1/*	$NetBSD: n_jn.c,v 1.8 2018/03/05 23:00:55 christos Exp $	*/
2/*-
3 * Copyright (c) 1992, 1993
4 *	The Regents of the University of California.  All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 *    notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 *    notice, this list of conditions and the following disclaimer in the
13 *    documentation and/or other materials provided with the distribution.
14 * 3. Neither the name of the University nor the names of its contributors
15 *    may be used to endorse or promote products derived from this software
16 *    without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
21 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * SUCH DAMAGE.
29 */
30
31#ifndef lint
32#if 0
33static char sccsid[] = "@(#)jn.c	8.2 (Berkeley) 11/30/93";
34#endif
35#endif /* not lint */
36
37/*
38 * 16 December 1992
39 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture.
40 */
41
42/*
43 * ====================================================
44 * Copyright (C) 1992 by Sun Microsystems, Inc.
45 *
46 * Developed at SunPro, a Sun Microsystems, Inc. business.
47 * Permission to use, copy, modify, and distribute this
48 * software is freely granted, provided that this notice
49 * is preserved.
50 * ====================================================
51 *
52 * ******************* WARNING ********************
53 * This is an alpha version of SunPro's FDLIBM (Freely
54 * Distributable Math Library) for IEEE double precision
55 * arithmetic. FDLIBM is a basic math library written
56 * in C that runs on machines that conform to IEEE
57 * Standard 754/854. This alpha version is distributed
58 * for testing purpose. Those who use this software
59 * should report any bugs to
60 *
61 *		fdlibm-comments@sunpro.eng.sun.com
62 *
63 * -- K.C. Ng, Oct 12, 1992
64 * ************************************************
65 */
66
67/*
68 * jn(int n, double x), yn(int n, double x)
69 * floating point Bessel's function of the 1st and 2nd kind
70 * of order n
71 *
72 * Special cases:
73 *	y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
74 *	y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
75 * Note 2. About jn(n,x), yn(n,x)
76 *	For n=0, j0(x) is called,
77 *	for n=1, j1(x) is called,
78 *	for n<x, forward recursion us used starting
79 *	from values of j0(x) and j1(x).
80 *	for n>x, a continued fraction approximation to
81 *	j(n,x)/j(n-1,x) is evaluated and then backward
82 *	recursion is used starting from a supposed value
83 *	for j(n,x). The resulting value of j(0,x) is
84 *	compared with the actual value to correct the
85 *	supposed value of j(n,x).
86 *
87 *	yn(n,x) is similar in all respects, except
88 *	that forward recursion is used for all
89 *	values of n>1.
90 *
91 */
92
93#include "mathimpl.h"
94#include <float.h>
95#include <errno.h>
96
97#if defined(__vax__) || defined(tahoe)
98#define _IEEE	0
99#else
100#define _IEEE	1
101#define infnan(x) (0.0)
102#endif
103
104static const double
105#if _IEEE
106invsqrtpi= 5.641895835477562869480794515607725858441e-0001,
107#endif
108two  = 2.0,
109zero = 0.0,
110one  = 1.0;
111
112double
113jn(int n, double x)
114{
115	int i, sgn;
116	double a, b, temp;
117	double z, w;
118
119    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
120     * Thus, J(-n,x) = J(n,-x)
121     */
122    /* if J(n,NaN) is NaN */
123#if _IEEE
124	if (snan(x)) return x+x;
125#endif
126	if (n<0){
127		n = -n;
128		x = -x;
129	}
130	if (n==0) return(j0(x));
131	if (n==1) return(j1(x));
132	sgn = (n&1)&(x < zero);		/* even n -- 0, odd n -- sign(x) */
133	x = fabs(x);
134	if (x == 0 || !finite (x)) 	/* if x is 0 or inf */
135	    b = zero;
136	else if ((double) n <= x) {
137			/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
138#if _IEEE
139	    if (x >= 8.148143905337944345e+090) {
140					/* x >= 2**302 */
141    /* (x >> n**2)
142     *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
143     *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
144     *	    Let s=sin(x), c=cos(x),
145     *		xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
146     *
147     *		   n	sin(xn)*sqt2	cos(xn)*sqt2
148     *		----------------------------------
149     *		   0	 s-c		 c+s
150     *		   1	-s-c 		-c+s
151     *		   2	-s+c		-c-s
152     *		   3	 s+c		 c-s
153     */
154		switch(n&3) {
155		    case 0: temp =  cos(x)+sin(x); break;
156		    case 1: temp = -cos(x)+sin(x); break;
157		    case 2: temp = -cos(x)-sin(x); break;
158		    case 3: temp =  cos(x)-sin(x); break;
159		}
160		b = invsqrtpi*temp/sqrt(x);
161	    } else
162#endif
163	    {
164	        a = j0(x);
165	        b = j1(x);
166	        for(i=1;i<n;i++){
167		    temp = b;
168		    b = b*((double)(i+i)/x) - a; /* avoid underflow */
169		    a = temp;
170	        }
171	    }
172	} else {
173	    if (x < 1.86264514923095703125e-009) { /* x < 2**-29 */
174    /* x is tiny, return the first Taylor expansion of J(n,x)
175     * J(n,x) = 1/n!*(x/2)^n  - ...
