1 2 /* @(#)e_jn.c 5.1 93/09/24 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunPro, 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)=y1(0)=yn(n,0) = -inf with division by zero signal; 21 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 22 * Note 2. About jn(n,x), yn(n,x) 23 * For n=0, j0(x) is called, 24 * for n=1, j1(x) is called, 25 * for n<x, forward recursion us used starting 26 * from values of j0(x) and 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 #ifndef _DOUBLE_IS_32BITS 43 44 #ifdef __STDC__ 45 static const double 46 #else 47 static double 48 #endif 49 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 50 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 51 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 52 53 #ifdef __STDC__ 54 static const double zero = 0.00000000000000000000e+00; 55 #else 56 static double zero = 0.00000000000000000000e+00; 57 #endif 58 59 #ifdef __STDC__ __ieee754_jn(int n,double x)60 double __ieee754_jn(int n, double x) 61 #else 62 double __ieee754_jn(n,x) 63 int n; double x; 64 #endif 65 { 66 __int32_t i,hx,ix,lx, sgn; 67 double a, b, temp, di; 68 double z, w; 69 70 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 71 * Thus, J(-n,x) = J(n,-x) 72 */ 73 EXTRACT_WORDS(hx,lx,x); 74 ix = 0x7fffffff&hx; 75 /* if J(n,NaN) is NaN */ 76 if((ix|((__uint32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 77 if(n<0){ 78 n = -n; 79 x = -x; 80 hx ^= 0x80000000; 81 } 82 if(n==0) return(__ieee754_j0(x)); 83 if(n==1) return(__ieee754_j1(x)); 84 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 85 x = fabs(x); 86 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 87 b = zero; 88 else if((double)n<=x) { 89 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 90 if(ix>=0x52D00000) { /* x > 2**302 */ 91 /* (x >> n**2) 92 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 93 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 94 * Let s=sin(x), c=cos(x), 95 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 96 * 97 * n sin(xn)*sqt2 cos(xn)*sqt2 98 * ---------------------------------- 99 * 0 s-c c+s 100 * 1 -s-c -c+s 101 * 2 -s+c -c-s 102 * 3 s+c c-s 103 */ 104 switch(n&3) { 105 case 0: temp = cos(x)+sin(x); break; 106 case 1: temp = -cos(x)+sin(x); break; 107 case 2: temp = -cos(x)-sin(x); break; 108 case 3: temp = cos(x)-sin(x); break; 109 } 110 b = invsqrtpi*temp/__ieee754_sqrt(x); 111 } else { 112 a = __ieee754_j0(x); 113 b = __ieee754_j1(x); 114 for(i=1;i<n;i++){ 115 temp = b; 116 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 117 a = temp; 118 } 119 } 120 } else { 121 if(ix<0x3e100000) { /* x < 2**-29 */ 122 /* x is tiny, return the first Taylor expansion of J(n,x) 123 * J(n,x) = 1/n!*(x/2)^n - ... 124 */ 125 if(n>33) /* underflow */ 126 b = zero; 127 else { 128 temp = x*0.5; b = temp; 129 for (a=one,i=2;i<=n;i++) { 130 a *= (double)i; /* a = n! */ 131 b *= temp; /* b = (x/2)^n */ 132 } 133 b = b/a; 134 } 135 } else { 136 /* use backward recurrence */ 137 /* x x^2 x^2 138 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 139 * 2n - 2(n+1) - 2(n+2) 140 * 141 * 1 1 1 142 * (for large x) = ---- ------ ------ ..... 143 * 2n 2(n+1) 2(n+2) 144 * -- - ------ - ------ - 145 * x x x 146 * 147 * Let w = 2n/x and h=2/x, then the above quotient 148 * is equal to the continued fraction: 149 * 1 150 * = ----------------------- 151 * 1 152 * w - ----------------- 153 * 1 154 * w+h - --------- 155 * w+2h - ... 