xref: /illumos-gate/usr/src/lib/libm/common/C/jn.c (revision 685c1a21304711e8d0a21bade51b783487d53044)
1 /*
2  * CDDL HEADER START
3  *
4  * The contents of this file are subject to the terms of the
5  * Common Development and Distribution License (the "License").
6  * You may not use this file except in compliance with the License.
7  *
8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9  * or http://www.opensolaris.org/os/licensing.
10  * See the License for the specific language governing permissions
11  * and limitations under the License.
12  *
13  * When distributing Covered Code, include this CDDL HEADER in each
14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15  * If applicable, add the following below this CDDL HEADER, with the
16  * fields enclosed by brackets "[]" replaced with your own identifying
17  * information: Portions Copyright [yyyy] [name of copyright owner]
18  *
19  * CDDL HEADER END
20  */
21 
22 /*
23  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
24  */
25 /*
26  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
27  * Use is subject to license terms.
28  */
29 
30 #pragma weak __jn = jn
31 #pragma weak __yn = yn
32 
33 /*
34  * floating point Bessel's function of the 1st and 2nd kind
35  * of order n: jn(n,x),yn(n,x);
36  *
37  * Special cases:
38  *	y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
39  *	y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
40  * Note 2. About jn(n,x), yn(n,x)
41  *	For n=0, j0(x) is called,
42  *	for n=1, j1(x) is called,
43  *	for n<x, forward recursion us used starting
44  *	from values of j0(x) and j1(x).
45  *	for n>x, a continued fraction approximation to
46  *	j(n,x)/j(n-1,x) is evaluated and then backward
47  *	recursion is used starting from a supposed value
48  *	for j(n,x). The resulting value of j(0,x) is
49  *	compared with the actual value to correct the
50  *	supposed value of j(n,x).
51  *
52  *	yn(n,x) is similar in all respects, except
53  *	that forward recursion is used for all
54  *	values of n>1.
55  *
56  */
57 
58 #include "libm.h"
59 #include <float.h>	/* DBL_MIN */
60 #include <values.h>	/* X_TLOSS */
61 #include "xpg6.h"	/* __xpg6 */
62 
63 #define	GENERIC double
64 
65 static const GENERIC
66 	invsqrtpi = 5.641895835477562869480794515607725858441e-0001,
67 	two	= 2.0,
68 	zero	= 0.0,
69 	one	= 1.0;
70 
71 GENERIC
jn(int n,GENERIC x)72 jn(int n, GENERIC x)
73 {
74 	int i, sgn;
75 	GENERIC a, b, temp = 0;
76 	GENERIC z, w, ox, on;
77 
78 	/*
79 	 * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
80 	 * Thus, J(-n,x) = J(n,-x)
81 	 */
82 	ox = x;
83 	on = (GENERIC)n;
84 
85 	if (n < 0) {
86 		n = -n;
87 		x = -x;
88 	}
89 	if (isnan(x))
90 		return (x*x);	/* + -> * for Cheetah */
91 	if (!((int)_lib_version == libm_ieee ||
92 	    (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
93 		if (fabs(x) > X_TLOSS)
94 			return (_SVID_libm_err(on, ox, 38));
95 	}
96 	if (n == 0)
97 		return (j0(x));
98 	if (n == 1)
99 		return (j1(x));
100 	if ((n&1) == 0)
101 		sgn = 0;			/* even n */
102 	else
103 		sgn = signbit(x);	/* old n  */
104 	x = fabs(x);
105 	if (x == zero||!finite(x)) b = zero;
106 	else if ((GENERIC)n <= x) {
107 					/*
108 					 * Safe to use
109 					 *  J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
110 					 */
111 		if (x > 1.0e91) {
112 				/*
113 				 * x >> n**2
114 				 *    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
115 				 *   Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
116 				 *   Let s=sin(x), c=cos(x),
117 				 *	xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
118 				 *
119 				 *	   n	sin(xn)*sqt2	cos(xn)*sqt2
120 				 *	----------------------------------
121 				 *	   0	 s-c		 c+s
122 				 *	   1	-s-c		-c+s
123 				 *	   2	-s+c		-c-s
124 				 *	   3	 s+c		 c-s
125 				 */
126 			switch (n&3) {
127 			case 0:
128 				temp =  cos(x)+sin(x);
129 				break;
130 			case 1:
131 				temp = -cos(x)+sin(x);
132 				break;
133 			case 2:
134 				temp = -cos(x)-sin(x);
135 				break;
136 			case 3:
137 				temp =  cos(x)-sin(x);
138 				break;
139 			}
140 			b = invsqrtpi*temp/sqrt(x);
141 		} else {
142 			a = j0(x);
143 			b = j1(x);
144 			for (i = 1; i < n; i++) {
145 				temp = b;
146 				/* avoid underflow */
147 				b = b*((GENERIC)(i+i)/x) - a;
148 				a = temp;
149 			}
150 		}
151 	} else {
152 		if (x < 1e-9) {	/* use J(n,x) = 1/n!*(x/2)^n */
153 			b = pow(0.5*x, (GENERIC) n);
154 			if (b != zero) {
155 				for (a = one, i = 1; i <= n; i++)
156 					a *= (GENERIC)i;
157 				b = b/a;
158 			}
159 		} else {
160 			/*
161 			 * use backward recurrence
162 			 *			x	  x^2	  x^2
163 			 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
164 			 *			2n  - 2(n+1) - 2(n+2)
165 			 *
166 			 *			1	  1	    1
167 			 *  (for large x)   =  ----  ------   ------   .....
