x)) | |
263 | (if (or (math-negp (calcFunc-re a)) | |
264 | (math-lessp-float (calcFunc-re x) | |
265 | (math-add-float (math-abs-approx a) | |
266 | '(float 1 0)))) | |
267 | (math-sub (or math-current-gamma-value (calcFunc-gamma a)) | |
268 | (math-inc-gamma-series a x)) | |
bf77c646 | 269 | (math-inc-gamma-cfrac a x))))) |
136211a9 EZ |
270 | |
271 | (defun math-inc-gamma-series (a x) | |
272 | (if (Math-zerop x) | |
273 | '(float 0 0) | |
274 | (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x)) | |
275 | (math-with-extra-prec 2 | |
276 | (let ((start (math-div '(float 1 0) a))) | |
bf77c646 | 277 | (math-inc-gamma-series-step start start a x)))))) |
136211a9 EZ |
278 | |
279 | (defun math-inc-gamma-series-step (sum term a x) | |
280 | (math-working "gamma" sum) | |
281 | (setq a (math-add a '(float 1 0)) | |
282 | term (math-div (math-mul term x) a)) | |
283 | (let ((next (math-add sum term))) | |
284 | (if (math-nearly-equal sum next) | |
285 | next | |
bf77c646 | 286 | (math-inc-gamma-series-step next term a x)))) |
136211a9 EZ |
287 | |
288 | (defun math-inc-gamma-cfrac (a x) | |
289 | (if (Math-zerop x) | |
290 | (or math-current-gamma-value (calcFunc-gamma a)) | |
291 | (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x)) | |
292 | (math-inc-gamma-cfrac-step '(float 1 0) x | |
293 | '(float 0 0) '(float 1 0) | |
294 | '(float 1 0) '(float 1 0) '(float 0 0) | |
bf77c646 | 295 | a x)))) |
136211a9 EZ |
296 | |
297 | (defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x) | |
298 | (let ((ana (math-sub n a)) | |
299 | (anf (math-mul n fac))) | |
300 | (setq n (math-add n '(float 1 0)) | |
301 | a0 (math-mul (math-add a1 (math-mul a0 ana)) fac) | |
302 | b0 (math-mul (math-add b1 (math-mul b0 ana)) fac) | |
303 | a1 (math-add (math-mul x a0) (math-mul anf a1)) | |
304 | b1 (math-add (math-mul x b0) (math-mul anf b1))) | |
305 | (if (math-zerop a1) | |
306 | (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x) | |
307 | (setq fac (math-div '(float 1 0) a1)) | |
308 | (let ((next (math-mul b1 fac))) | |
309 | (math-working "gamma" next) | |
310 | (if (math-nearly-equal next g) | |
311 | next | |
bf77c646 | 312 | (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x)))))) |
136211a9 EZ |
313 | |
314 | ||
315 | ;;; Error function. | |
316 | ||
317 | (defun calcFunc-erf (x) | |
318 | (if (equal x '(var inf var-inf)) | |
319 | '(float 1 0) | |
320 | (if (equal x '(neg (var inf var-inf))) | |
321 | '(float -1 0) | |
322 | (if (Math-zerop x) | |
323 | x | |
324 | (let ((math-current-gamma-value (math-sqrt-pi))) | |
325 | (math-to-same-complex-quad | |
326 | (math-div (calcFunc-gammag '(float 5 -1) | |
327 | (math-sqr (math-to-complex-quad-one x))) | |
328 | math-current-gamma-value) | |
bf77c646 | 329 | x)))))) |
136211a9 EZ |
330 | |
331 | (defun calcFunc-erfc (x) | |
332 | (if (equal x '(var inf var-inf)) | |
333 | '(float 0 0) | |
334 | (if (math-posp x) | |
335 | (let ((math-current-gamma-value (math-sqrt-pi))) | |
336 | (math-div (calcFunc-gammaG '(float 5 -1) (math-sqr x)) | |
337 | math-current-gamma-value)) | |
bf77c646 | 338 | (math-sub 1 (calcFunc-erf x))))) |
136211a9 EZ |
339 | |
340 | (defun math-to-complex-quad-one (x) | |
341 | (if (eq (car-safe x) 'polar) (setq x (math-complex x))) | |
342 | (if (eq (car-safe x) 'cplx) | |
343 | (list 'cplx (math-abs (nth 1 x)) (math-abs (nth 2 x))) | |
bf77c646 | 344 | x)) |
136211a9 EZ |
345 | |
346 | (defun math-to-same-complex-quad (x y) | |
347 | (if (eq (car-safe y) 'cplx) | |
348 | (if (eq (car-safe x) 'cplx) | |
349 | (list 'cplx | |
350 | (if (math-negp (nth 1 y)) (math-neg (nth 1 x)) (nth 1 x)) | |
