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 /* 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 us 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 value of j(0,x) is 30 * compared with the actual value 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 39 #include "math.h" 40 #include "math_private.h" 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, 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 switch(n&3) { 91 case 0: temp = cos(x)+sin(x); break; 92 case 1: temp = -cos(x)+sin(x); break; 93 case 2: temp = -cos(x)-sin(x); break; 94 case 3: temp = cos(x)-sin(x); break; 95 } 96 b = invsqrtpi*temp/sqrt(x); 97 } else { 98 a = j0(x); 99 b = j1(x); 100 for(i=1;i<n;i++){ 101 temp = b; 102 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 103 a = temp; 104 } 105 } 106 } else { 107 if(ix<0x3e100000) { /* x < 2**-29 */ 108 /* x is tiny, return the first Taylor expansion of J(n,x) 109 * J(n,x) = 1/n!*(x/2)^n - ... 110 */ 111 if(n>33) /* underflow */ 112 b = zero; 113 else { 114 temp = x*0.5; b = temp; 115 for (a=one,i=2;i<=n;i++) { 116 a *= (double)i; /* a = n! */ 117 b *= temp; /* b = (x/2)^n */ 118 } 119 b = b/a; 120 } 121 } else { 122 /* use backward recurrence */ 123 /* x x^2 x^2 124 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 125 * 2n - 2(n+1) - 2(n+2) 126 * 127 * 1 1 1 128 * (for large x) = ---- ------ ------ ..... 129 * 2n 2(n+1) 2(n+2) 130 * -- - ------ - ------ - 131 * x x x 132 * 133 * Let w = 2n/x and h=2/x, then the above quotient 134 * is equal to the continued fraction: 135 * 1 136 * = ----------------------- 137 * 1 138 * w - ----------------- 139 * 1 140 * w+h - --------- 141 * w+2h - ... 142 * 143 * To determine how many terms needed, let 144 * Q(0) = w, Q(1) = w(w+h) - 1, 145 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 146 * When Q(k) > 1e4 good for single 147 * When Q(k) > 1e9 good for double 148 * When Q(k) > 1e17 good for quadruple 149 */ 150 /* determine k */ 151 double t,v; 152 double q0,q1,h,tmp; int32_t k,m; 153 w = (n+n)/(double)x; h = 2.0/(double)x; 154 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 155 while(q1<1.0e9) { 156 k += 1; z += h; 157 tmp = z*q1 - q0; 158 q0 = q1; 159 q1 = tmp; 160 } 161 m = n+n; 162 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 163 a = t; 164 b = one; 165 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 166 * Hence, if n*(log(2n/x)) > ... 167 * single 8.8722839355e+01 168 * double 7.09782712893383973096e+02 169 * long double 1.1356523406294143949491931077970765006170e+04 170 * then recurrent value may overflow and the result is 171 * likely underflow to zero 172 */ 173 tmp = n; 174 v = two/x; 175 tmp = tmp*log(fabs(v*tmp)); 176 if(tmp<7.09782712893383973096e+02) { 177 for(i=n-1,di=(double)(i+i);i>0;i--){ 178 temp = b; 179 b *= di; 180 b = b/x - a; 181 a = temp; 182 di -= two; 183 } 184 } else { 185 for(i=n-1,di=(double)(i+i);i>0;i--){ 186 temp = b; 187 b *= di; 188 b = b/x - a; 189 a = temp; 190 di -= two; 191 /* scale b to avoid spurious overflow */ 192 if(b>1e100) { 193 a /= b; 194 t /= b; 195 b = one; 196 } 197 } 198 } 199 b = (t*j0(x)/b); 200 } 201 } 202 if(sgn==1) return -b; else return b; 203 } 204 205 double 206 yn(int n, double x) 207 { 208 int32_t i,hx,ix,lx; 209 int32_t sign; 210 double a, b, temp; 211 212 EXTRACT_WORDS(hx,lx,x); 213 ix = 0x7fffffff&hx; 214 /* if Y(n,NaN) is NaN */ 215 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 216 if((ix|lx)==0) return -one/zero; 217 if(hx<0) return zero/zero; 218 sign = 1; 219 if(n<0){ 220 n = -n; 221 sign = 1 - ((n&1)<<1); 222 } 223 if(n==0) return(y0(x)); 224 if(n==1) return(sign*y1(x)); 225 if(ix==0x7ff00000) return zero; 226 if(ix>=0x52D00000) { /* x > 2**302 */ 227 /* (x >> n**2) 228 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 229 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 230 * Let s=sin(x), c=cos(x), 231 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 232 * 233 * n sin(xn)*sqt2 cos(xn)*sqt2 234 * ---------------------------------- 235 * 0 s-c c+s 236 * 1 -s-c -c+s 237 * 2 -s+c -c-s 238 * 3 s+c c-s 239 */ 240 switch(n&3) { 241 case 0: temp = sin(x)-cos(x); break; 242 case 1: temp = -sin(x)-cos(x); break; 243 case 2: temp = -sin(x)+cos(x); break; 244 case 3: temp = sin(x)+cos(x); break; 245 } 246 b = invsqrtpi*temp/sqrt(x); 247 } else { 248 u_int32_t high; 249 a = y0(x); 250 b = y1(x); 251 /* quit if b is -inf */ 252 GET_HIGH_WORD(high,b); 253 for(i=1;i<n&&high!=0xfff00000;i++){ 254 temp = b; 255 b = ((double)(i+i)/x)*b - a; 256 GET_HIGH_WORD(high,b); 257 a = temp; 258 } 259 } 260 if(sign>0) return b; else return -b; 261 } 262