e_jn.c revision 97409
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#ifndef lint 14static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_jn.c 97409 2002-05-28 17:51:46Z alfred $"; 15#endif 16 17/* 18 * __ieee754_jn(n, x), __ieee754_yn(n, x) 19 * floating point Bessel's function of the 1st and 2nd kind 20 * of order n 21 * 22 * Special cases: 23 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 24 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 25 * Note 2. About jn(n,x), yn(n,x) 26 * For n=0, j0(x) is called, 27 * for n=1, j1(x) is called, 28 * for n<x, forward recursion us used starting 29 * from values of j0(x) and j1(x). 30 * for n>x, a continued fraction approximation to 31 * j(n,x)/j(n-1,x) is evaluated and then backward 32 * recursion is used starting from a supposed value 33 * for j(n,x). The resulting value of j(0,x) is 34 * compared with the actual value to correct the 35 * supposed value of j(n,x). 36 * 37 * yn(n,x) is similar in all respects, except 38 * that forward recursion is used for all 39 * values of n>1. 40 * 41 */ 42 43#include "math.h" 44#include "math_private.h" 45 46static const double 47invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 48two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 49one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 50 51static const double zero = 0.00000000000000000000e+00; 52 53 double __ieee754_jn(int n, double x) 54{ 55 int32_t i,hx,ix,lx, sgn; 56 double a, b, temp, di; 57 double z, w; 58 59 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 60 * Thus, J(-n,x) = J(n,-x) 61 */ 62 EXTRACT_WORDS(hx,lx,x); 63 ix = 0x7fffffff&hx; 64 /* if J(n,NaN) is NaN */ 65 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 66 if(n<0){ 67 n = -n; 68 x = -x; 69 hx ^= 0x80000000; 70 } 71 if(n==0) return(__ieee754_j0(x)); 72 if(n==1) return(__ieee754_j1(x)); 73 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 74 x = fabs(x); 75 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 76 b = zero; 77 else if((double)n<=x) { 78 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 79 if(ix>=0x52D00000) { /* x > 2**302 */ 80 /* (x >> n**2) 81 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 82 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 83 * Let s=sin(x), c=cos(x), 84 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 85 * 86 * n sin(xn)*sqt2 cos(xn)*sqt2 87 * ---------------------------------- 88 * 0 s-c c+s 89 * 1 -s-c -c+s 90 * 2 -s+c -c-s 91 * 3 s+c c-s 92 */ 93 switch(n&3) { 94 case 0: temp = cos(x)+sin(x); break; 95 case 1: temp = -cos(x)+sin(x); break; 96 case 2: temp = -cos(x)-sin(x); break; 97 case 3: temp = cos(x)-sin(x); break; 98 } 99 b = invsqrtpi*temp/sqrt(x); 100 } else { 101 a = __ieee754_j0(x); 102 b = __ieee754_j1(x); 103 for(i=1;i<n;i++){ 104 temp = b; 105 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 106 a = temp; 107 } 108 } 109 } else { 110 if(ix<0x3e100000) { /* x < 2**-29 */ 111 /* x is tiny, return the first Taylor expansion of J(n,x) 112 * J(n,x) = 1/n!*(x/2)^n - ... 113 */ 114 if(n>33) /* underflow */ 115 b = zero; 116 else { 117 temp = x*0.5; b = temp; 118 for (a=one,i=2;i<=n;i++) { 119 a *= (double)i; /* a = n! */ 120 b *= temp; /* b = (x/2)^n */ 121 } 122 b = b/a; 123 } 124 } else { 125 /* use backward recurrence */ 126 /* x x^2 x^2 127 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 128 * 2n - 2(n+1) - 2(n+2) 129 * 130 * 1 1 1 131 * (for large x) = ---- ------ ------ ..... 