1 ;;; calc-funcs.el --- well-known functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
4 ;; 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
6 ;; Author: David Gillespie <daveg@synaptics.com>
7 ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
9 ;; This file is part of GNU Emacs.
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.
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.
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/>.
28 ;; This file is autoloaded from calc-ext.el.
33 (defun calc-inc-gamma (arg)
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
)))))
48 (calc-unary-op "erfc" 'calcFunc-erfc arg
)
49 (calc-unary-op "erf" 'calcFunc-erf arg
))))
51 (defun calc-erfc (arg)
56 (defun calc-beta (arg)
59 (calc-binary-op "beta" 'calcFunc-beta arg
)))
61 (defun calc-inc-beta ()
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))))))
68 (defun calc-bessel-J (arg)
71 (calc-binary-op "besJ" 'calcFunc-besJ arg
)))
73 (defun calc-bessel-Y (arg)
76 (calc-binary-op "besY" 'calcFunc-besY arg
)))
78 (defun calc-bernoulli-number (arg)
81 (if (calc-is-hyperbolic)
82 (calc-binary-op "bern" 'calcFunc-bern arg
)
83 (calc-unary-op "bern" 'calcFunc-bern arg
))))
85 (defun calc-euler-number (arg)
88 (if (calc-is-hyperbolic)
89 (calc-binary-op "eulr" 'calcFunc-euler arg
)
90 (calc-unary-op "eulr" 'calcFunc-euler arg
))))
92 (defun calc-stirling-number (arg)
95 (if (calc-is-hyperbolic)
96 (calc-binary-op "str2" 'calcFunc-stir2 arg
)
97 (calc-binary-op "str1" 'calcFunc-stir1 arg
))))
101 (calc-prob-dist "b" 3))
105 (calc-prob-dist "c" 2))
109 (calc-prob-dist "f" 3))
113 (calc-prob-dist "n" 3))
117 (calc-prob-dist "p" 2))
121 (calc-prob-dist "t" 2))
123 (defun calc-prob-dist (letter nargs
)
125 (if (calc-is-inverse)
126 (calc-enter-result nargs
(concat "ltp" letter
)
127 (append (list (intern (concat "calcFunc-ltp" letter
))
129 (calc-top-list-n (1- nargs
) 2)))
130 (calc-enter-result nargs
(concat "utp" letter
)
131 (append (list (intern (concat "calcFunc-utp" letter
))
133 (calc-top-list-n (1- nargs
) 2))))))
138 ;;; Sources: Numerical Recipes, Press et al;
139 ;;; Handbook of Mathematical Functions, Abramowitz & Stegun.
144 (defun calcFunc-gamma (x)
145 (or (math-numberp x
) (math-reject-arg x
'numberp
))
146 (calcFunc-fact (math-add x -
1)))
148 (defun math-gammap1-raw (x &optional fprec nfprec
)
149 "Compute gamma(1+X) to the appropriate precision."
151 (setq fprec
(math-float calc-internal-prec
)
152 nfprec
(math-float (- calc-internal-prec
))))
153 (cond ((math-lessp-float (calcFunc-re x
) fprec
)
154 (if (math-lessp-float (calcFunc-re x
) nfprec
)
157 (math-mul (math-gammap1-raw
158 (math-add (math-neg x
)
162 (math-mul (math-pi) x
)))))
163 (let ((xplus1 (math-add x
'(float 1 0))))
164 (math-div (math-gammap1-raw xplus1 fprec nfprec
) xplus1
))))
166 (math-lessp-float '(float 736276 0) x
))
168 (t ; re(x) now >= 10.0
169 (let ((xinv (math-div 1 x
))
170 (lnx (math-ln-raw x
)))
171 (math-mul (math-sqrt-two-pi)
174 (math-sub (math-mul (math-add x
'(float 5 -
1))
182 (defun math-gamma-series (sum x xinvsqr oterm n
)
183 (math-working "gamma" sum
)
184 (let* ((bn (math-bernoulli-number n
))
185 (term (math-mul (math-div-float (math-float (nth 1 bn
))
186 (math-float (* (nth 2 bn
)
189 (next (math-add sum term
)))
190 (if (math-nearly-equal sum next
)
192 (if (> n
(* 2 calc-internal-prec
))
194 ;; Need this because series eventually diverges for large enough n.
