2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 * 3. All advertising materials mentioning features or use of this software
14 * must display the following acknowledgement:
15 * This product includes software developed by the University of
16 * California, Berkeley and its contributors.
17 * 4. Neither the name of the University nor the names of its contributors
18 * may be used to endorse or promote products derived from this software
19 * without specific prior written permission.
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
35 static char sccsid[] = "@(#)jn.c 8.2 (Berkeley) 11/30/93";
40 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture.
44 * ====================================================
45 * Copyright (C) 1992 by Sun Microsystems, Inc.
47 * Developed at SunPro, a Sun Microsystems, Inc. business.
48 * Permission to use, copy, modify, and distribute this
49 * software is freely granted, provided that this notice
51 * ====================================================
53 * ******************* WARNING ********************
54 * This is an alpha version of SunPro's FDLIBM (Freely
55 * Distributable Math Library) for IEEE double precision
56 * arithmetic. FDLIBM is a basic math library written
57 * in C that runs on machines that conform to IEEE
58 * Standard 754/854. This alpha version is distributed
59 * for testing purpose. Those who use this software
60 * should report any bugs to
62 * fdlibm-comments@sunpro.eng.sun.com
64 * -- K.C. Ng, Oct 12, 1992
65 * ************************************************
69 * jn(int n, double x), yn(int n, double x)
70 * floating point Bessel's function of the 1st and 2nd kind
74 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
75 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
76 * Note 2. About jn(n,x), yn(n,x)
77 * For n=0, j0(x) is called,
78 * for n=1, j1(x) is called,
79 * for n<x, forward recursion us used starting
80 * from values of j0(x) and j1(x).
81 * for n>x, a continued fraction approximation to
82 * j(n,x)/j(n-1,x) is evaluated and then backward
83 * recursion is used starting from a supposed value
84 * for j(n,x). The resulting value of j(0,x) is
85 * compared with the actual value to correct the
86 * supposed value of j(n,x).
88 * yn(n,x) is similar in all respects, except
89 * that forward recursion is used for all
98 #if defined(vax) || defined(tahoe)
102 #define infnan(x) (0.0)
106 invsqrtpi= 5.641895835477562869480794515607725858441e-0001,
118 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
119 * Thus, J(-n,x) = J(n,-x)
121 /* if J(n,NaN) is NaN */
122 if (_IEEE && isnan(x)) return x+x;
127 if (n==0) return(j0(x));
128 if (n==1) return(j1(x));
129 sgn = (n&1)&(x < zero); /* even n -- 0, odd n -- sign(x) */
131 if (x == 0 || !finite (x)) /* if x is 0 or inf */
133 else if ((double) n <= x) {
134 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
135 if (_IEEE && x >= 8.148143905337944345e+090) {
138 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
139 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
140 * Let s=sin(x), c=cos(x),
141 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
143 * n sin(xn)*sqt2 cos(xn)*sqt2
144 * ----------------------------------
151 case 0: temp = cos(x)+sin(x); break;
152 case 1: temp = -cos(x)+sin(x); break;
153 case 2: temp = -cos(x)-sin(x); break;
154 case 3: temp = cos(x)-sin(x); break;
156 b = invsqrtpi*temp/sqrt(x);
162 b = b*((double)(i+i)/x) - a; /* avoid underflow */
167 if (x < 1.86264514923095703125e-009) { /* x < 2**-29 */
168 /* x is tiny, return the first Taylor expansion of J(n,x)
169 * J(n,x) = 1/n!*(x/2)^n - ...
171 if (n > 33) /* underflow */
174 temp = x*0.5; b = temp;
175 for (a=one,i=2;i<=n;i++) {
176 a *= (double)i; /* a = n! */
177 b *= temp; /* b = (x/2)^n */
182 /* use backward recurrence */
184 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
185 * 2n - 2(n+1) - 2(n+2)
188 * (for large x) = ---- ------ ------ .....
190 * -- - ------ - ------ -
193 * Let w = 2n/x and h=2/x, then the above quotient
194 * is equal to the continued fraction:
196 * = -----------------------
198 * w - -----------------
203 * To determine how many terms needed, let
204 * Q(0) = w, Q(1) = w(w+h) - 1,
205 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
206 * When Q(k) > 1e4 good for single
207 * When Q(k) > 1e9 good for double
208 * When Q(k) > 1e17 good for quadruple
212 double q0,q1,h,tmp; int k,m;
213 w = (n+n)/(double)x; h = 2.0/(double)x;
214 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
222 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
225 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
226 * Hence, if n*(log(2n/x)) > ...
227 * single 8.8722839355e+01
228 * double 7.09782712893383973096e+02
229 * long double 1.1356523406294143949491931077970765006170e+04
230 * then recurrent value may overflow and the result will
231 * likely underflow to zero
235 tmp = tmp*log(fabs(v*tmp));
240 /* scale b to avoid spurious overflow */
241 # if defined(vax) || defined(tahoe)
245 # endif /* defined(vax) || defined(tahoe) */
255 return ((sgn == 1) ? -b : b);
263 /* Y(n,NaN), Y(n, x < 0) is NaN */
264 if (x <= 0 || (_IEEE && x != x))
265 if (_IEEE && x < 0) return zero/zero;
266 else if (x < 0) return (infnan(EDOM));
267 else if (_IEEE) return -one/zero;
268 else return(infnan(-ERANGE));
269 else if (!finite(x)) return(0);
273 sign = 1 - ((n&1)<<2);
275 if (n == 0) return(y0(x));
276 if (n == 1) return(sign*y1(x));
277 if(_IEEE && x >= 8.148143905337944345e+090) { /* x > 2**302 */
279 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
280 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
281 * Let s=sin(x), c=cos(x),
282 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
284 * n sin(xn)*sqt2 cos(xn)*sqt2
285 * ----------------------------------
292 case 0: temp = sin(x)-cos(x); break;
293 case 1: temp = -sin(x)-cos(x); break;
294 case 2: temp = -sin(x)+cos(x); break;
295 case 3: temp = sin(x)+cos(x); break;
297 b = invsqrtpi*temp/sqrt(x);
301 /* quit if b is -inf */
302 for (i = 1; i < n && !finite(b); i++){
304 b = ((double)(i+i)/x)*b - a;
308 if (!_IEEE && !finite(b))
309 return (infnan(-sign * ERANGE));
310 return ((sign > 0) ? b : -b);