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 #ifndef lint 15 static char rcsid[] = "$FreeBSD$"; 16 #endif 17 18 /* 19 * __ieee754_jn(n, x), __ieee754_yn(n, x) 20 * floating point Bessel's function of the 1st and 2nd kind 21 * of order n 22 * 23 * Special cases: 24 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 25 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 26 * Note 2. About jn(n,x), yn(n,x) 27 * For n=0, j0(x) is called, 28 * for n=1, j1(x) is called, 29 * for n<x, forward recursion us used starting 30 * from values of j0(x) and j1(x). 31 * for n>x, a continued fraction approximation to 32 * j(n,x)/j(n-1,x) is evaluated and then backward 33 * recursion is used starting from a supposed value 34 * for j(n,x). The resulting value of j(0,x) is 35 * compared with the actual value to correct the 36 * supposed value of j(n,x). 37 * 38 * yn(n,x) is similar in all respects, except 39 * that forward recursion is used for all 40 * values of n>1. 41 * 42 */ 43 44 #include "math.h" 45 #include "math_private.h" 46 47 static const double 48 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 49 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 50 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 51 52 static const double zero = 0.00000000000000000000e+00; 53 54 double 55 __ieee754_jn(int n, double x) 56 { 57 int32_t i,hx,ix,lx, sgn; 58 double a, b, temp, di; 59 double z, w; 60 61 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 62 * Thus, J(-n,x) = J(n,-x) 63 */ 64 EXTRACT_WORDS(hx,lx,x); 65 ix = 0x7fffffff&hx; 66 /* if J(n,NaN) is NaN */ 67 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 68 if(n<0){ 69 n = -n; 70 x = -x; 71 hx ^= 0x80000000; 72 } 73 if(n==0) return(__ieee754_j0(x)); 74 if(n==1) return(__ieee754_j1(x)); 75 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 76 x = fabs(x); 77 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 78 b = zero; 79 else if((double)n<=x) { 80 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 81 if(ix>=0x52D00000) { /* x > 2**302 */ 82 /* (x >> n**2) 83 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 84 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 85 * Let s=sin(x), c=cos(x), 86 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 87 * 88 * n sin(xn)*sqt2 cos(xn)*sqt2 89 * ---------------------------------- 90 * 0 s-c c+s 91 * 1 -s-c -c+s 92 * 2 -s+c -c-s 93 * 3 s+c c-s 94 */ 95 switch(n&3) { 96 case 0: temp = cos(x)+sin(x); break; 97 case 1: temp = -cos(x)+sin(x); break; 98 case 2: temp = -cos(x)-sin(x); break; 99 case 3: temp = cos(x)-sin(x); break; 100 } 101 b = invsqrtpi*temp/sqrt(x); 102 } else { 103 a = __ieee754_j0(x); 104 b = __ieee754_j1(x); 105 for(i=1;i<n;i++){ 106 temp = b; 107 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 108 a = temp; 109 } 110 } 111 } else { 112 if(ix<0x3e100000) { /* x < 2**-29 */ 113 /* x is tiny, return the first Taylor expansion of J(n,x) 114 * J(n,x) = 1/n!*(x/2)^n - ... 115 */ 116 if(n>33) /* underflow */ 117 b = zero; 118 else { 119 temp = x*0.5; b = temp; 120 for (a=one,i=2;i<=n;i++) { 121 a *= (double)i; /* a = n! */ 122 b *= temp; /* b = (x/2)^n */ 123 } 124 b = b/a; 125 } 126 } else { 127 /* use backward recurrence */ 128 /* x x^2 x^2 129 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 130 * 2n - 2(n+1) - 2(n+2) 131 * 132 * 1 1 1 133 * (for large x) = ---- ------ ------ ..... 