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Diffstat (limited to 'lisp/calc/calc-funcs.el')
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diff --git a/lisp/calc/calc-funcs.el b/lisp/calc/calc-funcs.el new file mode 100644 index 00000000000..90b4761a8a0 --- /dev/null +++ b/lisp/calc/calc-funcs.el @@ -0,0 +1,1034 @@ +;; Calculator for GNU Emacs, part II [calc-funcs.el] +;; Copyright (C) 1990, 1991, 1992, 1993 Free Software Foundation, Inc. +;; Written by Dave Gillespie, daveg@synaptics.com. + +;; This file is part of GNU Emacs. + +;; GNU Emacs is distributed in the hope that it will be useful, +;; but WITHOUT ANY WARRANTY. No author or distributor +;; accepts responsibility to anyone for the consequences of using it +;; or for whether it serves any particular purpose or works at all, +;; unless he says so in writing. Refer to the GNU Emacs General Public +;; License for full details. + +;; Everyone is granted permission to copy, modify and redistribute +;; GNU Emacs, but only under the conditions described in the +;; GNU Emacs General Public License. A copy of this license is +;; supposed to have been given to you along with GNU Emacs so you +;; can know your rights and responsibilities. It should be in a +;; file named COPYING. Among other things, the copyright notice +;; and this notice must be preserved on all copies. + + + +;; This file is autoloaded from calc-ext.el. +(require 'calc-ext) + +(require 'calc-macs) + +(defun calc-Need-calc-funcs () nil) + + +(defun calc-inc-gamma (arg) + (interactive "P") + (calc-slow-wrapper + (if (calc-is-inverse) + (if (calc-is-hyperbolic) + (calc-binary-op "gamG" 'calcFunc-gammaG arg) + (calc-binary-op "gamQ" 'calcFunc-gammaQ arg)) + (if (calc-is-hyperbolic) + (calc-binary-op "gamg" 'calcFunc-gammag arg) + (calc-binary-op "gamP" 'calcFunc-gammaP arg)))) +) + +(defun calc-erf (arg) + (interactive "P") + (calc-slow-wrapper + (if (calc-is-inverse) + (calc-unary-op "erfc" 'calcFunc-erfc arg) + (calc-unary-op "erf" 'calcFunc-erf arg))) +) + +(defun calc-erfc (arg) + (interactive "P") + (calc-invert-func) + (calc-erf arg) +) + +(defun calc-beta (arg) + (interactive "P") + (calc-slow-wrapper + (calc-binary-op "beta" 'calcFunc-beta arg)) +) + +(defun calc-inc-beta () + (interactive) + (calc-slow-wrapper + (if (calc-is-hyperbolic) + (calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3))) + (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3))))) +) + +(defun calc-bessel-J (arg) + (interactive "P") + (calc-slow-wrapper + (calc-binary-op "besJ" 'calcFunc-besJ arg)) +) + +(defun calc-bessel-Y (arg) + (interactive "P") + (calc-slow-wrapper + (calc-binary-op "besY" 'calcFunc-besY arg)) +) + +(defun calc-bernoulli-number (arg) + (interactive "P") + (calc-slow-wrapper + (if (calc-is-hyperbolic) + (calc-binary-op "bern" 'calcFunc-bern arg) + (calc-unary-op "bern" 'calcFunc-bern arg))) +) + +(defun calc-euler-number (arg) + (interactive "P") + (calc-slow-wrapper + (if (calc-is-hyperbolic) + (calc-binary-op "eulr" 'calcFunc-euler arg) + (calc-unary-op "eulr" 'calcFunc-euler arg))) +) + +(defun calc-stirling-number (arg) + (interactive "P") + (calc-slow-wrapper + (if (calc-is-hyperbolic) + (calc-binary-op "str2" 'calcFunc-stir2 arg) + (calc-binary-op "str1" 'calcFunc-stir1 arg))) +) + +(defun calc-utpb () + (interactive) + (calc-prob-dist "b" 3) +) + +(defun calc-utpc () + (interactive) + (calc-prob-dist "c" 2) +) + +(defun calc-utpf () + (interactive) + (calc-prob-dist "f" 3) +) + +(defun calc-utpn () + (interactive) + (calc-prob-dist "n" 3) +) + +(defun calc-utpp () + (interactive) + (calc-prob-dist "p" 