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/* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*- |
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* |
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* ***** BEGIN LICENSE BLOCK ***** |
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* Version: MPL 1.1/GPL 2.0/LGPL 2.1 |
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* |
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* The contents of this file are subject to the Mozilla Public License Version |
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* 1.1 (the "License"); you may not use this file except in compliance with |
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* the License. You may obtain a copy of the License at |
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* http://www.mozilla.org/MPL/ |
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* |
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* Software distributed under the License is distributed on an "AS IS" basis, |
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* WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License |
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* for the specific language governing rights and limitations under the |
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* License. |
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* |
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* The Original Code is Mozilla Communicator client code, released |
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* March 31, 1998. |
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* |
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* The Initial Developer of the Original Code is |
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* Sun Microsystems, Inc. |
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* Portions created by the Initial Developer are Copyright (C) 1998 |
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* the Initial Developer. All Rights Reserved. |
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* |
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* Contributor(s): |
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* |
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* Alternatively, the contents of this file may be used under the terms of |
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* either of the GNU General Public License Version 2 or later (the "GPL"), |
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* or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"), |
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* in which case the provisions of the GPL or the LGPL are applicable instead |
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* of those above. If you wish to allow use of your version of this file only |
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* under the terms of either the GPL or the LGPL, and not to allow others to |
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* use your version of this file under the terms of the MPL, indicate your |
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* decision by deleting the provisions above and replace them with the notice |
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* and other provisions required by the GPL or the LGPL. If you do not delete |
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* the provisions above, a recipient may use your version of this file under |
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* the terms of any one of the MPL, the GPL or the LGPL. |
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* |
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* ***** END LICENSE BLOCK ***** */ |
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|
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/* @(#)e_jn.c 1.4 95/01/18 */ |
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/* |
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* ==================================================== |
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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* |
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* Developed at SunSoft, a Sun Microsystems, Inc. business. |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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*/ |
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|
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/* |
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* __ieee754_jn(n, x), __ieee754_yn(n, x) |
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* floating point Bessel's function of the 1st and 2nd kind |
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* of order n |
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* |
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* Special cases: |
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* y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; |
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* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. |
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* Note 2. About jn(n,x), yn(n,x) |
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* For n=0, j0(x) is called, |
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* for n=1, j1(x) is called, |
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* for n<x, forward recursion us used starting |
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* from values of j0(x) and j1(x). |
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* for n>x, a continued fraction approximation to |
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* j(n,x)/j(n-1,x) is evaluated and then backward |
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* recursion is used starting from a supposed value |
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* for j(n,x). The resulting value of j(0,x) is |
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* compared with the actual value to correct the |
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* supposed value of j(n,x). |
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* |
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* yn(n,x) is similar in all respects, except |
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* that forward recursion is used for all |
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* values of n>1. |
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* |
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*/ |
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|
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#include "fdlibm.h" |
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|
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#ifdef __STDC__ |
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static const double |
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#else |
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static double |
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#endif |
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invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ |
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two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ |
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one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ |
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|
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static double zero = 0.00000000000000000000e+00; |
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|
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#ifdef __STDC__ |
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double __ieee754_jn(int n, double x) |
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#else |
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double __ieee754_jn(n,x) |
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int n; double x; |
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#endif |
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{ |
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fd_twoints u; |
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int i,hx,ix,lx, sgn; |
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double a, b, temp, di; |
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double z, w; |
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|
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/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) |
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* Thus, J(-n,x) = J(n,-x) |
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*/ |
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u.d = x; |
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hx = __HI(u); |
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ix = 0x7fffffff&hx; |
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lx = __LO(u); |
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/* if J(n,NaN) is NaN */ |
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if((ix|((unsigned)(lx|-lx))>>31)>0x7ff00000) return x+x; |
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if(n<0){ |
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n = -n; |
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x = -x; |
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hx ^= 0x80000000; |
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} |
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if(n==0) return(__ieee754_j0(x)); |
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if(n==1) return(__ieee754_j1(x)); |
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sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ |
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x = fd_fabs(x); |
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if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ |
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b = zero; |
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else if((double)n<=x) { |
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/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ |
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if(ix>=0x52D00000) { /* x > 2**302 */ |
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/* (x >> n**2) |
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
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* Let s=sin(x), c=cos(x), |
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then |
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* |
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* n sin(xn)*sqt2 cos(xn)*sqt2 |
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* ---------------------------------- |
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* 0 s-c c+s |
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* 1 -s-c -c+s |
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* 2 -s+c -c-s |
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* 3 s+c c-s |
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*/ |
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switch(n&3) { |
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case 0: temp = fd_cos(x)+fd_sin(x); break; |
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case 1: temp = -fd_cos(x)+fd_sin(x); break; |
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case 2: temp = -fd_cos(x)-fd_sin(x); break; |
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case 3: temp = fd_cos(x)-fd_sin(x); break; |
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} |
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b = invsqrtpi*temp/fd_sqrt(x); |
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} else { |
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a = __ieee754_j0(x); |
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b = __ieee754_j1(x); |
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for(i=1;i<n;i++){ |
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temp = b; |