132 * 2n 2(n+1) 2(n+2) 133 * -- - ------ - ------ - 134 * x x x 135 * 136 * Let w = 2n/x and h=2/x, then the above quotient 137 * is equal to the continued fraction: 138 * 1 139 * = ----------------------- 140 * 1 141 * w - ----------------- 142 * 1 143 * w+h - --------- 144 * w+2h - ... 145 * 146 * To determine how many terms needed, let 147 * Q(0) = w, Q(1) = w(w+h) - 1, 148 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 149 * When Q(k) > 1e4 good for single 150 * When Q(k) > 1e9 good for double 151 * When Q(k) > 1e17 good for quadruple 152 */ 153 /* determine k */ 154 double t,v; 155 double q0,q1,h,tmp; int32_t k,m; 156 w = (n+n)/(double)x; h = 2.0/(double)x; 157 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 158 while(q1<1.0e9) { 159 k += 1; z += h; 160 tmp = z*q1 - q0; 161 q0 = q1; 162 q1 = tmp; 163 } 164 m = n+n; 165 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 166 a = t; 167 b = one; 168 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 169 * Hence, if n*(log(2n/x)) > ... 170 * single 8.8722839355e+01 171 * double 7.09782712893383973096e+02 172 * long double 1.1356523406294143949491931077970765006170e+04 173 * then recurrent value may overflow and the result is 174 * likely underflow to zero 175 */ 176 tmp = n; 177 v = two/x; 178 tmp = tmp*__ieee754_log(fabs(v*tmp)); 179 if(tmp<7.09782712893383973096e+02) { 180 for(i=n-1,di=(double)(i+i);i>0;i--){ 181 temp = b; 182 b *= di; 183 b = b/x - a; 184 a = temp; 185 di -= two; 186 } 187 } else { 188 for(i=n-1,di=(double)(i+i);i>0;i--){ 189 temp = b; 190 b *= di; 191 b = b/x - a; 192 a = temp; 193 di -= two; 194 /* scale b to avoid spurious overflow */ 195 if(b>1e100) { 196 a /= b; 197 t /= b; 198 b = one; 199 } 200 } 201 } 202 b = (t*__ieee754_j0(x)/b); 203 } 204 } 205 if(sgn==1) return -b; else return b; 206} 207 208 double __ieee754_yn(int n, double x) 209{ 210 int32_t i,hx,ix,lx; 211 int32_t sign; 212 double a, b, temp; 213 214 EXTRACT_WORDS(hx,lx,x); 215 ix = 0x7fffffff&hx; 216 /* if Y(n,NaN) is NaN */ 217 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 218 if((ix|lx)==0) return -one/zero; 219 if(hx<0) return zero/zero; 220 sign = 1; 221 if(n<0){ 222 n = -n; 223 sign = 1 - ((n&1)<<1); 224 } 225 if(n==0) return(__ieee754_y0(x)); 226 if(n==1) return(sign*__ieee754_y1(x)); 227 if(ix==0x7ff00000) return zero; 228 if(ix>=0x52D00000) { /* x > 2**302 */ 229 /* (x >> n**2) 230 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 231 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 232 * Let s=sin(x), c=cos(x), 233 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 234 * 235 * n sin(xn)*sqt2 cos(xn)*sqt2 236 * ---------------------------------- 237 * 0 s-c c+s 238 * 1 -s-c -c+s 239 * 2 -s+c -c-s 240 * 3 s+c c-s 241 */ 242 switch(n&3) { 243 case 0: temp = sin(x)-cos(x); break; 244 case 1: temp = -sin(x)-cos(x); break; 245 case 2: temp = -sin(x)+cos(x); break; 246 case 3: temp = sin(x)+cos(x); break; 247 } 248 b = invsqrtpi*temp/sqrt(x); 249 } else { 250 u_int32_t high; 251 a = __ieee754_y0(x); 252 b = __ieee754_y1(x); 253 /* quit if b is -inf */ 254 GET_HIGH_WORD(high,b); 255 for(i=1;i<n&&high!=0xfff00000;i++){ 256 temp = b; 257 b = ((double)(i+i)/x)*b - a; 258 GET_HIGH_WORD(high,b); 259 a = temp; 260 } 261 } 262 if(sign>0) return b; else return -b; 263} 264