2) +) + +(defun calc-utpt () + (interactive) + (calc-prob-dist "t" 2) +) + +(defun calc-prob-dist (letter nargs) + (calc-slow-wrapper + (if (calc-is-inverse) + (calc-enter-result nargs (concat "ltp" letter) + (append (list (intern (concat "calcFunc-ltp" letter)) + (calc-top-n 1)) + (calc-top-list-n (1- nargs) 2))) + (calc-enter-result nargs (concat "utp" letter) + (append (list (intern (concat "calcFunc-utp" letter)) + (calc-top-n 1)) + (calc-top-list-n (1- nargs) 2))))) +) + + + + +;;; Sources: Numerical Recipes, Press et al; +;;; Handbook of Mathematical Functions, Abramowitz & Stegun. + + +;;; Gamma function. + +(defun calcFunc-gamma (x) + (or (math-numberp x) (math-reject-arg x 'numberp)) + (calcFunc-fact (math-add x -1)) +) + +(defun math-gammap1-raw (x &optional fprec nfprec) ; compute gamma(1 + x) + (or fprec + (setq fprec (math-float calc-internal-prec) + nfprec (math-float (- calc-internal-prec)))) + (cond ((math-lessp-float (calcFunc-re x) fprec) + (if (math-lessp-float (calcFunc-re x) nfprec) + (math-neg (math-div + (math-pi) + (math-mul (math-gammap1-raw + (math-add (math-neg x) + '(float -1 0)) + fprec nfprec) + (math-sin-raw + (math-mul (math-pi) x))))) + (let ((xplus1 (math-add x '(float 1 0)))) + (math-div (math-gammap1-raw xplus1 fprec nfprec) xplus1)))) + ((and (math-realp x) + (math-lessp-float '(float 736276 0) x)) + (math-overflow)) + (t ; re(x) now >= 10.0 + (let ((xinv (math-div 1 x)) + (lnx (math-ln-raw x))) + (math-mul (math-sqrt-two-pi) + (math-exp-raw + (math-gamma-series + (math-sub (math-mul (math-add x '(float 5 -1)) + lnx) + x) + xinv + (math-sqr xinv) + '(float 0 0) + 2)))))) +) + +(defun math-gamma-series (sum x xinvsqr oterm n) + (math-working "gamma" sum) + (let* ((bn (math-bernoulli-number n)) + (term (math-mul (math-div-float (math-float (nth 1 bn)) + (math-float (* (nth 2 bn) + (* n (1- n))))) + x)) + (next (math-add sum term))) + (if (math-nearly-equal sum next) + next + (if (> n (* 2 calc-internal-prec)) + (progn + ;; Need this because series eventually diverges for large enough n. + (calc-record-why + "*Gamma computation stopped early, not all digits may be valid") + next) + (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2))))) +) + + +;;; Incomplete gamma function. + +(defun calcFunc-gammaP (a x) + (if (equal x '(var inf var-inf)) + '(float 1 0) + (math-inexact-result) + (or (Math-numberp a) (math-reject-arg a 'numberp)) + (or (math-numberp x) (math-reject-arg x 'numberp)) + (if (and (math-num-integerp a) + (integerp (setq a (math-trunc a))) + (> a 0) (< a 20)) + (math-sub 1 (calcFunc-gammaQ a x)) + (let ((math-current-gamma-value (calcFunc-gamma a))) + (math-div (calcFunc-gammag a x) math-current-gamma-value)))) +) + +(defun calcFunc-gammaQ (a x) + (if (equal x '(var inf var-inf)) + '(float 0 0) + (math-inexact-result) + (or (Math-numberp a) (math-reject-arg a 'numberp)) + (or (math-numberp x) (math-reject-arg x 'numberp)) + (if (and (math-num-integerp a) + (integerp (setq a (math-trunc a))) + (> a 0) (< a 20)) + (let ((n 0) + (sum '(float 1 0)) + (term '(float 1 0))) + (math-with-extra-prec 1 + (while (< (setq n (1+ n)) a) + (setq term (math-div (math-mul term x) n) + sum (math-add sum term)) + (math-working "gamma" sum)) + (math-mul sum (calcFunc-exp (math-neg x))))) + (let ((math-current-gamma-value (calcFunc-gamma a))) + (math-div (calcFunc-gammaG a x) math-current-gamma-value)))) +) + +(defun calcFunc-gammag (a x) + (if (equal x '(var inf var-inf)) + (calcFunc-gamma a) + (math-inexact-result) + (or (Math-numberp a) (math-reject-arg a 'numberp)) + (or (Math-numberp x) (math-reject-arg x 