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976 lines
30 KiB
EmacsLisp
976 lines
30 KiB
EmacsLisp
;;; calc-funcs.el --- well-known functions for Calc
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;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
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;; 2005, 2006, 2007 Free Software Foundation, Inc.
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;; Author: David Gillespie <daveg@synaptics.com>
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;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
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;; This file is part of GNU Emacs.
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;; GNU Emacs is free software; you can redistribute it and/or modify
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;; it under the terms of the GNU General Public License as published by
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;; the Free Software Foundation; either version 3, or (at your option)
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;; any later version.
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;; GNU Emacs is distributed in the hope that it will be useful,
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;; but WITHOUT ANY WARRANTY; without even the implied warranty of
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;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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;; GNU General Public License for more details.
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;; You should have received a copy of the GNU General Public License
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;; along with GNU Emacs; see the file COPYING. If not, write to the
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;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
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;; Boston, MA 02110-1301, USA.
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;;; Commentary:
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;;; Code:
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;; This file is autoloaded from calc-ext.el.
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(require 'calc-ext)
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(require 'calc-macs)
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(defun calc-inc-gamma (arg)
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(interactive "P")
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(calc-slow-wrapper
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(if (calc-is-inverse)
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(if (calc-is-hyperbolic)
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(calc-binary-op "gamG" 'calcFunc-gammaG arg)
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(calc-binary-op "gamQ" 'calcFunc-gammaQ arg))
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(if (calc-is-hyperbolic)
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(calc-binary-op "gamg" 'calcFunc-gammag arg)
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(calc-binary-op "gamP" 'calcFunc-gammaP arg)))))
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(defun calc-erf (arg)
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(interactive "P")
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(calc-slow-wrapper
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(if (calc-is-inverse)
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(calc-unary-op "erfc" 'calcFunc-erfc arg)
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(calc-unary-op "erf" 'calcFunc-erf arg))))
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(defun calc-erfc (arg)
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(interactive "P")
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(calc-invert-func)
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(calc-erf arg))
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(defun calc-beta (arg)
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(interactive "P")
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(calc-slow-wrapper
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(calc-binary-op "beta" 'calcFunc-beta arg)))
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(defun calc-inc-beta ()
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(interactive)
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(calc-slow-wrapper
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(if (calc-is-hyperbolic)
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(calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3)))
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(calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3))))))
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(defun calc-bessel-J (arg)
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(interactive "P")
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(calc-slow-wrapper
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(calc-binary-op "besJ" 'calcFunc-besJ arg)))
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(defun calc-bessel-Y (arg)
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(interactive "P")
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(calc-slow-wrapper
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(calc-binary-op "besY" 'calcFunc-besY arg)))
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(defun calc-bernoulli-number (arg)
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(interactive "P")
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(calc-slow-wrapper
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(if (calc-is-hyperbolic)
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(calc-binary-op "bern" 'calcFunc-bern arg)
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(calc-unary-op "bern" 'calcFunc-bern arg))))
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(defun calc-euler-number (arg)
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(interactive "P")
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(calc-slow-wrapper
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(if (calc-is-hyperbolic)
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(calc-binary-op "eulr" 'calcFunc-euler arg)
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(calc-unary-op "eulr" 'calcFunc-euler arg))))
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(defun calc-stirling-number (arg)
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(interactive "P")
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(calc-slow-wrapper
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(if (calc-is-hyperbolic)
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(calc-binary-op "str2" 'calcFunc-stir2 arg)
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(calc-binary-op "str1" 'calcFunc-stir1 arg))))
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(defun calc-utpb ()
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(interactive)
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(calc-prob-dist "b" 3))
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(defun calc-utpc ()
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(interactive)
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(calc-prob-dist "c" 2))
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(defun calc-utpf ()
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(interactive)
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(calc-prob-dist "f" 3))
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(defun calc-utpn ()
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(interactive)
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(calc-prob-dist "n" 3))
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(defun calc-utpp ()
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(interactive)
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(calc-prob-dist "p" 2))
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(defun calc-utpt ()
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(interactive)
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(calc-prob-dist "t" 2))
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(defun calc-prob-dist (letter nargs)
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(calc-slow-wrapper
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(if (calc-is-inverse)
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(calc-enter-result nargs (concat "ltp" letter)
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(append (list (intern (concat "calcFunc-ltp" letter))
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(calc-top-n 1))
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(calc-top-list-n (1- nargs) 2)))
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(calc-enter-result nargs (concat "utp" letter)
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(append (list (intern (concat "calcFunc-utp" letter))
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(calc-top-n 1))
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(calc-top-list-n (1- nargs) 2))))))
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;;; Sources: Numerical Recipes, Press et al;
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;;; Handbook of Mathematical Functions, Abramowitz & Stegun.
