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 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
78 * Thus, J(-n,x) = J(n,-x)
79 */
80 if(n<0){
81 n = -n;
82 x = -x;
83 }
84 if(n==0) return(j0l(x));
85 if(n==1) return(j1l(x));
86 if(x!=x) return x+x;
87 if((n&1)==0)
88 sgn=0; /* even n */
89 else
90 sgn = signbitl(x); /* old n */
91 x = fabsl(x);
92 if(x == zero||!finitel(x)) b = zero;
93 else if((GENERIC)n<=x) { /* Safe to use
94 J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
95 */
96 if(x>1.0e91L) { /* x >> n**2
97 Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
98 Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
99 Let s=sin(x), c=cos(x),
100 xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
101
102 n sin(xn)*sqt2 cos(xn)*sqt2
103 ----------------------------------
104 0 s-c c+s
105 1 -s-c -c+s
106 2 -s+c -c-s
107 3 s+c c-s
108 */
109 switch(n&3) {
110 case 0: temp = cosl(x)+sinl(x); break;
111 case 1: temp = -cosl(x)+sinl(x); break;
112 case 2: temp = -cosl(x)-sinl(x); break;
113 case 3: temp = cosl(x)-sinl(x); break;
114 }
115 b = invsqrtpi*temp/sqrtl(x);
116 } else {
117 a = j0l(x);
118 b = j1l(x);
119 for(i=1;i<n;i++){
120 temp = b;
121 b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */
122 a = temp;
123 }
124 }
125 } else {
126 if(x<1e-17L) { /* use J(n,x) = 1/n!*(x/2)^n */
127 b = powl(0.5L*x,(GENERIC) n);
128 if (b!=zero) {
129 for(a=one,i=1;i<=n;i++) a *= (GENERIC)i;
130 b = b/a;
131 }
132 } else {
133 /* use backward recurrence */
134 /* x x^2 x^2
135 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
136 * 2n - 2(n+1) - 2(n+2)
137 *
138 * 1 1 1
139 * (for large x) = ---- ------ ------ .....
140 * 2n 2(n+1) 2(n+2)
141 * -- - ------ - ------ -
142 * x x x
143 *
144 * Let w = 2n/x and h=2/x, then the above quotient
145 * is equal to the continued fraction:
146 * 1
147 * = -----------------------
148 * 1
149 * w - -----------------
150 * 1
151 * w+h - ---------
152 * w+2h - ...
153 *
154 * To determine how many terms needed, let
155 * Q(0) = w, Q(1) = w(w+h) - 1,
156 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
157 * When Q(k) > 1e4 good for single
158 * When Q(k) > 1e9 good for double
159 * When Q(k) > 1e17 good for quaduple
160 */
161 /* determin k */
162 GENERIC t,v;
163 double q0,q1,h,tmp; int k,m;
164 w = (n+n)/(double)x; h = 2.0/(double)x;
165 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
166 while(q1<1.0e17) {
167 k += 1; z += h;
168 tmp = z*q1 - q0;
169 q0 = q1;
170 q1 = tmp;
171 }
172 m = n+n;
173 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
174 a = t;
175 b = one;
176 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
177 hence, if n*(log(2n/x)) > ...
178 single 8.8722839355e+01
179 double 7.09782712893383973096e+02
180 long double 1.1356523406294143949491931077970765006170e+04
181 then recurrent value may overflow and the result is
182 likely underflow to zero
183 */
184 tmp = n;
185 v = two/x;
186 tmp = tmp*logl(fabsl(v*tmp));
187 if(tmp<1.1356523406294143949491931077970765e+04L) {
188 for(i=n-1;i>0;i--){
189 temp = b;
190 b = ((i+i)/x)*b - a;
191 a = temp;
192 }
193 } else {
194 for(i=n-1;i>0;i--){
195 temp = b;
196 b = ((i+i)/x)*b - a;
197 a = temp;
198 if(b>1e1000L) {
199 a /= b;
200 t /= b;
201 b = 1.0;
202 }
203 }
204 }
205 b = (t*j0l(x)/b);
206 }
207 }
208 if(sgn==1) return -b; else return b;
209 }
210
211 GENERIC ynl(n,x)
212 int n; GENERIC x;{
213 int i;
214 int sign;
215 GENERIC a, b, temp = 0;
216
217 if(x!=x)
218 return x+x;
219 if (x <= zero) {
220 if(x==zero)
221 return -one/zero;
222 else
223 return zero/zero;
224 }
225 sign = 1;
226 if(n<0){
227 n = -n;
228 if((n&1) == 1) sign = -1;
229 }
230 if(n==0) return(y0l(x));
231 if(n==1) return(sign*y1l(x));
232 if(!finitel(x)) return zero;
233
234 if(x>1.0e91L) { /* x >> n**2
235 Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
236 Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
237 Let s=sin(x), c=cos(x),
238 xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
239
240 n sin(xn)*sqt2 cos(xn)*sqt2
241 ----------------------------------
242 0 s-c c+s
243 1 -s-c -c+s
244 2 -s+c -c-s
245 3 s+c c-s
246 */
247 switch(n&3) {
248 case 0: temp = sinl(x)-cosl(x); break;
249 case 1: temp = -sinl(x)-cosl(x); break;
250 case 2: temp = -sinl(x)+cosl(x); break;
251 case 3: temp = sinl(x)+cosl(x); break;
252 }
253 b = invsqrtpi*temp/sqrtl(x);
254 } else {
255 a = y0l(x);
256 b = y1l(x);
257 /*
258 * fix 1262058 and take care of non-default rounding
259 */
260 for (i = 1; i < n; i++) {
261 temp = b;
262 b *= (GENERIC) (i + i) / x;
263 if (b <= -LDBL_MAX)
264 break;
265 b -= a;
266 a = temp;
267 }
268 }
269 if(sign>0) return b; else return -b;
270 }