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("$FreeBSD$"); 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 46 static const double 47 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 48 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 49 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 50 51 static const double zero = 0.00000000000000000000e+00; 52 53 double 54 __ieee754_jn(int n, double x) 55 { 56 int32_t i,hx,ix,lx, sgn; 57 double a, b, temp, di; 58 double z, w; 59 60 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 61 * Thus, J(-n,x) = J(n,-x) 62 */ 63 EXTRACT_WORDS(hx,lx,x); 64 ix = 0x7fffffff&hx; 65 /* if J(n,NaN) is NaN */ 66 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 67 if(n<0){ 68 n = -n; 69 x = -x; 70 hx ^= 0x80000000; 71 } 72 if(n==0) return(__ieee754_j0(x)); 73 if(n==1) return(__ieee754_j1(x)); 74 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 75 x = fabs(x); 76 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 77 b = zero; 78 else if((double)n<=x) { 79 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 80 if(ix>=0x52D00000) { /* x > 2**302 */ 81 /* (x >> n**2) 82 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 83 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 84 * Let s=sin(x), c=cos(x), 85 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 86 * 87 * n sin(xn)*sqt2 cos(xn)*sqt2 88 * ---------------------------------- 89 * 0 s-c c+s 90 * 1 -s-c -c+s 91 * 2 -s+c -c-s 92 * 3 s+c c-s 93 */ 94 switch(n&3) { 95 case 0: temp = cos(x)+sin(x); break; 96 case 1: temp = -cos(x)+sin(x); break; 97 case 2: temp = -cos(x)-sin(x); break; 98 case 3: temp = cos(x)-sin(x); break; 99 } 100 b = invsqrtpi*temp/sqrt(x); 101 } else { 102 a = __ieee754_j0(x); 103 b = __ieee754_j1(x); 104 for(i=1;i<n;i++){ 105 temp = b; 106 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 107 a = temp; 108 } 109 } 110 } else { 111 if(ix<0x3e100000) { /* x < 2**-29 */ 112 /* x is tiny, return the first Taylor expansion of J(n,x) 113 * J(n,x) = 1/n!*(x/2)^n - ... 114 */ 115 if(n>33) /* underflow */ 116 b = zero; 117 else { 118 temp = x*0.5; b = temp; 119 for (a=one,i=2;i<=n;i++) { 120 a *= (double)i; /* a = n! */ 121 b *= temp; /* b = (x/2)^n */ 122 } 123 b = b/a; 124 } 125 } else { 126 /* use backward recurrence */ 127 /* x x^2 x^2 128 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 129 * 2n - 2(n+1) - 2(n+2) 130 * 131 * 1 1 1 132 * (for large x) = ---- ------ ------ ..... 133 * 2n 2(n+1) 2(n+2) 134 * -- - ------ - ------ - 135 * x x x 136 * 137 * Let w = 2n/x and h=2/x, then the above quotient 138 * is equal to the continued fraction: 139 * 1 140 * = ----------------------- 141 * 1 142 * w - ----------------- 143 * 1 144 * w+h - --------- 145 * w+2h - ... 146 * 147 * To determine how many terms needed, let 148 * Q(0) = w, Q(1) = w(w+h) - 1, 149 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 150 * When Q(k) > 1e4 good for single 151 * When Q(k) > 1e9 good for double 152 * When Q(k) > 1e17 good for quadruple 153 */ 154 /* determine k */ 155 double t,v; 156 double q0,q1,h,tmp; int32_t k,m; 157 w = (n+n)/(double)x; h = 2.0/(double)x; 158 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 159 while(q1<1.0e9) { 160 k += 1; z += h; 161 tmp = z*q1 - q0; 162 q0 = q1; 163 q1 = tmp; 164 } 165 m = n+n; 166 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 167 a = t; 168 b = one; 169 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 170 * Hence, if n*(log(2n/x)) > ... 171 * single 8.8722839355e+01 172 * double 7.09782712893383973096e+02 173 * long double 1.1356523406294143949491931077970765006170e+04 174 * then recurrent value may overflow and the result is 175 * likely underflow to zero 176 */ 177 tmp = n; 178 v = two/x; 179 tmp = tmp*__ieee754_log(fabs(v*tmp)); 180 if(tmp<7.09782712893383973096e+02) { 181 for(i=n-1,di=(double)(i+i);i>0;i--){ 182 temp = b; 183 b *= di; 184 b = b/x - a; 185 a = temp; 186 di -= two; 187 } 188 } else { 189 for(i=n-1,di=(double)(i+i);i>0;i--){ 190 temp = b; 191 b *= di; 192 b = b/x - a; 193 a = temp; 194 di -= two; 195 /* scale b to avoid spurious overflow */ 196 if(b>1e100) { 197 a /= b; 198 t /= b; 199 b = one; 200 } 201 } 202 } 203 z = __ieee754_j0(x); 204 w = __ieee754_j1(x); 205 if (fabs(z) >= fabs(w)) 206 b = (t*z/b); 207 else 208 b = (t*w/a); 209 } 210 } 211 if(sgn==1) return -b; else return b; 212 } 213 214 double 215 __ieee754_yn(int n, double x) 216 { 217 int32_t i,hx,ix,lx; 218 int32_t sign; 219 double a, b, temp; 220 221 EXTRACT_WORDS(hx,lx,x); 222 ix = 0x7fffffff&hx; 223 /* if Y(n,NaN) is NaN */ 224 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 225 if((ix|lx)==0) return -one/zero; 226 if(hx<0) return zero/zero; 227 sign = 1; 228 if(n<0){ 229 n = -n; 230 sign = 1 - ((n&1)<<1); 231 } 232 if(n==0) return(__ieee754_y0(x)); 233 if(n==1) return(sign*__ieee754_y1(x)); 234 if(ix==0x7ff00000) return zero; 235 if(ix>=0x52D00000) { /* x > 2**302 */ 236 /* (x >> n**2) 237 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 238 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 239 * Let s=sin(x), c=cos(x), 240 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 241 * 242 * n sin(xn)*sqt2 cos(xn)*sqt2 243 * ---------------------------------- 244 * 0 s-c c+s 245 * 1 -s-c -c+s 246 * 2 -s+c -c-s 247 * 3 s+c c-s 248 */ 249 switch(n&3) { 250 case 0: temp = sin(x)-cos(x); break; 251 case 1: temp = -sin(x)-cos(x); break; 252 case 2: temp = -sin(x)+cos(x); break; 253 case 3: temp = sin(x)+cos(x); break; 254 } 255 b = invsqrtpi*temp/sqrt(x); 256 } else { 257 u_int32_t high; 258 a = __ieee754_y0(x); 259 b = __ieee754_y1(x); 260 /* quit if b is -inf */ 261 GET_HIGH_WORD(high,b); 262 for(i=1;i<n&&high!=0xfff00000;i++){ 263 temp = b; 264 b = ((double)(i+i)/x)*b - a; 265 GET_HIGH_WORD(high,b); 266 a = temp; 267 } 268 } 269 if(sign>0) return b; else return -b; 270 } 271