1 /* @(#)e_jn.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 #include <sys/cdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: e_jn.c,v 1.10 1997/10/09 11:29:18 lukem Exp $"); 16 #endif 17 18 /* 19 * __ieee754_jn(n, x), __ieee754_yn(n, x) 20 * floating point Bessel's function of the 1st and 2nd kind 21 * of order n 22 * 23 * Special cases: 24 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 25 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 26 * Note 2. About jn(n,x), yn(n,x) 27 * For n=0, j0(x) is called, 28 * for n=1, j1(x) is called, 29 * for n<x, forward recursion us used starting 30 * from values of j0(x) and j1(x). 31 * for n>x, a continued fraction approximation to 32 * j(n,x)/j(n-1,x) is evaluated and then backward 33 * recursion is used starting from a supposed value 34 * for j(n,x). The resulting value of j(0,x) is 35 * compared with the actual value to correct the 36 * supposed value of j(n,x). 37 * 38 * yn(n,x) is similar in all respects, except 39 * that forward recursion is used for all 40 * values of n>1. 41 * 42 */ 43 44 #include "math.h" 45 #include "math_private.h" 46 47 #ifdef __STDC__ 48 static const double 49 #else 50 static double 51 #endif 52 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 53 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 54 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 55 56 #ifdef __STDC__ 57 static const double zero = 0.00000000000000000000e+00; 58 #else 59 static double zero = 0.00000000000000000000e+00; 60 #endif 61 62 #ifdef __STDC__ 63 double __ieee754_jn(int n, double x) 64 #else 65 double __ieee754_jn(n,x) 66 int n; double x; 67 #endif 68 { 69 int32_t i,hx,ix,lx, sgn; 70 double a, b, temp, di; 71 double z, w; 72 73 temp = 0; 74 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 75 * Thus, J(-n,x) = J(n,-x) 76 */ 77 EXTRACT_WORDS(hx,lx,x); 78 ix = 0x7fffffff&hx; 79 /* if J(n,NaN) is NaN */ 80 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 81 if(n<0){ 82 n = -n; 83 x = -x; 84 hx ^= 0x80000000; 85 } 86 if(n==0) return(__ieee754_j0(x)); 87 if(n==1) return(__ieee754_j1(x)); 88 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 89 x = fabs(x); 90 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 91 b = zero; 92 else if((double)n<=x) { 93 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 94 if(ix>=0x52D00000) { /* x > 2**302 */ 95 /* (x >> n**2) 96 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 97 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 98 * Let s=sin(x), c=cos(x), 99 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 100 * 101 * n sin(xn)*sqt2 cos(xn)*sqt2 102 * ---------------------------------- 103 * 0 s-c c+s 104 * 1 -s-c -c+s 105 * 2 -s+c -c-s 106 * 3 s+c c-s 107 */ 108 switch(n&3) { 109 case 0: temp = cos(x)+sin(x); break; 110 case 1: temp = -cos(x)+sin(x); break; 111 case 2: temp = -cos(x)-sin(x); break; 112 case 3: temp = cos(x)-sin(x); break; 113 } 114 b = invsqrtpi*temp/sqrt(x); 115 } else { 116 a = __ieee754_j0(x); 117 b = __ieee754_j1(x); 118 for(i=1;i<n;i++){ 119 temp = b; 120 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 121 a = temp; 122 } 123 } 124 } else { 125 if(ix<0x3e100000) { /* x < 2**-29 */ 126 /* x is tiny, return the first Taylor expansion of J(n,x) 127 * J(n,x) = 1/n!*(x/2)^n - ... 128 */ 129 if(n>33) /* underflow */ 130 b = zero; 131 else { 132 temp = x*0.5; b = temp; 133 for (a=one,i=2;i<=n;i++) { 134 a *= (double)i; /* a = n! */ 135 b *= temp; /* b = (x/2)^n */ 136 } 137 b = b/a; 138 } 139 } else { 140 /* use backward recurrence */ 141 /* x x^2 x^2 142 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 143 * 2n - 2(n+1) - 2(n+2) 144 * 145 * 1 1 1 146 * (for large x) = ---- ------ ------ ..... 