n_jn.c revision 1.1
1 /* $NetBSD: n_jn.c,v 1.1 1995/10/10 23:36:54 ragge 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 static char sccsid[] = "@(#)jn.c 8.2 (Berkeley) 11/30/93"; 37 #endif /* not lint */ 38 39 /* 40 * 16 December 1992 41 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture. 42 */ 43 44 /* 45 * ==================================================== 46 * Copyright (C) 1992 by Sun Microsystems, Inc. 47 * 48 * Developed at SunPro, a Sun Microsystems, Inc. business. 49 * Permission to use, copy, modify, and distribute this 50 * software is freely granted, provided that this notice 51 * is preserved. 52 * ==================================================== 53 * 54 * ******************* WARNING ******************** 55 * This is an alpha version of SunPro's FDLIBM (Freely 56 * Distributable Math Library) for IEEE double precision 57 * arithmetic. FDLIBM is a basic math library written 58 * in C that runs on machines that conform to IEEE 59 * Standard 754/854. This alpha version is distributed 60 * for testing purpose. Those who use this software 61 * should report any bugs to 62 * 63 * fdlibm-comments (at) sunpro.eng.sun.com 64 * 65 * -- K.C. Ng, Oct 12, 1992 66 * ************************************************ 67 */ 68 69 /* 70 * jn(int n, double x), yn(int n, double x) 71 * floating point Bessel's function of the 1st and 2nd kind 72 * of order n 73 * 74 * Special cases: 75 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 76 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 77 * Note 2. About jn(n,x), yn(n,x) 78 * For n=0, j0(x) is called, 79 * for n=1, j1(x) is called, 80 * for n<x, forward recursion us used starting 81 * from values of j0(x) and j1(x). 82 * for n>x, a continued fraction approximation to 83 * j(n,x)/j(n-1,x) is evaluated and then backward 84 * recursion is used starting from a supposed value 85 * for j(n,x). The resulting value of j(0,x) is 86 * compared with the actual value to correct the 87 * supposed value of j(n,x). 88 * 89 * yn(n,x) is similar in all respects, except 90 * that forward recursion is used for all 91 * values of n>1. 92 * 93 */ 94 95 #include <math.h> 96 #include <float.h> 97 #include <errno.h> 98 99 #if defined(vax) || defined(tahoe) 100 #define _IEEE 0 101 #else 102 #define _IEEE 1 103 #define infnan(x) (0.0) 104 #endif 105 106 static double 107 invsqrtpi= 5.641895835477562869480794515607725858441e-0001, 108 two = 2.0, 109 zero = 0.0, 110 one = 1.0; 111 112 double jn(n,x) 113 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 && isnan(x)) return x+x; 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 && x >= 8.148143905337944345e+090) { 137 /* x >= 2**302 */ 138 /* (x >> n**2) 139 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 140 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 141 * Let s=sin(x), c=cos(x), 142 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 143 * 144 * n sin(xn)*sqt2 cos(xn)*sqt2 145 * ---------------------------------- 146 * 0 s-c c+s 147 * 1 -s-c -c+s 148 * 2 -s+c -c-s 149 * 3 s+c c-s 150 */ 151 switch(n&3) { 152 case 0: temp = cos(x)+sin(x); break; 153 case 1: temp = -cos(x)+sin(x); break; 154 case 2: temp = -cos(x)-sin(x); break; 155 case 3: temp = cos(x)-sin(x); break; 156 } 157 b = invsqrtpi*temp/sqrt(x); 158 } else { 159 a = j0(x); 160 b = j1(x); 161 for(i=1;i<n;i++){ 162 temp = b; 163 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 164 a = temp; 165 } 166 } 167 } else { 168 if (x < 1.86264514923095703125e-009) { /* x < 2**-29 */ 169 /* x is tiny, return the first Taylor expansion of J(n,x) 170 * J(n,x) = 1/n!*(x/2)^n - ... 