1 /* @(#)e_jn.c 1.4 95/01/18 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunSoft, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 #include <sys/cdefs.h> 14 /* 15 * jn(n, x), yn(n, x) 16 * floating point Bessel's function of the 1st and 2nd kind 17 * of order n 18 * 19 * Special cases: 20 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 21 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 22 * Note 2. About jn(n,x), yn(n,x) 23 * For n=0, j0(x) is called. 24 * For n=1, j1(x) is called. 25 * For n<x, forward recursion is used starting 26 * from values of j0(x) and j1(x). 27 * For n>x, a continued fraction approximation to 28 * j(n,x)/j(n-1,x) is evaluated and then backward 29 * recursion is used starting from a supposed value 30 * for j(n,x). The resulting values of j(0,x) or j(1,x) are 31 * compared with the actual values to correct the 32 * supposed value of j(n,x). 33 * 34 * yn(n,x) is similar in all respects, except 35 * that forward recursion is used for all 36 * values of n>1. 37 */ 38 39 #include "math.h" 40 #include "math_private.h" 41 42 static const volatile double vone = 1, vzero = 0; 43 44 static const double 45 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 46 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 47 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 48 49 static const double zero = 0.00000000000000000000e+00; 50 51 double 52 jn(int n, double x) 53 { 54 int32_t i,hx,ix,lx, sgn; 55 double a, b, c, s, temp, di; 56 double z, w; 57 58 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 59 * Thus, J(-n,x) = J(n,-x) 60 */ 61 EXTRACT_WORDS(hx,lx,x); 62 ix = 0x7fffffff&hx; 63 /* if J(n,NaN) is NaN */ 64 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 65 if(n<0){ 66 n = -n; 67 x = -x; 68 hx ^= 0x80000000; 69 } 70 if(n==0) return(j0(x)); 71 if(n==1) return(j1(x)); 72 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 73 x = fabs(x); 74 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 75 b = zero; 76 else if((double)n<=x) { 77 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 78 if(ix>=0x52D00000) { /* x > 2**302 */ 79 /* (x >> n**2) 80 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 81 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 82 * Let s=sin(x), c=cos(x), 83 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then 84 * 85 * n sin(xn)*sqt2 cos(xn)*sqt2 86 * ---------------------------------- 87 * 0 s-c c+s 88 * 1 -s-c -c+s 89 * 2 -s+c -c-s 90 * 3 s+c c-s 91 */ 92 sincos(x, &s, &c); 93 switch(n&3) { 94 case 0: temp = c+s; break; 95 case 1: temp = -c+s; break; 96 case 2: temp = -c-s; break; 97 case 3: temp = c-s; break; 98 } 99 b = invsqrtpi*temp/sqrt(x); 100 } else { 101 a = j0(x); 102 b = j1(x); 103 for(i=1;i<n;i++){ 104 temp = b; 105 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 106 a = temp; 107 } 108 } 109 } else { 110 if(ix<0x3e100000) { /* x < 2**-29 */ 111 /* x is tiny, return the first Taylor expansion of J(n,x) 112 * J(n,x) = 1/n!*(x/2)^n - ... 113 */ 114 if(n>33) /* underflow */ 115 b = zero; 116 else { 117 temp = x*0.5; b = temp; 118 for (a=one,i=2;i<=n;i++) { 119 a *= (double)i; /* a = n! */ 120 b *= temp; /* b = (x/2)^n */ 121 } 122 b = b/a; 123 } 124 } else { 125 /* use backward recurrence */ 126 /* x x^2 x^2 127 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 128 * 2n - 2(n+1) - 2(n+2) 129 * 130 * 1 1 1 131 * (for large x) = ---- ------ ------ ..... 