176     */
177		if (n > 33)	/* underflow */
178		    b = zero;
179		else {
180		    temp = x*0.5; b = temp;
181		    for (a=one,i=2;i<=n;i++) {
182			a *= (double)i;		/* a = n! */
183			b *= temp;		/* b = (x/2)^n */
184		    }
185		    b = b/a;
186		}
187	    } else {
188		/* use backward recurrence */
189		/* 			x      x^2      x^2
190		 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
191		 *			2n  - 2(n+1) - 2(n+2)
192		 *
193		 * 			1      1        1
194		 *  (for large x)   =  ----  ------   ------   .....
195		 *			2n   2(n+1)   2(n+2)
196		 *			-- - ------ - ------ -
197		 *			 x     x         x
198		 *
199		 * Let w = 2n/x and h=2/x, then the above quotient
200		 * is equal to the continued fraction:
201		 *		    1
202		 *	= -----------------------
203		 *		       1
204		 *	   w - -----------------
205		 *			  1
206		 * 	        w+h - ---------
207		 *		       w+2h - ...
208		 *
209		 * To determine how many terms needed, let
210		 * Q(0) = w, Q(1) = w(w+h) - 1,
211		 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
212		 * When Q(k) > 1e4	good for single
213		 * When Q(k) > 1e9	good for double
214		 * When Q(k) > 1e17	good for quadruple
215		 */
216	    /* determine k */
217		double t,v;
218		double q0,q1,h,tmp; int k,m;
219		w  = (n+n)/(double)x; h = 2.0/(double)x;
220		q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
221		while (q1<1.0e9) {
222			k += 1; z += h;
223			tmp = z*q1 - q0;
224			q0 = q1;
225			q1 = tmp;
226		}
227		m = n+n;
228		for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
229		a = t;
230		b = one;
231		/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
232		 *  Hence, if n*(log(2n/x)) > ...
233		 *  single 8.8722839355e+01
234		 *  double 7.09782712893383973096e+02
235		 *  long double 1.1356523406294143949491931077970765006170e+04
236		 *  then recurrent value may overflow and the result will
237		 *  likely underflow to zero
238		 */
239		tmp = n;
240		v = two/x;
241		tmp = tmp*log(fabs(v*tmp));
242	    	for (i=n-1;i>0;i--){
243		        temp = b;
244		        b = ((i+i)/x)*b - a;
245		        a = temp;
246		    /* scale b to avoid spurious overflow */
247#			if defined(__vax__) || defined(tahoe)
248#				define BMAX 1e13
249#			else
250#				define BMAX 1e100
251#			endif /* defined(__vax__) || defined(tahoe) */
252			if (b > BMAX) {
253				a /= b;
254				t /= b;
255				b = one;
256			}
257		}
258	    	b = (t*j0(x)/b);
259	    }
260	}
261	return ((sgn == 1) ? -b : b);
262}
263
264double
265yn(int n, double x)
266{
267	int i, sign;
268	double a, b, temp;
269
270    /* Y(n,NaN), Y(n, x < 0) is NaN */
271	if (x <= 0 || (_IEEE && x != x))
272		if (_IEEE && x < 0) return zero/zero;
273		else if (x < 0)     return (infnan(EDOM));
274		else if (_IEEE)     return -one/zero;
275		else		    return(infnan(-ERANGE));
276	else if (!finite(x)) return(0);
277	sign = 1;
278	if (n<0){
279		n = -n;
280		sign = 1 - ((n&1)<<2);
281	}
282	if (n == 0) return(y0(x));
283	if (n == 1) return(sign*y1(x));
284#if _IEEE
285	if(x >= 8.148143905337944345e+090) { /* x > 2**302 */
286    /* (x >> n**2)
287     *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
288     *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
289     *	    Let s=sin(x), c=cos(x),
290     *		xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
291     *
292     *		   n	sin(xn)*sqt2	cos(xn)*sqt2
293     *		----------------------------------
294     *		   0	 s-c		 c+s
295     *		   1	-s-c 		-c+s
296     *		   2	-s+c		-c-s
297     *		   3	 s+c		 c-s
298     */
299		switch (n&3) {
300		    case 0: temp =  sin(x)-cos(x); break;
301		    case 1: temp = -sin(x)-cos(x); break;
302		    case 2: temp = -sin(x)+cos(x); break;
303		    case 3: temp =  sin(x)+cos(x); break;
304		}
305		b = invsqrtpi*temp/sqrt(x);
306	} else
307#endif
308	{
309	    a = y0(x);
310	    b = y1(x);
311	/* quit if b is -inf */
312	    for (i = 1; i < n && !finite(b); i++){
313		temp = b;
314		b = ((double)(i+i)/x)*b - a;
315		a = temp;
316	    }
317	}
318	if (!_IEEE && !finite(b))
319		return (infnan(-sign * ERANGE));
320	return ((sign > 0) ? b : -b);
321}
322