1 /* $NetBSD: n_jn.c,v 1.6 2003/08/07 16:44:51 agc 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@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 && isnan(x)) return x+x; 122 if (n<0){ 123 n = -n; 124 x = -x; 125 } 126 if (n==0) return(j0(x)); 127 if (n==1) return(j1(x)); 128 sgn = (n&1)&(x < zero); /* even n -- 0, odd n -- sign(x) */ 129 x = fabs(x); 130 if (x == 0 || !finite (x)) /* if x is 0 or inf */ 131 b = zero; 132 else if ((double) n <= x) { 133 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 134 if (_IEEE && x >= 8.148143905337944345e+090) { 135 /* x >= 2**302 */ 136 /* (x >> n**2) 137 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 138 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 139 * Let s=sin(x), c=cos(x), 140 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 141 * 142 * n sin(xn)*sqt2 cos(xn)*sqt2 143 * ---------------------------------- 144 * 0 s-c c+s 145 * 1 -s-c -c+s 146 * 2 -s+c -c-s 147 * 3 s+c c-s 148 */ 149 switch(n&3) { 150 case 0: temp = cos(x)+sin(x); break; 151 case 1: temp = -cos(x)+sin(x); break; 152 case 2: temp = -cos(x)-sin(x); break; 153 case 3: temp = cos(x)-sin(x); break; 154 } 155 b = invsqrtpi*temp/sqrt(x); 156 } else { 157 a = j0(x); 158 b = j1(x); 159 for(i=1;i<n;i++){ 160 temp = b; 161 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 162 a = temp; 163 } 164 } 165 } else { 166 if (x < 1.86264514923095703125e-009) { /* x < 2**-29 */ 167 /* x is tiny, return the first Taylor expansion of J(n,x) 168 * J(n,x) = 1/n!*(x/2)^n - ... 169 */ 170 if (n > 33) /* underflow */ 171 b = zero; 172 else { 173 temp = x*0.5; b = temp; 174 for (a=one,i=2;i<=n;i++) { 175 a *= (double)i; /* a = n! */ 176 b *= temp; /* b = (x/2)^n */ 177 } 178 b = b/a; 179 } 180 } else { 181 /* use backward recurrence */ 182 /* x x^2 x^2 183 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 184 * 2n - 2(n+1) - 2(n+2) 185 * 186 * 1 1 1 187 * (for large x) = ---- ------ ------ ..... 188 * 2n 2(n+1) 2(n+2) 189 * -- - ------ - ------ - 190 * x x x 191 * 192 * Let w = 2n/x and h=2/x, then the above quotient 193 * is equal to the continued fraction: 194 * 1 195 * = ----------------------- 196 * 1 197 * w - ----------------- 198 * 1 199 * w+h - --------- 200 * w+2h - ... 201 * 202 * To determine how many terms needed, let 203 * Q(0) = w, Q(1) = w(w+h) - 1, 204 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 205 * When Q(k) > 1e4 good for single 206 * When Q(k) > 1e9 good for double 207 * When Q(k) > 1e17 good for quadruple 208 */ 209 /* determine k */ 210 double t,v; 211 double q0,q1,h,tmp; int k,m; 212 w = (n+n)/(double)x; h = 2.0/(double)x; 213 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 214 while (q1<1.0e9) { 215 k += 1; z += h; 216 tmp = z*q1 - q0; 217 q0 = q1; 218 q1 = tmp; 219 } 220 m = n+n; 221 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 222 a = t; 223 b = one; 224 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 225 * Hence, if n*(log(2n/x)) > ... 226 * single 8.8722839355e+01 227 * double 7.09782712893383973096e+02 228 * long double 1.1356523406294143949491931077970765006170e+04 229 * then recurrent value may overflow and the result will 230 * likely underflow to zero 231 */ 232 tmp = n; 233 v = two/x; 234 tmp = tmp*log(fabs(v*tmp)); 235 for (i=n-1;i>0;i--){ 236 temp = b; 237 b = ((i+i)/x)*b - a; 238 a = temp; 239 /* scale b to avoid spurious overflow */ 240 # if defined(__vax__) || defined(tahoe) 241 # define BMAX 1e13 242 # else 243 # define BMAX 1e100 244 # endif /* defined(__vax__) || defined(tahoe) */ 245 if (b > BMAX) { 246 a /= b; 247 t /= b; 248 b = one; 249 } 250 } 251 b = (t*j0(x)/b); 252 } 253 } 254 return ((sgn == 1) ? -b : b); 255 } 256 257 double 258 yn(int n, double x) 259 { 260 int i, sign; 261 double a, b, temp; 262 263 /* Y(n,NaN), Y(n, x < 0) is NaN */ 264 if (x <= 0 || (_IEEE && x != x)) 265 if (_IEEE && x < 0) return zero/zero; 266 else if (x < 0) return (infnan(EDOM)); 267 else if (_IEEE) return -one/zero; 268 else return(infnan(-ERANGE)); 269 else if (!finite(x)) return(0); 270 sign = 1; 271 if (n<0){ 272 n = -n; 273 sign = 1 - ((n&1)<<2); 274 } 275 if (n == 0) return(y0(x)); 276 if (n == 1) return(sign*y1(x)); 277 if(_IEEE && x >= 8.148143905337944345e+090) { /* x > 2**302 */ 278 /* (x >> n**2) 279 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 280 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 281 * Let s=sin(x), c=cos(x), 282 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 283 * 284 * n sin(xn)*sqt2 cos(xn)*sqt2 285 * ---------------------------------- 286 * 0 s-c c+s 287 * 1 -s-c -c+s 288 * 2 -s+c -c-s 289 * 3 s+c c-s 290 */ 291 switch (n&3) { 292 case 0: temp = sin(x)-cos(x); break; 293 case 1: temp = -sin(x)-cos(x); break; 294 case 2: temp = -sin(x)+cos(x); break; 295 case 3: temp = sin(x)+cos(x); break; 296 } 297 b = invsqrtpi*temp/sqrt(x); 298 } else { 299 a = y0(x); 300 b = y1(x); 301 /* quit if b is -inf */ 302 for (i = 1; i < n && !finite(b); i++){ 303 temp = b; 304 b = ((double)(i+i)/x)*b - a; 305 a = temp; 306 } 307 } 308 if (!_IEEE && !finite(b)) 309 return (infnan(-sign * ERANGE)); 310 return ((sign > 0) ? b : -b); 311 } 312