134 * 2n 2(n+1) 2(n+2) 135 * -- - ------ - ------ - 136 * x x x 137 * 138 * Let w = 2n/x and h=2/x, then the above quotient 139 * is equal to the continued fraction: 140 * 1 141 * = ----------------------- 142 * 1 143 * w - ----------------- 144 * 1 145 * w+h - --------- 146 * w+2h - ... 147 * 148 * To determine how many terms needed, let 149 * Q(0) = w, Q(1) = w(w+h) - 1, 150 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 151 * When Q(k) > 1e4 good for single 152 * When Q(k) > 1e9 good for double 153 * When Q(k) > 1e17 good for quadruple 154 */ 155 /* determine k */ 156 double t,v; 157 double q0,q1,h,tmp; int32_t k,m; 158 w = (n+n)/(double)x; h = 2.0/(double)x; 159 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 160 while(q1<1.0e9) { 161 k += 1; z += h; 162 tmp = z*q1 - q0; 163 q0 = q1; 164 q1 = tmp; 165 } 166 m = n+n; 167 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 168 a = t; 169 b = one; 170 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 171 * Hence, if n*(log(2n/x)) > ... 172 * single 8.8722839355e+01 173 * double 7.09782712893383973096e+02 174 * long double 1.1356523406294143949491931077970765006170e+04 175 * then recurrent value may overflow and the result is 176 * likely underflow to zero 177 */ 178 tmp = n; 179 v = two/x; 180 tmp = tmp*__ieee754_log(fabs(v*tmp)); 181 if(tmp<7.09782712893383973096e+02) { 182 for(i=n-1,di=(double)(i+i);i>0;i--){ 183 temp = b; 184 b *= di; 185 b = b/x - a; 186 a = temp; 187 di -= two; 188 } 189 } else { 190 for(i=n-1,di=(double)(i+i);i>0;i--){ 191 temp = b; 192 b *= di; 193 b = b/x - a; 194 a = temp; 195 di -= two; 196 /* scale b to avoid spurious overflow */ 197 if(b>1e100) { 198 a /= b; 199 t /= b; 200 b = one; 201 } 202 } 203 } 204 b = (t*__ieee754_j0(x)/b); 205 } 206 } 207 if(sgn==1) return -b; else return b; 208 } 209 210 double 211 __ieee754_yn(int n, double x) 212 { 213 int32_t i,hx,ix,lx; 214 int32_t sign; 215 double a, b, temp; 216 217 EXTRACT_WORDS(hx,lx,x); 218 ix = 0x7fffffff&hx; 219 /* if Y(n,NaN) is NaN */ 220 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 221 if((ix|lx)==0) return -one/zero; 222 if(hx<0) return zero/zero; 223 sign = 1; 224 if(n<0){ 225 n = -n; 226 sign = 1 - ((n&1)<<1); 227 } 228 if(n==0) return(__ieee754_y0(x)); 229 if(n==1) return(sign*__ieee754_y1(x)); 230 if(ix==0x7ff00000) return zero; 231 if(ix>=0x52D00000) { /* x > 2**302 */ 232 /* (x >> n**2) 233 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 234 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 235 * Let s=sin(x), c=cos(x), 236 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 237 * 238 * n sin(xn)*sqt2 cos(xn)*sqt2 239 * ---------------------------------- 240 * 0 s-c c+s 241 * 1 -s-c -c+s 242 * 2 -s+c -c-s 243 * 3 s+c c-s 244 */ 245 switch(n&3) { 246 case 0: temp = sin(x)-cos(x); break; 247 case 1: temp = -sin(x)-cos(x); break; 248 case 2: temp = -sin(x)+cos(x); break; 249 case 3: temp = sin(x)+cos(x); break; 250 } 251 b = invsqrtpi*temp/sqrt(x); 252 } else { 253 u_int32_t high; 254 a = __ieee754_y0(x); 255 b = __ieee754_y1(x); 256 /* quit if b is -inf */ 257 GET_HIGH_WORD(high,b); 258 for(i=1;i<n&&high!=0xfff00000;i++){ 259 temp = b; 260 b = ((double)(i+i)/x)*b - a; 261 GET_HIGH_WORD(high,b); 262 a = temp; 263 } 264 } 265 if(sign>0) return b; else return -b; 266 } 267