/[jscoverage]/trunk/js/src/fdlibm/e_jn.c
ViewVC logotype

Contents of /trunk/js/src/fdlibm/e_jn.c

Parent Directory Parent Directory | Revision Log Revision Log


Revision 2 - (show annotations)
Wed Aug 1 13:51:53 2007 UTC (14 years, 11 months ago) by siliconforks
File MIME type: text/plain
File size: 9092 byte(s)
Initial import.

1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
2 *
3 * ***** BEGIN LICENSE BLOCK *****
4 * Version: MPL 1.1/GPL 2.0/LGPL 2.1
5 *
6 * The contents of this file are subject to the Mozilla Public License Version
7 * 1.1 (the "License"); you may not use this file except in compliance with
8 * the License. You may obtain a copy of the License at
9 * http://www.mozilla.org/MPL/
10 *
11 * Software distributed under the License is distributed on an "AS IS" basis,
12 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
13 * for the specific language governing rights and limitations under the
14 * License.
15 *
16 * The Original Code is Mozilla Communicator client code, released
17 * March 31, 1998.
18 *
19 * The Initial Developer of the Original Code is
20 * Sun Microsystems, Inc.
21 * Portions created by the Initial Developer are Copyright (C) 1998
22 * the Initial Developer. All Rights Reserved.
23 *
24 * Contributor(s):
25 *
26 * Alternatively, the contents of this file may be used under the terms of
27 * either of the GNU General Public License Version 2 or later (the "GPL"),
28 * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
29 * in which case the provisions of the GPL or the LGPL are applicable instead
30 * of those above. If you wish to allow use of your version of this file only
31 * under the terms of either the GPL or the LGPL, and not to allow others to
32 * use your version of this file under the terms of the MPL, indicate your
33 * decision by deleting the provisions above and replace them with the notice
34 * and other provisions required by the GPL or the LGPL. If you do not delete
35 * the provisions above, a recipient may use your version of this file under
36 * the terms of any one of the MPL, the GPL or the LGPL.
37 *
38 * ***** END LICENSE BLOCK ***** */
39
40 /* @(#)e_jn.c 1.4 95/01/18 */
41 /*
42 * ====================================================
43 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
44 *
45 * Developed at SunSoft, a Sun Microsystems, Inc. business.
46 * Permission to use, copy, modify, and distribute this
47 * software is freely granted, provided that this notice
48 * is preserved.
49 * ====================================================
50 */
51
52 /*
53 * __ieee754_jn(n, x), __ieee754_yn(n, x)
54 * floating point Bessel's function of the 1st and 2nd kind
55 * of order n
56 *
57 * Special cases:
58 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
59 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
60 * Note 2. About jn(n,x), yn(n,x)
61 * For n=0, j0(x) is called,
62 * for n=1, j1(x) is called,
63 * for n<x, forward recursion us used starting
64 * from values of j0(x) and j1(x).
65 * for n>x, a continued fraction approximation to
66 * j(n,x)/j(n-1,x) is evaluated and then backward
67 * recursion is used starting from a supposed value
68 * for j(n,x). The resulting value of j(0,x) is
69 * compared with the actual value to correct the
70 * supposed value of j(n,x).
71 *
72 * yn(n,x) is similar in all respects, except
73 * that forward recursion is used for all
74 * values of n>1.
75 *
76 */
77
78 #include "fdlibm.h"
79
80 #ifdef __STDC__
81 static const double
82 #else
83 static double
84 #endif
85 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
86 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
87 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
88
89 static double zero = 0.00000000000000000000e+00;
90
91 #ifdef __STDC__
92 double __ieee754_jn(int n, double x)
93 #else
94 double __ieee754_jn(n,x)
95 int n; double x;
96 #endif
97 {
98 fd_twoints u;
99 int i,hx,ix,lx, sgn;
100 double a, b, temp, di;
101 double z, w;
102
103 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
104 * Thus, J(-n,x) = J(n,-x)
105 */
106 u.d = x;
107 hx = __HI(u);
108 ix = 0x7fffffff&hx;
109 lx = __LO(u);
110 /* if J(n,NaN) is NaN */
111 if((ix|((unsigned)(lx|-lx))>>31)>0x7ff00000) return x+x;
112 if(n<0){
113 n = -n;
114 x = -x;
115 hx ^= 0x80000000;
116 }
117 if(n==0) return(__ieee754_j0(x));
118 if(n==1) return(__ieee754_j1(x));
119 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
120 x = fd_fabs(x);
121 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
122 b = zero;
123 else if((double)n<=x) {
124 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
125 if(ix>=0x52D00000) { /* x > 2**302 */
126 /* (x >> n**2)
127 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
