267 lines
7.0 KiB
C
267 lines
7.0 KiB
C
|
|
/* @(#)e_jn.c 1.4 95/01/18 */
|
|
/*
|
|
* ====================================================
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* Developed at SunSoft, 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[] = "$FreeBSD: src/lib/msun/src/e_jn.c,v 1.9 2005/02/04 18:26:06 das 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"
|
|
|
|
static const double
|
|
invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
|
|
two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
|
|
one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
|
|
|
|
static const double zero = 0.00000000000000000000e+00;
|
|
|
|
double
|
|
__ieee754_jn(int n, double x)
|
|
{
|
|
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;
|
|
}
|
|
|
|
double
|
|
__ieee754_yn(int n, double x)
|
|
{
|
|
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;
|
|
}
|