/* $NetBSD: n_jn.c,v 1.8 2018/03/05 23:00:55 christos Exp $ */ /*- * Copyright (c) 1992, 1993 * The Regents of the University of California. All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * 3. Neither the name of the University nor the names of its contributors * may be used to endorse or promote products derived from this software * without specific prior written permission. * * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF * SUCH DAMAGE. */ #ifndef lint #if 0 static char sccsid[] = "@(#)jn.c 8.2 (Berkeley) 11/30/93"; #endif #endif /* not lint */ /* * 16 December 1992 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture. */ /* * ==================================================== * Copyright (C) 1992 by Sun Microsystems, Inc. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== * * ******************* WARNING ******************** * This is an alpha version of SunPro's FDLIBM (Freely * Distributable Math Library) for IEEE double precision * arithmetic. FDLIBM is a basic math library written * in C that runs on machines that conform to IEEE * Standard 754/854. This alpha version is distributed * for testing purpose. Those who use this software * should report any bugs to * * fdlibm-comments@sunpro.eng.sun.com * * -- K.C. Ng, Oct 12, 1992 * ************************************************ */ /* * jn(int n, double x), yn(int n, double x) * floating point Bessel's function of the 1st and 2nd kind * of order n * * Special cases: * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. * Note 2. About jn(n,x), yn(n,x) * For n=0, j0(x) is called, * for n=1, j1(x) is called, * for n<x, forward recursion us used starting * from values of j0(x) and j1(x). * for n>x, a continued fraction approximation to * j(n,x)/j(n-1,x) is evaluated and then backward * recursion is used starting from a supposed value * for j(n,x). The resulting value of j(0,x) is * compared with the actual value to correct the * supposed value of j(n,x). * * yn(n,x) is similar in all respects, except * that forward recursion is used for all * values of n>1. * */ #include "mathimpl.h" #include <float.h> #include <errno.h> #if defined(__vax__) || defined(tahoe) #define _IEEE 0 #else #define _IEEE 1 #define infnan(x) (0.0) #endif static const double #if _IEEE invsqrtpi= 5.641895835477562869480794515607725858441e-0001, #endif two = 2.0, zero = 0.0, one = 1.0; double jn(int n, double x) { int i, sgn; double a, b, temp; double z, w; /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) * Thus, J(-n,x) = J(n,-x) */ /* if J(n,NaN) is NaN */ #if _IEEE if (snan(x)) return x+x; #endif if (n<0){ n = -n; x = -x; } if (n==0) return(j0(x)); if (n==1) return(j1(x)); sgn = (n&1)&(x < zero); /* even n -- 0, odd n -- sign(x) */ x = fabs(x); if (x == 0 || !finite (x)) /* if x is 0 or inf */ b = zero; else if ((double) n <= x) { /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ #if _IEEE if (x >= 8.148143905337944345e+090) { /* x >= 2**302 */ /* (x >> n**2) * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) * Let s=sin(x), c=cos(x), * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then * * n sin(xn)*sqt2 cos(xn)*sqt2 * ---------------------------------- * 0 s-c c+s * 1 -s-c -c+s * 2 -s+c -c-s * 3 s+c c-s */ switch(n&3) { case 0: temp = cos(x)+sin(x); break; case 1: temp = -cos(x)+sin(x); break; case 2: temp = -cos(x)-sin(x); break; case 3: temp = cos(x)-sin(x); break; } b = invsqrtpi*temp/sqrt(x); } else #endif { a = j0(x); b = j1(x); for(i=1;i<n;i++){ temp = b; b = b*((double)(i+i)/x) - a; /* avoid underflow */ a = temp; } } } else { if (x < 1.86264514923095703125e-009) { /* x < 2**-29 */ /* x is tiny, return the first Taylor expansion of J(n,x) * J(n,x) = 1/n!*(x/2)^n - ... */ if (n > 33) /* underflow */ b = zero; else { temp = x*0.5; b = temp; for (a=one,i=2;i<=n;i++) { a *= (double)i; /* a = n! */ b *= temp; /* b = (x/2)^n */ } b = b/a; } } else { /* use backward recurrence */ /* x x^2 x^2 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... * 2n - 2(n+1) - 2(n+2) * * 1 1 1 * (for large x) = ---- ------ ------ ..... * 2n 2(n+1) 2(n+2) * -- - ------ - ------ - * x x x * * Let w = 2n/x and h=2/x, then the above quotient * is equal to the continued fraction: * 1 * = ----------------------- * 1 * w - ----------------- * 1 * w+h - --------- * w+2h - ... * * To determine how many terms needed, let * Q(0) = w, Q(1) = w(w+h) - 1, * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), * When Q(k) > 1e4 good for single * When Q(k) > 1e9 good for double * When Q(k) > 1e17 good for quadruple */ /* determine k */ double t,v; double q0,q1,h,tmp; int k,m; w = (n+n)/(double)x; h = 2.0/(double)x; q0 = w; z = w+h; q1 = w*z - 1.0; k=1; while (q1<1.0e9) { k += 1; z += h; tmp = z*q1 - q0; q0 = q1; q1 = tmp; } m = n+n; for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); a = t; b = one; /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) * Hence, if n*(log(2n/x)) > ... * single 8.8722839355e+01 * double 7.09782712893383973096e+02 * long double 1.1356523406294143949491931077970765006170e+04 * then recurrent value may overflow and the result will * likely underflow to zero */ tmp = n; v = two/x; tmp = tmp*log(fabs(v*tmp)); for (i=n-1;i>0;i--){ temp = b; b = ((i+i)/x)*b - a; a = temp; /* scale b to avoid spurious overflow */ # if defined(__vax__) || defined(tahoe) # define BMAX 1e13 # else # define BMAX 1e100 # endif /* defined(__vax__) || defined(tahoe) */ if (b > BMAX) { a /= b; t /= b; b = one; } } b = (t*j0(x)/b); } } return ((sgn == 1) ? -b : b); } double yn(int n, double x) { int i, sign; double a, b, temp; /* Y(n,NaN), Y(n, x < 0) is NaN */ if (x <= 0 || (_IEEE && x != x)) if (_IEEE && x < 0) return zero/zero; else if (x < 0) return (infnan(EDOM)); else if (_IEEE) return -one/zero; else return(infnan(-ERANGE)); else if (!finite(x)) return(0); sign = 1; if (n<0){ n = -n; sign = 1 - ((n&1)<<2); } if (n == 0) return(y0(x)); if (n == 1) return(sign*y1(x)); #if _IEEE if(x >= 8.148143905337944345e+090) { /* x > 2**302 */ /* (x >> n**2) * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) * Let s=sin(x), c=cos(x), * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then * * n sin(xn)*sqt2 cos(xn)*sqt2 * ---------------------------------- * 0 s-c c+s * 1 -s-c -c+s * 2 -s+c -c-s * 3 s+c c-s */ switch (n&3) { case 0: temp = sin(x)-cos(x); break; case 1: temp = -sin(x)-cos(x); break; case 2: temp = -sin(x)+cos(x); break; case 3: temp = sin(x)+cos(x); break; } b = invsqrtpi*temp/sqrt(x); } else #endif { a = y0(x); b = y1(x); /* quit if b is -inf */ for (i = 1; i < n && !finite(b); i++){ temp = b; b = ((double)(i+i)/x)*b - a; a = temp; } } if (!_IEEE && !finite(b)) return (infnan(-sign * ERANGE)); return ((sign > 0) ? b : -b); }
