1
2/* @(#)e_jn.c 1.4 95/01/18 */
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14#include <sys/cdefs.h>
15__FBSDID("$FreeBSD$");
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
46static const double
47invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
48two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
49one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
50
51static const double zero  =  0.00000000000000000000e+00;
52
53double
54__ieee754_jn(int n, double x)
55{
56	int32_t i,hx,ix,lx, sgn;
57	double a, b, temp, di;
58	double z, w;
59
60    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
61     * Thus, J(-n,x) = J(n,-x)
62     */
63	EXTRACT_WORDS(hx,lx,x);
64	ix = 0x7fffffff&hx;
65    /* if J(n,NaN) is NaN */
66	if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
67	if(n<0){
68		n = -n;
69		x = -x;
70		hx ^= 0x80000000;
71	}
72	if(n==0) return(__ieee754_j0(x));
73	if(n==1) return(__ieee754_j1(x));
74	sgn = (n&1)&(hx>>31);	/* even n -- 0, odd n -- sign(x) */
75	x = fabs(x);
76	if((ix|lx)==0||ix>=0x7ff00000) 	/* if x is 0 or inf */
77	    b = zero;
78	else if((double)n<=x) {
79		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
80	    if(ix>=0x52D00000) { /* x > 2**302 */
81    /* (x >> n**2)
82     *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
83     *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
84     *	    Let s=sin(x), c=cos(x),
85     *		xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
86     *
87     *		   n	sin(xn)*sqt2	cos(xn)*sqt2
88     *		----------------------------------
89     *		   0	 s-c		 c+s
90     *		   1	-s-c 		-c+s
91     *		   2	-s+c		-c-s
92     *		   3	 s+c		 c-s
93     */
94		switch(n&3) {
95		    case 0: temp =  cos(x)+sin(x); break;
96		    case 1: temp = -cos(x)+sin(x); break;
97		    case 2: temp = -cos(x)-sin(x); break;
98		    case 3: temp =  cos(x)-sin(x); break;
99		}
100		b = invsqrtpi*temp/sqrt(x);
101	    } else {
102	        a = __ieee754_j0(x);
103	        b = __ieee754_j1(x);
104	        for(i=1;i<n;i++){
105		    temp = b;
106		    b = b*((double)(i+i)/x) - a; /* avoid underflow */
107		    a = temp;
108	        }
109	    }
110	} else {
111	    if(ix<0x3e100000) {	/* x < 2**-29 */
112    /* x is tiny, return the first Taylor expansion of J(n,x)
113     * J(n,x) = 1/n!*(x/2)^n  - ...
114     */
115		if(n>33)	/* underflow */
116		    b = zero;
117		else {
118		    temp = x*0.5; b = temp;
119		    for (a=one,i=2;i<=n;i++) {
120			a *= (double)i;		/* a = n! */
121			b *= temp;		/* b = (x/2)^n */
122		    }
123		    b = b/a;
124		}
125	    } else {
126		/* use backward recurrence */
127		/* 			x      x^2      x^2
128		 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
129		 *			2n  - 2(n+1) - 2(n+2)
130		 *
131		 * 			1      1        1
132		 *  (for large x)   =  ----  ------   ------   .....
133		 *			2n   2(n+1)   2(n+2)
134		 *			-- - ------ - ------ -
135		 *			 x     x         x
136		 *
137		 * Let w = 2n/x and h=2/x, then the above quotient
138		 * is equal to the continued fraction:
139		 *		    1
140		 *	= -----------------------
141		 *		       1
142		 *	   w - -----------------
143		 *			  1
144		 * 	        w+h - ---------
145		 *		       w+2h - ...
