2 /* @(#)e_jn.c 1.4 95/01/18 */
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
11 * ====================================================
15 static char rcsid
[] = "$FreeBSD: src/lib/msun/src/e_jn.c,v 1.11 2010/11/13 10:54:10 uqs Exp $";
19 * __ieee754_jn(n, x), __ieee754_yn(n, x)
20 * floating point Bessel's function of the 1st and 2nd kind
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).
38 * yn(n,x) is similar in all respects, except
39 * that forward recursion is used for all
45 #include "math_private.h"
47 static const volatile double vone
= 1, vzero
= 0;
50 invsqrtpi
= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
51 two
= 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
52 one
= 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
54 static const double zero
= 0.00000000000000000000e+00;
57 __ieee754_jn(int n
, double x
)
59 int32_t i
,hx
,ix
,lx
, sgn
;
60 double a
, b
, temp
, di
;
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)
66 EXTRACT_WORDS(hx
,lx
,x
);
68 /* if J(n,NaN) is NaN */
69 if((ix
|((uint32_t)(lx
|-lx
))>>31)>0x7ff00000) return x
+x
;
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) */
79 if((ix
|lx
)==0||ix
>=0x7ff00000) /* if x is 0 or inf */
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 */
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
90 * n sin(xn)*sqt2 cos(xn)*sqt2
91 * ----------------------------------
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;
103 b
= invsqrtpi
*temp
/sqrt(x
);
109 b
= b
*((double)(i
+i
)/x
) - a
; /* avoid underflow */
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 - ...
118 if(n
>33) /* underflow */
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 */
129 /* use backward recurrence */
131 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
132 * 2n - 2(n+1) - 2(n+2)
135 * (for large x) = ---- ------ ------ .....
137 * -- - ------ - ------ -
140 * Let w = 2n/x and h=2/x, then the above quotient
141 * is equal to the continued fraction:
143 * = -----------------------
145 * w - -----------------
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
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;
169 for(t
=zero
, i
= 2*(n
+k
); i
>=m
; i
-= 2) t
= one
/(i
/x
-t
);
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
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
--){
192 for(i
=n
-1,di
=(double)(i
+i
);i
>0;i
--){
198 /* scale b to avoid spurious overflow */
208 if (fabs(z
) >= fabs(w
))
214 if(sgn
==1) return -b
; else return b
;
218 __ieee754_yn(int n
, double x
)
224 EXTRACT_WORDS(hx
,lx
,x
);
226 /* yn(n,NaN) = NaN */
227 if((ix
|((uint32_t)(lx
|-lx
))>>31)>0x7ff00000) return x
+x
;
228 /* yn(n,+-0) = -inf and raise divide-by-zero exception. */
229 if((ix
|lx
)==0) return -one
/vzero
;
230 /* yn(n,x<0) = NaN and raise invalid exception. */
231 if(hx
<0) return vzero
/vzero
;
235 sign
= 1 - ((n
&1)<<1);
237 if(n
==0) return(__ieee754_y0(x
));
238 if(n
==1) return(sign
*__ieee754_y1(x
));
239 if(ix
==0x7ff00000) return zero
;
240 if(ix
>=0x52D00000) { /* x > 2**302 */
242 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
243 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
244 * Let s=sin(x), c=cos(x),
245 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
247 * n sin(xn)*sqt2 cos(xn)*sqt2
248 * ----------------------------------
255 case 0: temp
= sin(x
)-cos(x
); break;
256 case 1: temp
= -sin(x
)-cos(x
); break;
257 case 2: temp
= -sin(x
)+cos(x
); break;
258 case 3: temp
= sin(x
)+cos(x
); break;
260 b
= invsqrtpi
*temp
/sqrt(x
);
265 /* quit if b is -inf */
266 GET_HIGH_WORD(high
,b
);
267 for(i
=1;i
<n
&&high
!=0xfff00000;i
++){
269 b
= ((double)(i
+i
)/x
)*b
- a
;
270 GET_HIGH_WORD(high
,b
);
274 if(sign
>0) return b
; else return -b
;