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b = b*((double)(i+i)/x) - a; /* avoid underflow */ |
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a = temp; |
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} |
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} |
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} else { |
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if(ix<0x3e100000) { /* x < 2**-29 */ |
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/* x is tiny, return the first Taylor expansion of J(n,x) |
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* J(n,x) = 1/n!*(x/2)^n - ... |
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*/ |
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if(n>33) /* underflow */ |
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b = zero; |
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else { |
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temp = x*0.5; b = temp; |
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for (a=one,i=2;i<=n;i++) { |
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a *= (double)i; /* a = n! */ |
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b *= temp; /* b = (x/2)^n */ |
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} |
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b = b/a; |
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} |
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} else { |
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/* use backward recurrence */ |
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/* x x^2 x^2 |
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* J(n,x)/J(n-1,x) = ---- ------ ------ ..... |
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* 2n - 2(n+1) - 2(n+2) |
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* |
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* 1 1 1 |
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* (for large x) = ---- ------ ------ ..... |
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* 2n 2(n+1) 2(n+2) |
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* -- - ------ - ------ - |
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* x x x |
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* |
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* Let w = 2n/x and h=2/x, then the above quotient |
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* is equal to the continued fraction: |
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* 1 |
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* = ----------------------- |
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* 1 |
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* w - ----------------- |
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* 1 |
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* w+h - --------- |
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* w+2h - ... |
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* |
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* To determine how many terms needed, let |
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* Q(0) = w, Q(1) = w(w+h) - 1, |
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* Q(k) = (w+k*h)*Q(k-1) - Q(k-2), |
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* When Q(k) > 1e4 good for single |
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* When Q(k) > 1e9 good for double |
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* When Q(k) > 1e17 good for quadruple |
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*/ |
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/* determine k */ |
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double t,v; |
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double q0,q1,h,tmp; int k,m; |
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w = (n+n)/(double)x; h = 2.0/(double)x; |
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q0 = w; z = w+h; q1 = w*z - 1.0; k=1; |
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while(q1<1.0e9) { |
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k += 1; z += h; |
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tmp = z*q1 - q0; |
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q0 = q1; |
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q1 = tmp; |
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} |
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m = n+n; |
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for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); |
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a = t; |
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b = one; |
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/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) |
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* Hence, if n*(log(2n/x)) > ... |
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* single 8.8722839355e+01 |
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* double 7.09782712893383973096e+02 |
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* long double 1.1356523406294143949491931077970765006170e+04 |
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* then recurrent value may overflow and the result is |
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* likely underflow to zero |
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*/ |
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tmp = n; |
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v = two/x; |
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tmp = tmp*__ieee754_log(fd_fabs(v*tmp)); |
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if(tmp<7.09782712893383973096e+02) { |
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for(i=n-1,di=(double)(i+i);i>0;i--){ |
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temp = b; |
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b *= di; |
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b = b/x - a; |
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a = temp; |
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di -= two; |
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} |
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} else { |
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for(i=n-1,di=(double)(i+i);i>0;i--){ |
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temp = b; |
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b *= di; |
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b = b/x - a; |
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a = temp; |
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di -= two; |
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/* scale b to avoid spurious overflow */ |
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if(b>1e100) { |
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a /= b; |
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t /= b; |
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b = one; |
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} |
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} |
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} |
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b = (t*__ieee754_j0(x)/b); |
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} |
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} |
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if(sgn==1) return -b; else return b; |
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} |
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|
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#ifdef __STDC__ |
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double __ieee754_yn(int n, double x) |
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#else |
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double __ieee754_yn(n,x) |
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int n; double x; |
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#endif |
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{ |
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fd_twoints u; |
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int i,hx,ix,lx; |
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int sign; |
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double a, b, temp; |
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|
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u.d = x; |
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hx = __HI(u); |
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ix = 0x7fffffff&hx; |
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lx = __LO(u); |
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/* if Y(n,NaN) is NaN */ |
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if((ix|((unsigned)(lx|-lx))>>31)>0x7ff00000) return x+x; |
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if((ix|lx)==0) return -one/zero; |
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if(hx<0) return zero/zero; |
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sign = 1; |
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if(n<0){ |
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n = -n; |
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sign = 1 - ((n&1)<<1); |
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} |
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if(n==0) return(__ieee754_y0(x)); |
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if(n==1) return(sign*__ieee754_y1(x)); |
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if(ix==0x7ff00000) return zero; |
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if(ix>=0x52D00000) { /* x > 2**302 */ |
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/* (x >> n**2) |
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
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* Let s=sin(x), c=cos(x), |
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then |
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* |
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* n sin(xn)*sqt2 cos(xn)*sqt2 |
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* ---------------------------------- |
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* 0 s-c c+s |
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* 1 -s-c -c+s |
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* 2 -s+c -c-s |
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* 3 s+c c-s |
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*/ |
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switch(n&3) { |
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case 0: temp = fd_sin(x)-fd_cos(x); break; |
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case 1: temp = -fd_sin(x)-fd_cos(x); break; |
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case 2: temp = -fd_sin(x)+fd_cos(x); break; |
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case 3: temp = fd_sin(x)+fd_cos(x); break; |
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} |
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b = invsqrtpi*temp/fd_sqrt(x); |
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} else { |
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a = __ieee754_y0(x); |
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b = __ieee754_y1(x); |
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/* quit if b is -inf */ |
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u.d = b; |
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for(i=1;i<n&&(__HI(u) != 0xfff00000);i++){ |
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temp = b; |
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b = ((double)(i+i)/x)*b - a; |
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a = temp; |
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} |
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} |
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if(sign>0) return b; else return -b; |
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} |
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