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1/* @(#)e_jn.c 5.1 93/09/24 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13
14
15#include <sys/cdefs.h>
16#if defined(LIBM_SCCS) && !defined(lint)
17__RCSID("$NetBSD: e_jn.c,v 1.11 1999/07/02 15:37:39 simonb Exp $");
18#endif
19
20/*
21 * __ieee754_jn(n, x), __ieee754_yn(n, x)
22 * floating point Bessel's function of the 1st and 2nd kind
23 * of order n
24 *
25 * Special cases:
26 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
27 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
28 * Note 2. About jn(n,x), yn(n,x)
29 * For n=0, j0(x) is called,
30 * for n=1, j1(x) is called,
31 * for n<x, forward recursion us used starting
32 * from values of j0(x) and j1(x).
33 * for n>x, a continued fraction approximation to
34 * j(n,x)/j(n-1,x) is evaluated and then backward
35 * recursion is used starting from a supposed value
36 * for j(n,x). The resulting value of j(0,x) is
37 * compared with the actual value to correct the
38 * supposed value of j(n,x).
39 *
40 * yn(n,x) is similar in all respects, except
41 * that forward recursion is used for all
42 * values of n>1.
43 *
44 */
45
46#include "math.h"
47#include "math_private.h"
48
49static const double
50invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
51two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
52one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
53
54static const double zero = 0.00000000000000000000e+00;
55
56#define __ieee754_j0 j0
57#define __ieee754_j1 j1
58#define __ieee754_y0 y0
59#define __ieee754_y1 y1
60#define __ieee754_log log
61
62double jn(int n, double x)
63{
64 int32_t i,hx,ix,lx, sgn;
65 double a, b, temp, di;
66 double z, w;
67
68 temp = 0;
69 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
70 * Thus, J(-n,x) = J(n,-x)
71 */
72 EXTRACT_WORDS(hx,lx,x);
73 ix = 0x7fffffff&hx;
74 /* if J(n,NaN) is NaN */
75 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
76 if(n<0){
77 n = -n;
78 x = -x;
79 hx ^= 0x80000000;
80 }
81 if(n==0) return(__ieee754_j0(x));
82 if(n==1) return(__ieee754_j1(x));
83 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
84 x = fabs(x);
85 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
86 b = zero;
87 else if((double)n<=x) {
88 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
89 if(ix>=0x52D00000) { /* x > 2**302 */
90 /* (x >> n**2)
91 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
92 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
93 * Let s=sin(x), c=cos(x),
94 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
95 *
96 * n sin(xn)*sqt2 cos(xn)*sqt2
97 * ----------------------------------
98 * 0 s-c c+s
99 * 1 -s-c -c+s
100 * 2 -s+c -c-s
101 * 3 s+c c-s
102 */
103 switch(n&3) {
104 case 0: temp = cos(x)+sin(x); break;
105 case 1: temp = -cos(x)+sin(x); break;
106 case 2: temp = -cos(x)-sin(x); break;
107 case 3: temp = cos(x)-sin(x); break;
108 }
109 b = invsqrtpi*temp/sqrt(x);
110 } else {
111 a = __ieee754_j0(x);
112 b = __ieee754_j1(x);
113 for(i=1;i<n;i++){
114 temp = b;
115 b = b*((double)(i+i)/x) - a; /* avoid underflow */
116 a = temp;
117 }
118 }
119 } else {
120 if(ix<0x3e100000) { /* x < 2**-29 */
121 /* x is tiny, return the first Taylor expansion of J(n,x)
122 * J(n,x) = 1/n!*(x/2)^n - ...
123 */
124 if(n>33) /* underflow */
125 b = zero;
126 else {
127 temp = x*0.5; b = temp;
128 for (a=one,i=2;i<=n;i++) {
129 a *= (double)i; /* a = n! */
130 b *= temp; /* b = (x/2)^n */
131 }
132 b = b/a;
133 }
134 } else {
135 /* use backward recurrence */
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 quadruple
162 */
163 /* determine k */
164 double t,v;
165 double q0,q1,h,tmp; int32_t 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<1.0e9) {
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 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
179 * Hence, if n*(log(2n/x)) > ...
180 * single 8.8722839355e+01
181 * double 7.09782712893383973096e+02
182 * long double 1.1356523406294143949491931077970765006170e+04
183 * then recurrent value may overflow and the result is
184 * likely underflow to zero
185 */
186 tmp = n;
187 v = two/x;
188 tmp = tmp*__ieee754_log(fabs(v*tmp));
189 if(tmp<7.09782712893383973096e+02) {
190 for(i=n-1,di=(double)(i+i);i>0;i--){
191 temp = b;
192 b *= di;
193 b = b/x - a;
194 a = temp;
195 di -= two;
196 }
197 } else {
198 for(i=n-1,di=(double)(i+i);i>0;i--){
199 temp = b;
200 b *= di;
201 b = b/x - a;
202 a = temp;
203 di -= two;
204 /* scale b to avoid spurious overflow */
205 if(b>1e100) {
206 a /= b;
207 t /= b;
208 b = one;
209 }
210 }
211 }
212 b = (t*__ieee754_j0(x)/b);
213 }
214 }
215 if(sgn==1) return -b; else return b;
216}
217
218double yn(int n, double x)
219{
220 int32_t i,hx,ix,lx;
221 int32_t sign;
222 double a, b, temp;
223
224 temp = 0;
225 EXTRACT_WORDS(hx,lx,x);
226 ix = 0x7fffffff&hx;
227 /* if Y(n,NaN) is NaN */
228 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
229 if((ix|lx)==0) return -one/zero;
230 if(hx<0) return zero/zero;
231 sign = 1;
232 if(n<0){
233 n = -n;
234 sign = 1 - ((n&1)<<1);
235 }
236 if(n==0) return(__ieee754_y0(x));
237 if(n==1) return(sign*__ieee754_y1(x));
238 if(ix==0x7ff00000) return zero;
239 if(ix>=0x52D00000) { /* x > 2**302 */
240 /* (x >> n**2)
241 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
242 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
243 * Let s=sin(x), c=cos(x),
244 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
245 *
246 * n sin(xn)*sqt2 cos(xn)*sqt2
247 * ----------------------------------
248 * 0 s-c c+s
249 * 1 -s-c -c+s
250 * 2 -s+c -c-s
251 * 3 s+c c-s
252 */
253 switch(n&3) {
254 case 0: temp = sin(x)-cos(x); break;
255 case 1: temp = -sin(x)-cos(x); break;
256 case 2: temp = -sin(x)+cos(x); break;
257 case 3: temp = sin(x)+cos(x); break;
258 }
259 b = invsqrtpi*temp/sqrt(x);
260 } else {
261 u_int32_t high;
262 a = __ieee754_y0(x);
263 b = __ieee754_y1(x);
264 /* quit if b is -inf */
265 GET_HIGH_WORD(high,b);
266 for(i=1;i<n&&high!=0xfff00000;i++){
267 temp = b;
268 b = ((double)(i+i)/x)*b - a;
269 GET_HIGH_WORD(high,b);
270 a = temp;
271 }
272 }
273 if(sign>0) return b; else return -b;
274}