2 /* @(#)e_jn.c 1.4 95/01/18 */
3 /* $FreeBSD: head/lib/msun/src/e_jn.c 215237 2010-11-13 10:54:10Z uqs $ */
5 * ====================================================
6 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
8 * Developed at SunSoft, a Sun Microsystems, Inc. business.
9 * Permission to use, copy, modify, and distribute this
10 * software is freely granted, provided that this notice
12 * ====================================================
16 * __ieee754_jn(n, x), __ieee754_yn(n, x)
17 * floating point Bessel's function of the 1st and 2nd kind
21 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
22 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
23 * Note 2. About jn(n,x), yn(n,x)
24 * For n=0, j0(x) is called,
25 * for n=1, j1(x) is called,
26 * for n<x, forward recursion us used starting
27 * from values of j0(x) and j1(x).
28 * for n>x, a continued fraction approximation to
29 * j(n,x)/j(n-1,x) is evaluated and then backward
30 * recursion is used starting from a supposed value
31 * for j(n,x). The resulting value of j(0,x) is
32 * compared with the actual value to correct the
33 * supposed value of j(n,x).
35 * yn(n,x) is similar in all respects, except
36 * that forward recursion is used for all
42 #include "math_private.h"
45 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
46 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
47 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
49 static const double zero = 0.00000000000000000000e+00;
52 __ieee754_jn(int n, double x)
54 int32_t i,hx,ix,lx, sgn;
55 double a, b, temp, di;
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)
61 EXTRACT_WORDS(hx,lx,x);
63 /* if J(n,NaN) is NaN */
64 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
70 if(n==0) return(__ieee754_j0(x));
71 if(n==1) return(__ieee754_j1(x));
72 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
74 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
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 */
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
85 * n sin(xn)*sqt2 cos(xn)*sqt2
86 * ----------------------------------
93 case 0: temp = cos(x)+sin(x); break;
94 case 1: temp = -cos(x)+sin(x); break;
95 case 2: temp = -cos(x)-sin(x); break;
96 case 3: temp = cos(x)-sin(x); 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*__ieee754_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;
213 __ieee754_yn(int n, double x)
219 EXTRACT_WORDS(hx,lx,x);
221 /* if Y(n,NaN) is NaN */
222 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
223 if((ix|lx)==0) return -one/zero;
224 if(hx<0) return zero/zero;
228 sign = 1 - ((n&1)<<1);
230 if(n==0) return(__ieee754_y0(x));
231 if(n==1) return(sign*__ieee754_y1(x));
232 if(ix==0x7ff00000) return zero;
233 if(ix>=0x52D00000) { /* x > 2**302 */
235 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
236 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
237 * Let s=sin(x), c=cos(x),
238 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
240 * n sin(xn)*sqt2 cos(xn)*sqt2
241 * ----------------------------------
248 case 0: temp = sin(x)-cos(x); break;
249 case 1: temp = -sin(x)-cos(x); break;
250 case 2: temp = -sin(x)+cos(x); break;
251 case 3: temp = sin(x)+cos(x); break;
253 b = invsqrtpi*temp/sqrt(x);
258 /* quit if b is -inf */
259 GET_HIGH_WORD(high,b);
260 for(i=1;i<n&&high!=0xfff00000;i++){
262 b = ((double)(i+i)/x)*b - a;
263 GET_HIGH_WORD(high,b);
267 if(sign>0) return b; else return -b;