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;;; Gamma function.
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(defun calcFunc-gamma (x)
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(or (math-numberp x) (math-reject-arg x 'numberp))
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(calcFunc-fact (math-add x -1)))
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(defun math-gammap1-raw (x &optional fprec nfprec) ; compute gamma(1 + x)
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(or fprec
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(setq fprec (math-float calc-internal-prec)
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nfprec (math-float (- calc-internal-prec))))
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(cond ((math-lessp-float (calcFunc-re x) fprec)
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(if (math-lessp-float (calcFunc-re x) nfprec)
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(math-neg (math-div
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(math-pi)
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(math-mul (math-gammap1-raw
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(math-add (math-neg x)
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'(float -1 0))
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fprec nfprec)
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(math-sin-raw
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(math-mul (math-pi) x)))))
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(let ((xplus1 (math-add x '(float 1 0))))
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(math-div (math-gammap1-raw xplus1 fprec nfprec) xplus1))))
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((and (math-realp x)
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(math-lessp-float '(float 736276 0) x))
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(math-overflow))
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(t ; re(x) now >= 10.0
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(let ((xinv (math-div 1 x))
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(lnx (math-ln-raw x)))
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(math-mul (math-sqrt-two-pi)
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(math-exp-raw
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(math-gamma-series
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(math-sub (math-mul (math-add x '(float 5 -1))
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lnx)
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x)
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xinv
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(math-sqr xinv)
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'(float 0 0)
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2)))))))
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(defun math-gamma-series (sum x xinvsqr oterm n)
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(math-working "gamma" sum)
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(let* ((bn (math-bernoulli-number n))
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(term (math-mul (math-div-float (math-float (nth 1 bn))
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(math-float (* (nth 2 bn)
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(* n (1- n)))))
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x))
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(next (math-add sum term)))
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(if (math-nearly-equal sum next)
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next
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(if (> n (* 2 calc-internal-prec))
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(progn
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;; Need this because series eventually diverges for large enough n.
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(calc-record-why
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"*Gamma computation stopped early, not all digits may be valid")
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next)
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(math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2))))))
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;;; Incomplete gamma function.
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(defvar math-current-gamma-value nil)
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(defun calcFunc-gammaP (a x)
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(if (equal x '(var inf var-inf))
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'(float 1 0)
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(math-inexact-result)
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(or (Math-numberp a) (math-reject-arg a 'numberp))
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(or (math-numberp x) (math-reject-arg x 'numberp))
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(if (and (math-num-integerp a)
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(integerp (setq a (math-trunc a)))
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(> a 0) (< a 20))
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(math-sub 1 (calcFunc-gammaQ a x))
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(let ((math-current-gamma-value (calcFunc-gamma a)))
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(math-div (calcFunc-gammag a x) math-current-gamma-value)))))
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(defun calcFunc-gammaQ (a x)
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(if (equal x '(var inf var-inf))
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'(float 0 0)
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(math-inexact-result)
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(or (Math-numberp a) (math-reject-arg a 'numberp))
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(or (math-numberp x) (math-reject-arg x 'numberp))
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(if (and (math-num-integerp a)
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(integerp (setq a (math-trunc a)))
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(> a 0) (< a 20))
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(let ((n 0)
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(sum '(float 1 0))
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(term '(float 1 0)))
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(math-with-extra-prec 1
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(while (< (setq n (1+ n)) a)
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(setq term (math-div (math-mul term x) n)