156 * 157 * To determine how many terms needed, let 158 * Q(0) = w, Q(1) = w(w+h) - 1, 159 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 160 * When Q(k) > 1e4 good for single 161 * When Q(k) > 1e9 good for double 162 * When Q(k) > 1e17 good for quadruple 163 */ 164 /* determine k */ 165 double t,v; 166 double q0,q1,h,tmp; __int32_t k,m; 167 w = (n+n)/(double)x; h = 2.0/(double)x; 168 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 169 while(q1<1.0e9) { 170 k += 1; z += h; 171 tmp = z*q1 - q0; 172 q0 = q1; 173 q1 = tmp; 174 } 175 m = n+n; 176 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 177 a = t; 178 b = one; 179 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 180 * Hence, if n*(log(2n/x)) > ... 181 * single 8.8722839355e+01 182 * double 7.09782712893383973096e+02 183 * long double 1.1356523406294143949491931077970765006170e+04 184 * then recurrent value may overflow and the result is 185 * likely underflow to zero 186 */ 187 tmp = n; 188 v = two/x; 189 tmp = tmp*__ieee754_log(fabs(v*tmp)); 190 if(tmp<7.09782712893383973096e+02) { 191 for(i=n-1,di=(double)(i+i);i>0;i--){ 192 temp = b; 193 b *= di; 194 b = b/x - a; 195 a = temp; 196 di -= two; 197 } 198 } else { 199 for(i=n-1,di=(double)(i+i);i>0;i--){ 200 temp = b; 201 b *= di; 202 b = b/x - a; 203 a = temp; 204 di -= two; 205 /* scale b to avoid spurious overflow */ 206 if(b>1e100) { 207 a /= b; 208 t /= b; 209 b = one; 210 } 211 } 212 } 213 b = (t*__ieee754_j0(x)/b); 214 } 215 } 216 if(sgn==1) return -b; else return b; 217 } 218 219 #ifdef __STDC__ __ieee754_yn(int n,double x)220 double __ieee754_yn(int n, double x) 221 #else 222 double __ieee754_yn(n,x) 223 int n; double x; 224 #endif 225 { 226 __int32_t i,hx,ix,lx; 227 __int32_t sign; 228 double a, b, temp; 229 230 EXTRACT_WORDS(hx,lx,x); 231 ix = 0x7fffffff&hx; 232 /* if Y(n,NaN) is NaN */ 233 if((ix|((__uint32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 234 if((ix|lx)==0) return -one/zero; 235 if(hx<0) return zero/zero; 236 sign = 1; 237 if(n<0){ 238 n = -n; 239 sign = 1 - ((n&1)<<1); 240 } 241 if(n==0) return(__ieee754_y0(x)); 242 if(n==1) return(sign*__ieee754_y1(x)); 243 if(ix==0x7ff00000) return zero; 244 if(ix>=0x52D00000) { /* x > 2**302 */ 245 /* (x >> n**2) 246 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 247 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 248 * Let s=sin(x), c=cos(x), 249 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 250 * 251 * n sin(xn)*sqt2 cos(xn)*sqt2 252 * ---------------------------------- 253 * 0 s-c c+s 254 * 1 -s-c -c+s 255 * 2 -s+c -c-s 256 * 3 s+c c-s 257 */ 258 switch(n&3) { 259 case 0: temp = sin(x)-cos(x); break; 260 case 1: temp = -sin(x)-cos(x); break; 261 case 2: temp = -sin(x)+cos(x); break; 262 case 3: temp = sin(x)+cos(x); break; 263 } 264 b = invsqrtpi*temp/__ieee754_sqrt(x); 265 } else { 266 __uint32_t high; 267 a = __ieee754_y0(x); 268 b = __ieee754_y1(x); 269 /* quit if b is -inf */ 270 GET_HIGH_WORD(high,b); 271 for(i=1;i<n&&high!=0xfff00000;i++){ 272 temp = b; 273 b = ((double)(i+i)/x)*b - a; 274 GET_HIGH_WORD(high,b); 275 a = temp; 276 } 277 } 278 if(sign>0) return b; else return -b; 279 } 280 281 #endif /* defined(_DOUBLE_IS_32BITS) */ 282