e_jn.c revision 22993 
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#ifndef lint
14static char rcsid[] = "$Id$";
15#endif
16
17/*
18 * __ieee754_jn(n, x), __ieee754_yn(n, x)
19 * floating point Bessel's function of the 1st and 2nd kind
20 * of order n
21 *
22 * Special cases:
23 *	y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
24 *	y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
25 * Note 2. About jn(n,x), yn(n,x)
26 *	For n=0, j0(x) is called,
27 *	for n=1, j1(x) is called,
28 *	for n<x, forward recursion us used starting
29 *	from values of j0(x) and j1(x).
30 *	for n>x, a continued fraction approximation to
31 *	j(n,x)/j(n-1,x) is evaluated and then backward
32 *	recursion is used starting from a supposed value
33 *	for j(n,x). The resulting value of j(0,x) is
34 *	compared with the actual value to correct the
35 *	supposed value of j(n,x).
36 *
37 *	yn(n,x) is similar in all respects, except
38 *	that forward recursion is used for all
39 *	values of n>1.
40 *
41 */
42
43#include "math.h"
44#include "math_private.h"
45
46#ifdef __STDC__
47static const double
48#else
49static double
50#endif
51invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
52two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
53one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
54
55#ifdef __STDC__
56static const double zero  =  0.00000000000000000000e+00;
57#else
58static double zero  =  0.00000000000000000000e+00;
59#endif
60
61#ifdef __STDC__
62	double __ieee754_jn(int n, double x)
63#else
64	double __ieee754_jn(n,x)
65	int n; double x;
66#endif
67{
68	int32_t i,hx,ix,lx, sgn;
69	double a, b, temp, di;
70	double z, w;
71
72    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
73     * Thus, J(-n,x) = J(n,-x)
74     */
75	EXTRACT_WORDS(hx,lx,x);
76	ix = 0x7fffffff&hx;
77    /* if J(n,NaN) is NaN */
78	if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
79	if(n<0){
80		n = -n;
81		x = -x;
82		hx ^= 0x80000000;
83	}
84	if(n==0) return(__ieee754_j0(x));
85	if(n==1) return(__ieee754_j1(x));
86	sgn = (n&1)&(hx>>31);	/* even n -- 0, odd n -- sign(x) */
87	x = fabs(x);
88	if((ix|lx)==0||ix>=0x7ff00000) 	/* if x is 0 or inf */
89	    b = zero;
90	else if((double)n<=x) {
91		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
92	    if(ix>=0x52D00000) { /* x > 2**302 */
93    /* (x >> n**2)
94     *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
95     *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
96     *	    Let s=sin(x), c=cos(x),
97     *		xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
98     *
99     *		   n	sin(xn)*sqt2	cos(xn)*sqt2
100     *		----------------------------------
101     *		   0	 s-c		 c+s
102     *		   1	-s-c 		-c+s
103     *		   2	-s+c		-c-s
104     *		   3	 s+c		 c-s
105     */
106		switch(n&3) {
107		    case 0: temp =  cos(x)+sin(x); break;
108		    case 1: temp = -cos(x)+sin(x); break;
109		    case 2: temp = -cos(x)-sin(x); break;
110		    case 3: temp =  cos(x)-sin(x); break;
111		}
112		b = invsqrtpi*temp/sqrt(x);
113	    } else {
114	        a = __ieee754_j0(x);
115	        b = __ieee754_j1(x);
116	        for(i=1;i<n;i++){
117		    temp = b;
118		    b = b*((double)(i+i)/x) - a; /* avoid underflow */
119		    a = temp;
120	        }
121	    }
122	} else {
123	    if(ix<0x3e100000) {	/* x < 2**-29 */
124    /* x is tiny, return the first Taylor expansion of J(n,x)
125     * J(n,x) = 1/n!*(x/2)^n  - ...
126     */
127		if(n>33)	/* underflow */
128		    b = zero;
129		else {
130		    temp = x*0.5; b = temp;
131		    for (a=one,i=2;i<=n;i++) {
132			a *= (double)i;		/* a = n! */
133			b *= temp;		/* b = (x/2)^n */
134		    }
135		    b = b/a;
136		}
137	    } else {
138		/* use backward recurrence */
139		/* 			x      x^2      x^2
140		 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
141		 *			2n  - 2(n+1) - 2(n+2)
142		 *
143		 * 			1      1        1
144		 *  (for large x)   =  ----  ------   ------   .....