'numberp)) + (math-with-extra-prec 2 + (setq a (math-float a)) + (setq x (math-float x)) + (if (or (math-negp (calcFunc-re a)) + (math-lessp-float (calcFunc-re x) + (math-add-float (calcFunc-re a) + '(float 1 0)))) + (math-inc-gamma-series a x) + (math-sub (or math-current-gamma-value (calcFunc-gamma a)) + (math-inc-gamma-cfrac a x))))) +) +(setq math-current-gamma-value nil) + +(defun calcFunc-gammaG (a x) + (if (equal x '(var inf var-inf)) + '(float 0 0) + (math-inexact-result) + (or (Math-numberp a) (math-reject-arg a 'numberp)) + (or (Math-numberp x) (math-reject-arg x 'numberp)) + (math-with-extra-prec 2 + (setq a (math-float a)) + (setq x (math-float x)) + (if (or (math-negp (calcFunc-re a)) + (math-lessp-float (calcFunc-re x) + (math-add-float (math-abs-approx a) + '(float 1 0)))) + (math-sub (or math-current-gamma-value (calcFunc-gamma a)) + (math-inc-gamma-series a x)) + (math-inc-gamma-cfrac a x)))) +) + +(defun math-inc-gamma-series (a x) + (if (Math-zerop x) + '(float 0 0) + (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x)) + (math-with-extra-prec 2 + (let ((start (math-div '(float 1 0) a))) + (math-inc-gamma-series-step start start a x))))) +) + +(defun math-inc-gamma-series-step (sum term a x) + (math-working "gamma" sum) + (setq a (math-add a '(float 1 0)) + term (math-div (math-mul term x) a)) + (let ((next (math-add sum term))) + (if (math-nearly-equal sum next) + next + (math-inc-gamma-series-step next term a x))) +) + +(defun math-inc-gamma-cfrac (a x) + (if (Math-zerop x) + (or math-current-gamma-value (calcFunc-gamma a)) + (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x)) + (math-inc-gamma-cfrac-step '(float 1 0) x + '(float 0 0) '(float 1 0) + '(float 1 0) '(float 1 0) '(float 0 0) + a x))) +) + +(defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x) + (let ((ana (math-sub n a)) + (anf (math-mul n fac))) + (setq n (math-add n '(float 1 0)) + a0 (math-mul (math-add a1 (math-mul a0 ana)) fac) + b0 (math-mul (math-add b1 (math-mul b0 ana)) fac) + a1 (math-add (math-mul x a0) (math-mul anf a1)) + b1 (math-add (math-mul x b0) (math-mul anf b1))) + (if (math-zerop a1) + (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x) + (setq fac (math-div '(float 1 0) a1)) + (let ((next (math-mul b1 fac))) + (math-working "gamma" next) + (if (math-nearly-equal next g) + next + (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x))))) +) + + +;;; Error function. + +(defun calcFunc-erf (x) + (if (equal x '(var inf var-inf)) + '(float 1 0) + (if (equal x '(neg (var inf var-inf))) + '(float -1 0) + (if (Math-zerop x) + x + (let ((math-current-gamma-value (math-sqrt-pi))) + (math-to-same-complex-quad + (math-div (calcFunc-gammag '(float 5 -1) + (math-sqr (math-to-complex-quad-one x))) + math-current-gamma-value) + x))))) +) + +(defun calcFunc-erfc (x) + (if (equal x '(var inf var-inf)) + '(float 0 0) + (if (math-posp x) + (let ((math-current-gamma-value (math-sqrt-pi))) + (math-div (calcFunc-gammaG '(float 5 -1) (math-sqr x)) + math-current-gamma-value)) + (math-sub 1 (calcFunc-erf x)))) +) + +(defun math-to-complex-quad-one (x) + (if (eq (car-safe x) 'polar) (setq x (math-complex x))) + (if (eq (car-safe x) 'cplx) + (list 'cplx (math-abs (nth 1 x)) (math-abs (nth 2 x))) + x) +) + +(defun math-to-same-complex-quad (x y) + (if (eq (car-safe y) 'cplx) + (if (eq (car-safe x) 'cplx) + (list 'cplx + (if (math-negp (nth 1 y)) (math-neg (nth 1 x)) (nth 1 x)) + (if (math-negp (nth 2 y)) (math-neg (nth 2 x)) (nth 2 x))) + (if (math-negp (nth 1 y)) (math-neg x) x)) + (if (math-negp y) + (if (eq (car-safe x) 'cplx) + (list 'cplx (math-neg (nth 