351 | (if (math-negp (nth 2 y)) (math-neg (nth 2 x)) (nth 2 x))) | |
352 | (if (math-negp (nth 1 y)) (math-neg x) x)) | |
353 | (if (math-negp y) | |
354 | (if (eq (car-safe x) 'cplx) | |
355 | (list 'cplx (math-neg (nth 1 x)) (nth 2 x)) | |
356 | (math-neg x)) | |
bf77c646 | 357 | x))) |
136211a9 EZ |
358 | |
359 | ||
360 | ;;; Beta function. | |
361 | ||
362 | (defun calcFunc-beta (a b) | |
363 | (if (math-num-integerp a) | |
364 | (let ((am (math-add a -1))) | |
365 | (or (math-numberp b) (math-reject-arg b 'numberp)) | |
366 | (math-div 1 (math-mul b (calcFunc-choose (math-add b am) am)))) | |
367 | (if (math-num-integerp b) | |
368 | (calcFunc-beta b a) | |
369 | (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b)) | |
bf77c646 | 370 | (calcFunc-gamma (math-add a b)))))) |
136211a9 EZ |
371 | |
372 | ||
373 | ;;; Incomplete beta function. | |
374 | ||
12f1951a | 375 | (defvar math-current-beta-value nil) |
136211a9 EZ |
376 | (defun calcFunc-betaI (x a b) |
377 | (cond ((math-zerop x) | |
378 | '(float 0 0)) | |
379 | ((math-equal-int x 1) | |
380 | '(float 1 0)) | |
381 | ((or (math-zerop a) | |
382 | (and (math-num-integerp a) | |
383 | (math-negp a))) | |
384 | (if (or (math-zerop b) | |
385 | (and (math-num-integerp b) | |
386 | (math-negp b))) | |
387 | (math-reject-arg b 'range) | |
388 | '(float 1 0))) | |
389 | ((or (math-zerop b) | |
390 | (and (math-num-integerp b) | |
391 | (math-negp b))) | |
392 | '(float 0 0)) | |
393 | ((not (math-numberp a)) (math-reject-arg a 'numberp)) | |
394 | ((not (math-numberp b)) (math-reject-arg b 'numberp)) | |
395 | ((math-inexact-result)) | |
396 | (t (let ((math-current-beta-value (calcFunc-beta a b))) | |
bf77c646 | 397 | (math-div (calcFunc-betaB x a b) math-current-beta-value))))) |
136211a9 EZ |
398 | |
399 | (defun calcFunc-betaB (x a b) | |
400 | (cond | |
401 | ((math-zerop x) | |
402 | '(float 0 0)) | |
403 | ((math-equal-int x 1) | |
404 | (calcFunc-beta a b)) | |
405 | ((not (math-numberp x)) (math-reject-arg x 'numberp)) | |
406 | ((not (math-numberp a)) (math-reject-arg a 'numberp)) | |
407 | ((not (math-numberp b)) (math-reject-arg b 'numberp)) | |
408 | ((math-zerop a) (math-reject-arg a 'nonzerop)) | |
409 | ((math-zerop b) (math-reject-arg b 'nonzerop)) | |
410 | ((and (math-num-integerp b) | |
411 | (if (math-negp b) | |
412 | (math-reject-arg b 'range) | |
413 | (Math-natnum-lessp (setq b (math-trunc b)) 20))) | |
414 | (and calc-symbolic-mode (or (math-floatp a) (math-floatp b)) | |
415 | (math-inexact-result)) | |
416 | (math-mul | |
417 | (math-with-extra-prec 2 | |
418 | (let* ((i 0) | |
419 | (term 1) | |
420 | (sum (math-div term a))) | |
421 | (while (< (setq i (1+ i)) b) | |
422 | (setq term (math-mul (math-div (math-mul term (- i b)) i) x) | |
423 | sum (math-add sum (math-div term (math-add a i)))) | |
424 | (math-working "beta" sum)) | |
425 | sum)) | |
426 | (math-pow x a))) | |
427 | ((and (math-num-integerp a) | |
428 | (if (math-negp a) | |
429 | (math-reject-arg a 'range) | |
430 | (Math-natnum-lessp (setq a (math-trunc a)) 20))) | |
431 | (math-sub (or math-current-beta-value (calcFunc-beta a b)) | |
432 | (calcFunc-betaB (math-sub 1 x) b a))) | |
433 | (t | |
434 | (math-inexact-result) | |
435 | (math-with-extra-prec 2 | |
436 | (setq x (math-float x)) | |
437 | (setq a (math-float a)) | |
438 | (setq b (math-float b)) | |
439 | (let ((bt (math-exp-raw (math-add (math-mul a (math-ln-raw x)) | |
440 | (math-mul b (math-ln-raw | |
441 | (math-sub '(float 1 0) | |
442 | x))))))) | |
443 | (if (Math-lessp x (math-div (math-add a '(float 1 0)) | |
444 | (math-add (math-add a b) '(float 2 0)))) | |