196 "*Gamma computation stopped early, not all digits may be valid")
198 (math-gamma-series next
(math-mul x xinvsqr
) xinvsqr term
(+ n
2))))))
201 ;;; Incomplete gamma function.
203 (defvar math-current-gamma-value nil
)
204 (defun calcFunc-gammaP (a x
)
205 (if (equal x
'(var inf var-inf
))
207 (math-inexact-result)
208 (or (Math-numberp a
) (math-reject-arg a
'numberp
))
209 (or (math-numberp x
) (math-reject-arg x
'numberp
))
210 (if (and (math-num-integerp a
)
211 (integerp (setq a
(math-trunc a
)))
213 (math-sub 1 (calcFunc-gammaQ a x
))
214 (let ((math-current-gamma-value (calcFunc-gamma a
)))
215 (math-div (calcFunc-gammag a x
) math-current-gamma-value
)))))
217 (defun calcFunc-gammaQ (a x
)
218 (if (equal x
'(var inf var-inf
))
220 (math-inexact-result)
221 (or (Math-numberp a
) (math-reject-arg a
'numberp
))
222 (or (math-numberp x
) (math-reject-arg x
'numberp
))
223 (if (and (math-num-integerp a
)
224 (integerp (setq a
(math-trunc a
)))
229 (math-with-extra-prec 1
230 (while (< (setq n
(1+ n
)) a
)
231 (setq term
(math-div (math-mul term x
) n
)
232 sum
(math-add sum term
))
233 (math-working "gamma" sum
))
234 (math-mul sum
(calcFunc-exp (math-neg x
)))))
235 (let ((math-current-gamma-value (calcFunc-gamma a
)))
236 (math-div (calcFunc-gammaG a x
) math-current-gamma-value
)))))
238 (defun calcFunc-gammag (a x
)
239 (if (equal x
'(var inf var-inf
))
241 (math-inexact-result)
242 (or (Math-numberp a
) (math-reject-arg a
'numberp
))
243 (or (Math-numberp x
) (math-reject-arg x
'numberp
))
244 (math-with-extra-prec 2
245 (setq a
(math-float a
))
246 (setq x
(math-float x
))
247 (if (or (math-negp (calcFunc-re a
))
248 (math-lessp-float (calcFunc-re x
)
249 (math-add-float (calcFunc-re a
)
251 (math-inc-gamma-series a x
)
252 (math-sub (or math-current-gamma-value
(calcFunc-gamma a
))
253 (math-inc-gamma-cfrac a x
))))))
255 (defun calcFunc-gammaG (a x
)
256 (if (equal x
'(var inf var-inf
))
258 (math-inexact-result)
259 (or (Math-numberp a
) (math-reject-arg a
'numberp
))
260 (or (Math-numberp x
) (math-reject-arg x
'numberp
))
261 (math-with-extra-prec 2
262 (setq a
(math-float a
))
263 (setq x
(math-float x
))
264 (if (or (math-negp (calcFunc-re a
))
265 (math-lessp-float (calcFunc-re x
)
266 (math-add-float (math-abs-approx a
)
268 (math-sub (or math-current-gamma-value
(calcFunc-gamma a
))
269 (math-inc-gamma-series a x
))
270 (math-inc-gamma-cfrac a x
)))))
272 (defun math-inc-gamma-series (a x
)
275 (math-mul (math-exp-raw (math-sub (math-mul a
(math-ln-raw x
)) x
))
276 (math-with-extra-prec 2
277 (let ((start (math-div '(float 1 0) a
)))
278 (math-inc-gamma-series-step start start a x
))))))
280 (defun math-inc-gamma-series-step (sum term a x
)
