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, 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 /* use backward recurrence */
163 /* BEGIN CSTYLED */
164 /*
165 * x x^2 x^2
166 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
167 * 2n - 2(n+1) - 2(n+2)
168 *
169 * 1 1 1
170 * (for large x) = ---- ------ ------ .....
171 * 2n 2(n+1) 2(n+2)
172 * -- - ------ - ------ -
173 * x x x
174 *
175 * Let w = 2n/x and h=2/x, then the above quotient
176 * is equal to the continued fraction:
177 * 1
178 * = -----------------------
179 * 1
180 * w - -----------------
181 * 1
182 * w+h - ---------
183 * w+2h - ...
184 *
185 * To determine how many terms needed, let
186 * Q(0) = w, Q(1) = w(w+h) - 1,
187 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
188 * When Q(k) > 1e4 good for single
189 * When Q(k) > 1e9 good for double
190 * When Q(k) > 1e17 good for quaduple
191 */
192 /* END CSTYLED */
193 /* determine k */
194 GENERIC t, v;
195 double q0, q1, h, tmp;
196 int k, m;
197
198 w = (n + n) / (double)x;
199 h = 2.0 / (double)x;
200 q0 = w;
201 z = w + h;
202 q1 = w * z - 1.0;
203 k = 1;
204
205 while (q1 < 1.0e17) {
206 k += 1;
207 z += h;
208 tmp = z * q1 - q0;
209 q0 = q1;
210 q1 = tmp;
211 }
212
213 m = n + n;
214
215 for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
216 t = one / (i / x - t);
217
218 a = t;
219 b = one;
220
221 /*
222 * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
223 * hence, if n*(log(2n/x)) > ...
224 * single:
225 * 8.8722839355e+01
226 * double:
227 * 7.09782712893383973096e+02
228 * long double:
229 * 1.1356523406294143949491931077970765006170e+04
230 * then recurrent value may overflow and the result is
231 * likely underflow to zero
232 */
233 tmp = n;
234 v = two / x;
235 tmp = tmp * logl(fabsl(v * tmp));
236
237 if (tmp < 1.1356523406294143949491931077970765e+04L) {
238 for (i = n - 1; i > 0; i--) {
239 temp = b;
240 b = ((i + i) / x) * b - a;
241 a = temp;
242 }
243 } else {
244 for (i = n - 1; i > 0; i--) {
245 temp = b;
246 b = ((i + i) / x) * b - a;
247 a = temp;
248
249 if (b > 1e1000L) {
250 a /= b;
251 t /= b;
252 b = 1.0;
253 }
254 }
255 }
256
257 b = (t * j0l(x) / b);
258 }
259 }
260
261 if (sgn != 0)
262 return (-b);
263 else
264 return (b);
265 }
266
267 GENERIC
268 ynl(int n, GENERIC x)
269 {
270 int i;
271 int sign;
272 GENERIC a, b, temp;
273
274 if (x != x)
275 return (x + x);
276
277 if (x <= zero) {
278 if (x == zero)
279 return (-one / zero);
280 else
281 return (zero / zero);
282 }
283
284 sign = 1;
285
286 if (n < 0) {
287 n = -n;
288
289 if ((n & 1) == 1)
290 sign = -1;
291 }
292
293 if (n == 0)
294 return (y0l(x));
295
296 if (n == 1)
297 return (sign * y1l(x));
298
299 if (!finitel(x))
300 return (zero);
301
302 if (x > 1.0e91L) {
303 /*
304 * x >> n**2
305 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
306 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
307 * Let s = sin(x), c = cos(x),
308 * xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
309 *
310 * n sin(xn)*sqt2 cos(xn)*sqt2
311 * ----------------------------------
312 * 0 s-c c+s
313 * 1 -s-c -c+s
314 * 2 -s+c -c-s
315 * 3 s+c c-s
316 */
317 switch (n & 3) {
318 case 0:
319 temp = sinl(x) - cosl(x);
320 break;
321 case 1:
322 temp = -sinl(x) - cosl(x);
323 break;
324 case 2:
325 temp = -sinl(x) + cosl(x);
326 break;
327 case 3:
328 temp = sinl(x) + cosl(x);
329 break;
330 }
331
332 b = invsqrtpi * temp / sqrtl(x);
333 } else {
334 a = y0l(x);
335 b = y1l(x);
336
337 /*
338 * fix 1262058 and take care of non-default rounding
339 */
340 for (i = 1; i < n; i++) {
341 temp = b;
342 b *= (GENERIC)(i + i) / x;
343
344 if (b <= -LDBL_MAX)
345 break;
346
347 b -= a;
348 a = temp;
349 }
350 }
351
352 if (sign > 0)
353 return (b);
354 else
355 return (-b);
356 }