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#include <sys/cdefs.h> 15__FBSDID("$FreeBSD: head/lib/msun/src/e_jn.c 279856 2015-03-10 17:10:54Z kargl $"); 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 volatile double vone = 1, vzero = 0; 47 48static const double 49invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 50two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 51one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 52 53static const double zero = 0.00000000000000000000e+00; 54 55double 56__ieee754_jn(int n, double x) 57{ 58 int32_t i,hx,ix,lx, sgn; 59 double a, b, temp, di; 60 double z, w; 61 62 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 63 * Thus, J(-n,x) = J(n,-x) 64 */ 65 EXTRACT_WORDS(hx,lx,x); 66 ix = 0x7fffffff&hx; 67 /* if J(n,NaN) is NaN */ 68 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 69 if(n<0){ 70 n = -n; 71 x = -x; 72 hx ^= 0x80000000; 73 } 74 if(n==0) return(__ieee754_j0(x)); 75 if(n==1) return(__ieee754_j1(x)); 76 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 77 x = fabs(x); 78 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 79 b = zero; 80 else if((double)n<=x) { 81 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 82 if(ix>=0x52D00000) { /* x > 2**302 */ 83 /* (x >> n**2) 84 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 85 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 86 * Let s=sin(x), c=cos(x), 87 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 88 * 89 * n sin(xn)*sqt2 cos(xn)*sqt2 90 * ---------------------------------- 91 * 0 s-c c+s 92 * 1 -s-c -c+s 93 * 2 -s+c -c-s 94 * 3 s+c c-s 95 */ 96 switch(n&3) { 97 case 0: temp = cos(x)+sin(x); break; 98 case 1: temp = -cos(x)+sin(x); break; 99 case 2: temp = -cos(x)-sin(x); break; 100 case 3: temp = cos(x)-sin(x); break; 101 } 102 b = invsqrtpi*temp/sqrt(x); 103 } else { 104 a = __ieee754_j0(x); 105 b = __ieee754_j1(x); 106 for(i=1;i<n;i++){ 107 temp = b; 108 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 109 a = temp; 110 } 111 } 112 } else { 113 if(ix<0x3e100000) { /* x < 2**-29 */ 114 /* x is tiny, return the first Taylor expansion of J(n,x) 115 * J(n,x) = 1/n!*(x/2)^n - ... 116 */ 117 if(n>33) /* underflow */ 118 b = zero; 119 else { 120 temp = x*0.5; b = temp; 121 for (a=one,i=2;i<=n;i++) { 122 a *= (double)i; /* a = n! */ 123 b *= temp; /* b = (x/2)^n */ 124 } 125 b = b/a; 126 } 127 } else { 128 /* use backward recurrence */ 129 /* x x^2 x^2 130 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 131 * 2n - 2(n+1) - 2(n+2) 132 * 133 * 1 1 1 134 * (for large x) = ---- ------ ------ ..... 135 * 2n 2(n+1) 2(n+2) 136 * -- - ------ - ------ - 137 * x x x 138 * 139 * Let w = 2n/x and h=2/x, then the above quotient 140 * is equal to the continued fraction: 141 * 1 142 * = ----------------------- 143 * 1 144 * w - ----------------- 145 * 1 146 * w+h - --------- 147 * w+2h - ... 148 * 149 * To determine how many terms needed, let 150 * Q(0) = w, Q(1) = w(w+h) - 1, 151 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 152 * When Q(k) > 1e4 good for single 153 * When Q(k) > 1e9 good for double 154 * When Q(k) > 1e17 good for quadruple 155 */ 156 /* determine k */ 157 double t,v; 158 double q0,q1,h,tmp; int32_t k,m; 159 w = (n+n)/(double)x; h = 2.0/(double)x; 160 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 161 while(q1<1.0e9) { 162 k += 1; z += h; 163 tmp = z*q1 - q0; 164 q0 = q1; 165 q1 = tmp; 166 } 167 m = n+n; 168 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 169 a = t; 170 b = one; 171 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 172 * Hence, if n*(log(2n/x)) > ... 173 * single 8.8722839355e+01 174 * double 7.09782712893383973096e+02 175 * long double 1.1356523406294143949491931077970765006170e+04 176 * then recurrent value may overflow and the result is 177 * likely underflow to zero 178 */ 179 tmp = n; 180 v = two/x; 181 tmp = tmp*__ieee754_log(fabs(v*tmp)); 182 if(tmp<7.09782712893383973096e+02) { 183 for(i=n-1,di=(double)(i+i);i>0;i--){ 184 temp = b; 185 b *= di; 186 b = b/x - a; 187 a = temp; 188 di -= two; 189 } 190 } else { 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 /* scale b to avoid spurious overflow */ 198 if(b>1e100) { 199 a /= b; 200 t /= b; 201 b = one; 202 } 203 } 204 } 205 z = __ieee754_j0(x); 206 w = __ieee754_j1(x); 207 if (fabs(z) >= fabs(w)) 208 b = (t*z/b); 209 else 210 b = (t*w/a); 211 } 212 } 213 if(sgn==1) return -b; else return b; 214} 215 216double 217__ieee754_yn(int n, double x) 218{ 219 int32_t i,hx,ix,lx; 220 int32_t sign; 221 double a, b, temp; 222 223 EXTRACT_WORDS(hx,lx,x); 224 ix = 0x7fffffff&hx; 225 /* yn(n,NaN) = NaN */ 226 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 227 /* yn(n,+-0) = -inf and raise divide-by-zero exception. */ 228 if((ix|lx)==0) return -one/vzero; 229 /* yn(n,x<0) = NaN and raise invalid exception. */ 230 if(hx<0) return vzero/vzero; 231 sign = 1; 232 if(n<0){ 233 n = -n; 234 sign = 1 - ((n&1)<<1); 235 } 236 if(n==0) return(__ieee754_y0(x)); 237 if(n==1) return(sign*__ieee754_y1(x)); 238 if(ix==0x7ff00000) return zero; 239 if(ix>=0x52D00000) { /* x > 2**302 */ 240 /* (x >> n**2) 241 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 242 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 243 * Let s=sin(x), c=cos(x), 244 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 245 * 246 * n sin(xn)*sqt2 cos(xn)*sqt2 247 * ---------------------------------- 248 * 0 s-c c+s 249 * 1 -s-c -c+s 250 * 2 -s+c -c-s 251 * 3 s+c c-s 252 */ 253 switch(n&3) { 254 case 0: temp = sin(x)-cos(x); break; 255 case 1: temp = -sin(x)-cos(x); break; 256 case 2: temp = -sin(x)+cos(x); break; 257 case 3: temp = sin(x)+cos(x); break; 258 } 259 b = invsqrtpi*temp/sqrt(x); 260 } else { 261 u_int32_t high; 262 a = __ieee754_y0(x); 263 b = __ieee754_y1(x); 264 /* quit if b is -inf */ 265 GET_HIGH_WORD(high,b); 266 for(i=1;i<n&&high!=0xfff00000;i++){ 267 temp = b; 268 b = ((double)(i+i)/x)*b - a; 269 GET_HIGH_WORD(high,b); 270 a = temp; 271 } 272 } 273 if(sign>0) return b; else return -b; 274} 275