2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
9 * ====================================================
12 #include <sys/cdefs.h>
15 * floating point Bessel's function of the 1st and 2nd kind
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).
33 * yn(n,x) is similar in all respects, except
34 * that forward recursion is used for all
39 #include "math_private.h"
41 static const volatile double vone = 1, vzero = 0;
44 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
45 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
46 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
48 static const double zero = 0.00000000000000000000e+00;
53 int32_t i,hx,ix,lx, sgn;
54 double a, b, c, s, temp, di;
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)
60 EXTRACT_WORDS(hx,lx,x);
62 /* if J(n,NaN) is NaN */
63 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
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) */
73 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
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 */
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
84 * n sin(xn)*sqt2 cos(xn)*sqt2
85 * ----------------------------------
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;
98 b = invsqrtpi*temp/sqrt(x);
104 b = b*((double)(i+i)/x) - a; /* avoid underflow */
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 - ...
113 if(n>33) /* underflow */
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 */
124 /* use backward recurrence */
126 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
127 * 2n - 2(n+1) - 2(n+2)
130 * (for large x) = ---- ------ ------ .....
132 * -- - ------ - ------ -
135 * Let w = 2n/x and h=2/x, then the above quotient
136 * is equal to the continued fraction:
138 * = -----------------------
140 * w - -----------------
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
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;
164 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
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
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--){
187 for(i=n-1,di=(double)(i+i);i>0;i--){
193 /* scale b to avoid spurious overflow */
203 if (fabs(z) >= fabs(w))
209 if(sgn==1) return -b; else return b;
217 double a, b, c, s, temp;
219 EXTRACT_WORDS(hx,lx,x);
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;
230 sign = 1 - ((n&1)<<1);
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 */
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
242 * n sin(xn)*sqt2 cos(xn)*sqt2
243 * ----------------------------------
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;
256 b = invsqrtpi*temp/sqrt(x);
261 /* quit if b is -inf */
262 GET_HIGH_WORD(high,b);
263 for(i=1;i<n&&high!=0xfff00000;i++){
265 b = ((double)(i+i)/x)*b - a;
266 GET_HIGH_WORD(high,b);
270 if(sign>0) return b; else return -b;