168 			 *			2n   2(n+1)   2(n+2)
169 			 *			-- - ------ - ------ -
170 			 *			 x	 x		 x
171 			 *
172 			 * Let w = 2n/x and h = 2/x, then the above quotient
173 			 * is equal to the continued fraction:
174 			 *		    1
175 			 *	= -----------------------
176 			 *			   1
177 			 *	   w - -----------------
178 			 *			  1
179 			 *			w+h - ---------
180 			 *			   w+2h - ...
181 			 *
182 			 * To determine how many terms needed, let
183 			 * Q(0) = w, Q(1) = w(w+h) - 1,
184 			 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
185 			 * When Q(k) > 1e4	good for single
186 			 * When Q(k) > 1e9	good for double
187 			 * When Q(k) > 1e17	good for quaduple
188 			 */
189 			/* determine k */
190 			GENERIC t, v;
191 			double q0, q1, h, tmp;
192 			int k, m;
193 			w  = (n+n)/(double)x;
194 			h = 2.0/(double)x;
195 			q0 = w;
196 			z = w + h;
197 			q1 = w*z - 1.0;
198 			k = 1;
199 
200 			while (q1 < 1.0e9) {
201 				k += 1;
202 				z += h;
203 				tmp = z*q1 - q0;
204 				q0 = q1;
205 				q1 = tmp;
206 			}
207 			m = n+n;
208 			for (t = zero, i = 2*(n+k); i >= m; i -= 2)
209 				t = one/(i/x-t);
210 			a = t;
211 			b = one;
212 			/*
213 			 * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
214 			 * hence, if n*(log(2n/x)) > ...
215 			 *  single:
216 			 *    8.8722839355e+01
217 			 *  double:
218 			 *    7.09782712893383973096e+02
219 			 *  long double:
220 			 *    1.1356523406294143949491931077970765006170e+04
221 			 * then recurrent value may overflow and the result is
222 			 * likely underflow to zero
223 			 */
224 			tmp = n;
225 			v = two/x;
226 			tmp = tmp*log(fabs(v*tmp));
227 			if (tmp < 7.09782712893383973096e+02) {
228 				for (i = n-1; i > 0; i--) {
229 					temp = b;
230 					b = ((i+i)/x)*b - a;
231 					a = temp;
232 				}
233 			} else {
234 				for (i = n-1; i > 0; i--) {
235 					temp = b;
236 					b = ((i+i)/x)*b - a;
237 					a = temp;
238 					if (b > 1e100) {
239 						a /= b;
240 						t /= b;
241 						b  = 1.0;
242 					}
243 				}
244 			}
245 			b = (t*j0(x)/b);
246 		}
247 	}
248 	if (sgn != 0)
249 		return (-b);
250 	else
251 		return (b);
252 }
253 
254 GENERIC
yn(int n,GENERIC x)255 yn(int n, GENERIC x)
256 {
257 	int i;
258 	int sign;
259 	GENERIC a, b, temp = 0, ox, on;
260 
261 	ox = x;
262 	on = (GENERIC)n;
263 	if (isnan(x))
264 		return (x*x);	/* + -> * for Cheetah */
265 	if (x <= zero) {
266 		if (x == zero) {
267 			/* return -one/zero; */
268 			return (_SVID_libm_err((GENERIC)n, x, 12));
269 		} else {
270 			/* return zero/zero; */
271 			return (_SVID_libm_err((GENERIC)n, x, 13));
272 		}
273 	}
274 	if (!((int)_lib_version == libm_ieee ||
275 	    (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
276 		if (x > X_TLOSS)
277 			return (_SVID_libm_err(on, ox, 39));
278 	}
279 	sign = 1;
280 	if (n < 0) {
281 		n = -n;
282 		if ((n&1) == 1) sign = -1;
283 	}
284 	if (n == 0)
285 		return (y0(x));
286 	if (n == 1)
287 		return (sign*y1(x));
288 	if (!finite(x))
289 		return (zero);
290 
291 	if (x > 1.0e91) {
292 				/*
293 				 * x >> n**2
294 				 *  Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
295 				 *  Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
296 				 *  Let s = sin(x), c = cos(x),
297 				 *  xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
298 				 *
299 				 *    n	sin(xn)*sqt2	cos(xn)*sqt2
300 				 *	----------------------------------
301 				 *	 0	 s-c		 c+s
302 				 *	 1	-s-c		-c+s
303 				 *	 2	-s+c		-c-s
304 				 *	 3	 s+c		 c-s
305 				 */
306 		switch (n&3) {
307 		case 0:
308 			temp =  sin(x)-cos(x);
309 			break;
310 		case 1:
311 			temp = -sin(x)-cos(x);
312 			break;
313 		case 2:
314 			temp = -sin(x)+cos(x);
315 			break;
316 		case 3:
317 			temp =  sin(x)+cos(x);
318 			break;
319 		}
320 		b = invsqrtpi*temp/sqrt(x);
321 	} else {
322 		a = y0(x);
323 		b = y1(x);
324 		/*
325 		 * fix 1262058 and take care of non-default rounding
326 		 */
327 		for (i = 1; i < n; i++) {
328 			temp = b;
329 			b *= (GENERIC) (i + i) / x;
330 			if (b <= -DBL_MAX)
331 				break;
332 			b -= a;
333 			a = temp;
334 		}
335 	}
336 	if (sign > 0)
337 		return (b);
338 	else
339 		return (-b);
340 }
341