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 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __jnl = jnl
32 #pragma weak __ynl = ynl
33
34 /*
35 * floating point Bessel's function of the 1st and 2nd kind
36 * of order n: jn(n,x),yn(n,x);
37 *
38 * Special cases:
39 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
40 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
41 * Note 2. About jn(n,x), yn(n,x)
42 * For n=0, j0(x) is called,
43 * for n=1, j1(x) is called,
44 * for n<x, forward recursion us used starting
45 * from values of j0(x) and j1(x).
46 * for n>x, a continued fraction approximation to
47 * j(n,x)/j(n-1,x) is evaluated and then backward
48 * recursion is used starting from a supposed value
49 * for j(n,x). The resulting value of j(0,x) is
50 * compared with the actual value to correct the
51 * supposed value of j(n,x).
52 *
53 * yn(n,x) is similar in all respects, except
54 * that forward recursion is used for all
55 * values of n>1.
56 *
57 */
58
59 #include "libm.h"
60 #include "longdouble.h"
61 #include <float.h> /* LDBL_MAX */
62
63 #define GENERIC long double
64
65 static const GENERIC
66 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
67 two = 2.0L,
68 zero = 0.0L,
69 one = 1.0L;
70
71 GENERIC
72 jnl(int n, GENERIC x)
73 {
74 int i, sgn;
75 GENERIC a, b, temp = 0, z, w;
76
77 /*
78 * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
79 * Thus, J(-n,x) = J(n,-x)
80 */
81 if (n < 0) {
82 n = -n;
83 x = -x;
84 }
85
86 if (n == 0)
87 return (j0l(x));
88
89 if (n == 1)
90 return (j1l(x));
91
92 if (x != x)
93 return (x + x);
94
95 if ((n & 1) == 0)
96 sgn = 0; /* even n */
97 else
98 sgn = signbitl(x); /* old n */
99
100 x = fabsl(x);
101
102 if (x == zero || !finitel(x)) {
103 b = zero;
104 } else if ((GENERIC)n <= x) {
105 /*
106 * Safe to use
107 * J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
108 */
109 if (x > 1.0e91L) {
110 /*
111 * x >> n**2
112 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
113 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
114 * Let s=sin(x), c=cos(x),
115 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
116 *
117 * n sin(xn)*sqt2 cos(xn)*sqt2
118 * ----------------------------------
119 * 0 s-c c+s
120 * 1 -s-c -c+s
121 * 2 -s+c -c-s
122 * 3 s+c c-s
123 */
124 switch (n & 3) {
125 case 0:
126 temp = cosl(x) + sinl(x);
127 break;
128 case 1:
129 temp = -cosl(x) + sinl(x);
130 break;
131 case 2:
132 temp = -cosl(x) - sinl(x);
133 break;
134 case 3:
135 temp = cosl(x) - sinl(x);
136 break;
137 }
138
139 b = invsqrtpi * temp / sqrtl(x);
140 } else {
141 a = j0l(x);
142 b = j1l(x);
143
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-17L) { /* use J(n,x) = 1/n!*(x/2)^n */
153 b = powl(0.5L * x, (GENERIC)n);
154
155 if (b != zero) {
156 for (a = one, i = 1; i <= n; i++)
157 a *= (GENERIC)i;
158
159 b = b / a;
160 }
161 } else {
162 /* BEGIN CSTYLED */
163
164 /*
165 * use backward recurrence
166 * x x^2 x^2
167 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
168 * 2n - 2(n+1) - 2(n+2)
169 *
170 * 1 1 1
171 * (for large x) = ---- ------ ------ .....
172 * 2n 2(n+1) 2(n+2)
173 * -- - ------ - ------ -
174 * x x x
175 *
176 * Let w = 2n/x and h=2/x, then the above quotient
177 * is equal to the continued fraction:
178 * 1
179 * = -----------------------
180 * 1
181 * w - -----------------
182 * 1
183 * w+h - ---------
184 * w+2h - ...
