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 __jnl = jnl
31 #pragma weak __ynl = ynl
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 "longdouble.h"
60 #include <float.h> /* LDBL_MAX */
61
62 #define GENERIC long double
63
64 static const GENERIC
65 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
66 two = 2.0L,
67 zero = 0.0L,
68 one = 1.0L;
69
70 GENERIC
71 jnl(int n, GENERIC x)
72 {
73 int i, sgn;
74 GENERIC a, b, temp, z, w;
75
76 /*
77 * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
78 * Thus, J(-n,x) = J(n,-x)
79 */
80 if (n < 0) {
81 n = -n;
82 x = -x;
83 }
84 if (n == 0)
85 return (j0l(x));
86 if (n == 1)
87 return (j1l(x));
88 if (x != x)
89 return (x+x);
90 if ((n&1) == 0)
91 sgn = 0; /* even n */
92 else
93 sgn = signbitl(x); /* old n */
94 x = fabsl(x);
95 if (x == zero || !finitel(x)) b = zero;
96 else if ((GENERIC)n <= x) {
97 /*
98 * Safe to use
99 * J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
100 */
101 if (x > 1.0e91L) {
102 /*
103 * x >> n**2
104 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
105 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
106 * Let s=sin(x), c=cos(x),
107 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
108 *
109 * n sin(xn)*sqt2 cos(xn)*sqt2
110 * ----------------------------------
111 * 0 s-c c+s
112 * 1 -s-c -c+s
113 * 2 -s+c -c-s
114 * 3 s+c c-s
115 */
116 switch (n&3) {
117 case 0:
118 temp = cosl(x)+sinl(x);
119 break;
120 case 1:
121 temp = -cosl(x)+sinl(x);
122 break;
123 case 2:
124 temp = -cosl(x)-sinl(x);
125 break;
126 case 3:
127 temp = cosl(x)-sinl(x);
128 break;
129 }
130 b = invsqrtpi*temp/sqrtl(x);
131 } else {
132 a = j0l(x);
133 b = j1l(x);
134 for (i = 1; i < n; i++) {
135 temp = b;
136 /* avoid underflow */
137 b = b*((GENERIC)(i+i)/x) - a;
138 a = temp;
139 }
140 }
141 } else {
142 if (x < 1e-17L) { /* use J(n,x) = 1/n!*(x/2)^n */
143 b = powl(0.5L*x, (GENERIC)n);
144 if (b != zero) {
145 for (a = one, i = 1; i <= n; i++)
146 a *= (GENERIC)i;
147 b = b/a;
148 }
149 } else {
150 /* use backward recurrence */
151 /*
152 * x x^2 x^2
153 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
154 * 2n - 2(n+1) - 2(n+2)
155 *
156 * 1 1 1
157 * (for large x) = ---- ------ ------ .....
158 * 2n 2(n+1) 2(n+2)
159 * -- - ------ - ------ -
160 * x x x
161 *
162 * Let w = 2n/x and h=2/x, then the above quotient
163 * is equal to the continued fraction:
164 * 1
165 * = -----------------------
166 * 1
167 * w - -----------------
168 * 1
169 * w+h - ---------
170 * w+2h - ...
171 *
172 * To determine how many terms needed, let
173 * Q(0) = w, Q(1) = w(w+h) - 1,
174 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
175 * When Q(k) > 1e4 good for single
176 * When Q(k) > 1e9 good for double
177 * When Q(k) > 1e17 good for quaduple
178 */
179 /* determine k */
180 GENERIC t, v;
181 double q0, q1, h, tmp;
182 int k, m;
183 w = (n+n)/(double)x;
184 h = 2.0/(double)x;
185 q0 = w;
186 z = w+h;
187 q1 = w*z - 1.0;
188 k = 1;
189 while (q1 < 1.0e17) {
190 k += 1;
191 z += h;
192 tmp = z*q1 - q0;
193 q0 = q1;
194 q1 = tmp;
195 }
196 m = n+n;
197 for (t = zero, i = 2*(n+k); i >= m; i -= 2)
198 t = one/(i/x-t);
199 a = t;
200 b = one;
201 /*
202 * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
203 * hence, if n*(log(2n/x)) > ...
204 * single 8.8722839355e+01
205 * double 7.09782712893383973096e+02
206 * long double 1.1356523406294143949491931077970765006170e+04
207 * then recurrent value may overflow and the result is
208 * likely underflow to zero
209 */
210 tmp = n;
211 v = two/x;
212 tmp = tmp*logl(fabsl(v*tmp));
213 if (tmp < 1.1356523406294143949491931077970765e+04L) {
214 for (i = n-1; i > 0; i--) {
215 temp = b;
216 b = ((i+i)/x)*b - a;
217 a = temp;
218 }
219 } else {
220 for (i = n-1; i > 0; i--) {
221 temp = b;
222 b = ((i+i)/x)*b - a;
223 a = temp;
224 if (b > 1e1000L) {
225 a /= b;
226 t /= b;
227 b = 1.0;
228 }
229 }
230 }
231 b = (t*j0l(x)/b);
232 }
233 }
234 if (sgn != 0)
235 return (-b);
236 else
237 return (b);
238 }
239
240 GENERIC
241 ynl(int n, GENERIC x)
242 {
243 int i;
244 int sign;
245 GENERIC a, b, temp;
246
247 if (x != x)
248 return (x+x);
249 if (x <= zero) {
250 if (x == zero)
251 return (-one/zero);
252 else
253 return (zero/zero);
254 }
255 sign = 1;
256 if (n < 0) {
257 n = -n;
258 if ((n&1) == 1) sign = -1;
259 }
260 if (n == 0)
261 return (y0l(x));
262 if (n == 1)
263 return (sign*y1l(x));
264 if (!finitel(x))
265 return (zero);
266
267 if (x > 1.0e91L) {
268 /*
269 * x >> n**2
270 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
271 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
272 * Let s = sin(x), c = cos(x),
273 * xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
274 *
275 * n sin(xn)*sqt2 cos(xn)*sqt2
276 * ----------------------------------
277 * 0 s-c c+s
278 * 1 -s-c -c+s
279 * 2 -s+c -c-s
280 * 3 s+c c-s
281 */
282 switch (n&3) {
283 case 0:
284 temp = sinl(x)-cosl(x);
285 break;
286 case 1:
287 temp = -sinl(x)-cosl(x);
288 break;
289 case 2:
290 temp = -sinl(x)+cosl(x);
291 break;
292 case 3:
293 temp = sinl(x)+cosl(x);
294 break;
295 }
296 b = invsqrtpi*temp/sqrtl(x);
297 } else {
298 a = y0l(x);
299 b = y1l(x);
300 /*
301 * fix 1262058 and take care of non-default rounding
302 */
303 for (i = 1; i < n; i++) {
304 temp = b;
305 b *= (GENERIC) (i + i) / x;
306 if (b <= -LDBL_MAX)
307 break;
308 b -= a;
309 a = temp;
310 }
311 }
312 if (sign > 0)
313 return (b);
314 else
315 return (-b);
316 }