281 (math-working "gamma" sum
)
282 (setq a
(math-add a
'(float 1 0))
283 term
(math-div (math-mul term x
) a
))
284 (let ((next (math-add sum term
)))
285 (if (math-nearly-equal sum next
)
287 (math-inc-gamma-series-step next term a x
))))
289 (defun math-inc-gamma-cfrac (a x
)
291 (or math-current-gamma-value
(calcFunc-gamma a
))
292 (math-mul (math-exp-raw (math-sub (math-mul a
(math-ln-raw x
)) x
))
293 (math-inc-gamma-cfrac-step '(float 1 0) x
294 '(float 0 0) '(float 1 0)
295 '(float 1 0) '(float 1 0) '(float 0 0)
298 (defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x
)
299 (let ((ana (math-sub n a
))
300 (anf (math-mul n fac
)))
301 (setq n
(math-add n
'(float 1 0))
302 a0
(math-mul (math-add a1
(math-mul a0 ana
)) fac
)
303 b0
(math-mul (math-add b1
(math-mul b0 ana
)) fac
)
304 a1
(math-add (math-mul x a0
) (math-mul anf a1
))
305 b1
(math-add (math-mul x b0
) (math-mul anf b1
)))
307 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x
)
308 (setq fac
(math-div '(float 1 0) a1
))
309 (let ((next (math-mul b1 fac
)))
310 (math-working "gamma" next
)
311 (if (math-nearly-equal next g
)
313 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x
))))))
318 (defun calcFunc-erf (x)
319 (if (equal x
'(var inf var-inf
))
321 (if (equal x
'(neg (var inf var-inf
)))
325 (let ((math-current-gamma-value (math-sqrt-pi)))
326 (math-to-same-complex-quad
327 (math-div (calcFunc-gammag '(float 5 -
1)
328 (math-sqr (math-to-complex-quad-one x
)))
329 math-current-gamma-value
)
332 (defun calcFunc-erfc (x)
333 (if (equal x
'(var inf var-inf
))
336 (let ((math-current-gamma-value (math-sqrt-pi)))
337 (math-div (calcFunc-gammaG '(float 5 -
1) (math-sqr x
))
338 math-current-gamma-value
))
339 (math-sub 1 (calcFunc-erf x
)))))
341 (defun math-to-complex-quad-one (x)
342 (if (eq (car-safe x
) 'polar
) (setq x
(math-complex x
)))
343 (if (eq (car-safe x
) 'cplx
)
344 (list 'cplx
(math-abs (nth 1 x
)) (math-abs (nth 2 x
)))
347 (defun math-to-same-complex-quad (x y
)
348 (if (eq (car-safe y
) 'cplx
)
349 (if (eq (car-safe x
) 'cplx
)
351 (if (math-negp (nth 1 y
)) (math-neg (nth 1 x
)) (nth 1 x
))
352 (if (math-negp (nth 2 y
)) (math-neg (nth 2 x
)) (nth 2 x
)))
353 (if (math-negp (nth 1 y
)) (math-neg x
) x
))
355 (if (eq (car-safe x
) 'cplx
)
356 (list 'cplx
(math-neg (nth 1 x
)) (nth 2 x
))
363 (defun calcFunc-beta (a b
)
364 (if (math-num-integerp a
)
365 (let ((am (math-add a -
1)))
366 (or (math-numberp b
) (math-reject-arg b
'numberp
))
367 (math-div 1 (math-mul b
(calcFunc-choose (math-add b am
) am
))))
368 (if (math-num-integerp b
)
370 (math-div (math-mul (calcFunc-gamma a
) (calcFunc-gamma b
))
371 (calcFunc-gamma (math-add a b
))))))
374 ;;; Incomplete beta function.
