1 /* $NetBSD: n_jn.c,v 1.4 1999/07/02 15:37:37 simonb 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. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 */ 34 35 #ifndef lint 36 #if 0 37 static char sccsid[] = "@(#)jn.c 8.2 (Berkeley) 11/30/93"; 38 #endif 39 #endif /* not lint */ 40 41 /* 42 * 16 December 1992 43 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture. 44 */ 45 46 /* 47 * ==================================================== 48 * Copyright (C) 1992 by Sun Microsystems, Inc. 49 * 50 * Developed at SunPro, a Sun Microsystems, Inc. business. 51 * Permission to use, copy, modify, and distribute this 52 * software is freely granted, provided that this notice 53 * is preserved. 54 * ==================================================== 55 * 56 * ******************* WARNING ******************** 57 * This is an alpha version of SunPro's FDLIBM (Freely 58 * Distributable Math Library) for IEEE double precision 59 * arithmetic. FDLIBM is a basic math library written 60 * in C that runs on machines that conform to IEEE 61 * Standard 754/854. This alpha version is distributed 62 * for testing purpose. Those who use this software 63 * should report any bugs to 64 * 65 * fdlibm-comments@sunpro.eng.sun.com 66 * 67 * -- K.C. Ng, Oct 12, 1992 68 * ************************************************ 69 */ 70 71 /* 72 * jn(int n, double x), yn(int n, double x) 73 * floating point Bessel's function of the 1st and 2nd kind 74 * of order n 75 * 76 * Special cases: 77 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 78 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 79 * Note 2. About jn(n,x), yn(n,x) 80 * For n=0, j0(x) is called, 81 * for n=1, j1(x) is called, 82 * for n<x, forward recursion us used starting 83 * from values of j0(x) and j1(x). 84 * for n>x, a continued fraction approximation to 85 * j(n,x)/j(n-1,x) is evaluated and then backward 86 * recursion is used starting from a supposed value 87 * for j(n,x). The resulting value of j(0,x) is 88 * compared with the actual value to correct the 89 * supposed value of j(n,x). 90 * 91 * yn(n,x) is similar in all respects, except 92 * that forward recursion is used for all 93 * values of n>1. 94 * 95 */ 96 97 #include "mathimpl.h" 98 #include <float.h> 99 #include <errno.h> 100 101 #if defined(__vax__) || defined(tahoe) 102 #define _IEEE 0 103 #else 104 #define _IEEE 1 105 #define infnan(x) (0.0) 106 #endif 107 108 static double 109 invsqrtpi= 5.641895835477562869480794515607725858441e-0001, 110 two = 2.0, 111 zero = 0.0, 112 one = 1.0; 113 114 double jn(n,x) 115 int n; double x; 116 { 117 int i, sgn; 118 double a, b, temp; 119 double z, w; 120 121 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 122 * Thus, J(-n,x) = J(n,-x) 123 */ 124 /* if J(n,NaN) is NaN */ 125 if (_IEEE && isnan(x)) return x+x; 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 && x >= 8.148143905337944345e+090) { 139 /* x >= 2**302 */ 140 /* (x >> n**2) 141 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 142 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 143 * Let s=sin(x), c=cos(x), 144 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 145 * 146 * n sin(xn)*sqt2 cos(xn)*sqt2 147 * ---------------------------------- 148 * 0 s-c c+s 149 * 1 -s-c -c+s 150 * 2 -s+c -c-s 151 * 3 s+c c-s 152 */ 153 switch(n&3) { 154 case 0: temp = cos(x)+sin(x); break; 155 case 1: temp = -cos(x)+sin(x); break; 156 case 2: temp = -cos(x)-sin(x); break; 157 case 3: temp = cos(x)-sin(x); break; 158 } 159 b = invsqrtpi*temp/sqrt(x); 160 } else { 161 a = j0(x); 162 b = j1(x); 163 for(i=1;i<n;i++){ 164 temp = b; 165 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 166 a = temp; 167 } 168 } 169 } else { 170 if (x < 1.86264514923095703125e-009) { /* x < 2**-29 */ 171 /* x is tiny, return the first Taylor expansion of J(n,x) 172 * J(n,x) = 1/n!