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 #include <sys/cdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: e_jn.c,v 1.14 2010/11/29 15:10:06 drochner Exp $");
16 #endif
17
18 /*
19 * __ieee754_jn(n, x), __ieee754_yn(n, x)
20 * floating point Bessel's function of the 1st and 2nd kind
21 * of order n
22 *
23 * Special cases:
24 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
25 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
26 * Note 2. About jn(n,x), yn(n,x)
27 * For n=0, j0(x) is called,
28 * for n=1, j1(x) is called,
29 * for n<x, forward recursion us used starting
30 * from values of j0(x) and j1(x).
31 * for n>x, a continued fraction approximation to
32 * j(n,x)/j(n-1,x) is evaluated and then backward
33 * recursion is used starting from a supposed value
34 * for j(n,x). The resulting value of j(0,x) is
35 * compared with the actual value to correct the
36 * supposed value of j(n,x).
37 *
38 * yn(n,x) is similar in all respects, except
39 * that forward recursion is used for all
40 * values of n>1.
41 *
42 */
43
44 #include "namespace.h"
45 #include "math.h"
46 #include "math_private.h"
47
48 static const double
49 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
50 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
51 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
52
53 static const double zero = 0.00000000000000000000e+00;
54
55 double
__ieee754_jn(int n,double x)56 __ieee754_jn(int n, double x)
57 {
58 int32_t i,hx,ix,lx, sgn;
59 double a, b, temp, di;
60 double z, w;
61
62 temp = 0;
63 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
64 * Thus, J(-n,x) = J(n,-x)
65 */
66 EXTRACT_WORDS(hx,lx,x);
67 ix = 0x7fffffff&hx;
68 /* if J(n,NaN) is NaN */
69 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
70 if(n<0){
71 n = -n;
72 x = -x;
73 hx ^= 0x80000000;
74 }
75 if(n==0) return(__ieee754_j0(x));
76 if(n==1) return(__ieee754_j1(x));
77 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
78 x = fabs(x);
79 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
80 b = zero;
81 else if((double)n<=x) {
82 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
83 if(ix>=0x52D00000) { /* x > 2**302 */
84 /* (x >> n**2)
85 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
86 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
87 * Let s=sin(x), c=cos(x),
88 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
89 *
90 * n sin(xn)*sqt2 cos(xn)*sqt2
91 * ----------------------------------
92 * 0 s-c c+s
93 * 1 -s-c -c+s
94 * 2 -s+c -c-s
95 * 3 s+c c-s
96 */
97 switch(n&3) {
98 case 0: temp = cos(x)+sin(x); break;
99 case 1: temp = -cos(x)+sin(x); break;
100 case 2: temp = -cos(x)-sin(x); break;
101 case 3: temp = cos(x)-sin(x); break;
102 }
103 b = invsqrtpi*temp/sqrt(x);
104 } else {
105 a = __ieee754_j0(x);
106 b = __ieee754_j1(x);
107 for(i=1;i<n;i++){
108 temp = b;
109 b = b*((double)(i+i)/x) - a; /* avoid underflow */
110 a = temp;
111 }
112 }
113 } else {
114 if(ix<0x3e100000) { /* x < 2**-29 */
115 /* x is tiny, return the first Taylor expansion of J(n,x)
116 * J(n,x) = 1/n!*(x/2)^n - ...
117 */
118 if(n>33) /* underflow */
119 b = zero;
120 else {
121 temp = x*0.5; b = temp;
122 for (a=one,i=2;i<=n;i++) {
123 a *= (double)i; /* a = n! */
124 b *= temp; /* b = (x/2)^n */
125 }
126 b = b/a;
127 }
128 } else {
129 /* use backward recurrence */
130 /* x x^2 x^2
131 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
132 * 2n - 2(n+1) - 2(n+2)
133 *
134 * 1 1 1
135 * (for large x) = ---- ------ ------ .....
136 * 2n 2(n+1) 2(n+2)
137 * -- - ------ - ------ -
138 * x x x
139 *
140 * Let w = 2n/x and h=2/x, then the above quotient
141 * is equal to the continued fraction:
142 * 1
143 * = -----------------------
144 * 1
145 * w - -----------------
146 * 1
147 * w+h - ---------
148 * w+2h - ...
149 *
150 * To determine how many terms needed, let
151 * Q(0) = w, Q(1) = w(w+h) - 1,
152 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
153 * When Q(k) > 1e4 good for single
154 * When Q(k) > 1e9 good for double
155 * When Q(k) > 1e17 good for quadruple
156 */
157 /* determine k */
158 double t,v;
159 double q0,q1,h,tmp; int32_t k,m;
160 w = (n+n)/(double)x; h = 2.0/(double)x;
161 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
162 while(q1<1.0e9) {
163 k += 1; z += h;
164 tmp = z*q1 - q0;
165 q0 = q1;
166 q1 = tmp;
167 }
168 m = n+n;
169 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
170 a = t;
171 b = one;
172 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
173 * Hence, if n*(log(2n/x)) > ...
174 * single 8.8722839355e+01
175 * double 7.09782712893383973096e+02
176 * long double 1.1356523406294143949491931077970765006170e+04
177 * then recurrent value may overflow and the result is
178 * likely underflow to zero
179 */
180 tmp = n;
181 v = two/x;
182 tmp = tmp*__ieee754_log(fabs(v*tmp));
183 if(tmp<7.09782712893383973096e+02) {
184 for(i=n-1,di=(double)(i+i);i>0;i--){
185 temp = b;
186 b *= di;
187 b = b/x - a;
188 a = temp;
189 di -= two;
190 }
191 } else {
192 for(i=n-1,di=(double)(i+i);i>0;i--){
193 temp = b;
194 b *= di;
195 b = b/x - a;
196 a = temp;
197 di -= two;
198 /* scale b to avoid spurious overflow */
199 if(b>1e100) {
200 a /= b;
201 t /= b;
202 b = one;
203 }
204 }
205 }
206 z = __ieee754_j0(x);
207 w = __ieee754_j1(x);
208 if (fabs(z) >= fabs(w))
209 b = (t*z/b);
210 else
211 b = (t*w/a);
212 }
213 }
214 if(sgn==1) return -b; else return b;
215 }
216
217 double
__ieee754_yn(int n,double x)218 __ieee754_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 }
275