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/* @(#)e_jn.c 5.1 93/09/24 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#ifndef lint
static char rcsid
[] = "$\Id: e_jn.c,v 1.3 1995/05/30 05:48:24 rgrimes Exp $";
#endif
/*
* __ieee754_jn(n, x), __ieee754_yn(n, x)
* floating point Bessel's function of the 1st and 2nd kind
* of order n
*
* Special cases:
* y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
* Note 2. About jn(n,x), yn(n,x)
* For n=0, j0(x) is called,
* for n=1, j1(x) is called,
* for n<x, forward recursion us used starting
* from values of j0(x) and j1(x).
* for n>x, a continued fraction approximation to
* j(n,x)/j(n-1,x) is evaluated and then backward
* recursion is used starting from a supposed value
* for j(n,x). The resulting value of j(0,x) is
* compared with the actual value to correct the
* supposed value of j(n,x).
*
* yn(n,x) is similar in all respects, except
* that forward recursion is used for all
* values of n>1.
*
*/
#include "math.h"
#include "math_private.h"
#ifdef __STDC__
static const double
#else
static double
#endif
invsqrtpi
= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
two
= 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
one
= 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
#ifdef __STDC__
static const double zero
= 0.00000000000000000000e+00;
#else
static double zero
= 0.00000000000000000000e+00;
#endif
#ifdef __STDC__
double __ieee754_jn
(int n
, double x
)
#else
double __ieee754_jn
(n
,x
)
int n
; double x
;
#endif
{
int32_t i
,hx
,ix
,lx
, sgn
;
double a
, b
, temp
, di
;
double z
, w
;
/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
* Thus, J(-n,x) = J(n,-x)
*/
EXTRACT_WORDS
(hx
,lx
,x
);
ix
= 0x7fffffff&hx
;
/* if J(n,NaN) is NaN */
if((ix
|((u_int32_t
)(lx
|-lx
))>>31)>0x7ff00000) return x
+x
;
if(n
<0){
n
= -n
;
x
= -x
;
hx
^= 0x80000000;
}
if(n
==0) return(__ieee754_j0
(x
));
if(n
==1) return(__ieee754_j1
(x
));
sgn
= (n
&1)&(hx
>>31); /* even n -- 0, odd n -- sign(x) */
x
= fabs(x
);
if((ix
|lx
)==0||ix
>=0x7ff00000) /* if x is 0 or inf */
b
= zero
;
else if((double)n
<=x
) {
/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
if(ix
>=0x52D00000) { /* x > 2**302 */
/* (x >> n**2)
* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Let s=sin(x), c=cos(x),
* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
*
* n sin(xn)*sqt2 cos(xn)*sqt2
* ----------------------------------
* 0 s-c c+s
* 1 -s-c -c+s
* 2 -s+c -c-s
* 3 s+c c-s
*/
switch(n
&3) {
case 0: temp
= cos(x
)+sin(x
); break;
case 1: temp
= -cos(x
)+sin(x
); break;
case 2: temp
= -cos(x
)-sin(x
); break;
case 3: temp
= cos(x
)-sin(x
); break;
}
b
= invsqrtpi
*temp
/sqrt(x
);
} else {
a
= __ieee754_j0
(x
);
b
= __ieee754_j1
(x
);
for(i
=1;i
<n
;i
++){
temp
= b
;
b
= b
*((double)(i
+i
)/x
) - a
; /* avoid underflow */
a
= temp
;
}
}
} else {
if(ix
<0x3e100000) { /* x < 2**-29 */
/* x is tiny, return the first Taylor expansion of J(n,x)
* J(n,x) = 1/n!*(x/2)^n - ...
*/
if(n
>33) /* underflow */
b
= zero
;
else {
temp
= x
*0.5; b
= temp
;
for (a
=one
,i
=2;i
<=n
;i
++) {
a
*= (double)i
; /* a = n! */
b
*= temp
; /* b = (x/2)^n */
}
b
= b
/a
;
}
} else {
/* use backward recurrence */
/* x x^2 x^2
* J(n,x)/J(n-1,x) = ---- ------ ------ .....
* 2n - 2(n+1) - 2(n+2)
*
* 1 1 1
* (for large x) = ---- ------ ------ .....