1 x)) (nth 2 x)) + (math-neg x)) + x)) +) + + +;;; Beta function. + +(defun calcFunc-beta (a b) + (if (math-num-integerp a) + (let ((am (math-add a -1))) + (or (math-numberp b) (math-reject-arg b 'numberp)) + (math-div 1 (math-mul b (calcFunc-choose (math-add b am) am)))) + (if (math-num-integerp b) + (calcFunc-beta b a) + (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b)) + (calcFunc-gamma (math-add a b))))) +) + + +;;; Incomplete beta function. + +(defun calcFunc-betaI (x a b) + (cond ((math-zerop x) + '(float 0 0)) + ((math-equal-int x 1) + '(float 1 0)) + ((or (math-zerop a) + (and (math-num-integerp a) + (math-negp a))) + (if (or (math-zerop b) + (and (math-num-integerp b) + (math-negp b))) + (math-reject-arg b 'range) + '(float 1 0))) + ((or (math-zerop b) + (and (math-num-integerp b) + (math-negp b))) + '(float 0 0)) + ((not (math-numberp a)) (math-reject-arg a 'numberp)) + ((not (math-numberp b)) (math-reject-arg b 'numberp)) + ((math-inexact-result)) + (t (let ((math-current-beta-value (calcFunc-beta a b))) + (math-div (calcFunc-betaB x a b) math-current-beta-value)))) +) + +(defun calcFunc-betaB (x a b) + (cond + ((math-zerop x) + '(float 0 0)) + ((math-equal-int x 1) + (calcFunc-beta a b)) + ((not (math-numberp x)) (math-reject-arg x 'numberp)) + ((not (math-numberp a)) (math-reject-arg a 'numberp)) + ((not (math-numberp b)) (math-reject-arg b 'numberp)) + ((math-zerop a) (math-reject-arg a 'nonzerop)) + ((math-zerop b) (math-reject-arg b 'nonzerop)) + ((and (math-num-integerp b) + (if (math-negp b) + (math-reject-arg b 'range) + (Math-natnum-lessp (setq b (math-trunc b)) 20))) + (and calc-symbolic-mode (or (math-floatp a) (math-floatp b)) + (math-inexact-result)) + (math-mul + (math-with-extra-prec 2 + (let* ((i 0) + (term 1) + (sum (math-div term a))) + (while (< (setq i (1+ i)) b) + (setq term (math-mul (math-div (math-mul term (- i b)) i) x) + sum (math-add sum (math-div term (math-add a i)))) + (math-working "beta" sum)) + sum)) + (math-pow x a))) + ((and (math-num-integerp a) + (if (math-negp a) + (math-reject-arg a 'range) + (Math-natnum-lessp (setq a (math-trunc a)) 20))) + (math-sub (or math-current-beta-value (calcFunc-beta a b)) + (calcFunc-betaB (math-sub 1 x) b a))) + (t + (math-inexact-result) + (math-with-extra-prec 2 + (setq x (math-float x)) + (setq a (math-float a)) + (setq b (math-float b)) + (let ((bt (math-exp-raw (math-add (math-mul a (math-ln-raw x)) + (math-mul b (math-ln-raw + (math-sub '(float 1 0) + x))))))) + (if (Math-lessp x (math-div (math-add a '(float 1 0)) + (math-add (math-add a b) '(float 2 0)))) + (math-div (math-mul bt (math-beta-cfrac a b x)) a) + (math-sub (or math-current-beta-value (calcFunc-beta a b)) + (math-div (math-mul bt + (math-beta-cfrac b a (math-sub 1 x))) + b))))))) +) +(setq math-current-beta-value nil) + +(defun math-beta-cfrac (a b x) + (let ((qab (math-add a b)) + (qap (math-add a '(float 1 0))) + (qam (math-add a '(float -1 0)))) + (math-beta-cfrac-step '(float 1 0) + (math-sub '(float 1 0) + (math-div (math-mul qab x) qap)) + '(float 1 0) '(float 1 0) + '(float 1 0) + qab qap qam a b x)) +) + +(defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x) + (let* ((two-m (math-mul m '(float 2 0))) + (d (math-div (math-mul (math-mul (math-sub b m) m) x) + (math-mul (math-add qam two-m) (math-add a two-m)))) + (ap (math-add az (math-mul d am))) + (bp (math-add bz (math-mul d bm))) + (d2 (math-neg + (math-div (math-mul (math-mul (math-add a m) (math-add qab m)) x) + (math-mul (math-add qap two-m) (math-add a two-m))))) + (app (math-add ap (math-mul d2 az))) + (bpp (math-add bp (math-mul d2 bz))) + (next (math-div app bpp))) + (math-working "beta" next) + (if (math-nearly-equal next az) + next + (math-beta-cfrac-step next '(float 1 0) + (math-div ap bpp) (math-div bp bpp) + (math-add m '(float 1 0)) + qab qap qam a b x))) +) + + +;;; Bessel functions. + +;;; Should generalize this to handle arbitrary precision! + +(defun calcFunc-besJ (v x) + (or (math-numberp v) (math-reject-arg v 'numberp)) + (or (math-numberp x) (math-reject-arg x 'numberp)) + (let ((calc-internal-prec (min 8 calc-internal-prec))) + (math-with-extra-prec 3 + (setq x (math-float (math-normalize x))) + (setq v (math-float (math-normalize v))) + (cond ((math-zerop x) + (if (math-zerop v) + '(float 1 0) + '(float 0 0))) + ((math-inexact-result)) + ((not (math-num-integerp v)) + (let ((start (math-div 1 (calcFunc-fact v)))) + (math-mul (math-besJ-series start start + 0 + (math-mul '(float -25 -2) + (math-sqr x)) + v) + (math-pow (math-div x 2) v)))) + ((math-negp (setq v (math-trunc v))) + (if (math-oddp v) + (math-neg (calcFunc-besJ (math-neg v) x)) + (calcFunc-besJ (math-neg v) x))) + ((eq v 0) + (math-besJ0 x)) + ((eq v 1) + (math-besJ1 x)) + ((Math-lessp v (math-abs-approx x)) + (let ((j 0) + (bjm (math-besJ0 x)) + (bj (math-besJ1 x)) + (two-over-x (math-div 2 x)) + bjp) + (while (< (setq j (1+ j)) v) + (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj) + bjm) + bjm bj + bj bjp)) + bj)) + (t + (if (Math-lessp 100 v) (math-reject-arg v 'range)) + (let* ((j (logior (+ v (math-isqrt-small (* 40 v))) 1)) + (two-over-x (math-div 2 x)) + (jsum nil) + (bjp '(float 0 0)) + (sum '(float 0 0)) + (bj '(float 1 0)) + bjm ans) + (while (> (setq j (1- j)) 0) + (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj) + bjp) + bjp bj + bj bjm) + (if (> (nth 2 (math-abs-approx bj)) 10) + (setq bj (math-mul bj '(float 1 -10)) + bjp (math-mul bjp '(float 1 -10)) + ans (and ans (math-mul ans '(float 1 -10))) + sum (math-mul sum '(float 1 -10)))) + (or (setq jsum (not jsum)) + (setq sum (math-add sum bj))) + (if (= j v) + (setq ans bjp))) + (math-div ans (math-sub (math-mul 2 sum) bj))))))) +) + +(defun math-besJ-series (sum term k zz vk) + (math-working "besJ" sum) + (setq k (1+ k) + vk (math-add 1 vk) + term (math-div (math-mul term zz) (math-mul k vk))) + (let ((next (math-add sum term))) + (if (math-nearly-equal next sum) + next + (math-besJ-series next term k zz vk))) +) + +(defun math-besJ0 (x &optional yflag) + (cond ((and (not yflag) (math-negp (calcFunc-re x))) + (math-besJ0 (math-neg x))) + ((Math-lessp '(float 8 0) (math-abs-approx x)) + (let* ((z (math-div '(float 8 0) x)) + (y (math-sqr z)) + (xx (math-add x '(float (bigneg 164 398 785) -9))) + (a1 (math-poly-eval y + '((float (bigpos 211 887 093 2) -16) + (float (bigneg 639 370 073 2) -15) + (float (bigpos 407 510 734 2) -14) + (float (bigneg 627 628 098 1) -12) + (float 1 0)))) + (a2 (math-poly-eval y + '((float (bigneg 152 935 934) -16) + (float (bigpos 161 095 621 7) -16) + (float (bigneg 651 147 911 6) -15) + (float (bigpos 765 488 430 1) -13) + (float (bigneg 995 499 562 1) -11)))) + (sc (math-sin-cos-raw xx))) + (if yflag + (setq sc (cons (math-neg (cdr sc)) (car sc)))) + (math-mul (math-sqrt + (math-div '(float (bigpos 722 619 636) -9) x)) + (math-sub (math-mul (cdr sc) a1) + (math-mul (car sc) (math-mul z a2)))))) + (t + (let ((y (math-sqr x))) + (math-div (math-poly-eval y + '((float (bigneg 456 052 849 1) -7) + (float (bigpos 017 233 739 7) -5) + (float (bigneg 418 442 121 1) -2) + (float (bigpos 407 196 516 6) -1) + (float (bigneg 354 590 362 13) 0) + (float (bigpos 574 490 568 57) 0))) + (math-poly-eval