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sum (math-add sum term))
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(math-working "gamma" sum))
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(math-mul sum (calcFunc-exp (math-neg x)))))
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(let ((math-current-gamma-value (calcFunc-gamma a)))
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(math-div (calcFunc-gammaG a x) math-current-gamma-value)))))
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(defun calcFunc-gammag (a x)
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(if (equal x '(var inf var-inf))
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(calcFunc-gamma a)
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(math-inexact-result)
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(or (Math-numberp a) (math-reject-arg a 'numberp))
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(or (Math-numberp x) (math-reject-arg x 'numberp))
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(math-with-extra-prec 2
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(setq a (math-float a))
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(setq x (math-float x))
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(if (or (math-negp (calcFunc-re a))
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(math-lessp-float (calcFunc-re x)
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(math-add-float (calcFunc-re a)
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'(float 1 0))))
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(math-inc-gamma-series a x)
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(math-sub (or math-current-gamma-value (calcFunc-gamma a))
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(math-inc-gamma-cfrac a x))))))
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(defun calcFunc-gammaG (a x)
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(if (equal x '(var inf var-inf))
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'(float 0 0)
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(math-inexact-result)
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(or (Math-numberp a) (math-reject-arg a 'numberp))
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(or (Math-numberp x) (math-reject-arg x 'numberp))
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(math-with-extra-prec 2
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(setq a (math-float a))
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(setq x (math-float x))
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(if (or (math-negp (calcFunc-re a))
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(math-lessp-float (calcFunc-re x)
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(math-add-float (math-abs-approx a)
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'(float 1 0))))
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(math-sub (or math-current-gamma-value (calcFunc-gamma a))
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(math-inc-gamma-series a x))
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(math-inc-gamma-cfrac a x)))))
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(defun math-inc-gamma-series (a x)
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(if (Math-zerop x)
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'(float 0 0)
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(math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
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(math-with-extra-prec 2
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(let ((start (math-div '(float 1 0) a)))
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(math-inc-gamma-series-step start start a x))))))
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(defun math-inc-gamma-series-step (sum term a x)
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(math-working "gamma" sum)
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(setq a (math-add a '(float 1 0))
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term (math-div (math-mul term x) a))
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(let ((next (math-add sum term)))
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(if (math-nearly-equal sum next)
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next
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(math-inc-gamma-series-step next term a x))))
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(defun math-inc-gamma-cfrac (a x)
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(if (Math-zerop x)
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(or math-current-gamma-value (calcFunc-gamma a))
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(math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
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(math-inc-gamma-cfrac-step '(float 1 0) x
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'(float 0 0) '(float 1 0)
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'(float 1 0) '(float 1 0) '(float 0 0)
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a x))))
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(defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x)
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(let ((ana (math-sub n a))
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(anf (math-mul n fac)))
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(setq n (math-add n '(float 1 0))
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a0 (math-mul (math-add a1 (math-mul a0 ana)) fac)
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b0 (math-mul (math-add b1 (math-mul b0 ana)) fac)
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a1 (math-add (math-mul x a0) (math-mul anf a1))
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b1 (math-add (math-mul x b0) (math-mul anf b1)))
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(if (math-zerop a1)
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(math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x)
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(setq fac (math-div '(float 1 0) a1))
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(let ((next (math-mul b1 fac)))
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(math-working "gamma" next)
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(if (math-nearly-equal next g)
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next
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(math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x))))))
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;;; Error function.
