1/* $NetBSD: n_jn.c,v 1.8 2018/03/05 23:00:55 christos Exp $ */ 2/*- 3 * Copyright (c) 1992, 1993 4 * The Regents of the University of California. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 3. Neither the name of the University nor the names of its contributors 15 * may be used to endorse or promote products derived from this software 16 * without specific prior written permission. 17 * 18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 28 * SUCH DAMAGE. 29 */ 30 31#ifndef lint 32#if 0 33static char sccsid[] = "@(#)jn.c 8.2 (Berkeley) 11/30/93"; 34#endif 35#endif /* not lint */ 36 37/* 38 * 16 December 1992 39 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture. 40 */ 41 42/* 43 * ==================================================== 44 * Copyright (C) 1992 by Sun Microsystems, Inc. 45 * 46 * Developed at SunPro, a Sun Microsystems, Inc. business. 47 * Permission to use, copy, modify, and distribute this 48 * software is freely granted, provided that this notice 49 * is preserved. 50 * ==================================================== 51 * 52 * ******************* WARNING ******************** 53 * This is an alpha version of SunPro's FDLIBM (Freely 54 * Distributable Math Library) for IEEE double precision 55 * arithmetic. FDLIBM is a basic math library written 56 * in C that runs on machines that conform to IEEE 57 * Standard 754/854. This alpha version is distributed 58 * for testing purpose. Those who use this software 59 * should report any bugs to 60 * 61 * fdlibm-comments@sunpro.eng.sun.com 62 * 63 * -- K.C. Ng, Oct 12, 1992 64 * ************************************************ 65 */ 66 67/* 68 * jn(int n, double x), yn(int n, double x) 69 * floating point Bessel's function of the 1st and 2nd kind 70 * of order n 71 * 72 * Special cases: 73 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 74 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 75 * Note 2. About jn(n,x), yn(n,x) 76 * For n=0, j0(x) is called, 77 * for n=1, j1(x) is called, 78 * for n<x, forward recursion us used starting 79 * from values of j0(x) and j1(x). 80 * for n>x, a continued fraction approximation to 81 * j(n,x)/j(n-1,x) is evaluated and then backward 82 * recursion is used starting from a supposed value 83 * for j(n,x). The resulting value of j(0,x) is 84 * compared with the actual value to correct the 85 * supposed value of j(n,x). 86 * 87 * yn(n,x) is similar in all respects, except 88 * that forward recursion is used for all 89 * values of n>1. 90 * 91 */ 92 93#include "mathimpl.h" 94#include <float.h> 95#include <errno.h> 96 97#if defined(__vax__) || defined(tahoe) 98#define _IEEE 0 99#else 100#define _IEEE 1 101#define infnan(x) (0.0) 102#endif 103 104static const double 105#if _IEEE 106invsqrtpi= 5.641895835477562869480794515607725858441e-0001, 107#endif 108two = 2.0, 109zero = 0.0, 110one = 1.0; 111 112double 113jn(int n, double x) 114{ 115 int i, sgn; 116 double a, b, temp; 117 double z, w; 118 119 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 120 * Thus, J(-n,x) = J(n,-x) 121 */ 122 /* if J(n,NaN) is NaN */ 123#if _IEEE 124 if (snan(x)) return x+x; 125#endif 126 if (n<0){ 127 n = -n; 128 x = -x; 129 } 130 if (n==0) return(j0(x)); 131 if (n==1) return(j1(x)); 132 sgn = (n&1)&(x < zero); /* even n -- 0, odd n -- sign(x) */ 133 x = fabs(x); 134 if (x == 0 || !finite (x)) /* if x is 0 or inf */ 135 b = zero; 136 else if ((double) n <= x) { 137 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 138#if _IEEE 139 if (x >= 8.148143905337944345e+090) { 140 /* x >= 2**302 */ 141 /* (x >> n**2) 142 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 143 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 144 * Let s=sin(x), c=cos(x), 145 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 146 * 147 * n sin(xn)*sqt2 cos(xn)*sqt2 148 * ---------------------------------- 149 * 0 s-c c+s 150 * 1 -s-c -c+s 151 * 2 -s+c -c-s 152 * 3 s+c c-s 153 */ 154 switch(n&3) { 155 case 0: temp = cos(x)+sin(x); break; 156 case 1: temp = -cos(x)+sin(x); break; 157 case 2: temp = -cos(x)-sin(x); break; 158 case 3: temp = cos(x)-sin(x); break; 159 } 160 b = invsqrtpi*temp/sqrt(x); 161 } else 162#endif 163 { 164 a = j0(x); 165 b = j1(x); 166 for(i=1;i<n;i++){ 167 temp = b; 168 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 169 a = temp; 170 } 171 } 172 } else { 173 if (x < 1.86264514923095703125e-009) { /* x < 2**-29 */ 174 /* x is tiny, return the first Taylor expansion of J(n,x) 175 * J(n,x) = 1/n!