445 | (math-div (math-mul bt (math-beta-cfrac a b x)) a) | |
446 | (math-sub (or math-current-beta-value (calcFunc-beta a b)) | |
447 | (math-div (math-mul bt | |
448 | (math-beta-cfrac b a (math-sub 1 x))) | |
bf77c646 | 449 | b)))))))) |
136211a9 EZ |
450 | |
451 | (defun math-beta-cfrac (a b x) | |
452 | (let ((qab (math-add a b)) | |
453 | (qap (math-add a '(float 1 0))) | |
454 | (qam (math-add a '(float -1 0)))) | |
455 | (math-beta-cfrac-step '(float 1 0) | |
456 | (math-sub '(float 1 0) | |
457 | (math-div (math-mul qab x) qap)) | |
458 | '(float 1 0) '(float 1 0) | |
459 | '(float 1 0) | |
bf77c646 | 460 | qab qap qam a b x))) |
136211a9 EZ |
461 | |
462 | (defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x) | |
463 | (let* ((two-m (math-mul m '(float 2 0))) | |
464 | (d (math-div (math-mul (math-mul (math-sub b m) m) x) | |
465 | (math-mul (math-add qam two-m) (math-add a two-m)))) | |
466 | (ap (math-add az (math-mul d am))) | |
467 | (bp (math-add bz (math-mul d bm))) | |
468 | (d2 (math-neg | |
469 | (math-div (math-mul (math-mul (math-add a m) (math-add qab m)) x) | |
470 | (math-mul (math-add qap two-m) (math-add a two-m))))) | |
471 | (app (math-add ap (math-mul d2 az))) | |
472 | (bpp (math-add bp (math-mul d2 bz))) | |
473 | (next (math-div app bpp))) | |
474 | (math-working "beta" next) | |
475 | (if (math-nearly-equal next az) | |
476 | next | |
477 | (math-beta-cfrac-step next '(float 1 0) | |
478 | (math-div ap bpp) (math-div bp bpp) | |
479 | (math-add m '(float 1 0)) | |
bf77c646 | 480 | qab qap qam a b x)))) |
136211a9 EZ |
481 | |
482 | ||
483 | ;;; Bessel functions. | |
484 | ||
485 | ;;; Should generalize this to handle arbitrary precision! | |
486 | ||
487 | (defun calcFunc-besJ (v x) | |
488 | (or (math-numberp v) (math-reject-arg v 'numberp)) | |
489 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
490 | (let ((calc-internal-prec (min 8 calc-internal-prec))) | |
491 | (math-with-extra-prec 3 | |
492 | (setq x (math-float (math-normalize x))) | |
493 | (setq v (math-float (math-normalize v))) | |
494 | (cond ((math-zerop x) | |
495 | (if (math-zerop v) | |
496 | '(float 1 0) | |
497 | '(float 0 0))) | |
498 | ((math-inexact-result)) | |
499 | ((not (math-num-integerp v)) | |
500 | (let ((start (math-div 1 (calcFunc-fact v)))) | |
501 | (math-mul (math-besJ-series start start | |
502 | 0 | |
503 | (math-mul '(float -25 -2) | |
504 | (math-sqr x)) | |
505 | v) | |
506 | (math-pow (math-div x 2) v)))) | |
507 | ((math-negp (setq v (math-trunc v))) | |
508 | (if (math-oddp v) | |
509 | (math-neg (calcFunc-besJ (math-neg v) x)) | |
510 | (calcFunc-besJ (math-neg v) x))) | |
511 | ((eq v 0) | |
512 | (math-besJ0 x)) | |
513 | ((eq v 1) | |
514 | (math-besJ1 x)) | |
515 | ((Math-lessp v (math-abs-approx x)) | |
516 | (let ((j 0) | |
517 | (bjm (math-besJ0 x)) | |
518 | (bj (math-besJ1 x)) | |
519 | (two-over-x (math-div 2 x)) | |
520 | bjp) | |
521 | (while (< (setq j (1+ j)) v) | |
522 | (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj) | |
523 | bjm) | |
524 | bjm bj | |
525 | bj bjp)) | |
526 | bj)) | |
527 | (t | |
528 | (if (Math-lessp 100 v) (math-reject-arg v 'range)) | |
529 | (let* ((j (logior (+ v (math-isqrt-small (* 40 v))) 1)) | |
530 | (two-over-x (math-div 2 x)) | |
531 | (jsum nil) | |
532 | (bjp '(float 0 0)) | |
533 | (sum '(float 0 0)) | |
534 | (bj '(float 1 0)) | |
535 | bjm ans) | |
536 | (while (> (setq j (1- j)) 0) | |
537 | (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj) | |
538 | bjp) | |
539 | bjp bj | |
540 | bj bjm) | |
541 | (if (> (nth 2 (math-abs-approx bj)) 10) | |
542 | (setq bj (math-mul bj '(float 1 -10)) | |