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(defun calcFunc-erf (x)
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(if (equal x '(var inf var-inf))
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'(float 1 0)
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(if (equal x '(neg (var inf var-inf)))
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'(float -1 0)
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(if (Math-zerop x)
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x
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(let ((math-current-gamma-value (math-sqrt-pi)))
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(math-to-same-complex-quad
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(math-div (calcFunc-gammag '(float 5 -1)
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(math-sqr (math-to-complex-quad-one x)))
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math-current-gamma-value)
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x))))))
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(defun calcFunc-erfc (x)
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(if (equal x '(var inf var-inf))
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'(float 0 0)
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(if (math-posp x)
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(let ((math-current-gamma-value (math-sqrt-pi)))
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(math-div (calcFunc-gammaG '(float 5 -1) (math-sqr x))
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math-current-gamma-value))
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(math-sub 1 (calcFunc-erf x)))))
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(defun math-to-complex-quad-one (x)
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(if (eq (car-safe x) 'polar) (setq x (math-complex x)))
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(if (eq (car-safe x) 'cplx)
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(list 'cplx (math-abs (nth 1 x)) (math-abs (nth 2 x)))
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x))
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(defun math-to-same-complex-quad (x y)
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(if (eq (car-safe y) 'cplx)
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(if (eq (car-safe x) 'cplx)
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(list 'cplx
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(if (math-negp (nth 1 y)) (math-neg (nth 1 x)) (nth 1 x))
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(if (math-negp (nth 2 y)) (math-neg (nth 2 x)) (nth 2 x)))
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(if (math-negp (nth 1 y)) (math-neg x) x))
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(if (math-negp y)
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(if (eq (car-safe x) 'cplx)
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(list 'cplx (math-neg (nth 1 x)) (nth 2 x))
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(math-neg x))
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x)))
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;;; Beta function.
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(defun calcFunc-beta (a b)
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(if (math-num-integerp a)
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(let ((am (math-add a -1)))
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(or (math-numberp b) (math-reject-arg b 'numberp))
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(math-div 1 (math-mul b (calcFunc-choose (math-add b am) am))))
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(if (math-num-integerp b)
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(calcFunc-beta b a)
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(math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b))
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(calcFunc-gamma (math-add a b))))))
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;;; Incomplete beta function.
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(defvar math-current-beta-value nil)
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(defun calcFunc-betaI (x a b)
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(cond ((math-zerop x)
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'(float 0 0))
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((math-equal-int x 1)
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'(float 1 0))
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((or (math-zerop a)
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(and (math-num-integerp a)
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(math-negp a)))
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(if (or (math-zerop b)
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(and (math-num-integerp b)
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(math-negp b)))
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(math-reject-arg b 'range)
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'(float 1 0)))
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((or (math-zerop b)
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(and (math-num-integerp b)
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(math-negp b)))
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'(float 0 0))
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((not (math-numberp a)) (math-reject-arg a 'numberp))
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((not (math-numberp b)) (math-reject-arg b 'numberp))
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((math-inexact-result))
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(t (let ((math-current-beta-value (calcFunc-beta a b)))
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(math-div (calcFunc-betaB x a b) math-current-beta-value)))))
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(defun calcFunc-betaB (x a b)
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(cond
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((math-zerop x)
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'(float 0 0))
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((math-equal-int x 1)
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(calcFunc-beta a b))
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((not (math-numberp x)) (math-reject-arg x 'numberp))
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((not (math-numberp a)) (math-reject-arg a 'numberp))
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((not (math-numberp b)) (math-reject-arg b 'numberp))
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((math-zerop a) (math-reject-arg a 'nonzerop))
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((math-zerop b) (math-reject-arg b 'nonzerop))
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((and (math-num-integerp b)
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(if (math-negp b)
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(math-reject-arg b 'range)
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(Math-natnum-lessp (setq b (math-trunc b)) 20)))
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(and calc-symbolic-mode (or (math-floatp a) (math-floatp b))
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(math-inexact-result))
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(math-mul
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(math-with-extra-prec 2
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(let* ((i 0)
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(term 1)
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(sum (math-div term a)))
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(while (< (setq i (1+ i)) b)
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(setq term (math-mul (math-div (math-mul term (- i b)) i) x)
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sum (math-add sum (math-div term (math-add a i))))
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(math-working "beta" sum))
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sum))
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(math-pow x a)))
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((and (math-num-integerp a)
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(if (math-negp a)
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(math-reject-arg a 'range)
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(Math-natnum-lessp (setq a (math-trunc a)) 20)))
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(math-sub (or math-current-beta-value (calcFunc-beta a b))
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(calcFunc-betaB (math-sub 1 x) b a)))
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(t
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(math-inexact-result)
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(math-with-extra-prec 2
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(setq x (math-float x))
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(setq a (math-float a))
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(setq b (math-float b))
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(let ((bt (math-exp-raw (math-add (math-mul a (math-ln-raw x))
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(math-mul b (math-ln-raw
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(math-sub '(float 1 0)
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x)))))))
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(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 '(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))))))
|
|
|
|
(defvar 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 ))
|
|
|
|
(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)!