132 * 2n 2(n+1) 2(n+2) 133 * -- - ------ - ------ - 134 * x x x 135 * 136 * Let w = 2n/x and h=2/x, then the above quotient 137 * is equal to the continued fraction: 138 * 1 139 * = ----------------------- 140 * 1 141 * w - ----------------- 142 * 1 143 * w+h - --------- 144 * w+2h - ... 145 * 146 * To determine how many terms needed, let 147 * Q(0) = w, Q(1) = w(w+h) - 1, 148 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 149 * When Q(k) > 1e4 good for single 150 * When Q(k) > 1e9 good for double 151 * When Q(k) > 1e17 good for quadruple 152 */ 153 /* determine k */ 154 double t,v; 155 double q0,q1,h,tmp; int32_t k,m; 156 w = (n+n)/(double)x; h = 2.0/(double)x; 157 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 158 while(q1<1.0e9) { 159 k += 1; z += h; 160 tmp = z*q1 - q0; 161 q0 = q1; 162 q1 = tmp; 163 } 164 m = n+n; 165 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 166 a = t; 167 b = one; 168 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 169 * Hence, if n*(log(2n/x)) > ... 170 * single 8.8722839355e+01 171 * double 7.09782712893383973096e+02 172 * long double 1.1356523406294143949491931077970765006170e+04 173 * then recurrent value may overflow and the result is 174 * likely underflow to zero 175 */ 176 tmp = n; 177 v = two/x; 178 tmp = tmp*log(fabs(v*tmp)); 179 if(tmp<7.09782712893383973096e+02) { 180 for(i=n-1,di=(double)(i+i);i>0;i--){ 181 temp = b; 182 b *= di; 183 b = b/x - a; 184 a = temp; 185 di -= two; 186 } 187 } else { 188 for(i=n-1,di=(double)(i+i);i>0;i--){ 189 temp = b; 190 b *= di; 191 b = b/x - a; 192 a = temp; 193 di -= two; 194 /* scale b to avoid spurious overflow */ 195 if(b>1e100) { 196 a /= b; 197 t /= b; 198 b = one; 199 } 200 } 201 } 202 z = j0(x); 203 w = j1(x); 204 if (fabs(z) >= fabs(w)) 205 b = (t*z/b); 206 else 207 b = (t*w/a); 208 } 209 } 210 if(sgn==1) return -b; else return b; 211 } 212 213 double 214 yn(int n, double x) 215 { 216 int32_t i,hx,ix,lx; 217 int32_t sign; 218 double a, b, c, s, temp; 219 220 EXTRACT_WORDS(hx,lx,x); 221 ix = 0x7fffffff&hx; 222 /* yn(n,NaN) = NaN */ 223 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 224 /* yn(n,+-0) = -inf and raise divide-by-zero exception. */ 225 if((ix|lx)==0) return -one/vzero; 226 /* yn(n,x<0) = NaN and raise invalid exception. */ 227 if(hx<0) return vzero/vzero; 228 sign = 1; 229 if(n<0){ 230 n = -n; 231 sign = 1 - ((n&1)<<1); 232 } 233 if(n==0) return(y0(x)); 234 if(n==1) return(sign*y1(x)); 235 if(ix==0x7ff00000) return zero; 236 if(ix>=0x52D00000) { /* x > 2**302 */ 237 /* (x >> n**2) 238 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 239 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 240 * Let s=sin(x), c=cos(x), 241 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then 242 * 243 * n sin(xn)*sqt2 cos(xn)*sqt2 244 * ---------------------------------- 245 * 0 s-c c+s 246 * 1 -s-c -c+s 247 * 2 -s+c -c-s 248 * 3 s+c c-s 249 */ 250 sincos(x, &s, &c); 251 switch(n&3) { 252 case 0: temp = s-c; break; 253 case 1: temp = -s-c; break; 254 case 2: temp = -s+c; break; 255 case 3: temp = s+c; break; 256 } 257 b = invsqrtpi*temp/sqrt(x); 258 } else { 259 u_int32_t high; 260 a = y0(x); 261 b = y1(x); 262 /* quit if b is -inf */ 263 GET_HIGH_WORD(high,b); 264 for(i=1;i<n&&high!=0xfff00000;i++){ 265 temp = b; 266 b = ((double)(i+i)/x)*b - a; 267 GET_HIGH_WORD(high,b); 268 a = temp; 269 } 270 } 271 if(sign>0) return b; else return -b; 272 } 273