543 | bjp (math-mul bjp '(float 1 -10)) | |
544 | ans (and ans (math-mul ans '(float 1 -10))) | |
545 | sum (math-mul sum '(float 1 -10)))) | |
546 | (or (setq jsum (not jsum)) | |
547 | (setq sum (math-add sum bj))) | |
548 | (if (= j v) | |
549 | (setq ans bjp))) | |
bf77c646 | 550 | (math-div ans (math-sub (math-mul 2 sum) bj)))))))) |
136211a9 EZ |
551 | |
552 | (defun math-besJ-series (sum term k zz vk) | |
553 | (math-working "besJ" sum) | |
554 | (setq k (1+ k) | |
555 | vk (math-add 1 vk) | |
556 | term (math-div (math-mul term zz) (math-mul k vk))) | |
557 | (let ((next (math-add sum term))) | |
558 | (if (math-nearly-equal next sum) | |
559 | next | |
bf77c646 | 560 | (math-besJ-series next term k zz vk)))) |
136211a9 EZ |
561 | |
562 | (defun math-besJ0 (x &optional yflag) | |
563 | (cond ((and (not yflag) (math-negp (calcFunc-re x))) | |
564 | (math-besJ0 (math-neg x))) | |
565 | ((Math-lessp '(float 8 0) (math-abs-approx x)) | |
566 | (let* ((z (math-div '(float 8 0) x)) | |
567 | (y (math-sqr z)) | |
a8775b6d | 568 | (xx (math-add x |
c6254f53 | 569 | (math-read-number-simple "-0.785398164"))) |
136211a9 | 570 | (a1 (math-poly-eval y |
a8775b6d | 571 | (list |
f83b9601 JB |
572 | (math-read-number-simple "0.0000002093887211") |
573 | (math-read-number-simple "-0.000002073370639") | |
574 | (math-read-number-simple "0.00002734510407") | |
575 | (math-read-number-simple "-0.001098628627") | |
c6254f53 | 576 | '(float 1 0)))) |
136211a9 | 577 | (a2 (math-poly-eval y |
c6254f53 JB |
578 | (list |
579 | (math-read-number-simple "-0.0000000934935152") | |
580 | (math-read-number-simple "0.0000007621095161") | |
581 | (math-read-number-simple "-0.000006911147651") | |
582 | (math-read-number-simple "0.0001430488765") | |
583 | (math-read-number-simple "-0.01562499995")))) | |
136211a9 EZ |
584 | (sc (math-sin-cos-raw xx))) |
585 | (if yflag | |
586 | (setq sc (cons (math-neg (cdr sc)) (car sc)))) | |
587 | (math-mul (math-sqrt | |
c6254f53 JB |
588 | (math-div (math-read-number-simple "0.636619722") |
589 | x)) | |
136211a9 EZ |
590 | (math-sub (math-mul (cdr sc) a1) |
591 | (math-mul (car sc) (math-mul z a2)))))) | |
592 | (t | |
593 | (let ((y (math-sqr x))) | |
594 | (math-div (math-poly-eval y | |
c6254f53 JB |
595 | (list |
596 | (math-read-number-simple "-184.9052456") | |
597 | (math-read-number-simple "77392.33017") | |
598 | (math-read-number-simple "-11214424.18") | |
599 | (math-read-number-simple "651619640.7") | |
600 | (math-read-number-simple "-13362590354.0") | |
601 | (math-read-number-simple "57568490574.0"))) | |
136211a9 | 602 | (math-poly-eval y |
c6254f53 JB |
603 | (list |
604 | '(float 1 0) | |
605 | (math-read-number-simple "267.8532712") | |
606 | (math-read-number-simple "59272.64853") | |
607 | (math-read-number-simple "9494680.718") | |
608 | (math-read-number-simple "1029532985.0") | |
609 | (math-read-number-simple "57568490411.0")))))))) | |
136211a9 EZ |
610 | |
611 | (defun math-besJ1 (x &optional yflag) | |
612 | (cond ((and (math-negp (calcFunc-re x)) (not yflag)) | |
613 | (math-neg (math-besJ1 (math-neg x)))) | |
614 | ((Math-lessp '(float 8 0) (math-abs-approx x)) | |
615 | (let* ((z (math-div '(float 8 0) x)) | |
616 | (y (math-sqr z)) | |
c6254f53 | 617 | (xx (math-add x (math-read-number-simple "-2.356194491"))) |
136211a9 | 618 | (a1 (math-poly-eval y |
c6254f53 JB |
619 | (list |
620 | (math-read-number-simple "-0.000000240337019") | |
621 | (math-read-number-simple "0.000002457520174") | |
622 | (math-read-number-simple "-0.00003516396496") | |
623 | '(float 183105 -8) | |
624 | '(float 1 0)))) | |