y + '((float 1 0) + (float (bigpos 712 532 678 2) -7) + (float (bigpos 853 264 927 5) -5) + (float (bigpos 718 680 494 9) -3) + (float (bigpos 985 532 029 1) 0) + (float (bigpos 411 490 568 57) 0))))))) +) + +(defun math-besJ1 (x &optional yflag) + (cond ((and (math-negp (calcFunc-re x)) (not yflag)) + (math-neg (math-besJ1 (math-neg x)))) + ((Math-lessp '(float 8 0) (math-abs-approx x)) + (let* ((z (math-div '(float 8 0) x)) + (y (math-sqr z)) + (xx (math-add x '(float (bigneg 491 194 356 2) -9))) + (a1 (math-poly-eval y + '((float (bigneg 019 337 240) -15) + (float (bigpos 174 520 457 2) -15) + (float (bigneg 496 396 516 3) -14) + (float 183105 -8) + (float 1 0)))) + (a2 (math-poly-eval y + '((float (bigpos 412 787 105) -15) + (float (bigneg 987 228 88) -14) + (float (bigpos 096 199 449 8) -15) + (float (bigneg 873 690 002 2) -13) + (float (bigpos 995 499 687 4) -11)))) + (sc (math-sin-cos-raw xx))) + (if yflag + (setq sc (cons (math-neg (cdr sc)) (car sc))) + (if (math-negp x) + (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc)))))) + (math-mul (math-sqrt (math-div '(float (bigpos 722 619 636) -9) x)) + (math-sub (math-mul (cdr sc) a1) + (math-mul (car sc) (math-mul z a2)))))) + (t + (let ((y (math-sqr x))) + (math-mul + x + (math-div (math-poly-eval y + '((float (bigneg 606 036 016 3) -8) + (float (bigpos 826 044 157) -4) + (float (bigneg 439 611 972 2) -3) + (float (bigpos 531 968 423 2) -1) + (float (bigneg 235 059 895 7) 0) + (float (bigpos 232 614 362 72) 0))) + (math-poly-eval y + '((float 1 0) + (float (bigpos 397 991 769 3) -7) + (float (bigpos 394 743 944 9) -5) + (float (bigpos 474 330 858 1) -2) + (float (bigpos 178 535 300 2) 0) + (float (bigpos 442 228 725 144) + 0)))))))) +) + +(defun calcFunc-besY (v x) + (math-inexact-result) + (or (math-numberp v) (math-reject-arg v 'numberp)) + (or (math-numberp x) (math-reject-arg x 'numberp)) + (let ((calc-internal-prec (min 8 calc-internal-prec))) + (math-with-extra-prec 3 + (setq x (math-float (math-normalize x))) + (setq v (math-float (math-normalize v))) + (cond ((not (math-num-integerp v)) + (let ((sc (math-sin-cos-raw (math-mul v (math-pi))))) + (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc)) + (calcFunc-besJ (math-neg v) x)) + (car sc)))) + ((math-negp (setq v (math-trunc v))) + (if (math-oddp v) + (math-neg (calcFunc-besY (math-neg v) x)) + (calcFunc-besY (math-neg v) x))) + ((eq v 0) + (math-besY0 x)) + ((eq v 1) + (math-besY1 x)) + (t + (let ((j 0) + (bym (math-besY0 x)) + (by (math-besY1 x)) + (two-over-x (math-div 2 x)) + byp) + (while (< (setq j (1+ j)) v) + (setq byp (math-sub (math-mul (math-mul j two-over-x) by) + bym) + bym by + by byp)) + by))))) +) + +(defun math-besY0 (x) + (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) + (let ((y (math-sqr x))) + (math-add + (math-div (math-poly-eval y + '((float (bigpos 733 622 284 2) -7) + (float (bigneg 757 792 632 8) -5) + (float (bigpos 129 988 087 1) -2) + (float (bigneg 036 598 123 5) -1) + (float (bigpos 065 834 062 7) 0) + (float (bigneg 389 821 957 2) 0))) + (math-poly-eval y + '((float 1 0) + (float (bigpos 244 030 261 2) -7) + (float (bigpos 647 472 474) -4) + (float (bigpos 438 466 189 7) -3) + (float (bigpos 648 499 452 7) -1) + (float (bigpos 269 544 076 40) 0)))) + (math-mul '(float (bigpos 772 619 636) -9) + (math-mul (math-besJ0 x) (math-ln-raw x)))))) + ((math-negp (calcFunc-re x)) + (math-add (math-besJ0 (math-neg x) t) + (math-mul '(cplx 0 2) + (math-besJ0 (math-neg x))))) + (t + (math-besJ0 x t))) +) + +(defun math-besY1 (x) + (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) + (let ((y (math-sqr x))) + (math-add + (math-mul + x + (math-div (math-poly-eval y + '((float (bigpos 935 937 511 8) -6) + (float (bigneg 726 922 237 4) -3) + (float (bigpos 551 264 349 7) -1) + (float (bigneg 139 438 153 5) 1) + (float (bigpos 439 527 127) 4) + (float (bigneg 943 604 900 4) 3))) + (math-poly-eval y + '((float 1 0) + (float (bigpos 885 632 549 3) -7) + (float (bigpos 605 042 102) -3) + (float (bigpos 002 904 245 2) -2) + (float (bigpos 367 650 733 3) 0) + (float (bigpos 664 419 244 4) 2) + (float (bigpos 057 958 249) 5))))) + (math-mul '(float (bigpos 772 619 636) -9) + (math-sub (math-mul (math-besJ1 x) (math-ln-raw x)) + (math-div 1 x)))))) + ((math-negp (calcFunc-re x)) + (math-neg + (math-add (math-besJ1 (math-neg x) t) + (math-mul '(cplx 0 2) + (math-besJ1 (math-neg x)))))) + (t + (math-besJ1 x t))) +) + +(defun math-poly-eval (x coefs) + (let ((accum (car coefs))) + (while (setq coefs (cdr coefs)) + (setq accum (math-add (car coefs) (math-mul accum x)))) + accum) +) + + +;;;; Bernoulli and Euler polynomials and numbers. + +(defun calcFunc-bern (n &optional x) + (if (and x (not (math-zerop x))) + (if (and calc-symbolic-mode (math-floatp x)) + (math-inexact-result) + (math-build-polynomial-expr (math-bernoulli-coefs n) x)) + (or (math-num-natnump n) (math-reject-arg n 'natnump)) + (if (consp n) + (progn + (math-inexact-result) + (math-float (math-bernoulli-number (math-trunc n)))) + (math-bernoulli-number n))) +) + +(defun calcFunc-euler (n &optional x) + (or (math-num-natnump n) (math-reject-arg n 'natnump)) + (if x + (let* ((n1 (math-add n 1)) + (coefs (math-bernoulli-coefs n1)) + (fac (math-div (math-pow 2 n1) n1)) + (k -1) + (x1 (math-div (math-add x 1) 2)) + (x2 (math-div x 2))) + (if (math-numberp x) + (if (and calc-symbolic-mode (math-floatp x)) + (math-inexact-result) + (math-mul fac + (math-sub (math-build-polynomial-expr coefs x1) + (math-build-polynomial-expr coefs x2)))) + (calcFunc-collect + (math-reduce-vec + 'math-add + (cons 'vec + (mapcar (function + (lambda (c) + (setq k (1+ k)) + (math-mul (math-mul fac c) + (math-sub (math-pow x1 k) + (math-pow x2 k))))) + coefs))) + x))) + (math-mul (math-pow 2 n) + (if (consp n) + (progn + (math-inexact-result) + (calcFunc-euler n '(float 5 -1))) + (calcFunc-euler n '(frac 1 2))))) +) + +(defun math-bernoulli-coefs (n) + (let* ((coefs (list (calcFunc-bern n))) + (nn (math-trunc n)) + (k nn) + (term nn) + coef + (calc-prefer-frac (or (integerp n) calc-prefer-frac))) + (while (>= (setq k (1- k)) 0) + (setq term (math-div term (- nn k)) + coef (math-mul term (math-bernoulli-number k)) + coefs (cons (if (consp n) (math-float coef) coef) coefs) + term (math-mul term k))) + (nreverse coefs)) +) + +(defun math-bernoulli-number (n) + (if (= (% n 2) 1) + (if (= n 1) + '(frac -1 2) + 0) + (setq n (/ n 2)) + (while (>= n math-bernoulli-cache-size) + (let* ((sum 0) + (nk 1) ; nk = n-k+1 + (fact 1) ; fact = (n-k+1)! + ofact + (p math-bernoulli-b-cache) + (calc-prefer-frac t)) + (math-working "bernoulli B" (* 2 math-bernoulli-cache-size)) + (while p + (setq nk (+ nk 2) + ofact fact + fact (math-mul fact (* nk (1- nk))) + sum (math-add sum (math-div (car p) fact)) + p (cdr p))) + (setq ofact (math-mul ofact (1- nk)) + sum (math-sub (math-div '(frac 1 2) ofact) sum) + math-bernoulli-b-cache (cons sum math-bernoulli-b-cache) + math-bernoulli-B-cache (cons (math-mul sum ofact) + math-bernoulli-B-cache) + math-bernoulli-cache-size (1+ math-bernoulli-cache-size)))) + (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache)) +) + +;;; Bn = n! bn +;;; bn = - sum_k=0^n-1 bk / (n-k+1)! + +;;; A