*(x/2)^n - ... 173 */ 174 if (n > 33) /* underflow */ 175 b = zero; 176 else { 177 temp = x*0.5; b = temp; 178 for (a=one,i=2;i<=n;i++) { 179 a *= (double)i; /* a = n! */ 180 b *= temp; /* b = (x/2)^n */ 181 } 182 b = b/a; 183 } 184 } else { 185 /* use backward recurrence */ 186 /* x x^2 x^2 187 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 188 * 2n - 2(n+1) - 2(n+2) 189 * 190 * 1 1 1 191 * (for large x) = ---- ------ ------ ..... 192 * 2n 2(n+1) 2(n+2) 193 * -- - ------ - ------ - 194 * x x x 195 * 196 * Let w = 2n/x and h=2/x, then the above quotient 197 * is equal to the continued fraction: 198 * 1 199 * = ----------------------- 200 * 1 201 * w - ----------------- 202 * 1 203 * w+h - --------- 204 * w+2h - ... 205 * 206 * To determine how many terms needed, let 207 * Q(0) = w, Q(1) = w(w+h) - 1, 208 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 209 * When Q(k) > 1e4 good for single 210 * When Q(k) > 1e9 good for double 211 * When Q(k) > 1e17 good for quadruple 212 */ 213 /* determine k */ 214 double t,v; 215 double q0,q1,h,tmp; int k,m; 216 w = (n+n)/(double)x; h = 2.0/(double)x; 217 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 218 while (q1<1.0e9) { 219 k += 1; z += h; 220 tmp = z*q1 - q0; 221 q0 = q1; 222 q1 = tmp; 223 } 224 m = n+n; 225 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 226 a = t; 227 b = one; 228 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 229 * Hence, if n*(log(2n/x)) > ... 230 * single 8.8722839355e+01 231 * double 7.09782712893383973096e+02 232 * long double 1.1356523406294143949491931077970765006170e+04 233 * then recurrent value may overflow and the result will 234 * likely underflow to zero 235 */ 236 tmp = n; 237 v = two/x; 238 tmp = tmp*log(fabs(v*tmp)); 239 for (i=n-1;i>0;i--){ 240 temp = b; 241 b = ((i+i)/x)*b - a; 242 a = temp; 243 /* scale b to avoid spurious overflow */ 244 # if defined(__vax__) || defined(tahoe) 245 # define BMAX 1e13 246 # else 247 # define BMAX 1e100 248 # endif /* defined(__vax__) || defined(tahoe) */ 249 if (b > BMAX) { 250 a /= b; 251 t /= b; 252 b = one; 253 } 254 } 255 b = (t*j0(x)/b); 256 } 257 } 258 return ((sgn == 1) ? -b : b); 259 } 260 double yn(n,x) 261 int n; double x; 262 { 263 int i, sign; 264 double a, b, temp; 265 266 /* Y(n,NaN), Y(n, x < 0) is NaN */ 267 if (x <= 0 || (_IEEE && x != x)) 268 if (_IEEE && x < 0) return zero/zero; 269 else if (x < 0) return (infnan(EDOM)); 270 else if (_IEEE) return -one/zero; 271 else return(infnan(-ERANGE)); 272 else if (!finite(x)) return(0); 273 sign = 1; 274 if (n<0){ 275 n = -n; 276 sign = 1 - ((n&1)<<2); 277 } 278 if (n == 0) return(y0(x)); 279 if (n == 1) return(sign*y1(x)); 280 if(_IEEE && x >= 8.148143905337944345e+090) { /* x > 2**302 */ 281 /* (x >> n**2) 282 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 283 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 284 * Let s=sin(x), c=cos(x), 285 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 286 * 287 * n sin(xn)*sqt2 cos(xn)*sqt2 288 * ---------------------------------- 289 * 0 s-c c+s 290 * 1 -s-c -c+s 291 * 2 -s+c -c-s 292 * 3 s+c c-s 293 */ 294 switch (n&3) { 295 case 0: temp = sin(x)-cos(x); break; 296 case 1: temp = -sin(x)-cos(x); break; 297 case 2: temp = -sin(x)+cos(x); break; 298 case 3: temp = sin(x)+cos(x); break; 299 } 300 b = invsqrtpi*temp/sqrt(x); 301 } else { 302 a = y0(x); 303 b = y1(x); 304 /* quit if b is -inf */ 305 for (i = 1; i < n && !finite(b); i++){ 306 temp = b; 307 b = ((double)(i+i)/x)*b - a; 308 a = temp; 309 } 310 } 311 if (!_IEEE && !finite(b)) 312 return (infnan(-sign * ERANGE)); 313 return ((sign > 0) ? b : -b); 314 } 315