136211a9 | 625 | (a2 (math-poly-eval y |
c6254f53 JB |
626 | (list |
627 | (math-read-number-simple "0.000000105787412") | |
628 | (math-read-number-simple "-0.00000088228987") | |
629 | (math-read-number-simple "0.000008449199096") | |
630 | (math-read-number-simple "-0.0002002690873") | |
631 | (math-read-number-simple "0.04687499995")))) | |
136211a9 EZ |
632 | (sc (math-sin-cos-raw xx))) |
633 | (if yflag | |
634 | (setq sc (cons (math-neg (cdr sc)) (car sc))) | |
635 | (if (math-negp x) | |
636 | (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc)))))) | |
a8775b6d | 637 | (math-mul (math-sqrt (math-div |
c6254f53 | 638 | (math-read-number-simple "0.636619722") |
a8775b6d | 639 | x)) |
136211a9 EZ |
640 | (math-sub (math-mul (cdr sc) a1) |
641 | (math-mul (car sc) (math-mul z a2)))))) | |
642 | (t | |
643 | (let ((y (math-sqr x))) | |
644 | (math-mul | |
645 | x | |
646 | (math-div (math-poly-eval y | |
c6254f53 JB |
647 | (list |
648 | (math-read-number-simple "-30.16036606") | |
649 | (math-read-number-simple "15704.4826") | |
650 | (math-read-number-simple "-2972611.439") | |
651 | (math-read-number-simple "242396853.1") | |
652 | (math-read-number-simple "-7895059235.0") | |
653 | (math-read-number-simple "72362614232.0"))) | |
136211a9 | 654 | (math-poly-eval y |
c6254f53 JB |
655 | (list |
656 | '(float 1 0) | |
657 | (math-read-number-simple "376.9991397") | |
658 | (math-read-number-simple "99447.43394") | |
659 | (math-read-number-simple "18583304.74") | |
660 | (math-read-number-simple "2300535178.0") | |
661 | (math-read-number-simple "144725228442.0"))))))))) | |
136211a9 EZ |
662 | |
663 | (defun calcFunc-besY (v x) | |
664 | (math-inexact-result) | |
665 | (or (math-numberp v) (math-reject-arg v 'numberp)) | |
666 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
667 | (let ((calc-internal-prec (min 8 calc-internal-prec))) | |
668 | (math-with-extra-prec 3 | |
669 | (setq x (math-float (math-normalize x))) | |
670 | (setq v (math-float (math-normalize v))) | |
671 | (cond ((not (math-num-integerp v)) | |
672 | (let ((sc (math-sin-cos-raw (math-mul v (math-pi))))) | |
673 | (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc)) | |
674 | (calcFunc-besJ (math-neg v) x)) | |
675 | (car sc)))) | |
676 | ((math-negp (setq v (math-trunc v))) | |
677 | (if (math-oddp v) | |
678 | (math-neg (calcFunc-besY (math-neg v) x)) | |
679 | (calcFunc-besY (math-neg v) x))) | |
680 | ((eq v 0) | |
681 | (math-besY0 x)) | |
682 | ((eq v 1) | |
683 | (math-besY1 x)) | |
684 | (t | |
685 | (let ((j 0) | |
686 | (bym (math-besY0 x)) | |
687 | (by (math-besY1 x)) | |
688 | (two-over-x (math-div 2 x)) | |
689 | byp) | |
690 | (while (< (setq j (1+ j)) v) | |
691 | (setq byp (math-sub (math-mul (math-mul j two-over-x) by) | |
692 | bym) | |
693 | bym by | |
694 | by byp)) | |
bf77c646 | 695 | by)))))) |
136211a9 EZ |
696 | |
697 | (defun math-besY0 (x) | |
698 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
699 | (let ((y (math-sqr x))) | |
c6254f53 | 700 | (math-add |
136211a9 | 701 | (math-div (math-poly-eval y |
c6254f53 JB |
702 | (list |
703 | (math-read-number-simple "228.4622733") | |
704 | (math-read-number-simple "-86327.92757") | |
705 | (math-read-number-simple "10879881.29") | |
706 | (math-read-number-simple "-512359803.6") | |
707 | (math-read-number-simple "7062834065.0") | |
708 | (math-read-number-simple "-2957821389.0"))) | |
136211a9 | 709 | (math-poly-eval y |
c6254f53 JB |
710 | (list |
711 | '(float 1 0) | |
712 | (math-read-number-simple "226.1030244") | |
713 | (math-read-number-simple "47447.2647") | |
714 | (math-read-number-simple "7189466.438") | |