376 (defvar math-current-beta-value nil
)
377 (defun calcFunc-betaI (x a b
)
378 (cond ((math-zerop x
)
380 ((math-equal-int x
1)
383 (and (math-num-integerp a
)
385 (if (or (math-zerop b
)
386 (and (math-num-integerp b
)
388 (math-reject-arg b
'range
)
391 (and (math-num-integerp b
)
394 ((not (math-numberp a
)) (math-reject-arg a
'numberp
))
395 ((not (math-numberp b
)) (math-reject-arg b
'numberp
))
396 ((math-inexact-result))
397 (t (let ((math-current-beta-value (calcFunc-beta a b
)))
398 (math-div (calcFunc-betaB x a b
) math-current-beta-value
)))))
400 (defun calcFunc-betaB (x a b
)
404 ((math-equal-int x
1)
406 ((not (math-numberp x
)) (math-reject-arg x
'numberp
))
407 ((not (math-numberp a
)) (math-reject-arg a
'numberp
))
408 ((not (math-numberp b
)) (math-reject-arg b
'numberp
))
409 ((math-zerop a
) (math-reject-arg a
'nonzerop
))
410 ((math-zerop b
) (math-reject-arg b
'nonzerop
))
411 ((and (math-num-integerp b
)
413 (math-reject-arg b
'range
)
414 (Math-natnum-lessp (setq b
(math-trunc b
)) 20)))
415 (and calc-symbolic-mode
(or (math-floatp a
) (math-floatp b
))
416 (math-inexact-result))
418 (math-with-extra-prec 2
421 (sum (math-div term a
)))
422 (while (< (setq i
(1+ i
)) b
)
423 (setq term
(math-mul (math-div (math-mul term
(- i b
)) i
) x
)
424 sum
(math-add sum
(math-div term
(math-add a i
))))
425 (math-working "beta" sum
))
428 ((and (math-num-integerp a
)
430 (math-reject-arg a
'range
)
431 (Math-natnum-lessp (setq a
(math-trunc a
)) 20)))
432 (math-sub (or math-current-beta-value
(calcFunc-beta a b
))
433 (calcFunc-betaB (math-sub 1 x
) b a
)))
435 (math-inexact-result)
436 (math-with-extra-prec 2
437 (setq x
(math-float x
))
438 (setq a
(math-float a
))
439 (setq b
(math-float b
))
440 (let ((bt (math-exp-raw (math-add (math-mul a
(math-ln-raw x
))
441 (math-mul b
(math-ln-raw
442 (math-sub '(float 1 0)
444 (if (Math-lessp x
(math-div (math-add a
'(float 1 0))
445 (math-add (math-add a b
) '(float 2 0))))
446 (math-div (math-mul bt
(math-beta-cfrac a b x
)) a
)
447 (math-sub (or math-current-beta-value
(calcFunc-beta a b
))
448 (math-div (math-mul bt
449 (math-beta-cfrac b a
(math-sub 1 x
)))
452 (defun math-beta-cfrac (a b x
)
453 (let ((qab (math-add a b
))
454 (qap (math-add a
'(float 1 0)))
455 (qam (math-add a
'(float -
1 0))))
456 (math-beta-cfrac-step '(float 1 0)
457 (math-sub '(float 1 0)
458 (math-div (math-mul qab x
) qap
))
459 '(float 1 0) '(float 1 0)
463 (defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x
)
464 (let* ((two-m (math-mul m
'(float 2 0)))
465 (d (math-div (math-mul (math-mul (math-sub b m
) m
) x
)
466 (math-mul (math-add qam two-m
) (math-add a two-m
))))
467 (ap (math-add az
(math-mul d am
)))
468 (bp (math-add bz
(math-mul d bm
)))
470 (math-div (math-mul (math-mul (math-add a m
) (math-add qab m
)) x
)
471 (math-mul (math-add qap two-m
) (math-add a two-m
)))))
472 (app (math-add ap
(math-mul d2 az
)))
473 (bpp (math-add bp
(math-mul d2 bz
)))
474 (next (math-div app bpp
)))
475 (math-working "beta" next
)
476 (if (math-nearly-equal next az
)
478 (math-beta-cfrac-step next
'(float 1 0)
479 (math-div ap bpp
) (math-div bp bpp
)
480 (math-add m
'(float 1 0))
481 qab qap qam a b x
))))
484 ;;; Bessel functions.
486 ;;; Should generalize this to handle arbitrary precision!