faster method would be to use "tangent numbers", c.f., Concrete +;;; Mathematics pg. 273. + +(setq math-bernoulli-b-cache '( (frac -174611 + (bigpos 0 200 291 698 662 857 802)) + (frac 43867 (bigpos 0 944 170 217 94 109 5)) + (frac -3617 (bigpos 0 880 842 622 670 10)) + (frac 1 (bigpos 600 249 724 74)) + (frac -691 (bigpos 0 368 674 307 1)) + (frac 1 (bigpos 160 900 47)) + (frac -1 (bigpos 600 209 1)) + (frac 1 30240) (frac -1 720) + (frac 1 12) 1 )) + +(setq math-bernoulli-B-cache '( (frac -174611 330) (frac 43867 798) + (frac -3617 510) (frac 7 6) (frac -691 2730) + (frac 5 66) (frac -1 30) (frac 1 42) + (frac -1 30) (frac 1 6) 1 )) + +(setq math-bernoulli-cache-size 11) + + + +;;; Probability distributions. + +;;; Binomial. +(defun calcFunc-utpb (x n p) + (if math-expand-formulas + (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1))) + (calcFunc-betaI p x (math-add (math-sub n x) 1))) +) +(put 'calcFunc-utpb 'math-expandable t) + +(defun calcFunc-ltpb (x n p) + (math-sub 1 (calcFunc-utpb x n p)) +) +(put 'calcFunc-ltpb 'math-expandable t) + +;;; Chi-square. +(defun calcFunc-utpc (chisq v) + (if math-expand-formulas + (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2))) + (calcFunc-gammaQ (math-div v 2) (math-div chisq 2))) +) +(put 'calcFunc-utpc 'math-expandable t) + +(defun calcFunc-ltpc (chisq v) + (if math-expand-formulas + (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2))) + (calcFunc-gammaP (math-div v 2) (math-div chisq 2))) +) +(put 'calcFunc-ltpc 'math-expandable t) + +;;; F-distribution. +(defun calcFunc-utpf (f v1 v2) + (if math-expand-formulas + (math-normalize (list 'calcFunc-betaI + (list '/ v2 (list '+ v2 (list '* v1 f))) + (list '/ v2 2) + (list '/ v1 2))) + (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f))) + (math-div v2 2) + (math-div v1 2))) +) +(put 'calcFunc-utpf 'math-expandable t) + +(defun calcFunc-ltpf (f v1 v2) + (math-sub 1 (calcFunc-utpf f v1 v2)) +) +(put 'calcFunc-ltpf 'math-expandable t) + +;;; Normal. +(defun calcFunc-utpn (x mean sdev) + (if math-expand-formulas + (math-normalize + (list '/ + (list '+ 1 + (list 'calcFunc-erf + (list '/ (list '- mean x) + (list '* sdev (list 'calcFunc-sqrt 2))))) + 2)) + (math-mul (math-add '(float 1 0) + (calcFunc-erf + (math-div (math-sub mean x) + (math-mul sdev (math-sqrt-2))))) + '(float 5 -1))) +) +(put 'calcFunc-utpn 'math-expandable t) + +(defun calcFunc-ltpn (x mean sdev) + (if math-expand-formulas + (math-normalize + (list '/ + (list '+ 1 + (list 'calcFunc-erf + (list '/ (list '- x mean) + (list '* sdev (list 'calcFunc-sqrt 2))))) + 2)) + (math-mul (math-add '(float 1 0) + (calcFunc-erf + (math-div (math-sub x mean) + (math-mul sdev (math-sqrt-2))))) + '(float 5 -1))) +) +(put 'calcFunc-ltpn 'math-expandable t) + +;;; Poisson. +(defun calcFunc-utpp (n x) + (if math-expand-formulas + (math-normalize (list 'calcFunc-gammaP x n)) + (calcFunc-gammaP x n)) +) +(put 'calcFunc-utpp 'math-expandable t) + +(defun calcFunc-ltpp (n x) + (if math-expand-formulas + (math-normalize (list 'calcFunc-gammaQ x n)) + (calcFunc-gammaQ x n)) +) +(put 'calcFunc-ltpp 'math-expandable t) + +;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.) +(defun calcFunc-utpt (tt v) + (if math-expand-formulas + (math-normalize (list 'calcFunc-betaI + (list '/ v (list '+ v (list '^ tt 2))) + (list '/ v 2) + '(float 5 -1))) + (calcFunc-betaI (math-div v (math-add v (math-sqr tt))) + (math-div v 2) + '(float 5 -1))) +) +(put 'calcFunc-utpt 'math-expandable t) + +(defun calcFunc-ltpt (tt v) + (math-sub 1 (calcFunc-utpt tt v)) +) +(put 'calcFunc-ltpt 'math-expandable t) + + + + |