715 | (math-read-number-simple "745249964.8") | |
716 | (math-read-number-simple "40076544269.0")))) | |
717 | (math-mul (math-read-number-simple "0.636619772") | |
136211a9 EZ |
718 | (math-mul (math-besJ0 x) (math-ln-raw x)))))) |
719 | ((math-negp (calcFunc-re x)) | |
720 | (math-add (math-besJ0 (math-neg x) t) | |
721 | (math-mul '(cplx 0 2) | |
722 | (math-besJ0 (math-neg x))))) | |
723 | (t | |
bf77c646 | 724 | (math-besJ0 x t)))) |
136211a9 EZ |
725 | |
726 | (defun math-besY1 (x) | |
727 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
728 | (let ((y (math-sqr x))) | |
729 | (math-add | |
730 | (math-mul | |
731 | x | |
732 | (math-div (math-poly-eval y | |
c6254f53 JB |
733 | (list |
734 | (math-read-number-simple "8511.937935") | |
735 | (math-read-number-simple "-4237922.726") | |
736 | (math-read-number-simple "734926455.1") | |
737 | (math-read-number-simple "-51534381390.0") | |
738 | (math-read-number-simple "1275274390000.0") | |
739 | (math-read-number-simple "-4900604943000.0"))) | |
136211a9 | 740 | (math-poly-eval y |
c6254f53 JB |
741 | (list |
742 | '(float 1 0) | |
743 | (math-read-number-simple "354.9632885") | |
744 | (math-read-number-simple "102042.605") | |
745 | (math-read-number-simple "22459040.02") | |
746 | (math-read-number-simple "3733650367.0") | |
747 | (math-read-number-simple "424441966400.0") | |
748 | (math-read-number-simple "24995805700000.0"))))) | |
749 | (math-mul (math-read-number-simple "0.636619772") | |
a8775b6d | 750 | (math-sub (math-mul (math-besJ1 x) (math-ln-raw x)) |
136211a9 EZ |
751 | (math-div 1 x)))))) |
752 | ((math-negp (calcFunc-re x)) | |
753 | (math-neg | |
754 | (math-add (math-besJ1 (math-neg x) t) | |
755 | (math-mul '(cplx 0 2) | |
756 | (math-besJ1 (math-neg x)))))) | |
757 | (t | |
bf77c646 | 758 | (math-besJ1 x t)))) |
136211a9 EZ |
759 | |
760 | (defun math-poly-eval (x coefs) | |
761 | (let ((accum (car coefs))) | |
762 | (while (setq coefs (cdr coefs)) | |
763 | (setq accum (math-add (car coefs) (math-mul accum x)))) | |
bf77c646 | 764 | accum)) |
136211a9 EZ |
765 | |
766 | ||
767 | ;;;; Bernoulli and Euler polynomials and numbers. | |
768 | ||
769 | (defun calcFunc-bern (n &optional x) | |
770 | (if (and x (not (math-zerop x))) | |
771 | (if (and calc-symbolic-mode (math-floatp x)) | |
772 | (math-inexact-result) | |
773 | (math-build-polynomial-expr (math-bernoulli-coefs n) x)) | |
774 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
775 | (if (consp n) | |
776 | (progn | |
777 | (math-inexact-result) | |
778 | (math-float (math-bernoulli-number (math-trunc n)))) | |
bf77c646 | 779 | (math-bernoulli-number n)))) |
136211a9 EZ |
780 | |
781 | (defun calcFunc-euler (n &optional x) | |
782 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
783 | (if x | |
784 | (let* ((n1 (math-add n 1)) | |
785 | (coefs (math-bernoulli-coefs n1)) | |
786 | (fac (math-div (math-pow 2 n1) n1)) | |
787 | (k -1) | |
788 | (x1 (math-div (math-add x 1) 2)) | |
789 | (x2 (math-div x 2))) | |
790 | (if (math-numberp x) | |
791 | (if (and calc-symbolic-mode (math-floatp x)) | |
792 | (math-inexact-result) | |
793 | (math-mul fac | |
794 | (math-sub (math-build-polynomial-expr coefs x1) | |
795 | (math-build-polynomial-expr coefs x2)))) | |
796 | (calcFunc-collect | |
797 | (math-reduce-vec | |
798 | 'math-add | |
799 | (cons 'vec | |
800 | (mapcar (function | |
801 | (lambda (c) | |
802 | (setq k (1+ k)) | |
803 | (math-mul (math-mul fac c) | |
804 | (math-sub (math-pow x1 k) | |
805 | (math-pow x2 k))))) | |
806 | coefs))) | |
807 | x))) | |
808 | (math-mul (math-pow 2 n) | |
809 | (if (consp n) | |
810 | (progn | |