488 (defun calcFunc-besJ (v x
)
489 (or (math-numberp v
) (math-reject-arg v
'numberp
))
490 (or (math-numberp x
) (math-reject-arg x
'numberp
))
491 (let ((calc-internal-prec (min 8 calc-internal-prec
)))
492 (math-with-extra-prec 3
493 (setq x
(math-float (math-normalize x
)))
494 (setq v
(math-float (math-normalize v
)))
495 (cond ((math-zerop x
)
499 ((math-inexact-result))
500 ((not (math-num-integerp v
))
501 (let ((start (math-div 1 (calcFunc-fact v
))))
502 (math-mul (math-besJ-series start start
504 (math-mul '(float -
25 -
2)
507 (math-pow (math-div x
2) v
))))
508 ((math-negp (setq v
(math-trunc v
)))
510 (math-neg (calcFunc-besJ (math-neg v
) x
))
511 (calcFunc-besJ (math-neg v
) x
)))
516 ((Math-lessp v
(math-abs-approx x
))
520 (two-over-x (math-div 2 x
))
522 (while (< (setq j
(1+ j
)) v
)
523 (setq bjp
(math-sub (math-mul (math-mul j two-over-x
) bj
)
529 (if (Math-lessp 100 v
) (math-reject-arg v
'range
))
530 (let* ((j (logior (+ v
(math-isqrt-small (* 40 v
))) 1))
531 (two-over-x (math-div 2 x
))
537 (while (> (setq j
(1- j
)) 0)
538 (setq bjm
(math-sub (math-mul (math-mul j two-over-x
) bj
)
542 (if (> (nth 2 (math-abs-approx bj
)) 10)
543 (setq bj
(math-mul bj
'(float 1 -
10))
544 bjp
(math-mul bjp
'(float 1 -
10))
545 ans
(and ans
(math-mul ans
'(float 1 -
10)))
546 sum
(math-mul sum
'(float 1 -
10))))
547 (or (setq jsum
(not jsum
))
548 (setq sum
(math-add sum bj
)))
551 (math-div ans
(math-sub (math-mul 2 sum
) bj
))))))))
553 (defun math-besJ-series (sum term k zz vk
)
554 (math-working "besJ" sum
)
557 term
(math-div (math-mul term zz
) (math-mul k vk
)))
558 (let ((next (math-add sum term
)))
559 (if (math-nearly-equal next sum
)
561 (math-besJ-series next term k zz vk
))))
563 (defun math-besJ0 (x &optional yflag
)
564 (cond ((and (not yflag
) (math-negp (calcFunc-re x
)))
565 (math-besJ0 (math-neg x
)))
566 ((Math-lessp '(float 8 0) (math-abs-approx x
))
567 (let* ((z (math-div '(float 8 0) x
))
570 (math-read-number-simple "-0.785398164")))
571 (a1 (math-poly-eval y
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")
578 (a2 (math-poly-eval y
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
)))
587 (setq sc
(cons (math-neg (cdr sc
)) (car sc
))))
589 (math-div (math-read-number-simple "0.636619722")
591 (math-sub (math-mul (cdr sc
) a1
)
592 (math-mul (car sc
) (math-mul z a2
))))))
594 (let ((y (math-sqr x
)))
595 (math-div (math-poly-eval y
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")))
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"))))))))
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
))
618 (xx (math-add x
(math-read-number-simple "-2.356194491")))
619 (a1 (math-poly-eval y
621 (math-read-number-simple "-0.000000240337019")
622 (math-read-number-simple "0.000002457520174")
623 (math-read-number-simple "-0.00003516396496")
626 (a2 (math-poly-eval y
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
)))
635 (setq sc
(cons (math-neg (cdr sc
)) (car sc
)))
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")
641 (math-sub (math-mul (cdr sc
) a1
)
642 (math-mul (car sc
) (math-mul z a2
))))))
644 (let ((y (math-sqr x
)))
647 (math-div (math-poly-eval y
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")))
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")))))))))
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
))
677 ((math-negp (setq v
(math-trunc v
)))
679 (math-neg (calcFunc-besY (math-neg v
) x
))
680 (calcFunc-besY (math-neg v
) x
)))
689 (two-over-x (math-div 2 x
))
691 (while (< (setq j
(1+ j
)) v
)
692 (setq byp
(math-sub (math-mul (math-mul j two-over-x
) by
)
698 (defun math-besY0 (x)
699 (cond ((Math-lessp (math-abs-approx x
) '(float 8 0))
700 (let ((y (math-sqr x
)))
702 (math-div (math-poly-eval y
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")))
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
)))))
727 (defun math-besY1 (x)
728 (cond ((Math-lessp (math-abs-approx x
) '(float 8 0))
729 (let ((y (math-sqr x
)))
733 (math-div (math-poly-eval y
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")))
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
))
753 ((math-negp (calcFunc-re x
))
755 (math-add (math-besJ1 (math-neg x
) t
)
756 (math-mul '(cplx 0 2)
757 (math-besJ1 (math-neg x
))))))
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
))))
768 ;;;; Bernoulli and Euler polynomials and numbers.