147 * 2n 2(n+1) 2(n+2) 148 * -- - ------ - ------ - 149 * x x x 150 * 151 * Let w = 2n/x and h=2/x, then the above quotient 152 * is equal to the continued fraction: 153 * 1 154 * = ----------------------- 155 * 1 156 * w - ----------------- 157 * 1 158 * w+h - --------- 159 * w+2h - ... 160 * 161 * To determine how many terms needed, let 162 * Q(0) = w, Q(1) = w(w+h) - 1, 163 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 164 * When Q(k) > 1e4 good for single 165 * When Q(k) > 1e9 good for double 166 * When Q(k) > 1e17 good for quadruple 167 */ 168 /* determine k */ 169 double t,v; 170 double q0,q1,h,tmp; int32_t k,m; 171 w = (n+n)/(double)x; h = 2.0/(double)x; 172 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 173 while(q1<1.0e9) { 174 k += 1; z += h; 175 tmp = z*q1 - q0; 176 q0 = q1; 177 q1 = tmp; 178 } 179 m = n+n; 180 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 181 a = t; 182 b = one; 183 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 184 * Hence, if n*(log(2n/x)) > ... 185 * single 8.8722839355e+01 186 * double 7.09782712893383973096e+02 187 * long double 1.1356523406294143949491931077970765006170e+04 188 * then recurrent value may overflow and the result is 189 * likely underflow to zero 190 */ 191 tmp = n; 192 v = two/x; 193 tmp = tmp*__ieee754_log(fabs(v*tmp)); 194 if(tmp<7.09782712893383973096e+02) { 195 for(i=n-1,di=(double)(i+i);i>0;i--){ 196 temp = b; 197 b *= di; 198 b = b/x - a; 199 a = temp; 200 di -= two; 201 } 202 } else { 203 for(i=n-1,di=(double)(i+i);i>0;i--){ 204 temp = b; 205 b *= di; 206 b = b/x - a; 207 a = temp; 208 di -= two; 209 /* scale b to avoid spurious overflow */ 210 if(b>1e100) { 211 a /= b; 212 t /= b; 213 b = one; 214 } 215 } 216 } 217 b = (t*__ieee754_j0(x)/b); 218 } 219 } 220 if(sgn==1) return -b; else return b; 221 } 222 223 #ifdef __STDC__ 224 double __ieee754_yn(int n, double x) 225 #else 226 double __ieee754_yn(n,x) 227 int n; double x; 228 #endif 229 { 230 int32_t i,hx,ix,lx; 231 int32_t sign; 232 double a, b, temp; 233 234 temp = 0; 235 EXTRACT_WORDS(hx,lx,x); 236 ix = 0x7fffffff&hx; 237 /* if Y(n,NaN) is NaN */ 238 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 239 if((ix|lx)==0) return -one/zero; 240 if(hx<0) return zero/zero; 241 sign = 1; 242 if(n<0){ 243 n = -n; 244 sign = 1 - ((n&1)<<1); 245 } 246 if(n==0) return(__ieee754_y0(x)); 247 if(n==1) return(sign*__ieee754_y1(x)); 248 if(ix==0x7ff00000) return zero; 249 if(ix>=0x52D00000) { /* x > 2**302 */ 250 /* (x >> n**2) 251 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 252 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 253 * Let s=sin(x), c=cos(x), 254 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 255 * 256 * n sin(xn)*sqt2 cos(xn)*sqt2 257 * ---------------------------------- 258 * 0 s-c c+s 259 * 1 -s-c -c+s 260 * 2 -s+c -c-s 261 * 3 s+c c-s 262 */ 263 switch(n&3) { 264 case 0: temp = sin(x)-cos(x); break; 265 case 1: temp = -sin(x)-cos(x); break; 266 case 2: temp = -sin(x)+cos(x); break; 267 case 3: temp = sin(x)+cos(x); break; 268 } 269 b = invsqrtpi*temp/sqrt(x); 270 } else { 271 u_int32_t high; 272 a = __ieee754_y0(x); 273 b = __ieee754_y1(x); 274 /* quit if b is -inf */ 275 GET_HIGH_WORD(high,b); 276 for(i=1;i<n&&high!=0xfff00000;i++){ 277 temp = b; 278 b = ((double)(i+i)/x)*b - a; 279 GET_HIGH_WORD(high,b); 280 a = temp; 281 } 282 } 283 if(sign>0) return b; else return -b; 284 } 285