1 /* @(#)e_jn.c 5.1 93/09/24 */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
10 * ====================================================
12 * $FreeBSD: src/lib/msun/src/e_jn.c,v 1.6 1999/08/28 00:06:33 peter Exp $
13 * $DragonFly: src/lib/msun/src/Attic/e_jn.c,v 1.2 2003/06/17 04:26:52 dillon Exp $
17 * __ieee754_jn(n, x), __ieee754_yn(n, x)
18 * floating point Bessel's function of the 1st and 2nd kind
22 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
23 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
24 * Note 2. About jn(n,x), yn(n,x)
25 * For n=0, j0(x) is called,
26 * for n=1, j1(x) is called,
27 * for n<x, forward recursion us used starting
28 * from values of j0(x) and j1(x).
29 * for n>x, a continued fraction approximation to
30 * j(n,x)/j(n-1,x) is evaluated and then backward
31 * recursion is used starting from a supposed value
32 * for j(n,x). The resulting value of j(0,x) is
33 * compared with the actual value to correct the
34 * supposed value of j(n,x).
36 * yn(n,x) is similar in all respects, except
37 * that forward recursion is used for all
43 #include "math_private.h"
50 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
51 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
52 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
55 static const double zero = 0.00000000000000000000e+00;
57 static double zero = 0.00000000000000000000e+00;
61 double __ieee754_jn(int n, double x)
63 double __ieee754_jn(n,x)
67 int32_t i,hx,ix,lx, sgn;
68 double a, b, temp, di;
71 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
72 * Thus, J(-n,x) = J(n,-x)
74 EXTRACT_WORDS(hx,lx,x);
76 /* if J(n,NaN) is NaN */
77 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
83 if(n==0) return(__ieee754_j0(x));
84 if(n==1) return(__ieee754_j1(x));
85 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
87 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
89 else if((double)n<=x) {
90 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
91 if(ix>=0x52D00000) { /* x > 2**302 */
93 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
94 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
95 * Let s=sin(x), c=cos(x),
96 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
98 * n sin(xn)*sqt2 cos(xn)*sqt2
99 * ----------------------------------
106 case 0: temp = cos(x)+sin(x); break;
107 case 1: temp = -cos(x)+sin(x); break;
108 case 2: temp = -cos(x)-sin(x); break;
109 case 3: temp = cos(x)-sin(x); break;
111 b = invsqrtpi*temp/sqrt(x);
117 b = b*((double)(i+i)/x) - a; /* avoid underflow */
122 if(ix<0x3e100000) { /* x < 2**-29 */
123 /* x is tiny, return the first Taylor expansion of J(n,x)
124 * J(n,x) = 1/n!*(x/2)^n - ...
126 if(n>33) /* underflow */
129 temp = x*0.5; b = temp;
130 for (a=one,i=2;i<=n;i++) {
131 a *= (double)i; /* a = n! */
132 b *= temp; /* b = (x/2)^n */
137 /* use backward recurrence */
139 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
140 * 2n - 2(n+1) - 2(n+2)
143 * (for large x) = ---- ------ ------ .....
145 * -- - ------ - ------ -
148 * Let w = 2n/x and h=2/x, then the above quotient
149 * is equal to the continued fraction:
151 * = -----------------------
153 * w - -----------------
158 * To determine how many terms needed, let
159 * Q(0) = w, Q(1) = w(w+h) - 1,
160 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
161 * When Q(k) > 1e4 good for single
162 * When Q(k) > 1e9 good for double
163 * When Q(k) > 1e17 good for quadruple
167 double q0,q1,h,tmp; int32_t k,m;
168 w = (n+n)/(double)x; h = 2.0/(double)x;
169 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
177 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
180 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
181 * Hence, if n*(log(2n/x)) > ...
182 * single 8.8722839355e+01
183 * double 7.09782712893383973096e+02
184 * long double 1.1356523406294143949491931077970765006170e+04
185 * then recurrent value may overflow and the result is
186 * likely underflow to zero
190 tmp = tmp*__ieee754_log(fabs(v*tmp));
191 if(tmp<7.09782712893383973096e+02) {
192 for(i=n-1,di=(double)(i+i);i>0;i--){
200 for(i=n-1,di=(double)(i+i);i>0;i--){
206 /* scale b to avoid spurious overflow */
214 b = (t*__ieee754_j0(x)/b);
217 if(sgn==1) return -b; else return b;
221 double __ieee754_yn(int n, double x)
223 double __ieee754_yn(n,x)
231 EXTRACT_WORDS(hx,lx,x);
233 /* if Y(n,NaN) is NaN */
234 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
235 if((ix|lx)==0) return -one/zero;
236 if(hx<0) return zero/zero;
240 sign = 1 - ((n&1)<<1);
242 if(n==0) return(__ieee754_y0(x));
243 if(n==1) return(sign*__ieee754_y1(x));
244 if(ix==0x7ff00000) return zero;
245 if(ix>=0x52D00000) { /* x > 2**302 */
247 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
248 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
249 * Let s=sin(x), c=cos(x),
250 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
252 * n sin(xn)*sqt2 cos(xn)*sqt2
253 * ----------------------------------
260 case 0: temp = sin(x)-cos(x); break;
261 case 1: temp = -sin(x)-cos(x); break;
262 case 2: temp = -sin(x)+cos(x); break;
263 case 3: temp = sin(x)+cos(x); break;
265 b = invsqrtpi*temp/sqrt(x);
270 /* quit if b is -inf */
271 GET_HIGH_WORD(high,b);
272 for(i=1;i<n&&high!=0xfff00000;i++){
274 b = ((double)(i+i)/x)*b - a;
275 GET_HIGH_WORD(high,b);
279 if(sign>0) return b; else return -b;