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
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 8.8722839355e+01
216 * double 7.09782712893383973096e+02
217 * long double 1.1356523406294143949491931077970765006170e+04
218 * then recurrent value may overflow and the result is
219 * likely underflow to zero
220 */
221 tmp = n;
222 v = two/x;
223 tmp = tmp*log(fabs(v*tmp));
224 if (tmp < 7.09782712893383973096e+02) {
225 for (i = n-1; i > 0; i--) {
226 temp = b;
227 b = ((i+i)/x)*b - a;
228 a = temp;
229 }
230 } else {
231 for (i = n-1; i > 0; i--) {
232 temp = b;
233 b = ((i+i)/x)*b - a;
234 a = temp;
235 if (b > 1e100) {
236 a /= b;
237 t /= b;
238 b = 1.0;
239 }
240 }
241 }
242 b = (t*j0(x)/b);
243 }
244 }
245 if (sgn != 0)
246 return (-b);
247 else
248 return (b);
249 }
250
251 GENERIC
252 yn(int n, GENERIC x)
253 {
254 int i;
255 int sign;
256 GENERIC a, b, temp = 0, ox, on;
257
258 ox = x;
259 on = (GENERIC)n;
260 if (isnan(x))
261 return (x*x); /* + -> * for Cheetah */
262 if (x <= zero) {
263 if (x == zero) {
264 /* return -one/zero; */
265 return (_SVID_libm_err((GENERIC)n, x, 12));
266 } else {
267 /* return zero/zero; */
268 return (_SVID_libm_err((GENERIC)n, x, 13));
269 }
270 }
271 if (!((int)_lib_version == libm_ieee ||
272 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
273 if (x > X_TLOSS)
274 return (_SVID_libm_err(on, ox, 39));
275 }
276 sign = 1;
277 if (n < 0) {
278 n = -n;
279 if ((n&1) == 1) sign = -1;
280 }
281 if (n == 0)
282 return (y0(x));
283 if (n == 1)
284 return (sign*y1(x));
285 if (!finite(x))
286 return (zero);
287
288 if (x > 1.0e91) {
289 /*
290 * x >> n**2
291 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
292 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
293 * Let s = sin(x), c = cos(x),
294 * xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
295 *
296 * n sin(xn)*sqt2 cos(xn)*sqt2
297 * ----------------------------------
298 * 0 s-c c+s
299 * 1 -s-c -c+s
300 * 2 -s+c -c-s
301 * 3 s+c c-s
302 */
303 switch (n&3) {
304 case 0:
305 temp = sin(x)-cos(x);
306 break;
307 case 1:
308 temp = -sin(x)-cos(x);
309 break;
310 case 2:
311 temp = -sin(x)+cos(x);
312 break;
313 case 3:
314 temp = sin(x)+cos(x);
315 break;
316 }
317 b = invsqrtpi*temp/sqrt(x);
318 } else {
319 a = y0(x);
320 b = y1(x);
321 /*
322 * fix 1262058 and take care of non-default rounding
323 */
324 for (i = 1; i < n; i++) {
325 temp = b;
326 b *= (GENERIC) (i + i) / x;
327 if (b <= -DBL_MAX)
328 break;
329 b -= a;
330 a = temp;
331 }
332 }
333 if (sign > 0)
334 return (b);
335 else
336 return (-b);
337 }