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 #include <sys/cdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: e_jn.c,v 1.14 2010/11/29 15:10:06 drochner Exp $"); 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 "namespace.h" 45 #include "math.h" 46 #include "math_private.h" 47 48 static const double 49 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 50 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 51 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 52 53 static const double zero = 0.00000000000000000000e+00; 54 55 double 56 __ieee754_jn(int n, double x) 57 { 58 int32_t i,hx,ix,lx, sgn; 59 double a, b, temp, di; 60 double z, w; 61 62 temp = 0; 63 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 64 * Thus, J(-n,x) = J(n,-x) 65 */ 66 EXTRACT_WORDS(hx,lx,x); 67 ix = 0x7fffffff&hx; 68 /* if J(n,NaN) is NaN */ 69 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 70 if(n<0){ 71 n = -n; 72 x = -x; 73 hx ^= 0x80000000; 74 } 75 if(n==0) return(__ieee754_j0(x)); 76 if(n==1) return(__ieee754_j1(x)); 77 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 78 x = fabs(x); 79 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 80 b = zero; 81 else if((double)n<=x) { 82 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 83 if(ix>=0x52D00000) { /* x > 2**302 */ 84 /* (x >> n**2) 85 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 86 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 87 * Let s=sin(x), c=cos(x), 88 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 89 * 90 * n sin(xn)*sqt2 cos(xn)*sqt2 91 * ---------------------------------- 92 * 0 s-c c+s 93 * 1 -s-c -c+s 94 * 2 -s+c -c-s 95 * 3 s+c c-s 96 */ 97 switch(n&3) { 98 case 0: temp = cos(x)+sin(x); break; 99 case 1: temp = -cos(x)+sin(x); break; 100 case 2: temp = -cos(x)-sin(x); break; 101 case 3: temp = cos(x)-sin(x); break; 102 } 103 b = invsqrtpi*temp/sqrt(x); 104 } else { 105 a = __ieee754_j0(x); 106 b = __ieee754_j1(x); 107 for(i=1;i<n;i++){ 108 temp = b; 109 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 110 a = temp; 111 } 112 } 113 } else { 114 if(ix<0x3e100000) { /* x < 2**-29 */ 115 /* x is tiny, return the first Taylor expansion of J(n,x) 116 * J(n,x) = 1/n!*(x/2)^n - ... 117 */ 118 if(n>33) /* underflow */ 119 b = zero; 120 else { 121 temp = x*0.5; b = temp; 122 for (a=one,i=2;i<=n;i++) { 123 a *= (double)i; /* a = n! */ 124 b *= temp; /* b = (x/2)^n */ 125 } 126 b = b/a; 127 } 128 } else { 129 /* use backward recurrence */ 130 /* x x^2 x^2 131 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 132 * 2n - 2(n+1) - 2(n+2) 133 * 134 * 1 1 1 135 * (for large x) = ---- ------ ------ ..... 136 * 2n 2(n+1) 2(n+2) 137 * -- - ------ - ------ - 138 * x x x 139 * 140 * Let w = 2n/x and h=2/x, then the above quotient 141 * is equal to the continued fraction: 142 * 1 143 * = ----------------------- 144 * 1 145 * w - ----------------- 146 * 1 147 * w+h - --------- 148 * w+2h - ... 149 * 150 * To determine how many terms needed, let 151 * Q(0) = w, Q(1) = w(w+h) - 1, 152 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 153 * When Q(k) > 1e4 good for single 154 * When Q(k) > 1e9 good for double 155 * When Q(k) > 1e17 good for quadruple 156 */ 157 /* determine k */ 158 double t,v; 159 double q0,q1,h,tmp; int32_t k,m; 160 w = (n+n)/(double)x; h = 2.0/(double)x; 161 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 162 while(q1<1.0e9) { 163 k += 1; z += h; 164 tmp = z*q1 - q0; 165 q0 = q1; 166 q1 = tmp; 167 } 168 m = n+n; 169 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 170 a = t; 171 b = one; 172 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 173 * Hence, if n*(log(2n/x)) > ... 174 * single 8.8722839355e+01 175 * double 7.09782712893383973096e+02 176 * long double 1.1356523406294143949491931077970765006170e+04 177 * then recurrent value may overflow and the result is 178 * likely underflow to zero 179 */ 180 tmp = n; 181 v = two/x; 182 tmp = tmp*__ieee754_log(fabs(v*tmp)); 183 if(tmp<7.09782712893383973096e+02) { 184 for(i=n-1,di=(double)(i+i);i>0;i--){ 185 temp = b; 186 b *= di; 187 b = b/x - a; 188 a = temp; 189 di -= two; 190 } 191 } else { 192 for(i=n-1,di=(double)(i+i);i>0;i--){ 193 temp = b; 194 b *= di; 195 b = b/x - a; 196 a = temp; 197 di -= two; 198 /* scale b to avoid spurious overflow */ 199 if(b>1e100) { 200 a /= b; 201 t /= b; 202 b = one; 203 } 204 } 205 } 206 z = __ieee754_j0(x); 207 w = __ieee754_j1(x); 208 if (fabs(z) >= fabs(w)) 209 b = (t*z/b); 210 else 211 b = (t*w/a); 212 } 213 } 214 if(sgn==1) return -b; else return b; 215 } 216 217 double 218 __ieee754_yn(int n, double x) 219 { 220 int32_t i,hx,ix,lx; 221 int32_t sign; 222 double a, b, temp; 223 224 temp = 0; 225 EXTRACT_WORDS(hx,lx,x); 226 ix = 0x7fffffff&hx; 227 /* if Y(n,NaN) is NaN */ 228 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 229 if((ix|lx)==0) return -one/zero; 230 if(hx<0) return zero/zero; 231 sign = 1; 232 if(n<0){ 233 n = -n; 234 sign = 1 - ((n&1)<<1); 235 } 236 if(n==0) return(__ieee754_y0(x)); 237 if(n==1) return(sign*__ieee754_y1(x)); 238 if(ix==0x7ff00000) return zero; 239 if(ix>=0x52D00000) { /* x > 2**302 */ 240 /* (x >> n**2) 241 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 242 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 243 * Let s=sin(x), c=cos(x), 244 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 245 * 246 * n sin(xn)*sqt2 cos(xn)*sqt2 247 * ---------------------------------- 248 * 0 s-c c+s 249 * 1 -s-c -c+s 250 * 2 -s+c -c-s 251 * 3 s+c c-s 252 */ 253 switch(n&3) { 254 case 0: temp = sin(x)-cos(x); break; 255 case 1: temp = -sin(x)-cos(x); break; 256 case 2: temp = -sin(x)+cos(x); break; 257 case 3: temp = sin(x)+cos(x); break; 258 } 259 b = invsqrtpi*temp/sqrt(x); 260 } else { 261 u_int32_t high; 262 a = __ieee754_y0(x); 263 b = __ieee754_y1(x); 264 /* quit if b is -inf */ 265 GET_HIGH_WORD(high,b); 266 for(i=1;i<n&&high!=0xfff00000;i++){ 267 temp = b; 268 b = ((double)(i+i)/x)*b - a; 269 GET_HIGH_WORD(high,b); 270 a = temp; 271 } 272 } 273 if(sign>0) return b; else return -b; 274 } 275