;;; calc-funcs.el --- well-known functions for Calc ;; Copyright (C) 1990-1993, 2001-2016 Free Software Foundation, Inc. ;; Author: David Gillespie ;; This file is part of GNU Emacs. ;; GNU Emacs is free software: you can redistribute it and/or modify ;; it under the terms of the GNU General Public License as published by ;; the Free Software Foundation, either version 3 of the License, or ;; (at your option) any later version. ;; GNU Emacs is distributed in the hope that it will be useful, ;; but WITHOUT ANY WARRANTY; without even the implied warranty of ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ;; GNU General Public License for more details. ;; You should have received a copy of the GNU General Public License ;; along with GNU Emacs. If not, see . ;;; Commentary: ;;; Code: ;; This file is autoloaded from calc-ext.el. (require 'calc-ext) (require 'calc-macs) (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) to the appropriate precision." (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. (defvar math-current-gamma-value nil) (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)))))) (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. (defvar math-current-beta-value nil) (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)))))))) (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 (math-read-number-simple "-0.785398164"))) (a1 (math-poly-eval y (list (math-read-number-simple "0.0000002093887211") (math-read-number-simple "-0.000002073370639") (math-read-number-simple "0.00002734510407") (math-read-number-simple "-0.001098628627") '(float 1 0)))) (a2 (math-poly-eval y (list (math-read-number-simple "-0.0000000934935152") (math-read-number-simple "0.0000007621095161") (math-read-number-simple "-0.000006911147651") (math-read-number-simple "0.0001430488765") (math-read-number-simple "-0.01562499995")))) (sc (math-sin-cos-raw xx))) (if yflag (setq sc (cons (math-neg (cdr sc)) (car sc)))) (math-mul (math-sqrt (math-div (math-read-number-simple "0.636619722") 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 (list (math-read-number-simple "-184.9052456") (math-read-number-simple "77392.33017") (math-read-number-simple "-11214424.18") (math-read-number-simple "651619640.7") (math-read-number-simple "-13362590354.0") (math-read-number-simple "57568490574.0"))) (math-poly-eval y (list '(float 1 0) (math-read-number-simple "267.8532712") (math-read-number-simple "59272.64853") (math-read-number-simple "9494680.718") (math-read-number-simple "1029532985.0") (math-read-number-simple "57568490411.0")))))))) (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 (math-read-number-simple "-2.356194491"))) (a1 (math-poly-eval y (list (math-read-number-simple "-0.000000240337019") (math-read-number-simple "0.000002457520174") (math-read-number-simple "-0.00003516396496") '(float 183105 -8) '(float 1 0)))) (a2 (math-poly-eval y (list (math-read-number-simple "0.000000105787412") (math-read-number-simple "-0.00000088228987") (math-read-number-simple "0.000008449199096") (math-read-number-simple "-0.0002002690873") (math-read-number-simple "0.04687499995")))) (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 (math-read-number-simple "0.636619722") 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 (list (math-read-number-simple "-30.16036606") (math-read-number-simple "15704.4826") (math-read-number-simple "-2972611.439") (math-read-number-simple "242396853.1") (math-read-number-simple "-7895059235.0") (math-read-number-simple "72362614232.0"))) (math-poly-eval y (list '(float 1 0) (math-read-number-simple "376.9991397") (math-read-number-simple "99447.43394") (math-read-number-simple "18583304.74") (math-read-number-simple "2300535178.0") (math-read-number-simple "144725228442.0"))))))))) (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 (list (math-read-number-simple "228.4622733") (math-read-number-simple "-86327.92757") (math-read-number-simple "10879881.29") (math-read-number-simple "-512359803.6") (math-read-number-simple "7062834065.0") (math-read-number-simple "-2957821389.0"))) (math-poly-eval y (list '(float 1 0) (math-read-number-simple "226.1030244") (math-read-number-simple "47447.2647") (math-read-number-simple "7189466.438") (math-read-number-simple "745249964.8") (math-read-number-simple "40076544269.0")))) (math-mul (math-read-number-simple "0.636619772") (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 (list (math-read-number-simple "8511.937935") (math-read-number-simple "-4237922.726") (math-read-number-simple "734926455.1") (math-read-number-simple "-51534381390.0") (math-read-number-simple "1275274390000.0") (math-read-number-simple "-4900604943000.0"))) (math-poly-eval y (list '(float 1 0) (math-read-number-simple "354.9632885") (math-read-number-simple "102042.605") (math-read-number-simple "22459040.02") (math-read-number-simple "3733650367.0") (math-read-number-simple "424441966400.0") (math-read-number-simple "24995805700000.0"))))) (math-mul (math-read-number-simple "0.636619772") (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)))))) (defvar math-bernoulli-b-cache (list (list 'frac -174611 (math-read-number-simple "802857662698291200000")) (list 'frac 43867 (math-read-number-simple "5109094217170944000")) (list 'frac -3617 (math-read-number-simple "10670622842880000")) (list 'frac 1 (math-read-number-simple "74724249600")) (list 'frac -691 (math-read-number-simple "1307674368000")) (list 'frac 1 (math-read-number-simple "47900160")) (list 'frac -1 (math-read-number-simple "1209600")) (list 'frac 1 30240) (list 'frac -1 720) (list 'frac 1 12) 1 )) (defvar 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 )) (defvar math-bernoulli-cache-size 11) (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. ;;; 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) (provide 'calc-funcs) ;;; calc-funcs.el ends here