146		 *
147		 * To determine how many terms needed, let
148		 * Q(0) = w, Q(1) = w(w+h) - 1,
149		 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
150		 * When Q(k) > 1e4	good for single
151		 * When Q(k) > 1e9	good for double
152		 * When Q(k) > 1e17	good for quadruple
153		 */
154	    /* determine k */
155		double t,v;
156		double q0,q1,h,tmp; int32_t k,m;
157		w  = (n+n)/(double)x; h = 2.0/(double)x;
158		q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
159		while(q1<1.0e9) {
160			k += 1; z += h;
161			tmp = z*q1 - q0;
162			q0 = q1;
163			q1 = tmp;
164		}
165		m = n+n;
166		for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
167		a = t;
168		b = one;
169		/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
170		 *  Hence, if n*(log(2n/x)) > ...
171		 *  single 8.8722839355e+01
172		 *  double 7.09782712893383973096e+02
173		 *  long double 1.1356523406294143949491931077970765006170e+04
174		 *  then recurrent value may overflow and the result is
175		 *  likely underflow to zero
176		 */
177		tmp = n;
178		v = two/x;
179		tmp = tmp*__ieee754_log(fabs(v*tmp));
180		if(tmp<7.09782712893383973096e+02) {
181	    	    for(i=n-1,di=(double)(i+i);i>0;i--){
182		        temp = b;
183			b *= di;
184			b  = b/x - a;
185		        a = temp;
186			di -= two;
187	     	    }
188		} else {
189	    	    for(i=n-1,di=(double)(i+i);i>0;i--){
190		        temp = b;
191			b *= di;
192			b  = b/x - a;
193		        a = temp;
194			di -= two;
195		    /* scale b to avoid spurious overflow */
196			if(b>1e100) {
197			    a /= b;
198			    t /= b;
199			    b  = one;
200			}
201	     	    }
202		}
203		z = __ieee754_j0(x);
204		w = __ieee754_j1(x);
205		if (fabs(z) >= fabs(w))
206		    b = (t*z/b);
207		else
208		    b = (t*w/a);
209	    }
210	}
211	if(sgn==1) return -b; else return b;
212}
213
214double
215__ieee754_yn(int n, double x)
216{
217	int32_t i,hx,ix,lx;
218	int32_t sign;
219	double a, b, temp;
220
221	EXTRACT_WORDS(hx,lx,x);
222	ix = 0x7fffffff&hx;
223    /* if Y(n,NaN) is NaN */
224	if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
225	if((ix|lx)==0) return -one/zero;
226	if(hx<0) return zero/zero;
227	sign = 1;
228	if(n<0){
229		n = -n;
230		sign = 1 - ((n&1)<<1);
231	}
232	if(n==0) return(__ieee754_y0(x));
233	if(n==1) return(sign*__ieee754_y1(x));
234	if(ix==0x7ff00000) return zero;
235	if(ix>=0x52D00000) { /* x > 2**302 */
236    /* (x >> n**2)
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
241     *
242     *		   n	sin(xn)*sqt2	cos(xn)*sqt2
243     *		----------------------------------
244     *		   0	 s-c		 c+s
245     *		   1	-s-c 		-c+s
246     *		   2	-s+c		-c-s
247     *		   3	 s+c		 c-s
248     */
249		switch(n&3) {
250		    case 0: temp =  sin(x)-cos(x); break;
251		    case 1: temp = -sin(x)-cos(x); break;
252		    case 2: temp = -sin(x)+cos(x); break;
253		    case 3: temp =  sin(x)+cos(x); break;
254		}
255		b = invsqrtpi*temp/sqrt(x);
256	} else {
257	    u_int32_t high;
258	    a = __ieee754_y0(x);
259	    b = __ieee754_y1(x);
260	/* quit if b is -inf */
261	    GET_HIGH_WORD(high,b);
262	    for(i=1;i<n&&high!=0xfff00000;i++){
263		temp = b;
264		b = ((double)(i+i)/x)*b - a;
265		GET_HIGH_WORD(high,b);
266		a = temp;
267	    }
268	}
269	if(sign>0) return b; else return -b;
270}
271