n_jn.c revision 1.7
1 /* $NetBSD: n_jn.c,v 1.7 2011/11/02 02:34:56 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 33 static 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 (at) 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 103 104 static const double 105 invsqrtpi= 5.641895835477562869480794515607725858441e-0001, 106 two = 2.0, 107 zero = 0.0, 108 one = 1.0; 109 110 double 111 jn(int n, double x) 112 { 113 int i, sgn; 114 double a, b, temp; 115 double z, w; 116 117 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 118 * Thus, J(-n,x) = J(n,-x) 119 */ 120 /* if J(n,NaN) is NaN */ 121 #if _IEEE 122 if (snan(x)) return x+x; 123 #endif 124 if (n<0){ 125 n = -n; 126 x = -x; 127 } 128 if (n==0) return(j0(x)); 129 if (n==1) return(j1(x)); 130 sgn = (n&1)&(x < zero); /* even n -- 0, odd n -- sign(x) */ 131 x = fabs(x); 132 if (x == 0 || !finite (x)) /* if x is 0 or inf */ 133 b = zero; 134 else if ((double) n <= x) { 135 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 136 #if _IEEE 137 if (x >= 8.148143905337944345e+090) { 138 /* x >= 2**302 */ 139 /* (x >> n**2) 140 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 141 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 142 * Let s=sin(x), c=cos(x), 143 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 144 * 145 * n sin(xn)*sqt2 cos(xn)*sqt2 146 * ---------------------------------- 147 * 0 s-c c+s 148 * 1 -s-c -c+s 149 * 2 -s+c -c-s 150 * 3 s+c c-s 151 */ 152 switch(n&3) { 153 case 0: temp = cos(x)+sin(x); break; 154 case 1: temp = -cos(x)+sin(x); break; 155 case 2: temp = -cos(x)-sin(x); break; 156 case 3: temp = cos(x)-sin(x); break; 157 } 158 b = invsqrtpi*temp/sqrt(x); 159 } else 160 #endif 161 { 162 a = j0(x); 163 b = j1(x); 164 for(i=1;i<n;i++){ 165 temp = b; 166 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 167 a = temp; 168 } 169 } 170 } else { 171 if (x < 1.86264514923095703125e-009) { /* x < 2**-29 */ 172 /* x is tiny, return the first Taylor expansion of J(n,x) 173 * J(n,x) = 1/n!*(x/2)^n - ... 174 */ 175 if (n > 33) /* underflow */ 176 b = zero; 177 else { 178 temp = x*0.5; b = temp; 179 for (a=one,i=2;i<=n;i++) { 180 a *= (double)i; /* a = n! */ 181 b *= temp; /* b = (x/2)^n */ 182 } 183 b = b/a; 184 } 185 } else { 186 /* use backward recurrence */ 187 /* x x^2 x^2 188 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 189 * 2n - 2(n+1) - 2(n+2) 190 * 191 * 1 1 1 192 * (for large x) = ---- ------ ------ ..... 193 * 2n 2(n+1) 2(n+2) 194 * -- - ------ - ------ - 195 * x x x 196 * 197 * Let w = 2n/x and h=2/x, then the above quotient 198 * is equal to the continued fraction: 199 * 1 200 * = ----------------------- 201 * 1 202 * w - ----------------- 203 * 1 204 * w+h - --------- 205 * w+2h - ... 206 * 207 * To determine how many terms needed, let 208 * Q(0) = w, Q(1) = w(w+h) - 1, 209 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 210 * When Q(k) > 1e4 good for single 211 * When Q(k) > 1e9 good for double 212 * When Q(k) > 1e17 good for quadruple 213 */ 214 /* determine k */ 215 double t,v; 216 double q0,q1,h,tmp; int k,m; 217 w = (n+n)/(double)x; h = 2.0/(double)x; 218 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 219 while (q1<1.0e9) { 220 k += 1; z += h; 221 tmp = z*q1 - q0; 222 q0 = q1; 223 q1 = tmp; 224 } 225 m = n+n; 226 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 227 a = t; 228 b = one; 229 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 230 * Hence, if n*(log(2n/x)) > ... 231 * single 8.8722839355e+01 232 * double 7.09782712893383973096e+02 233 * long double 1.1356523406294143949491931077970765006170e+04 234 * then recurrent value may overflow and the result will 235 * likely underflow to zero 236 */ 237 tmp = n; 238 v = two/x; 239 tmp = tmp*log(fabs(v*tmp)); 240 for (i=n-1;i>0;i--){ 241 temp = b; 242 b = ((i+i)/x)*b - a; 243 a = temp; 244 /* scale b to avoid spurious overflow */ 245 # if defined(__vax__) || defined(tahoe) 246 # define BMAX 1e13 247 # else 248 # define BMAX 1e100 249 # endif /* defined(__vax__) || defined(tahoe) */ 250 if (b > BMAX) { 251 a /= b; 252 t /= b; 253 b = one; 254 } 255 } 256 b = (t*j0(x)/b); 257 } 258 } 259 return ((sgn == 1) ? -b : b); 260 } 261 262 double 263 yn(int n, double x) 264 { 265 int i, sign; 266 double a, b, temp; 267 268 /* Y(n,NaN), Y(n, x < 0) is NaN */ 269 if (x <= 0 || (_IEEE && x != x)) 270 if (_IEEE && x < 0) return zero/zero; 271 else if (x < 0) return (infnan(EDOM)); 272 else if (_IEEE) return -one/zero; 273 else return(infnan(-ERANGE)); 274 else if (!finite(x)) return(0); 275 sign = 1; 276 if (n<0){ 277 n = -n; 278 sign = 1 - ((n&1)<<2); 279 } 280 if (n == 0) return(y0(x)); 281 if (n == 1) return(sign*y1(x)); 282 #if _IEEE 283 if(x >= 8.148143905337944345e+090) { /* x > 2**302 */ 284 /* (x >> n**2) 285 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 286 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 287 * Let s=sin(x), c=cos(x), 288 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 289 * 290 * n sin(xn)*sqt2 cos(xn)*sqt2 291 * ---------------------------------- 292 * 0 s-c c+s 293 * 1 -s-c -c+s 294 * 2 -s+c -c-s 295 * 3 s+c c-s 296 */ 297 switch (n&3) { 298 case 0: temp = sin(x)-cos(x); break; 299 case 1: temp = -sin(x)-cos(x); break; 300 case 2: temp = -sin(x)+cos(x); break; 301 case 3: temp = sin(x)+cos(x); break; 302 } 303 b = invsqrtpi*temp/sqrt(x); 304 } else 305 #endif 306 { 307 a = y0(x); 308 b = y1(x); 309 /* quit if b is -inf */ 310 for (i = 1; i < n && !finite(b); i++){ 311 temp = b; 312 b = ((double)(i+i)/x)*b - a; 313 a = temp; 314 } 315 } 316 if (!_IEEE && !finite(b)) 317 return (infnan(-sign * ERANGE)); 318 return ((sign > 0) ? b : -b); 319 } 320
Indexes created Wed Sep 24 15:09:51 GMT 2025