1 2 /* @(#)e_jn.c 1.4 95/01/18 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 /* 15 * __ieee754_jn(n, x), __ieee754_yn(n, x) 16 * floating point Bessel's function of the 1st and 2nd kind 17 * of order n 18 * 19 * Special cases: 20 * y0(0)=ieee_y1(0)=ieee_yn(n,0) = -inf with division by zero signal; 21 * y0(-ve)=ieee_y1(-ve)=ieee_yn(n,-ve) are NaN with invalid signal. 22 * Note 2. About ieee_jn(n,x), ieee_yn(n,x) 23 * For n=0, ieee_j0(x) is called, 24 * for n=1, ieee_j1(x) is called, 25 * for n<x, forward recursion us used starting 26 * from values of ieee_j0(x) and ieee_j1(x). 27 * for n>x, a continued fraction approximation to 28 * j(n,x)/j(n-1,x) is evaluated and then backward 29 * recursion is used starting from a supposed value 30 * for j(n,x). The resulting value of j(0,x) is 31 * compared with the actual value to correct the 32 * supposed value of j(n,x). 33 * 34 * yn(n,x) is similar in all respects, except 35 * that forward recursion is used for all 36 * values of n>1. 37 * 38 */ 39 40 #include "fdlibm.h" 41 42 #ifdef __STDC__ 43 static const double 44 #else 45 static double 46 #endif 47 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 48 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 49 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 50 51 static double zero = 0.00000000000000000000e+00; 52 53 #ifdef __STDC__ 54 double __ieee754_jn(int n, double x) 55 #else 56 double __ieee754_jn(n,x) 57 int n; double x; 58 #endif 59 { 60 int i,hx,ix,lx, sgn; 61 double a, b, temp, di; 62 double z, w; 63 64 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 65 * Thus, J(-n,x) = J(n,-x) 66 */ 67 hx = __HI(x); 68 ix = 0x7fffffff&hx; 69 lx = __LO(x); 70 /* if J(n,NaN) is NaN */ 71 if((ix|((unsigned)(lx|-lx))>>31)>0x7ff00000) return x+x; 72 if(n<0){ 73 n = -n; 74 x = -x; 75 hx ^= 0x80000000; 76 } 77 if(n==0) return(__ieee754_j0(x)); 78 if(n==1) return(__ieee754_j1(x)); 79 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 80 x = ieee_fabs(x); 81 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 82 b = zero; 83 else if((double)n<=x) { 84 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 85 if(ix>=0x52D00000) { /* x > 2**302 */ 86 /* (x >> n**2) 87 * Jn(x) = ieee_cos(x-(2n+1)*pi/4)*ieee_sqrt(2/x*pi) 88 * Yn(x) = ieee_sin(x-(2n+1)*pi/4)*ieee_sqrt(2/x*pi) 89 * Let s=ieee_sin(x), c=ieee_cos(x), 90 * xn=x-(2n+1)*pi/4, sqt2 = ieee_sqrt(2),then 91 * 92 * n sin(xn)*sqt2 cos(xn)*sqt2 93 * ---------------------------------- 94 * 0 s-c c+s 95 * 1 -s-c -c+s 96 * 2 -s+c -c-s 97 * 3 s+c c-s 98 */ 99 switch(n&3) { 100 case 0: temp = ieee_cos(x)+ieee_sin(x); break; 101 case 1: temp = -ieee_cos(x)+ieee_sin(x); break; 102 case 2: temp = -ieee_cos(x)-ieee_sin(x); break; 103 case 3: temp = ieee_cos(x)-ieee_sin(x); break; 104 } 105 b = invsqrtpi*temp/ieee_sqrt(x); 106 } else { 107 a = __ieee754_j0(x); 108 b = __ieee754_j1(x); 109 for(i=1;i<n;i++){ 110 temp = b; 111 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 112 a = temp; 113 } 114 } 115 } else { 116 if(ix<0x3e100000) { /* x < 2**-29 */ 117 /* x is tiny, return the first Taylor expansion of J(n,x) 118 * J(n,x) = 1/n!*(x/2)^n - ... 119 */ 120 if(n>33) /* underflow */ 121 b = zero; 122 else { 123 temp = x*0.5; b = temp; 124 for (a=one,i=2;i<=n;i++) { 125 a *= (double)i; /* a = n! */ 126 b *= temp; /* b = (x/2)^n */ 127 } 128 b = b/a; 129 } 130 } else { 131 /* use backward recurrence */ 132 /* x x^2 x^2 133 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 134 * 2n - 2(n+1) - 2(n+2) 135 * 136 * 1 1 1 137 * (for large x) = ---- ------ ------ ..... 