1 /* @(#)e_jn.c 5.1 93/09/24 */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
10 * ====================================================
14 static char rcsid[] = "$FreeBSD$";
18 * __ieee754_jn(n, x), __ieee754_yn(n, x)
19 * floating point Bessel's function of the 1st and 2nd kind
23 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
24 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
25 * Note 2. About jn(n,x), yn(n,x)
26 * For n=0, j0(x) is called,
27 * for n=1, j1(x) is called,
28 * for n<x, forward recursion us used starting
29 * from values of j0(x) and j1(x).
30 * for n>x, a continued fraction approximation to
31 * j(n,x)/j(n-1,x) is evaluated and then backward
32 * recursion is used starting from a supposed value
33 * for j(n,x). The resulting value of j(0,x) is
34 * compared with the actual value to correct the
35 * supposed value of j(n,x).
37 * yn(n,x) is similar in all respects, except
38 * that forward recursion is used for all
44 #include "math_private.h"
51 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
52 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
53 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
56 static const double zero = 0.00000000000000000000e+00;
58 static double zero = 0.00000000000000000000e+00;
62 double __ieee754_jn(int n, double x)
64 double __ieee754_jn(n,x)
68 int32_t i,hx,ix,lx, sgn;
69 double a, b, temp, di;
72 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
73 * Thus, J(-n,x) = J(n,-x)
75 EXTRACT_WORDS(hx,lx,x);
77 /* if J(n,NaN) is NaN */
78 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
84 if(n==0) return(__ieee754_j0(x));
85 if(n==1) return(__ieee754_j1(x));
86 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
88 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
90 else if((double)n<=x) {
91 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
92 if(ix>=0x52D00000) { /* x > 2**302 */
94 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
95 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
96 * Let s=sin(x), c=cos(x),
97 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
99 * n sin(xn)*sqt2 cos(xn)*sqt2
100 * ----------------------------------
107 case 0: temp = cos(x)+sin(x); break;
108 case 1: temp = -cos(x)+sin(x); break;
109 case 2: temp = -cos(x)-sin(x); break;
110 case 3: temp = cos(x)-sin(x); break;
112 b = invsqrtpi*temp/sqrt(x);
118 b = b*((double)(i+i)/x) - a; /* avoid underflow */
123 if(ix<0x3e100000) { /* x < 2**-29 */
124 /* x is tiny, return the first Taylor expansion of J(n,x)
125 * J(n,x) = 1/n!*(x/2)^n - ...
127 if(n>33) /* underflow */
130 temp = x*0.5; b = temp;
131 for (a=one,i=2;i<=n;i++) {
132 a *= (double)i; /* a = n! */
133 b *= temp; /* b = (x/2)^n */
138 /* use backward recurrence */
140 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
141 * 2n - 2(n+1) - 2(n+2)
144 * (for large x) = ---- ------ ------ .....
146 * -- - ------ - ------ -
149 * Let w = 2n/x and h=2/x, then the above quotient
150 * is equal to the continued fraction:
152 * = -----------------------
154 * w - -----------------
159 * To determine how many terms needed, let
160 * Q(0) = w, Q(1) = w(w+h) - 1,
161 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
162 * When Q(k) > 1e4 good for single
163 * When Q(k) > 1e9 good for double
164 * When Q(k) > 1e17 good for quadruple
168 double q0,q1,h,tmp; int32_t k,m;
169 w = (n+n)/(double)x; h = 2.0/(double)x;
170 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
178 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
181 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
182 * Hence, if n*(log(2n/x)) > ...
183 * single 8.8722839355e+01
184 * double 7.09782712893383973096e+02
185 * long double 1.1356523406294143949491931077970765006170e+04
186 * then recurrent value may overflow and the result is
187 * likely underflow to zero
191 tmp = tmp*__ieee754_log(fabs(v*tmp));
192 if(tmp<7.09782712893383973096e+02) {
193 for(i=n-1,di=(double)(i+i);i>0;i--){
201 for(i=n-1,di=(double)(i+i);i>0;i--){
207 /* scale b to avoid spurious overflow */
215 b = (t*__ieee754_j0(x)/b);
218 if(sgn==1) return -b; else return b;
222 double __ieee754_yn(int n, double x)
224 double __ieee754_yn(n,x)
232 EXTRACT_WORDS(hx,lx,x);
234 /* if Y(n,NaN) is NaN */
235 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
236 if((ix|lx)==0) return -one/zero;
237 if(hx<0) return zero/zero;
241 sign = 1 - ((n&1)<<1);
243 if(n==0) return(__ieee754_y0(x));
244 if(n==1) return(sign*__ieee754_y1(x));
245 if(ix==0x7ff00000) return zero;
246 if(ix>=0x52D00000) { /* x > 2**302 */
248 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
249 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
250 * Let s=sin(x), c=cos(x),
251 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
253 * n sin(xn)*sqt2 cos(xn)*sqt2
254 * ----------------------------------
261 case 0: temp = sin(x)-cos(x); break;
262 case 1: temp = -sin(x)-cos(x); break;
263 case 2: temp = -sin(x)+cos(x); break;
264 case 3: temp = sin(x)+cos(x); break;
266 b = invsqrtpi*temp/sqrt(x);
271 /* quit if b is -inf */
272 GET_HIGH_WORD(high,b);
273 for(i=1;i<n&&high!=0xfff00000;i++){
275 b = ((double)(i+i)/x)*b - a;
276 GET_HIGH_WORD(high,b);
280 if(sign>0) return b; else return -b;