* 2n 2(n+1) 2(n+2)
* -- - ------ - ------ -
* x x x
*
* Let w = 2n/x and h=2/x, then the above quotient
* is equal to the continued fraction:
* 1
* = -----------------------
* 1
* w - -----------------
* 1
* w+h - ---------
* w+2h - ...
*
* To determine how many terms needed, let
* Q(0) = w, Q(1) = w(w+h) - 1,
* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
* When Q(k) > 1e4 good for single
* When Q(k) > 1e9 good for double
* When Q(k) > 1e17 good for quadruple
*/
/* determine k */
double t
,v
;
double q0
,q1
,h
,tmp
; int32_t k
,m
;
w
= (n
+n
)/(double)x
; h
= 2.0/(double)x
;
q0
= w
; z
= w
+h
; q1
= w
*z
- 1.0; k
=1;
while(q1
<1.0e9) {
k
+= 1; z
+= h
;
tmp
= z
*q1
- q0
;
q0
= q1
;
q1
= tmp
;
}
m
= n
+n
;
for(t
=zero
, i
= 2*(n
+k
); i
>=m
; i
-= 2) t
= one
/(i
/x
-t
);
a
= t
;
b
= one
;
/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
* Hence, if n*(log(2n/x)) > ...
* single 8.8722839355e+01
* double 7.09782712893383973096e+02
* long double 1.1356523406294143949491931077970765006170e+04
* then recurrent value may overflow and the result is
* likely underflow to zero
*/
tmp
= n
;
v
= two
/x
;
tmp
= tmp
*__ieee754_log
(fabs(v
*tmp
));
if(tmp
<7.09782712893383973096e+02) {
for(i
=n
-1,di
=(double)(i
+i
);i
>0;i
--){
temp
= b
;
b
*= di
;
b
= b
/x
- a
;
a
= temp
;
di
-= two
;
}
} else {
for(i
=n
-1,di
=(double)(i
+i
);i
>0;i
--){
temp
= b
;
b
*= di
;
b
= b
/x
- a
;
a
= temp
;
di
-= two
;
/* scale b to avoid spurious overflow */
if(b
>1e100) {
a
/= b
;
t
/= b
;
b
= one
;
}
}
}
b
= (t
*__ieee754_j0
(x
)/b
);
}
}
if(sgn
==1) return -b
; else return b
;
}
#ifdef __STDC__
double __ieee754_yn
(int n
, double x
)
#else
double __ieee754_yn
(n
,x
)
int n
; double x
;
#endif
{
int32_t i
,hx
,ix
,lx
;
int32_t sign
;
double a
, b
, temp
;
EXTRACT_WORDS
(hx
,lx
,x
);
ix
= 0x7fffffff&hx
;
/* if Y(n,NaN) is NaN */
if((ix
|((u_int32_t
)(lx
|-lx
))>>31)>0x7ff00000) return x
+x
;
if((ix
|lx
)==0) return -one
/zero
;
if(hx
<0) return zero
/zero
;
sign
= 1;
if(n
<0){
n
= -n
;
sign
= 1 - ((n
&1)<<1);
}
if(n
==0) return(__ieee754_y0
(x
));
if(n
==1) return(sign
*__ieee754_y1
(x
));
if(ix
==0x7ff00000) return zero
;
if(ix
>=0x52D00000) { /* x > 2**302 */
/* (x >> n**2)
* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Let s=sin(x), c=cos(x),
* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
*
* n sin(xn)*sqt2 cos(xn)*sqt2
* ----------------------------------
* 0 s-c c+s
* 1 -s-c -c+s
* 2 -s+c -c-s
* 3 s+c c-s
*/
switch(n
&3) {
case 0: temp
= sin(x
)-cos(x
); break;
case 1: temp
= -sin(x
)-cos(x
); break;
case 2: temp
= -sin(x
)+cos(x
); break;
case 3: temp
= sin(x
)+cos(x
); break;
}
b
= invsqrtpi
*temp
/sqrt(x
);
} else {
u_int32_t high
;
a
= __ieee754_y0
(x
);
b
= __ieee754_y1
(x
);
/* quit if b is -inf */
GET_HIGH_WORD
(high
,b
);
for(i
=1;i
<n
&&high
!=0xfff00000;i
++){
temp
= b
;
b
= ((double)(i
+i
)/x
)*b
- a
;
GET_HIGH_WORD
(high
,b
);
a
= temp
;
}
}
if(sign
>0) return b
; else return -b
;
}