138 * 2n 2(n+1) 2(n+2) 139 * -- - ------ - ------ - 140 * x x x 141 * 142 * Let w = 2n/x and h=2/x, then the above quotient 143 * is equal to the continued fraction: 144 * 1 145 * = ----------------------- 146 * 1 147 * w - ----------------- 148 * 1 149 * w+h - --------- 150 * w+2h - ... 151 * 152 * To determine how many terms needed, let 153 * Q(0) = w, Q(1) = w(w+h) - 1, 154 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 155 * When Q(k) > 1e4 good for single 156 * When Q(k) > 1e9 good for double 157 * When Q(k) > 1e17 good for quadruple 158 */ 159 /* determine k */ 160 double t,v; 161 double q0,q1,h,tmp; int k,m; 162 w = (n+n)/(double)x; h = 2.0/(double)x; 163 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 164 while(q1<1.0e9) { 165 k += 1; z += h; 166 tmp = z*q1 - q0; 167 q0 = q1; 168 q1 = tmp; 169 } 170 m = n+n; 171 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 172 a = t; 173 b = one; 174 /* estimate ieee_log((2/x)^n*n!) = n*ieee_log(2/x)+n*ln(n) 175 * Hence, if n*(ieee_log(2n/x)) > ... 176 * single 8.8722839355e+01 177 * double 7.09782712893383973096e+02 178 * long double 1.1356523406294143949491931077970765006170e+04 179 * then recurrent value may overflow and the result is 180 * likely underflow to zero 181 */ 182 tmp = n; 183 v = two/x; 184 tmp = tmp*__ieee754_log(ieee_fabs(v*tmp)); 185 if(tmp<7.09782712893383973096e+02) { 186 for(i=n-1,di=(double)(i+i);i>0;i--){ 187 temp = b; 188 b *= di; 189 b = b/x - a; 190 a = temp; 191 di -= two; 192 } 193 } else { 194 for(i=n-1,di=(double)(i+i);i>0;i--){ 195 temp = b; 196 b *= di; 197 b = b/x - a; 198 a = temp; 199 di -= two; 200 /* scale b to avoid spurious overflow */ 201 if(b>1e100) { 202 a /= b; 203 t /= b; 204 b = one; 205 } 206 } 207 } 208 b = (t*__ieee754_j0(x)/b); 209 } 210 } 211 if(sgn==1) return -b; else return b; 212 } 213 214 #ifdef __STDC__ 215 double __ieee754_yn(int n, double x) 216 #else 217 double __ieee754_yn(n,x) 218 int n; double x; 219 #endif 220 { 221 int i,hx,ix,lx; 222 int sign; 223 double a, b, temp; 224 225 hx = __HI(x); 226 ix = 0x7fffffff&hx; 227 lx = __LO(x); 228 /* if Y(n,NaN) is NaN */ 229 if((ix|((unsigned)(lx|-lx))>>31)>0x7ff00000) return x+x; 230 if((ix|lx)==0) return -one/zero; 231 if(hx<0) return zero/zero; 232 sign = 1; 233 if(n<0){ 234 n = -n; 235 sign = 1 - ((n&1)<<1); 236 } 237 if(n==0) return(__ieee754_y0(x)); 238 if(n==1) return(sign*__ieee754_y1(x)); 239 if(ix==0x7ff00000) return zero; 240 if(ix>=0x52D00000) { /* x > 2**302 */ 241 /* (x >> n**2) 242 * Jn(x) = ieee_cos(x-(2n+1)*pi/4)*ieee_sqrt(2/x*pi) 243 * Yn(x) = ieee_sin(x-(2n+1)*pi/4)*ieee_sqrt(2/x*pi) 244 * Let s=ieee_sin(x), c=ieee_cos(x), 245 * xn=x-(2n+1)*pi/4, sqt2 = ieee_sqrt(2),then 246 * 247 * n sin(xn)*sqt2 cos(xn)*sqt2 248 * ---------------------------------- 249 * 0 s-c c+s 250 * 1 -s-c -c+s 251 * 2 -s+c -c-s 252 * 3 s+c c-s 253 */ 254 switch(n&3) { 255 case 0: temp = ieee_sin(x)-ieee_cos(x); break; 256 case 1: temp = -ieee_sin(x)-ieee_cos(x); break; 257 case 2: temp = -ieee_sin(x)+ieee_cos(x); break; 258 case 3: temp = ieee_sin(x)+ieee_cos(x); break; 259 } 260 b = invsqrtpi*temp/ieee_sqrt(x); 261 } else { 262 a = __ieee754_y0(x); 263 b = __ieee754_y1(x); 264 /* quit if b is -inf */ 265 for(i=1;i<n&&(__HI(b) != 0xfff00000);i++){ 266 temp = b; 267 b = ((double)(i+i)/x)*b - a; 268 a = temp; 269 } 270 } 271 if(sign>0) return b; else return -b; 272 } 273
