1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunSoft, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* 13 * jn(n, x), yn(n, x) 14 * floating point Bessel's function of the 1st and 2nd kind 15 * of order n 16 * 17 * Special cases: 18 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 19 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 20 * Note 2. About jn(n,x), yn(n,x) 21 * For n=0, j0(x) is called. 22 * For n=1, j1(x) is called. 23 * For n<x, forward recursion is used starting 24 * from values of j0(x) and j1(x). 25 * For n>x, a continued fraction approximation to 26 * j(n,x)/j(n-1,x) is evaluated and then backward 27 * recursion is used starting from a supposed value 28 * for j(n,x). The resulting values of j(0,x) or j(1,x) are 29 * compared with the actual values to correct the 30 * supposed value of j(n,x). 31 * 32 * yn(n,x) is similar in all respects, except 33 * that forward recursion is used for all 34 * values of n>1. 35 */ 36 37 #include "math.h" 38 #include "math_private.h" 39 40 static const volatile double vone = 1, vzero = 0; 41 42 static const double 43 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 44 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 45 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 46 47 static const double zero = 0.00000000000000000000e+00; 48 49 double 50 jn(int n, double x) 51 { 52 int32_t i,hx,ix,lx, sgn; 53 double a, b, c, s, temp, di; 54 double z, w; 55 56 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 57 * Thus, J(-n,x) = J(n,-x) 58 */ 59 EXTRACT_WORDS(hx,lx,x); 60 ix = 0x7fffffff&hx; 61 /* if J(n,NaN) is NaN */ 62 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 63 if(n<0){ 64 n = -n; 65 x = -x; 66 hx ^= 0x80000000; 67 } 68 if(n==0) return(j0(x)); 69 if(n==1) return(j1(x)); 70 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 71 x = fabs(x); 72 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 73 b = zero; 74 else if((double)n<=x) { 75 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 76 if(ix>=0x52D00000) { /* x > 2**302 */ 77 /* (x >> n**2) 78 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 79 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 80 * Let s=sin(x), c=cos(x), 81 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then 82 * 83 * n sin(xn)*sqt2 cos(xn)*sqt2 84 * ---------------------------------- 85 * 0 s-c c+s 86 * 1 -s-c -c+s 87 * 2 -s+c -c-s 88 * 3 s+c c-s 89 */ 90 sincos(x, &s, &c); 91 switch(n&3) { 92 case 0: temp = c+s; break; 93 case 1: temp = -c+s; break; 94 case 2: temp = -c-s; break; 95 case 3: temp = c-s; break; 96 } 97 b = invsqrtpi*temp/sqrt(x); 98 } else { 99 a = j0(x); 100 b = j1(x); 101 for(i=1;i<n;i++){ 102 temp = b; 103 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 104 a = temp; 105 } 106 } 107 } else { 108 if(ix<0x3e100000) { /* x < 2**-29 */ 109 /* x is tiny, return the first Taylor expansion of J(n,x) 110 * J(n,x) = 1/n!*(x/2)^n - ... 111 */ 112 if(n>33) /* underflow */ 113 b = zero; 114 else { 115 temp = x*0.5; b = temp; 116 for (a=one,i=2;i<=n;i++) { 117 a *= (double)i; /* a = n! */ 118 b *= temp; /* b = (x/2)^n */ 119 } 120 b = b/a; 121 } 122 } else { 123 /* use backward recurrence */ 124 /* x x^2 x^2 125 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 126 * 2n - 2(n+1) - 2(n+2) 127 * 128 * 1 1 1 129 * (for large x) = ---- ------ ------ ..... 130 * 2n 2(n+1) 2(n+2) 131 * -- - ------ - ------ - 132 * x x x 133 * 134 * Let w = 2n/x and h=2/x, then the above quotient 135 * is equal to the continued fraction: 136 * 1 137 * = ----------------------- 138 * 1 139 * w - ----------------- 140 * 1 141 * w+h - --------- 142 * w+2h - ... 143 * 144 * To determine how many terms needed, let 145 * Q(0) = w, Q(1) = w(w+h) - 1, 146 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 147 * When Q(k) > 1e4 good for single 148 * When Q(k) > 1e9 good for double 149 * When Q(k) > 1e17 good for quadruple 150 */ 151 /* determine k */ 152 double t,v; 153 double q0,q1,h,tmp; int32_t k,m; 154 w = (n+n)/(double)x; h = 2.0/(double)x; 155 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 156 while(q1<1.0e9) { 157 k += 1; z += h; 158 tmp = z*q1 - q0; 159 q0 = q1; 160 q1 = tmp; 161 } 162 m = n+n; 163 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 164 a = t; 165 b = one; 166 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 167 * Hence, if n*(log(2n/x)) > ... 168 * single 8.8722839355e+01 169 * double 7.09782712893383973096e+02 170 * long double 1.1356523406294143949491931077970765006170e+04 171 * then recurrent value may overflow and the result is 172 * likely underflow to zero 173 */ 174 tmp = n; 175 v = two/x; 176 tmp = tmp*log(fabs(v*tmp)); 177 if(tmp<7.09782712893383973096e+02) { 178 for(i=n-1,di=(double)(i+i);i>0;i--){ 179 temp = b; 180 b *= di; 181 b = b/x - a; 182 a = temp; 183 di -= two; 184 } 185 } else { 186 for(i=n-1,di=(double)(i+i);i>0;i--){ 187 temp = b; 188 b *= di; 189 b = b/x - a; 190 a = temp; 191 di -= two; 192 /* scale b to avoid spurious overflow */ 193 if(b>1e100) { 194 a /= b; 195 t /= b; 196 b = one; 197 } 198 } 199 } 200 z = j0(x); 201 w = j1(x); 202 if (fabs(z) >= fabs(w)) 203 b = (t*z/b); 204 else 205 b = (t*w/a); 206 } 207 } 208 if(sgn==1) return -b; else return b; 209 } 210 211 double 212 yn(int n, double x) 213 { 214 int32_t i,hx,ix,lx; 215 int32_t sign; 216 double a, b, c, s, temp; 217 218 EXTRACT_WORDS(hx,lx,x); 219 ix = 0x7fffffff&hx; 220 /* yn(n,NaN) = NaN */ 221 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 222 /* yn(n,+-0) = -inf and raise divide-by-zero exception. */ 223 if((ix|lx)==0) return -one/vzero; 224 /* yn(n,x<0) = NaN and raise invalid exception. */ 225 if(hx<0) return vzero/vzero; 226 sign = 1; 227 if(n<0){ 228 n = -n; 229 sign = 1 - ((n&1)<<1); 230 } 231 if(n==0) return(y0(x)); 232 if(n==1) return(sign*y1(x)); 233 if(ix==0x7ff00000) return zero; 234 if(ix>=0x52D00000) { /* x > 2**302 */ 235 /* (x >> n**2) 236 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 237 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 238 * Let s=sin(x), c=cos(x), 239 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then 240 * 241 * n sin(xn)*sqt2 cos(xn)*sqt2 242 * ---------------------------------- 243 * 0 s-c c+s 244 * 1 -s-c -c+s 245 * 2 -s+c -c-s 246 * 3 s+c c-s 247 */ 248 sincos(x, &s, &c); 249 switch(n&3) { 250 case 0: temp = s-c; break; 251 case 1: temp = -s-c; break; 252 case 2: temp = -s+c; break; 253 case 3: temp = s+c; break; 254 } 255 b = invsqrtpi*temp/sqrt(x); 256 } else { 257 u_int32_t high; 258 a = y0(x); 259 b = 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