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 #if defined(ELFOBJ)
31 #pragma weak jnl = __jnl
32 #pragma weak ynl = __ynl
33 #endif
34
35 /*
36 * floating point Bessel's function of the 1st and 2nd kind
37 * of order n: jn(n,x),yn(n,x);
38 *
39 * Special cases:
40 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
41 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
42 * Note 2. About jn(n,x), yn(n,x)
43 * For n=0, j0(x) is called,
44 * for n=1, j1(x) is called,
45 * for n<x, forward recursion us used starting
46 * from values of j0(x) and j1(x).
47 * for n>x, a continued fraction approximation to
48 * j(n,x)/j(n-1,x) is evaluated and then backward
49 * recursion is used starting from a supposed value
50 * for j(n,x). The resulting value of j(0,x) is
51 * compared with the actual value to correct the
52 * supposed value of j(n,x).
53 *
54 * yn(n,x) is similar in all respects, except
55 * that forward recursion is used for all
56 * values of n>1.
57 *
58 */
59
60 #include "libm.h"
61 #include "longdouble.h"
62 #include <float.h> /* LDBL_MAX */
63
64 #define GENERIC long double
65
66 static const GENERIC
67 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
68 two = 2.0L,
69 zero = 0.0L,
70 one = 1.0L;
71
72 GENERIC
73 jnl(n, x) int n; GENERIC x; {
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 if (n == 0) return (j0l(x));
86 if (n == 1) return (j1l(x));
87 if (x != x) return x+x;
88 if ((n&1) == 0)
89 sgn = 0; /* even n */
90 else
91 sgn = signbitl(x); /* old n */
92 x = fabsl(x);
93 if (x == zero || !finitel(x)) b = zero;
94 else if ((GENERIC)n <= x) {
95 /*
96 * Safe to use
97 * J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
98 */
99 if (x > 1.0e91L) {
100 /*
101 * x >> n**2
102 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
103 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
104 * Let s=sin(x), c=cos(x),
105 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
106 *
107 * n sin(xn)*sqt2 cos(xn)*sqt2
108 * ----------------------------------
109 * 0 s-c c+s
110 * 1 -s-c -c+s
111 * 2 -s+c -c-s
112 * 3 s+c c-s
113 */
114 switch (n&3) {
115 case 0: temp = cosl(x)+sinl(x); break;
116 case 1: temp = -cosl(x)+sinl(x); break;
117 case 2: temp = -cosl(x)-sinl(x); break;
118 case 3: temp = cosl(x)-sinl(x); break;
119 }
120 b = invsqrtpi*temp/sqrtl(x);
121 } else {
122 a = j0l(x);
123 b = j1l(x);
124 for (i = 1; i < n; i++) {
125 temp = b;
126 b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */
127 a = temp;
128 }
129 }
130 } else {
131 if (x < 1e-17L) { /* use J(n,x) = 1/n!*(x/2)^n */
132 b = powl(0.5L*x, (GENERIC) n);
133 if (b != zero) {
134 for (a = one, i = 1; i <= n; i++) a *= (GENERIC)i;
135 b = b/a;
136 }
137 } else {
138 /*
139 * use backward recurrence
140 * x x^2 x^2
141 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
142 * 2n - 2(n+1) - 2(n+2)
143 *
144 * 1 1 1
145 * (for large x) = ---- ------ ------ .....
146 * 2n 2(n+1) 2(n+2)
147 * -- - ------ - ------ -
148 * x x x
149 *
150 * Let w = 2n/x and h=2/x, then the above quotient
151 * is equal to the continued fraction:
152 * 1
153 * = -----------------------
154 * 1
155 * w - -----------------
156 * 1
157 * w+h - ---------
158 * w+2h - ...
159 *
160 * To determine how many terms needed, let
161 * Q(0) = w, Q(1) = w(w+h) - 1,
162 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
163 * When Q(k) > 1e4 good for single
164 * When Q(k) > 1e9 good for double
165 * When Q(k) > 1e17 good for quaduple
166 */
167 /* determin k */
168 GENERIC t, v;
169 double q0, q1, h, tmp; int k, m;
170 w = (n+n)/(double)x; h = 2.0/(double)x;
171 q0 = w; z = w+h; q1 = w*z - 1.0; k = 1;
172 while (q1 < 1.0e17) {
173 k += 1; z += h;
174 tmp = z*q1 - q0;
175 q0 = q1;
176 q1 = tmp;
177 }
178 m = n+n;
179 for (t = zero, i = 2*(n+k); i >= m; i -= 2) t = one/(i/x-t);
180 a = t;
181 b = one;
182 /*
183 * Estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
184 * hence, if n*(log(2n/x)) > ...
185 * single 8.8722839355e+01
186 * double 7.09782712893383973096e+02
187 * long double 1.1356523406294143949491931077970765006170e+04
188 * then recurrent value may overflow and the result is
189 * likely underflow to zero.
190 */
191 tmp = n;
192 v = two/x;
193 tmp = tmp*logl(fabsl(v*tmp));
194 if (tmp < 1.1356523406294143949491931077970765e+04L) {
195 for (i = n-1; i > 0; i--) {
196 temp = b;
197 b = ((i+i)/x)*b - a;
198 a = temp;
199 }
200 } else {
201 for (i = n-1; i > 0; i--) {
202 temp = b;
203 b = ((i+i)/x)*b - a;
204 a = temp;
205 if (b > 1e1000L) {
206 a /= b;
207 t /= b;
208 b = 1.0;
209 }
210 }
211 }
212 b = (t*j0l(x)/b);
213 }
214 }
215 if (sgn == 1)
216 return -b;
217 else
218 return b;
219 }
220
221 GENERIC
222 ynl(n, x) int n; GENERIC x; {
223 int i;
224 int sign;
225 GENERIC a, b, temp = 0;
226
227 if (x != x)
228 return x+x;
229 if (x <= zero) {
230 if (x == zero)
231 return -one/zero;
232 else
233 return zero/zero;
234 }
235 sign = 1;
236 if (n < 0) {
237 n = -n;
238 if ((n&1) == 1) sign = -1;
239 }
240 if (n == 0) return (y0l(x));
241 if (n == 1) return (sign*y1l(x));
242 if (!finitel(x)) return zero;
243
244 if (x > 1.0e91L) {
245 /*
246 * x >> n**2
247 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
248 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
249 * Let s=sin(x), c=cos(x),
250 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
251 *
252 * n sin(xn)*sqt2 cos(xn)*sqt2
253 * ----------------------------------
254 * 0 s-c c+s
255 * 1 -s-c -c+s
256 * 2 -s+c -c-s
257 * 3 s+c c-s
258 */
259 switch (n&3) {
260 case 0: temp = sinl(x)-cosl(x); break;
261 case 1: temp = -sinl(x)-cosl(x); break;
262 case 2: temp = -sinl(x)+cosl(x); break;
263 case 3: temp = sinl(x)+cosl(x); break;
264 }
265 b = invsqrtpi*temp/sqrtl(x);
266 } else {
267 a = y0l(x);
268 b = y1l(x);
269 /*
270 * fix 1262058 and take care of non-default rounding
271 */
272 for (i = 1; i < n; i++) {
273 temp = b;
274 b *= (GENERIC) (i + i) / x;
275 if (b <= -LDBL_MAX)
276 break;
277 b -= a;
278 a = temp;
279 }
280 }
281 if (sign > 0)
282 return b;
283 else
284 return -b;
285 }