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;;; calc-funcs.el --- well-known functions for Calc
;; Copyright (C) 1990-1993, 2001-2015 Free Software Foundation, Inc.
;; Author: David Gillespie <daveg@synaptics.com>
;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
;; 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 <http://www.gnu.org/licenses/>.
;;; 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
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