*(x/2)^n - ... 176 */ 177 if (n > 33) /* underflow */ 178 b = zero; 179 else { 180 temp = x*0.5; b = temp; 181 for (a=one,i=2;i<=n;i++) { 182 a *= (double)i; /* a = n! */ 183 b *= temp; /* b = (x/2)^n */ 184 } 185 b = b/a; 186 } 187 } else { 188 /* use backward recurrence */ 189 /* x x^2 x^2 190 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 191 * 2n - 2(n+1) - 2(n+2) 192 * 193 * 1 1 1 194 * (for large x) = ---- ------ ------ ..... 195 * 2n 2(n+1) 2(n+2) 196 * -- - ------ - ------ - 197 * x x x 198 * 199 * Let w = 2n/x and h=2/x, then the above quotient 200 * is equal to the continued fraction: 201 * 1 202 * = ----------------------- 203 * 1 204 * w - ----------------- 205 * 1 206 * w+h - --------- 207 * w+2h - ... 208 * 209 * To determine how many terms needed, let 210 * Q(0) = w, Q(1) = w(w+h) - 1, 211 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 212 * When Q(k) > 1e4 good for single 213 * When Q(k) > 1e9 good for double 214 * When Q(k) > 1e17 good for quadruple 215 */ 216 /* determine k */ 217 double t,v; 218 double q0,q1,h,tmp; int k,m; 219 w = (n+n)/(double)x; h = 2.0/(double)x; 220 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 221 while (q1<1.0e9) { 222 k += 1; z += h; 223 tmp = z*q1 - q0; 224 q0 = q1; 225 q1 = tmp; 226 } 227 m = n+n; 228 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 229 a = t; 230 b = one; 231 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 232 * Hence, if n*(log(2n/x)) > ... 233 * single 8.8722839355e+01 234 * double 7.09782712893383973096e+02 235 * long double 1.1356523406294143949491931077970765006170e+04 236 * then recurrent value may overflow and the result will 237 * likely underflow to zero 238 */ 239 tmp = n; 240 v = two/x; 241 tmp = tmp*log(fabs(v*tmp)); 242 for (i=n-1;i>0;i--){ 243 temp = b; 244 b = ((i+i)/x)*b - a; 245 a = temp; 246 /* scale b to avoid spurious overflow */ 247# if defined(__vax__) || defined(tahoe) 248# define BMAX 1e13 249# else 250# define BMAX 1e100 251# endif /* defined(__vax__) || defined(tahoe) */ 252 if (b > BMAX) { 253 a /= b; 254 t /= b; 255 b = one; 256 } 257 } 258 b = (t*j0(x)/b); 259 } 260 } 261 return ((sgn == 1) ? -b : b); 262} 263 264double 265yn(int n, double x) 266{ 267 int i, sign; 268 double a, b, temp; 269 270 /* Y(n,NaN), Y(n, x < 0) is NaN */ 271 if (x <= 0 || (_IEEE && x != x)) 272 if (_IEEE && x < 0) return zero/zero; 273 else if (x < 0) return (infnan(EDOM)); 274 else if (_IEEE) return -one/zero; 275 else return(infnan(-ERANGE)); 276 else if (!finite(x)) return(0); 277 sign = 1; 278 if (n<0){ 279 n = -n; 280 sign = 1 - ((n&1)<<2); 281 } 282 if (n == 0) return(y0(x)); 283 if (n == 1) return(sign*y1(x)); 284#if _IEEE 285 if(x >= 8.148143905337944345e+090) { /* x > 2**302 */ 286 /* (x >> n**2) 287 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 288 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 289 * Let s=sin(x), c=cos(x), 290 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 291 * 292 * n sin(xn)*sqt2 cos(xn)*sqt2 293 * ---------------------------------- 294 * 0 s-c c+s 295 * 1 -s-c -c+s 296 * 2 -s+c -c-s 297 * 3 s+c c-s 298 */ 299 switch (n&3) { 300 case 0: temp = sin(x)-cos(x); break; 301 case 1: temp = -sin(x)-cos(x); break; 302 case 2: temp = -sin(x)+cos(x); break; 303 case 3: temp = sin(x)+cos(x); break; 304 } 305 b = invsqrtpi*temp/sqrt(x); 306 } else 307#endif 308 { 309 a = y0(x); 310 b = y1(x); 311 /* quit if b is -inf */ 312 for (i = 1; i < n && !finite(b); i++){ 313 temp = b; 314 b = ((double)(i+i)/x)*b - a; 315 a = temp; 316 } 317 } 318 if (!_IEEE && !finite(b)) 319 return (infnan(-sign * ERANGE)); 320 return ((sign > 0) ? b : -b); 321} 322