185 *
186 * To determine how many terms needed, let
187 * Q(0) = w, Q(1) = w(w+h) - 1,
188 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
189 * When Q(k) > 1e4 good for single
190 * When Q(k) > 1e9 good for double
191 * When Q(k) > 1e17 good for quaduple
192 */
193
194 /*
195 * END CSTYLED
196 * determine k
197 */
198 GENERIC t, v;
199 double q0, q1, h, tmp;
200 int k, m;
201
202 w = (n + n) / (double)x;
203 h = 2.0 / (double)x;
204 q0 = w;
205 z = w + h;
206 q1 = w * z - 1.0;
207 k = 1;
208
209 while (q1 < 1.0e17) {
210 k += 1;
211 z += h;
212 tmp = z * q1 - q0;
213 q0 = q1;
214 q1 = tmp;
215 }
216
217 m = n + n;
218
219 for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
220 t = one / (i / x - t);
221
222 a = t;
223 b = one;
224
225 /*
226 * Estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
227 * hence, if n*(log(2n/x)) > ...
228 * single:
229 * 8.8722839355e+01
230 * double:
231 * 7.09782712893383973096e+02
232 * long double:
233 * 1.1356523406294143949491931077970765006170e+04
234 * then recurrent value may overflow and the result is
235 * likely underflow to zero.
236 */
237 tmp = n;
238 v = two / x;
239 tmp = tmp * logl(fabsl(v * tmp));
240
241 if (tmp < 1.1356523406294143949491931077970765e+04L) {
242 for (i = n - 1; i > 0; i--) {
243 temp = b;
244 b = ((i + i) / x) * b - a;
245 a = temp;
246 }
247 } else {
248 for (i = n - 1; i > 0; i--) {
249 temp = b;
250 b = ((i + i) / x) * b - a;
251 a = temp;
252
253 if (b > 1e1000L) {
254 a /= b;
255 t /= b;
256 b = 1.0;
257 }
258 }
259 }
260
261 b = (t * j0l(x) / b);
262 }
263 }
264
265 if (sgn != 0)
266 return (-b);
267 else
268 return (b);
269 }
270
271 GENERIC
272 ynl(int n, GENERIC x)
273 {
274 int i;
275 int sign;
276 GENERIC a, b, temp = 0;
277
278 if (x != x)
279 return (x + x);
280
281 if (x <= zero) {
282 if (x == zero)
283 return (-one / zero);
284 else
285 return (zero / zero);
286 }
287
288 sign = 1;
289
290 if (n < 0) {
291 n = -n;
292
293 if ((n & 1) == 1)
294 sign = -1;
295 }
296
297 if (n == 0)
298 return (y0l(x));
299
300 if (n == 1)
301 return (sign * y1l(x));
302
303 if (!finitel(x))
304 return (zero);
305
306 if (x > 1.0e91L) {
307 /*
308 * x >> n**2
309 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
310 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
311 * Let s=sin(x), c=cos(x),
312 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
313 *
314 * n sin(xn)*sqt2 cos(xn)*sqt2
315 * ----------------------------------
316 * 0 s-c c+s
317 * 1 -s-c -c+s
318 * 2 -s+c -c-s
319 * 3 s+c c-s
320 */
321 switch (n & 3) {
322 case 0:
323 temp = sinl(x) - cosl(x);
324 break;
325 case 1:
326 temp = -sinl(x) - cosl(x);
327 break;
328 case 2:
329 temp = -sinl(x) + cosl(x);
330 break;
331 case 3:
332 temp = sinl(x) + cosl(x);
333 break;
334 }
335
336 b = invsqrtpi * temp / sqrtl(x);
337 } else {
338 a = y0l(x);
339 b = y1l(x);
340
341 /*
342 * fix 1262058 and take care of non-default rounding
343 */
344 for (i = 1; i < n; i++) {
345 temp = b;
346 b *= (GENERIC)(i + i) / x;
347
348 if (b <= -LDBL_MAX)
349 break;
350
351 b -= a;
352 a = temp;
353 }
354 }
355
356 if (sign > 0)
357 return (b);
358 else
359 return (-b);
360 }