145		 *			2n   2(n+1)   2(n+2)
146		 *			-- - ------ - ------ -
147		 *			 x     x         x
148		 *
149		 * Let w = 2n/x and h=2/x, then the above quotient
150		 * is equal to the continued fraction:
151		 *		    1
152		 *	= -----------------------
153		 *		       1
154		 *	   w - -----------------
155		 *			  1
156		 * 	        w+h - ---------
157		 *		       w+2h - ...
158		 *
159		 * To determine how many terms needed, let
160		 * Q(0) = w, Q(1) = w(w+h) - 1,
161		 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
162		 * When Q(k) > 1e4	good for single
163		 * When Q(k) > 1e9	good for double
164		 * When Q(k) > 1e17	good for quadruple
165		 */
166	    /* determine k */
167		double t,v;
168		double q0,q1,h,tmp; int32_t k,m;
169		w  = (n+n)/(double)x; h = 2.0/(double)x;
170		q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
171		while(q1<1.0e9) {
172			k += 1; z += h;
173			tmp = z*q1 - q0;
174			q0 = q1;
175			q1 = tmp;
176		}
177		m = n+n;
178		for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
179		a = t;
180		b = one;
181		/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
182		 *  Hence, if n*(log(2n/x)) > ...
183		 *  single 8.8722839355e+01
184		 *  double 7.09782712893383973096e+02
185		 *  long double 1.1356523406294143949491931077970765006170e+04
186		 *  then recurrent value may overflow and the result is
187		 *  likely underflow to zero
188		 */
189		tmp = n;
190		v = two/x;
191		tmp = tmp*__ieee754_log(fabs(v*tmp));
192		if(tmp<7.09782712893383973096e+02) {
193	    	    for(i=n-1,di=(double)(i+i);i>0;i--){
194		        temp = b;
195			b *= di;
196			b  = b/x - a;
197		        a = temp;
198			di -= two;
199	     	    }
200		} else {
201	    	    for(i=n-1,di=(double)(i+i);i>0;i--){
202		        temp = b;
203			b *= di;
204			b  = b/x - a;
205		        a = temp;
206			di -= two;
207		    /* scale b to avoid spurious overflow */
208			if(b>1e100) {
209			    a /= b;
210			    t /= b;
211			    b  = one;
212			}
213	     	    }
214		}
215	    	b = (t*__ieee754_j0(x)/b);
216	    }
217	}
218	if(sgn==1) return -b; else return b;
219}
220
221#ifdef __STDC__
222	double __ieee754_yn(int n, double x)
223#else
224	double __ieee754_yn(n,x)
225	int n; double x;
226#endif
227{
228	int32_t i,hx,ix,lx;
229	int32_t sign;
230	double a, b, temp;
231
232	EXTRACT_WORDS(hx,lx,x);
233	ix = 0x7fffffff&hx;
234    /* if Y(n,NaN) is NaN */
235	if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
236	if((ix|lx)==0) return -one/zero;
237	if(hx<0) return zero/zero;
238	sign = 1;
239	if(n<0){
240		n = -n;
241		sign = 1 - ((n&1)<<1);
242	}
243	if(n==0) return(__ieee754_y0(x));
244	if(n==1) return(sign*__ieee754_y1(x));
245	if(ix==0x7ff00000) return zero;
246	if(ix>=0x52D00000) { /* x > 2**302 */
247    /* (x >> n**2)
248     *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
249     *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
250     *	    Let s=sin(x), c=cos(x),
251     *		xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
252     *
253     *		   n	sin(xn)*sqt2	cos(xn)*sqt2
254     *		----------------------------------
255     *		   0	 s-c		 c+s
256     *		   1	-s-c 		-c+s
257     *		   2	-s+c		-c-s
258     *		   3	 s+c		 c-s
259     */
260		switch(n&3) {
261		    case 0: temp =  sin(x)-cos(x); break;
262		    case 1: temp = -sin(x)-cos(x); break;
263		    case 2: temp = -sin(x)+cos(x); break;
264		    case 3: temp =  sin(x)+cos(x); break;
265		}
266		b = invsqrtpi*temp/sqrt(x);
267	} else {
268	    u_int32_t high;
269	    a = __ieee754_y0(x);
270	    b = __ieee754_y1(x);
271	/* quit if b is -inf */
272	    GET_HIGH_WORD(high,b);
273	    for(i=1;i<n&&high!=0xfff00000;i++){
274		temp = b;
275		b = ((double)(i+i)/x)*b - a;
276		GET_HIGH_WORD(high,b);
277		a = temp;
278	    }
279	}
280	if(sign>0) return b; else return -b;
281}
282