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
))
778 (math-inexact-result)
779 (math-float (math-bernoulli-number (math-trunc n
))))
780 (math-bernoulli-number n
))))
782 (defun calcFunc-euler (n &optional x
)
783 (or (math-num-natnump n
) (math-reject-arg n
'natnump
))
785 (let* ((n1 (math-add n
1))
786 (coefs (math-bernoulli-coefs n1
))
787 (fac (math-div (math-pow 2 n1
) n1
))
789 (x1 (math-div (math-add x
1) 2))
792 (if (and calc-symbolic-mode
(math-floatp x
))
793 (math-inexact-result)
795 (math-sub (math-build-polynomial-expr coefs x1
)
796 (math-build-polynomial-expr coefs x2
))))
804 (math-mul (math-mul fac c
)
805 (math-sub (math-pow x1 k
)
809 (math-mul (math-pow 2 n
)
812 (math-inexact-result)
813 (calcFunc-euler n
'(float 5 -
1)))
814 (calcFunc-euler n
'(frac 1 2))))))
816 (defvar math-bernoulli-b-cache
820 (math-read-number-simple "802857662698291200000"))
823 (math-read-number-simple "5109094217170944000"))
826 (math-read-number-simple "10670622842880000"))
829 (math-read-number-simple "74724249600"))
832 (math-read-number-simple "1307674368000"))
835 (math-read-number-simple "47900160"))
838 (math-read-number-simple "1209600"))
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 ))
856 (defvar math-bernoulli-cache-size
11)
857 (defun math-bernoulli-coefs (n)
858 (let* ((coefs (list (calcFunc-bern n
)))
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
)))
871 (defun math-bernoulli-number (n)
877 (while (>= n math-bernoulli-cache-size
)
880 (fact 1) ; fact = (n-k+1)!
882 (p math-bernoulli-b-cache
)
883 (calc-prefer-frac t
))
884 (math-working "bernoulli B" (* 2 math-bernoulli-cache-size
))
888 fact
(math-mul fact
(* nk
(1- nk
)))
889 sum
(math-add sum
(math-div (car p
) fact
))
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
)))
900 ;;; bn = - sum_k=0^n-1 bk / (n-k+1)!
902 ;;; A faster method would be to use "tangent numbers", c.f., Concrete
903 ;;; Mathematics pg. 273.
906 ;;; Probability distributions.
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
)
915 (defun calcFunc-ltpb (x n p
)
916 (math-sub 1 (calcFunc-utpb x n p
)))
917 (put 'calcFunc-ltpb
'math-expandable t
)
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
)
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
)
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
)))
939 (calcFunc-betaI (math-div v2
(math-add v2
(math-mul v1 f
)))
942 (put 'calcFunc-utpf
'math-expandable t
)
944 (defun calcFunc-ltpf (f v1 v2
)
945 (math-sub 1 (calcFunc-utpf f v1 v2
)))
946 (put 'calcFunc-ltpf
'math-expandable t
)
949 (defun calcFunc-utpn (x mean sdev
)
950 (if math-expand-formulas
955 (list '/ (list '- mean x
)
956 (list '* sdev
(list 'calcFunc-sqrt
2)))))
958 (math-mul (math-add '(float 1 0)
960 (math-div (math-sub mean x
)
961 (math-mul sdev
(math-sqrt-2)))))
963 (put 'calcFunc-utpn
'math-expandable t
)
965 (defun calcFunc-ltpn (x mean sdev
)
966 (if math-expand-formulas
971 (list '/ (list '- x mean
)
972 (list '* sdev
(list 'calcFunc-sqrt
2)))))
974 (math-mul (math-add '(float 1 0)
976 (math-div (math-sub x mean
)
977 (math-mul sdev
(math-sqrt-2)))))
979 (put 'calcFunc-ltpn
'math-expandable t
)
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
)
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
)
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)))
1001 (calcFunc-betaI (math-div v
(math-add v
(math-sqr tt
)))
1004 (put 'calcFunc-utpt
'math-expandable t
)
1006 (defun calcFunc-ltpt (tt v
)
1007 (math-sub 1 (calcFunc-utpt tt v
)))
1008 (put 'calcFunc-ltpt
'math-expandable t
)
1010 (provide 'calc-funcs
)
1012 ;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a
1013 ;;; calc-funcs.el ends here