811 | (math-inexact-result) | |
812 | (calcFunc-euler n '(float 5 -1))) | |
bf77c646 | 813 | (calcFunc-euler n '(frac 1 2)))))) |
136211a9 | 814 | |
c6254f53 JB |
815 | (defvar math-bernoulli-b-cache |
816 | (list | |
817 | (list 'frac | |
818 | -174611 | |
819 | (math-read-number-simple "802857662698291200000")) | |
820 | (list 'frac | |
821 | 43867 | |
822 | (math-read-number-simple "5109094217170944000")) | |
823 | (list 'frac | |
824 | -3617 | |
825 | (math-read-number-simple "10670622842880000")) | |
826 | (list 'frac | |
827 | 1 | |
828 | (math-read-number-simple "74724249600")) | |
829 | (list 'frac | |
830 | -691 | |
831 | (math-read-number-simple "1307674368000")) | |
832 | (list 'frac | |
833 | 1 | |
834 | (math-read-number-simple "47900160")) | |
835 | (list 'frac | |
836 | -1 | |
837 | (math-read-number-simple "1209600")) | |
838 | (list 'frac | |
839 | 1 | |
840 | 30240) | |
841 | (list 'frac | |
842 | -1 | |
843 | 720) | |
844 | (list 'frac | |
845 | 1 | |
846 | 12) | |
847 | 1 )) | |
848 | ||
849 | (defvar math-bernoulli-B-cache | |
850 | '((frac -174611 330) (frac 43867 798) | |
851 | (frac -3617 510) (frac 7 6) (frac -691 2730) | |
852 | (frac 5 66) (frac -1 30) (frac 1 42) | |
853 | (frac -1 30) (frac 1 6) 1 )) | |
12f1951a CW |
854 | |
855 | (defvar math-bernoulli-cache-size 11) | |
136211a9 EZ |
856 | (defun math-bernoulli-coefs (n) |
857 | (let* ((coefs (list (calcFunc-bern n))) | |
858 | (nn (math-trunc n)) | |
859 | (k nn) | |
860 | (term nn) | |
861 | coef | |
862 | (calc-prefer-frac (or (integerp n) calc-prefer-frac))) | |
863 | (while (>= (setq k (1- k)) 0) | |
864 | (setq term (math-div term (- nn k)) | |
865 | coef (math-mul term (math-bernoulli-number k)) | |
866 | coefs (cons (if (consp n) (math-float coef) coef) coefs) | |
867 | term (math-mul term k))) | |
bf77c646 | 868 | (nreverse coefs))) |
136211a9 EZ |
869 | |
870 | (defun math-bernoulli-number (n) | |
871 | (if (= (% n 2) 1) | |
872 | (if (= n 1) | |
873 | '(frac -1 2) | |
874 | 0) | |
875 | (setq n (/ n 2)) | |
876 | (while (>= n math-bernoulli-cache-size) | |
877 | (let* ((sum 0) | |
878 | (nk 1) ; nk = n-k+1 | |
879 | (fact 1) ; fact = (n-k+1)! | |
880 | ofact | |
881 | (p math-bernoulli-b-cache) | |
882 | (calc-prefer-frac t)) | |
883 | (math-working "bernoulli B" (* 2 math-bernoulli-cache-size)) | |
884 | (while p | |
885 | (setq nk (+ nk 2) | |
886 | ofact fact | |
887 | fact (math-mul fact (* nk (1- nk))) | |
888 | sum (math-add sum (math-div (car p) fact)) | |
889 | p (cdr p))) | |
890 | (setq ofact (math-mul ofact (1- nk)) | |
891 | sum (math-sub (math-div '(frac 1 2) ofact) sum) | |
892 | math-bernoulli-b-cache (cons sum math-bernoulli-b-cache) | |
893 | math-bernoulli-B-cache (cons (math-mul sum ofact) | |
894 | math-bernoulli-B-cache) | |
895 | math-bernoulli-cache-size (1+ math-bernoulli-cache-size)))) | |
bf77c646 | 896 | (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache))) |
136211a9 EZ |
897 | |
898 | ;;; Bn = n! bn | |
899 | ;;; bn = - sum_k=0^n-1 bk / (n-k+1)! | |
900 | ||
901 | ;;; A faster method would be to use "tangent numbers", c.f., Concrete | |
902 | ;;; Mathematics pg. 273. | |
903 | ||
136211a9 EZ |
904 | |
905 | ;;; Probability distributions. | |
906 | ||
907 | ;;; Binomial. | |
908 | (defun calcFunc-utpb (x n p) | |
909 | (if math-expand-formulas | |
910 | (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1))) | |
bf77c646 | 911 | (calcFunc-betaI p x (math-add (math-sub n x) 1)))) |
136211a9 EZ |
912 | (put 'calcFunc-utpb 'math-expandable t) |
913 | ||
914 | (defun calcFunc-ltpb (x n p) | |
bf77c646 | 915 | (math-sub 1 (calcFunc-utpb x n p))) |
136211a9 EZ |