171 */ 172 if (n > 33) /* underflow */ 173 b = zero; 174 else { 175 temp = x*0.5; b = temp; 176 for (a=one,i=2;i<=n;i++) { 177 a *= (double)i; /* a = n! */ 178 b *= temp; /* b = (x/2)^n */ 179 } 180 b = b/a; 181 } 182 } else { 183 /* use backward recurrence */ 184 /* x x^2 x^2 185 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 186 * 2n - 2(n+1) - 2(n+2) 187 * 188 * 1 1 1 189 * (for large x) = ---- ------ ------ ..... 190 * 2n 2(n+1) 2(n+2) 191 * -- - ------ - ------ - 192 * x x x 193 * 194 * Let w = 2n/x and h=2/x, then the above quotient 195 * is equal to the continued fraction: 196 * 1 197 * = ----------------------- 198 * 1 199 * w - ----------------- 200 * 1 201 * w+h - --------- 202 * w+2h - ... 203 * 204 * To determine how many terms needed, let 205 * Q(0) = w, Q(1) = w(w+h) - 1, 206 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 207 * When Q(k) > 1e4 good for single 208 * When Q(k) > 1e9 good for double 209 * When Q(k) > 1e17 good for quadruple 210 */ 211 /* determine k */ 212 double t,v; 213 double q0,q1,h,tmp; int k,m; 214 w = (n+n)/(double)x; h = 2.0/(double)x; 215 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 216 while (q1<1.0e9) { 217 k += 1; z += h; 218 tmp = z*q1 - q0; 219 q0 = q1; 220 q1 = tmp; 221 } 222 m = n+n; 223 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 224 a = t; 225 b = one; 226 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 227 * Hence, if n*(log(2n/x)) > ... 228 * single 8.8722839355e+01 229 * double 7.09782712893383973096e+02 230 * long double 1.1356523406294143949491931077970765006170e+04 231 * then recurrent value may overflow and the result will 232 * likely underflow to zero 233 */ 234 tmp = n; 235 v = two/x; 236 tmp = tmp*log(fabs(v*tmp)); 237 for (i=n-1;i>0;i--){ 238 temp = b; 239 b = ((i+i)/x)*b - a; 240 a = temp; 241 /* scale b to avoid spurious overflow */ 242 # if defined(vax) || defined(tahoe) 243 # define BMAX 1e13 244 # else 245 # define BMAX 1e100 246 # endif /* defined(vax) || defined(tahoe) */ 247 if (b > BMAX) { 248 a /= b; 249 t /= b; 250 b = one; 251 } 252 } 253 b = (t*j0(x)/b); 254 } 255 } 256 return ((sgn == 1) ? -b : b); 257 } 258 double yn(n,x) 259 int n; double x; 260 { 261 int i, sign; 262 double a, b, temp; 263 264 /* Y(n,NaN), Y(n, x < 0) is NaN */ 265 if (x <= 0 || (_IEEE && x != x)) 266 if (_IEEE && x < 0) return zero/zero; 267 else if (x < 0) return (infnan(EDOM)); 268 else if (_IEEE) return -one/zero; 269 else return(infnan(-ERANGE)); 270 else if (!finite(x)) return(0); 271 sign = 1; 272 if (n<0){ 273 n = -n; 274 sign = 1 - ((n&1)<<2); 275 } 276 if (n == 0) return(y0(x)); 277 if (n == 1) return(sign*y1(x)); 278 if(_IEEE && x >= 8.148143905337944345e+090) { /* x > 2**302 */ 279 /* (x >> n**2) 280 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 281 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 282 * Let s=sin(x), c=cos(x), 283 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 284 * 285 * n sin(xn)*sqt2 cos(xn)*sqt2 286 * ---------------------------------- 287 * 0 s-c c+s 288 * 1 -s-c -c+s 289 * 2 -s+c -c-s 290 * 3 s+c c-s 291 */ 292 switch (n&3) { 293 case 0: temp = sin(x)-cos(x); break; 294 case 1: temp = -sin(x)-cos(x); break; 295 case 2: temp = -sin(x)+cos(x); break; 296 case 3: temp = sin(x)+cos(x); break; 297 } 298 b = invsqrtpi*temp/sqrt(x); 299 } else { 300 a = y0(x); 301 b = y1(x); 302 /* quit if b is -inf */ 303 for (i = 1; i < n && !finite(b); i++){ 304 temp = b; 305 b = ((double)(i+i)/x)*b - a; 306 a = temp; 307 } 308 } 309 if (!_IEEE && !finite(b)) 310 return (infnan(-sign * ERANGE)); 311 return ((sign > 0) ? b : -b); 312 } 313
Indexes created Wed Sep 24 05:09:52 GMT 2025