1 /* @(#)e_jn.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, 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 #ifndef lint 14 static char rcsid[] = "$FreeBSD$"; 15 #endif 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 #ifdef __STDC__ 47 static const double 48 #else 49 static double 50 #endif 51 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 52 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 53 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 54 55 #ifdef __STDC__ 56 static const double zero = 0.00000000000000000000e+00; 57 #else 58 static double zero = 0.00000000000000000000e+00; 59 #endif 60 61 #ifdef __STDC__ 62 double __ieee754_jn(int n, double x) 63 #else 64 double __ieee754_jn(n,x) 65 int n; double x; 66 #endif 67 { 68 int32_t i,hx,ix,lx, sgn; 69 double a, b, temp, di; 70 double z, w; 71 72 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 73 * Thus, J(-n,x) = J(n,-x) 74 */ 75 EXTRACT_WORDS(hx,lx,x); 76 ix = 0x7fffffff&hx; 77 /* if J(n,NaN) is NaN */ 78 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 79 if(n<0){ 80 n = -n; 81 x = -x; 82 hx ^= 0x80000000; 83 } 84 if(n==0) return(__ieee754_j0(x)); 85 if(n==1) return(__ieee754_j1(x)); 86 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 87 x = fabs(x); 88 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 89 b = zero; 90 else if((double)n<=x) { 91 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 92 if(ix>=0x52D00000) { /* x > 2**302 */ 93 /* (x >> n**2) 94 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 95 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 96 * Let s=sin(x), c=cos(x), 97 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 98 * 99 * n sin(xn)*sqt2 cos(xn)*sqt2 100 * ---------------------------------- 101 * 0 s-c c+s 102 * 1 -s-c -c+s 103 * 2 -s+c -c-s 104 * 3 s+c c-s 105 */ 106 switch(n&3) { 107 case 0: temp = cos(x)+sin(x); break; 108 case 1: temp = -cos(x)+sin(x); break; 109 case 2: temp = -cos(x)-sin(x); break; 110 case 3: temp = cos(x)-sin(x); break; 111 } 112 b = invsqrtpi*temp/sqrt(x); 113 } else { 114 a = __ieee754_j0(x); 115 b = __ieee754_j1(x); 116 for(i=1;i<n;i++){ 117 temp = b; 118 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 119 a = temp; 120 } 121 } 122 } else { 123 if(ix<0x3e100000) { /* x < 2**-29 */ 124 /* x is tiny, return the first Taylor expansion of J(n,x) 125 * J(n,x) = 1/n!*(x/2)^n - ... 126 */ 127 if(n>33) /* underflow */ 128 b = zero; 129 else { 130 temp = x*0.5; b = temp; 131 for (a=one,i=2;i<=n;i++) { 132 a *= (double)i; /* a = n! */ 133 b *= temp; /* b = (x/2)^n */ 134 } 135 b = b/a; 136 } 137 } else { 138 /* use backward recurrence */ 139 /* x x^2 x^2 140 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 141 * 2n - 2(n+1) - 2(n+2) 142 * 143 * 1 1 1 144 * (for large x) = ---- ------ ------ ..... 145 * 2n 2(n+1) 2(n+2) 146 * -- - ------ - ------ - 147 * x x x 148 * 149 * Let w = 2n/x and h=2/x, then the above quotient 150 * is equal to the continued fraction: 151 * 1 152 * = ----------------------- 153 * 1 154 * w - ----------------- 155 * 1 156 * w+h - --------- 157 * w+2h - ... 158 * 159 * To determine how many terms needed, let 160 * Q(0) = w, Q(1) = w(w+h) - 1, 161 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 162 * When Q(k) > 1e4 good for single 163 * When Q(k) > 1e9 good for double 164 * When Q(k) > 1e17 good for quadruple 165 */ 166 /* determine k */ 167 double t,v; 168 double q0,q1,h,tmp; int32_t k,m; 169 w = (n+n)/(double)x; h = 2.0/(double)x; 170 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 171 while(q1<1.0e9) { 172 k += 1; z += h; 173 tmp = z*q1 - q0; 174 q0 = q1; 175 q1 = tmp; 176 } 177 m = n+n; 178 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 179 a = t; 180 b = one; 181 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 182 * Hence, if n*(log(2n/x)) > ... 183 * single 8.8722839355e+01 184 * double 7.09782712893383973096e+02 185 * long double 1.1356523406294143949491931077970765006170e+04 186 * then recurrent value may overflow and the result is 187 * likely underflow to zero 188 */ 189 tmp = n; 190 v = two/x; 191 tmp = tmp*__ieee754_log(fabs(v*tmp)); 192 if(tmp<7.09782712893383973096e+02) { 193 for(i=n-1,di=(double)(i+i);i>0;i--){ 194 temp = b; 195 b *= di; 196 b = b/x - a; 197 a = temp; 198 di -= two; 199 } 200 } else { 201 for(i=n-1,di=(double)(i+i);i>0;i--){ 202 temp = b; 203 b *= di; 204 b = b/x - a; 205 a = temp; 206 di -= two; 207 /* scale b to avoid spurious overflow */ 208 if(b>1e100) { 209 a /= b; 210 t /= b; 211 b = one; 212 } 213 } 214 } 215 b = (t*__ieee754_j0(x)/b); 216 } 217 } 218 if(sgn==1) return -b; else return b; 219 } 220 221 #ifdef __STDC__ 222 double __ieee754_yn(int n, double x) 223 #else 224 double __ieee754_yn(n,x) 225 int n; double x; 226 #endif 227 { 228 int32_t i,hx,ix,lx; 229 int32_t sign; 230 double a, b, temp; 231 232 EXTRACT_WORDS(hx,lx,x); 233 ix = 0x7fffffff&hx; 234 /* if Y(n,NaN) is NaN */ 235 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 236 if((ix|lx)==0) return -one/zero; 237 if(hx<0) return zero/zero; 238 sign = 1; 239 if(n<0){ 240 n = -n; 241 sign = 1 - ((n&1)<<1); 242 } 243 if(n==0) return(__ieee754_y0(x)); 244 if(n==1) return(sign*__ieee754_y1(x)); 245 if(ix==0x7ff00000) return zero; 246 if(ix>=0x52D00000) { /* x > 2**302 */ 247 /* (x >> n**2) 248 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 249 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 250 * Let s=sin(x), c=cos(x), 251 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 252 * 253 * n sin(xn)*sqt2 cos(xn)*sqt2 254 * ---------------------------------- 255 * 0 s-c c+s 256 * 1 -s-c -c+s 257 * 2 -s+c -c-s 258 * 3 s+c c-s 259 */ 260 switch(n&3) { 261 case 0: temp = sin(x)-cos(x); break; 262 case 1: temp = -sin(x)-cos(x); break; 263 case 2: temp = -sin(x)+cos(x); break; 264 case 3: temp = sin(x)+cos(x); break; 265 } 266 b = invsqrtpi*temp/sqrt(x); 267 } else { 268 u_int32_t high; 269 a = __ieee754_y0(x); 270 b = __ieee754_y1(x); 271 /* quit if b is -inf */ 272 GET_HIGH_WORD(high,b); 273 for(i=1;i<n&&high!=0xfff00000;i++){ 274 temp = b; 275 b = ((double)(i+i)/x)*b - a; 276 GET_HIGH_WORD(high,b); 277 a = temp; 278 } 279 } 280 if(sign>0) return b; else return -b; 281 } 282