916 | (put 'calcFunc-ltpb 'math-expandable t) |
917 | ||
918 | ;;; Chi-square. | |
919 | (defun calcFunc-utpc (chisq v) | |
920 | (if math-expand-formulas | |
921 | (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2))) | |
bf77c646 | 922 | (calcFunc-gammaQ (math-div v 2) (math-div chisq 2)))) |
136211a9 EZ |
923 | (put 'calcFunc-utpc 'math-expandable t) |
924 | ||
925 | (defun calcFunc-ltpc (chisq v) | |
926 | (if math-expand-formulas | |
927 | (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2))) | |
bf77c646 | 928 | (calcFunc-gammaP (math-div v 2) (math-div chisq 2)))) |
136211a9 EZ |
929 | (put 'calcFunc-ltpc 'math-expandable t) |
930 | ||
931 | ;;; F-distribution. | |
932 | (defun calcFunc-utpf (f v1 v2) | |
933 | (if math-expand-formulas | |
934 | (math-normalize (list 'calcFunc-betaI | |
935 | (list '/ v2 (list '+ v2 (list '* v1 f))) | |
936 | (list '/ v2 2) | |
937 | (list '/ v1 2))) | |
938 | (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f))) | |
939 | (math-div v2 2) | |
bf77c646 | 940 | (math-div v1 2)))) |
136211a9 EZ |
941 | (put 'calcFunc-utpf 'math-expandable t) |
942 | ||
943 | (defun calcFunc-ltpf (f v1 v2) | |
bf77c646 | 944 | (math-sub 1 (calcFunc-utpf f v1 v2))) |
136211a9 EZ |
945 | (put 'calcFunc-ltpf 'math-expandable t) |
946 | ||
947 | ;;; Normal. | |
948 | (defun calcFunc-utpn (x mean sdev) | |
949 | (if math-expand-formulas | |
950 | (math-normalize | |
951 | (list '/ | |
952 | (list '+ 1 | |
953 | (list 'calcFunc-erf | |
954 | (list '/ (list '- mean x) | |
955 | (list '* sdev (list 'calcFunc-sqrt 2))))) | |
956 | 2)) | |
957 | (math-mul (math-add '(float 1 0) | |
958 | (calcFunc-erf | |
959 | (math-div (math-sub mean x) | |
960 | (math-mul sdev (math-sqrt-2))))) | |
bf77c646 | 961 | '(float 5 -1)))) |
136211a9 EZ |
962 | (put 'calcFunc-utpn 'math-expandable t) |
963 | ||
964 | (defun calcFunc-ltpn (x mean sdev) | |
965 | (if math-expand-formulas | |
966 | (math-normalize | |
967 | (list '/ | |
968 | (list '+ 1 | |
969 | (list 'calcFunc-erf | |
970 | (list '/ (list '- x mean) | |
971 | (list '* sdev (list 'calcFunc-sqrt 2))))) | |
972 | 2)) | |
973 | (math-mul (math-add '(float 1 0) | |
974 | (calcFunc-erf | |
975 | (math-div (math-sub x mean) | |
976 | (math-mul sdev (math-sqrt-2))))) | |
bf77c646 | 977 | '(float 5 -1)))) |
136211a9 EZ |
978 | (put 'calcFunc-ltpn 'math-expandable t) |
979 | ||
980 | ;;; Poisson. | |
981 | (defun calcFunc-utpp (n x) | |
982 | (if math-expand-formulas | |
983 | (math-normalize (list 'calcFunc-gammaP x n)) | |
bf77c646 | 984 | (calcFunc-gammaP x n))) |
136211a9 EZ |
985 | (put 'calcFunc-utpp 'math-expandable t) |
986 | ||
987 | (defun calcFunc-ltpp (n x) | |
988 | (if math-expand-formulas | |
989 | (math-normalize (list 'calcFunc-gammaQ x n)) | |
bf77c646 | 990 | (calcFunc-gammaQ x n))) |
136211a9 EZ |
991 | (put 'calcFunc-ltpp 'math-expandable t) |
992 | ||
993 | ;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.) | |
994 | (defun calcFunc-utpt (tt v) | |
995 | (if math-expand-formulas | |
996 | (math-normalize (list 'calcFunc-betaI | |
997 | (list '/ v (list '+ v (list '^ tt 2))) | |
998 | (list '/ v 2) | |
999 | '(float 5 -1))) | |
1000 | (calcFunc-betaI (math-div v (math-add v (math-sqr tt))) | |
1001 | (math-div v 2) | |
bf77c646 | 1002 | '(float 5 -1)))) |
136211a9 EZ |
1003 | (put 'calcFunc-utpt 'math-expandable t) |
1004 | ||
1005 | (defun calcFunc-ltpt (tt v) | |
bf77c646 | 1006 | (math-sub 1 (calcFunc-utpt tt v))) |
136211a9 EZ |
1007 | (put 'calcFunc-ltpt 'math-expandable t) |
1008 | ||
300f28d3 | 1009 | (provide 'calc-funcs) |
136211a9 | 1010 | |
bf77c646 | 1011 | ;;; calc-funcs.el ends here |