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 <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(n,x) int n; GENERIC x;{
73 int i, sgn;
74 GENERIC a, b, temp, z, w;
75
76 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
77 * Thus, J(-n,x) = J(n,-x)
78 */
79 if(n<0){
80 n = -n;
81 x = -x;
82 }
83 if(n==0) return(j0l(x));
84 if(n==1) return(j1l(x));
85 if(x!=x) return x+x;
86 if((n&1)==0)
87 sgn=0; /* even n */
88 else
89 sgn = signbitl(x); /* old n */
90 x = fabsl(x);
91 if(x == zero||!finitel(x)) b = zero;
92 else if((GENERIC)n<=x) { /* Safe to use
93 J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
94 */
95 if(x>1.0e91L) { /* x >> n**2
96 Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
97 Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
98 Let s=sin(x), c=cos(x),
99 xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
100
101 n sin(xn)*sqt2 cos(xn)*sqt2
102 ----------------------------------
103 0 s-c c+s
104 1 -s-c -c+s
105 2 -s+c -c-s
106 3 s+c c-s
107 */
108 switch(n&3) {
109 case 0: temp = cosl(x)+sinl(x); break;
110 case 1: temp = -cosl(x)+sinl(x); break;
111 case 2: temp = -cosl(x)-sinl(x); break;
112 case 3: temp = cosl(x)-sinl(x); break;
113 }
114 b = invsqrtpi*temp/sqrtl(x);
115 } else {
116 a = j0l(x);
117 b = j1l(x);
118 for(i=1;i<n;i++){
119 temp = b;
120 b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */
121 a = temp;
122 }
123 }
124 } else {
125 if(x<1e-17L) { /* use J(n,x) = 1/n!*(x/2)^n */
126 b = powl(0.5L*x,(GENERIC) n);
127 if (b!=zero) {
128 for(a=one,i=1;i<=n;i++) a *= (GENERIC)i;
129 b = b/a;
130 }
131 } else {
132 /* use backward recurrence */
133 /* x x^2 x^2
134 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
135 * 2n - 2(n+1) - 2(n+2)
136 *
137 * 1 1 1
138 * (for large x) = ---- ------ ------ .....
139 * 2n 2(n+1) 2(n+2)
140 * -- - ------ - ------ -
141 * x x x
142 *
143 * Let w = 2n/x and h=2/x, then the above quotient
144 * is equal to the continued fraction:
145 * 1
146 * = -----------------------
147 * 1
148 * w - -----------------
149 * 1
150 * w+h - ---------
151 * w+2h - ...
152 *
153 * To determine how many terms needed, let
154 * Q(0) = w, Q(1) = w(w+h) - 1,
155 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
156 * When Q(k) > 1e4 good for single
157 * When Q(k) > 1e9 good for double
158 * When Q(k) > 1e17 good for quaduple
159 */
160 /* determin k */
161 GENERIC t,v;
162 double q0,q1,h,tmp; int k,m;
163 w = (n+n)/(double)x; h = 2.0/(double)x;
164 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
165 while(q1<1.0e17) {
166 k += 1; z += h;
167 tmp = z*q1 - q0;
168 q0 = q1;
169 q1 = tmp;
170 }
171 m = n+n;
172 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
173 a = t;
174 b = one;
175 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
176 hence, if n*(log(2n/x)) > ...
177 single 8.8722839355e+01
178 double 7.09782712893383973096e+02
179 long double 1.1356523406294143949491931077970765006170e+04
180 then recurrent value may overflow and the result is
181 likely underflow to zero
182 */
183 tmp = n;
184 v = two/x;
185 tmp = tmp*logl(fabsl(v*tmp));
186 if(tmp<1.1356523406294143949491931077970765e+04L) {
187 for(i=n-1;i>0;i--){
188 temp = b;
189 b = ((i+i)/x)*b - a;
190 a = temp;
191 }
192 } else {
193 for(i=n-1;i>0;i--){
194 temp = b;
195 b = ((i+i)/x)*b - a;
196 a = temp;
197 if(b>1e1000L) {
198 a /= b;
199 t /= b;
200 b = 1.0;
201 }
202 }
203 }
204 b = (t*j0l(x)/b);
205 }
206 }
207 if(sgn==1) return -b; else return b;
208 }
209
210 GENERIC ynl(n,x)
211 int n; GENERIC x;{
212 int i;
213 int sign;
214 GENERIC a, b, temp;
215
216 if(x!=x) return x+x;
217 if (x <= zero)
218 if(x==zero)
219 return -one/zero;
220 else
221 return zero/zero;
222 sign = 1;
223 if(n<0){
224 n = -n;
225 if((n&1) == 1) sign = -1;
226 }
227 if(n==0) return(y0l(x));
228 if(n==1) return(sign*y1l(x));
229 if(!finitel(x)) return zero;
230
231 if(x>1.0e91L) { /* x >> n**2
232 Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
233 Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
234 Let s=sin(x), c=cos(x),
235 xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
236
237 n sin(xn)*sqt2 cos(xn)*sqt2
238 ----------------------------------
239 0 s-c c+s
240 1 -s-c -c+s
241 2 -s+c -c-s
242 3 s+c c-s
243 */
244 switch(n&3) {
245 case 0: temp = sinl(x)-cosl(x); break;
246 case 1: temp = -sinl(x)-cosl(x); break;
247 case 2: temp = -sinl(x)+cosl(x); break;
248 case 3: temp = sinl(x)+cosl(x); break;
249 }
250 b = invsqrtpi*temp/sqrtl(x);
251 } else {
252 a = y0l(x);
253 b = y1l(x);
254 /*
255 * fix 1262058 and take care of non-default rounding
256 */
257 for (i = 1; i < n; i++) {
258 temp = b;
259 b *= (GENERIC) (i + i) / x;
260 if (b <= -LDBL_MAX)
261 break;
262 b -= a;
263 a = temp;
264 }
265 }
266 if(sign>0) return b; else return -b;
267 }