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 #include <sys/cdefs.h> 13 /* 14 * jn(n, x), yn(n, x) 15 * floating point Bessel's function of the 1st and 2nd kind 16 * of order n 17 * 18 * Special cases: 19 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 20 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 21 * Note 2. About jn(n,x), yn(n,x) 22 * For n=0, j0(x) is called. 23 * For n=1, j1(x) is called. 24 * For n<x, forward recursion is used starting 25 * from values of j0(x) and j1(x). 26 * For n>x, a continued fraction approximation to 27 * j(n,x)/j(n-1,x) is evaluated and then backward 28 * recursion is used starting from a supposed value 29 * for j(n,x). The resulting values of j(0,x) or j(1,x) are 30 * compared with the actual values to correct the 31 * supposed value of j(n,x). 32 * 33 * yn(n,x) is similar in all respects, except 34 * that forward recursion is used for all 35 * values of n>1. 36 */ 37 38 #include "math.h" 39 #include "math_private.h" 40 41 static const volatile double vone = 1, vzero = 0; 42 43 static const double 44 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 45 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 46 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 47 48 static const double zero = 0.00000000000000000000e+00; 49 50 double 51 jn(int n, double x) 52 { 53 int32_t i,hx,ix,lx, sgn; 54 double a, b, c, s, temp, di; 55 double z, w; 56 57 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 58 * Thus, J(-n,x) = J(n,-x) 59 */ 60 EXTRACT_WORDS(hx,lx,x); 61 ix = 0x7fffffff&hx; 62 /* if J(n,NaN) is NaN */ 63 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 64 if(n<0){ 65 n = -n; 66 x = -x; 67 hx ^= 0x80000000; 68 } 69 if(n==0) return(j0(x)); 70 if(n==1) return(j1(x)); 71 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 72 x = fabs(x); 73 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 74 b = zero; 75 else if((double)n<=x) { 76 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 77 if(ix>=0x52D00000) { /* x > 2**302 */ 78 /* (x >> n**2) 79 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 80 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 81 * Let s=sin(x), c=cos(x), 82 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then 83 * 84 * n sin(xn)*sqt2 cos(xn)*sqt2 85 * ---------------------------------- 86 * 0 s-c c+s 87 * 1 -s-c -c+s 88 * 2 -s+c -c-s 89 * 3 s+c c-s 90 */ 91 sincos(x, &s, &c); 92 switch(n&3) { 93 case 0: temp = c+s; break; 94 case 1: temp = -c+s; break; 95 case 2: temp = -c-s; break; 96 case 3: temp = c-s; break; 97 } 98 b = invsqrtpi*temp/sqrt(x); 99 } else { 100 a = j0(x); 101 b = j1(x); 102 for(i=1;i<n;i++){ 103 temp = b; 104 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 105 a = temp; 106 } 107 } 108 } else { 109 if(ix<0x3e100000) { /* x < 2**-29 */ 110 /* x is tiny, return the first Taylor expansion of J(n,x) 111 * J(n,x) = 1/n!*(x/2)^n - ... 112 */ 113 if(n>33) /* underflow */ 114 b = zero; 115 else { 116 temp = x*0.5; b = temp; 117 for (a=one,i=2;i<=n;i++) { 118 a *= (double)i; /* a = n! */ 119 b *= temp; /* b = (x/2)^n */ 120 } 121 b = b/a; 122 } 123 } else { 124 /* use backward recurrence */ 125 /* x x^2 x^2 126 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 127 * 2n - 2(n+1) - 2(n+2) 128 * 129 * 1 1 1 130 * (for large x) = ---- ------ ------ ..... 131 * 2n 2(n+1) 2(n+2) 132 * -- - ------ - ------ - 133 * x x x 134 * 135 * Let w = 2n/x and h=2/x, then the above quotient 136 * is equal to the continued fraction: 137 * 1 138 * = ----------------------- 139 * 1 140 * w - ----------------- 141 * 1 142 * w+h - --------- 143 * w+2h - ... 144 * 145 * To determine how many terms needed, let 146 * Q(0) = w, Q(1) = w(w+h) - 1, 147 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 148 * When Q(k) > 1e4 good for single 149 * When Q(k) > 1e9 good for double 150 * When Q(k) > 1e17 good for quadruple 151 */ 152 /* determine k */ 153 double t,v; 154 double q0,q1,h,tmp; int32_t k,m; 155 w = (n+n)/(double)x; h = 2.0/(double)x; 156 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 157 while(q1<1.0e9) { 158 k += 1; z += h; 159 tmp = z*q1 - q0; 160 q0 = q1; 161 q1 = tmp; 162 } 163 m = n+n; 164 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 165 a = t; 166 b = one; 167 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 168 * Hence, if n*(log(2n/x)) > ... 169 * single 8.8722839355e+01 170 * double 7.09782712893383973096e+02 171 * long double 1.1356523406294143949491931077970765006170e+04 172 * then recurrent value may overflow and the result is 173 * likely underflow to zero 174 */ 175 tmp = n; 176 v = two/x; 177 tmp = tmp*log(fabs(v*tmp)); 178 if(tmp<7.09782712893383973096e+02) { 179 for(i=n-1,di=(double)(i+i);i>0;i--){ 180 temp = b; 181 b *= di; 182 b = b/x - a; 183 a = temp; 184 di -= two; 185 } 186 } else { 187 for(i=n-1,di=(double)(i+i);i>0;i--){ 188 temp = b; 189 b *= di; 190 b = b/x - a; 191 a = temp; 192 di -= two; 193 /* scale b to avoid spurious overflow */ 194 if(b>1e100) { 195 a /= b; 196 t /= b; 197 b = one; 198 } 199 } 200 } 201 z = j0(x); 202 w = j1(x); 203 if (fabs(z) >= fabs(w)) 204 b = (t*z/b); 205 else 206 b = (t*w/a); 207 } 208 } 209 if(sgn==1) return -b; else return b; 210 } 211 212 double 213 yn(int n, double x) 214 { 215 int32_t i,hx,ix,lx; 216 int32_t sign; 217 double a, b, c, s, temp; 218 219 EXTRACT_WORDS(hx,lx,x); 220 ix = 0x7fffffff&hx; 221 /* yn(n,NaN) = NaN */ 222 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 223 /* yn(n,+-0) = -inf and raise divide-by-zero exception. */ 224 if((ix|lx)==0) return -one/vzero; 225 /* yn(n,x<0) = NaN and raise invalid exception. */ 226 if(hx<0) return vzero/vzero; 227 sign = 1; 228 if(n<0){ 229 n = -n; 230 sign = 1 - ((n&1)<<1); 231 } 232 if(n==0) return(y0(x)); 233 if(n==1) return(sign*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 sincos(x, &s, &c); 250 switch(n&3) { 251 case 0: temp = s-c; break; 252 case 1: temp = -s-c; break; 253 case 2: temp = -s+c; break; 254 case 3: temp = s+c; break; 255 } 256 b = invsqrtpi*temp/sqrt(x); 257 } else { 258 u_int32_t high; 259 a = y0(x); 260 b = y1(x); 261 /* quit if b is -inf */ 262 GET_HIGH_WORD(high,b); 263 for(i=1;i<n&&high!=0xfff00000;i++){ 264 temp = b; 265 b = ((double)(i+i)/x)*b - a; 266 GET_HIGH_WORD(high,b); 267 a = temp; 268 } 269 } 270 if(sign>0) return b; else return -b; 271 } 272