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, 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) { /* Safe to use
95 J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
96 */
97 if(x>1.0e91L) { /* x >> n**2
98 Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
99 Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
100 Let s=sin(x), c=cos(x),
101 xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
102
103 n sin(xn)*sqt2 cos(xn)*sqt2
104 ----------------------------------
105 0 s-c c+s
106 1 -s-c -c+s
107 2 -s+c -c-s
108 3 s+c c-s
109 */
110 switch(n&3) {
111 case 0: temp = cosl(x)+sinl(x); break;
112 case 1: temp = -cosl(x)+sinl(x); break;
113 case 2: temp = -cosl(x)-sinl(x); break;
114 case 3: temp = cosl(x)-sinl(x); break;
115 }
116 b = invsqrtpi*temp/sqrtl(x);
117 } else {
118 a = j0l(x);
119 b = j1l(x);
120 for(i=1;i<n;i++){
121 temp = b;
122 b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */
123 a = temp;
124 }
125 }
126 } else {
127 if(x<1e-17L) { /* use J(n,x) = 1/n!*(x/2)^n */
128 b = powl(0.5L*x,(GENERIC) n);
129 if (b!=zero) {
130 for(a=one,i=1;i<=n;i++) a *= (GENERIC)i;
131 b = b/a;
132 }
133 } else {
134 /* use backward recurrence */
135 /*
136 * x x^2 x^2
137 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
138 * 2n - 2(n+1) - 2(n+2)
139 *
140 * 1 1 1
141 * (for large x) = ---- ------ ------ .....
142 * 2n 2(n+1) 2(n+2)
143 * -- - ------ - ------ -
144 * x x x
145 *
146 * Let w = 2n/x and h=2/x, then the above quotient
147 * is equal to the continued fraction:
148 * 1
149 * = -----------------------
150 * 1
151 * w - -----------------
152 * 1
153 * w+h - ---------
154 * w+2h - ...
155 *
156 * To determine how many terms needed, let
157 * Q(0) = w, Q(1) = w(w+h) - 1,
158 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
159 * When Q(k) > 1e4 good for single
160 * When Q(k) > 1e9 good for double
161 * When Q(k) > 1e17 good for quaduple
162 */
163 /* determin k */
164 GENERIC t,v;
165 double q0,q1,h,tmp; int k,m;
166 w = (n+n)/(double)x; h = 2.0/(double)x;
167 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
168 while(q1<1.0e17) {
169 k += 1; z += h;
170 tmp = z*q1 - q0;
171 q0 = q1;
172 q1 = tmp;
173 }
174 m = n+n;
175 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
176 a = t;
177 b = one;
178 /*
179 * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
180 * hence, if n*(log(2n/x)) > ...
181 * single 8.8722839355e+01
182 * double 7.09782712893383973096e+02
183 * long double 1.1356523406294143949491931077970765006170e+04
184 * then recurrent value may overflow and the result is
185 * likely underflow to zero
186 */
187 tmp = n;
188 v = two/x;
189 tmp = tmp*logl(fabsl(v*tmp));
190 if(tmp<1.1356523406294143949491931077970765e+04L) {
191 for(i=n-1;i>0;i--){
192 temp = b;
193 b = ((i+i)/x)*b - a;
194 a = temp;
195 }
196 } else {
197 for(i=n-1;i>0;i--){
198 temp = b;
199 b = ((i+i)/x)*b - a;
200 a = temp;
201 if(b>1e1000L) {
202 a /= b;
203 t /= b;
204 b = 1.0;
205 }
206 }
207 }
208 b = (t*j0l(x)/b);
209 }
210 }
211 if(sgn==1) return -b; else return b;
212 }
213
214 GENERIC ynl(n,x)
215 int n; GENERIC x;{
216 int i;
217 int sign;
218 GENERIC a, b, temp;
219
220 if(x!=x) return x+x;
221 if (x <= zero)
222 if(x==zero)
223 return -one/zero;
224 else
225 return zero/zero;
226 sign = 1;
227 if(n<0){
228 n = -n;
229 if((n&1) == 1) sign = -1;
230 }
231 if(n==0) return(y0l(x));
232 if(n==1) return(sign*y1l(x));
233 if(!finitel(x)) return zero;
234
235 if(x>1.0e91L) { /* x >> n**2
236 Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
237 Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
238 Let s=sin(x), c=cos(x),
239 xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
240
241 n sin(xn)*sqt2 cos(xn)*sqt2
242 ----------------------------------
243 0 s-c c+s
244 1 -s-c -c+s
245 2 -s+c -c-s
246 3 s+c c-s
247 */
248 switch(n&3) {
249 case 0: temp = sinl(x)-cosl(x); break;
250 case 1: temp = -sinl(x)-cosl(x); break;
251 case 2: temp = -sinl(x)+cosl(x); break;
252 case 3: temp = sinl(x)+cosl(x); break;
253 }
254 b = invsqrtpi*temp/sqrtl(x);
255 } else {
256 a = y0l(x);
257 b = y1l(x);
258 /*
259 * fix 1262058 and take care of non-default rounding
260 */
261 for (i = 1; i < n; i++) {
262 temp = b;
263 b *= (GENERIC) (i + i) / x;
264 if (b <= -LDBL_MAX)
265 break;
266 b -= a;
267 a = temp;
268 }
269 }
270 if(sign>0) return b; else return -b;
271 }