128 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
129 * Let s=sin(x), c=cos(x),
130 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
131 *
132 * n sin(xn)*sqt2 cos(xn)*sqt2
133 * ----------------------------------
134 * 0 s-c c+s
135 * 1 -s-c -c+s
136 * 2 -s+c -c-s
137 * 3 s+c c-s
138 */
139 switch(n&3) {
140 case 0: temp = fd_cos(x)+fd_sin(x); break;
141 case 1: temp = -fd_cos(x)+fd_sin(x); break;
142 case 2: temp = -fd_cos(x)-fd_sin(x); break;
143 case 3: temp = fd_cos(x)-fd_sin(x); break;
144 }
145 b = invsqrtpi*temp/fd_sqrt(x);
146 } else {
147 a = __ieee754_j0(x);
148 b = __ieee754_j1(x);
149 for(i=1;i<n;i++){
150 temp = b;
151 b = b*((double)(i+i)/x) - a; /* avoid underflow */
152 a = temp;
153 }
154 }
155 } else {
156 if(ix<0x3e100000) { /* x < 2**-29 */
157 /* x is tiny, return the first Taylor expansion of J(n,x)
158 * J(n,x) = 1/n!*(x/2)^n - ...
159 */
160 if(n>33) /* underflow */
161 b = zero;
162 else {
163 temp = x*0.5; b = temp;
164 for (a=one,i=2;i<=n;i++) {
165 a *= (double)i; /* a = n! */
166 b *= temp; /* b = (x/2)^n */
167 }
168 b = b/a;
169 }
170 } else {
171 /* use backward recurrence */
172 /* x x^2 x^2
173 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
174 * 2n - 2(n+1) - 2(n+2)
175 *
176 * 1 1 1
177 * (for large x) = ---- ------ ------ .....
178 * 2n 2(n+1) 2(n+2)
179 * -- - ------ - ------ -
180 * x x x
181 *
182 * Let w = 2n/x and h=2/x, then the above quotient
183 * is equal to the continued fraction:
184 * 1
185 * = -----------------------
186 * 1
187 * w - -----------------
188 * 1
189 * w+h - ---------
190 * w+2h - ...
191 *
192 * To determine how many terms needed, let
193 * Q(0) = w, Q(1) = w(w+h) - 1,
194 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
195 * When Q(k) > 1e4 good for single
196 * When Q(k) > 1e9 good for double
197 * When Q(k) > 1e17 good for quadruple
198 */
199 /* determine k */
200 double t,v;
201 double q0,q1,h,tmp; int k,m;
202 w = (n+n)/(double)x; h = 2.0/(double)x;
203 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
204 while(q1<1.0e9) {
205 k += 1; z += h;
206 tmp = z*q1 - q0;
207 q0 = q1;
208 q1 = tmp;
209 }
210 m = n+n;
211 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
212 a = t;
213 b = one;
214 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
215 * Hence, if n*(log(2n/x)) > ...
216 * single 8.8722839355e+01
217 * double 7.09782712893383973096e+02
218 * long double 1.1356523406294143949491931077970765006170e+04
219 * then recurrent value may overflow and the result is
220 * likely underflow to zero
221 */
222 tmp = n;
223 v = two/x;
224 tmp = tmp*__ieee754_log(fd_fabs(v*tmp));
225 if(tmp<7.09782712893383973096e+02) {
226 for(i=n-1,di=(double)(i+i);i>0;i--){
227 temp = b;
228 b *= di;
229 b = b/x - a;
230 a = temp;
231 di -= two;
232 }
233 } else {
234 for(i=n-1,di=(double)(i+i);i>0;i--){
235 temp = b;
236 b *= di;
237 b = b/x - a;
238 a = temp;
239 di -= two;
240 /* scale b to avoid spurious overflow */
241 if(b>1e100) {
242 a /= b;
243 t /= b;
244 b = one;
245 }
246 }
247 }
248 b = (t*__ieee754_j0(x)/b);
249 }
250 }
251 if(sgn==1) return -b; else return b;
252 }
253
254 #ifdef __STDC__
255 double __ieee754_yn(int n, double x)
256 #else
257 double __ieee754_yn(n,x)
258 int n; double x;
259 #endif
260 {
261 fd_twoints u;
262 int i,hx,ix,lx;
263 int sign;
264 double a, b, temp;
265
266 u.d = x;
267 hx = __HI(u);
268 ix = 0x7fffffff&hx;
269 lx = __LO(u);
270 /* if Y(n,NaN) is NaN */
271 if((ix|((unsigned)(lx|-lx))>>31)>0x7ff00000) return x+x;
272 if((ix|lx)==0) return -one/zero;
273 if(hx<0) return zero/zero;
274 sign = 1;
275 if(n<0){
276 n = -n;
277 sign = 1 - ((n&1)<<1);
278 }
279 if(n==0) return(__ieee754_y0(x));
280 if(n==1) return(sign*__ieee754_y1(x));
281 if(ix==0x7ff00000) return zero;
282 if(ix>=0x52D00000) { /* x > 2**302 */
283 /* (x >> n**2)
284 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
285 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
286 * Let s=sin(x), c=cos(x),
287 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
288 *
289 * n sin(xn)*sqt2 cos(xn)*sqt2
290 * ----------------------------------
291 * 0 s-c c+s
292 * 1 -s-c -c+s
293 * 2 -s+c -c-s
294 * 3 s+c c-s
295 */
296 switch(n&3) {
297 case 0: temp = fd_sin(x)-fd_cos(x); break;
298 case 1: temp = -fd_sin(x)-fd_cos(x); break;
299 case 2: temp = -fd_sin(x)+fd_cos(x); break;
300 case 3: temp = fd_sin(x)+fd_cos(x); break;
301 }
302 b = invsqrtpi*temp/fd_sqrt(x);
303 } else {
304 a = __ieee754_y0(x);
305 b = __ieee754_y1(x);
306 /* quit if b is -inf */
307 u.d = b;
308 for(i=1;i<n&&(__HI(u) != 0xfff00000);i++){
309 temp = b;
310 b = ((double)(i+i)/x)*b - a;
311 a = temp;
312 }
313 }
314 if(sign>0) return b; else return -b;
315 }

  ViewVC Help
Powered by ViewVC 1.1.24