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;;; A faster method would be to use "tangent numbers", c.f., Concrete
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;;; Mathematics pg. 273.
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;;; Probability distributions.
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;;; Binomial.
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(defun calcFunc-utpb (x n p)
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(if math-expand-formulas
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(math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1)))
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(calcFunc-betaI p x (math-add (math-sub n x) 1))))
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(put 'calcFunc-utpb 'math-expandable t)
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(defun calcFunc-ltpb (x n p)
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(math-sub 1 (calcFunc-utpb x n p)))
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(put 'calcFunc-ltpb 'math-expandable t)
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;;; Chi-square.
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(defun calcFunc-utpc (chisq v)
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(if math-expand-formulas
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(math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2)))
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(calcFunc-gammaQ (math-div v 2) (math-div chisq 2))))
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(put 'calcFunc-utpc 'math-expandable t)
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(defun calcFunc-ltpc (chisq v)
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(if math-expand-formulas
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(math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2)))
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(calcFunc-gammaP (math-div v 2) (math-div chisq 2))))
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(put 'calcFunc-ltpc 'math-expandable t)
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;;; F-distribution.
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(defun calcFunc-utpf (f v1 v2)
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(if math-expand-formulas
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(math-normalize (list 'calcFunc-betaI
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(list '/ v2 (list '+ v2 (list '* v1 f)))
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(list '/ v2 2)
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(list '/ v1 2)))
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(calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f)))
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(math-div v2 2)
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(math-div v1 2))))
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(put 'calcFunc-utpf 'math-expandable t)
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(defun calcFunc-ltpf (f v1 v2)
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(math-sub 1 (calcFunc-utpf f v1 v2)))
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(put 'calcFunc-ltpf 'math-expandable t)
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;;; Normal.
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(defun calcFunc-utpn (x mean sdev)
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(if math-expand-formulas
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(math-normalize
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(list '/
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(list '+ 1
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(list 'calcFunc-erf
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(list '/ (list '- mean x)
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(list '* sdev (list 'calcFunc-sqrt 2)))))
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2))
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(math-mul (math-add '(float 1 0)
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(calcFunc-erf
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(math-div (math-sub mean x)
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(math-mul sdev (math-sqrt-2)))))
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'(float 5 -1))))
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(put 'calcFunc-utpn 'math-expandable t)
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(defun calcFunc-ltpn (x mean sdev)
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(if math-expand-formulas
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(math-normalize
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(list '/
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(list '+ 1
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(list 'calcFunc-erf
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(list '/ (list '- x mean)
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(list '* sdev (list 'calcFunc-sqrt 2)))))
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2))
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(math-mul (math-add '(float 1 0)
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(calcFunc-erf
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(math-div (math-sub x mean)
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(math-mul sdev (math-sqrt-2)))))
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'(float 5 -1))))
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(put 'calcFunc-ltpn 'math-expandable t)
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;;; Poisson.
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(defun calcFunc-utpp (n x)
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(if math-expand-formulas
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(math-normalize (list 'calcFunc-gammaP x n))
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(calcFunc-gammaP x n)))
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(put 'calcFunc-utpp 'math-expandable t)
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(defun calcFunc-ltpp (n x)
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(if math-expand-formulas
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(math-normalize (list 'calcFunc-gammaQ x n))
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(calcFunc-gammaQ x n)))
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(put 'calcFunc-ltpp 'math-expandable t)
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;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.)
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(defun calcFunc-utpt (tt v)
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(if math-expand-formulas
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(math-normalize (list 'calcFunc-betaI
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(list '/ v (list '+ v (list '^ tt 2)))
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(list '/ v 2)
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'(float 5 -1)))
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(calcFunc-betaI (math-div v (math-add v (math-sqr tt)))
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(math-div v 2)
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'(float 5 -1))))
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(put 'calcFunc-utpt 'math-expandable t)
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(defun calcFunc-ltpt (tt v)
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(math-sub 1 (calcFunc-utpt tt v)))
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(put 'calcFunc-ltpt 'math-expandable t)
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(provide 'calc-funcs)
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;;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a
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;;; calc-funcs.el ends here
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