315 lines
9.0 KiB
C
315 lines
9.0 KiB
C
/*-
|
|
* Copyright (c) 1992, 1993
|
|
* The Regents of the University of California. All rights reserved.
|
|
*
|
|
* Redistribution and use in source and binary forms, with or without
|
|
* modification, are permitted provided that the following conditions
|
|
* are met:
|
|
* 1. Redistributions of source code must retain the above copyright
|
|
* notice, this list of conditions and the following disclaimer.
|
|
* 2. Redistributions in binary form must reproduce the above copyright
|
|
* notice, this list of conditions and the following disclaimer in the
|
|
* documentation and/or other materials provided with the distribution.
|
|
* 3. All advertising materials mentioning features or use of this software
|
|
* must display the following acknowledgement:
|
|
* This product includes software developed by the University of
|
|
* California, Berkeley and its contributors.
|
|
* 4. Neither the name of the University nor the names of its contributors
|
|
* may be used to endorse or promote products derived from this software
|
|
* without specific prior written permission.
|
|
*
|
|
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
|
|
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
|
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
|
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
|
|
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
|
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
|
|
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
|
|
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
|
|
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
|
|
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
|
|
* SUCH DAMAGE.
|
|
*/
|
|
|
|
#include <sys/cdefs.h>
|
|
__FBSDID("$FreeBSD$");
|
|
|
|
#ifndef lint
|
|
static char sccsid[] = "@(#)jn.c 8.2 (Berkeley) 11/30/93";
|
|
#endif /* not lint */
|
|
|
|
/*
|
|
* 16 December 1992
|
|
* Minor modifications by Peter McIlroy to adapt non-IEEE architecture.
|
|
*/
|
|
|
|
/*
|
|
* ====================================================
|
|
* Copyright (C) 1992 by Sun Microsystems, Inc.
|
|
*
|
|
* 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.
|
|
* ====================================================
|
|
*
|
|
* ******************* WARNING ********************
|
|
* This is an alpha version of SunPro's FDLIBM (Freely
|
|
* Distributable Math Library) for IEEE double precision
|
|
* arithmetic. FDLIBM is a basic math library written
|
|
* in C that runs on machines that conform to IEEE
|
|
* Standard 754/854. This alpha version is distributed
|
|
* for testing purpose. Those who use this software
|
|
* should report any bugs to
|
|
*
|
|
* fdlibm-comments@sunpro.eng.sun.com
|
|
*
|
|
* -- K.C. Ng, Oct 12, 1992
|
|
* ************************************************
|
|
*/
|
|
|
|
/*
|
|
* jn(int n, double x), yn(int n, double 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 <float.h>
|
|
#include <errno.h>
|
|
|
|
#if defined(vax) || defined(tahoe)
|
|
#define _IEEE 0
|
|
#else
|
|
#define _IEEE 1
|
|
#define infnan(x) (0.0)
|
|
#endif
|
|
|
|
static double
|
|
invsqrtpi= 5.641895835477562869480794515607725858441e-0001,
|
|
two = 2.0,
|
|
zero = 0.0,
|
|
one = 1.0;
|
|
|
|
double jn(n,x)
|
|
int n; double x;
|
|
{
|
|
int i, sgn;
|
|
double a, b, temp;
|
|
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)
|
|
*/
|
|
/* if J(n,NaN) is NaN */
|
|
if (_IEEE && isnan(x)) return x+x;
|
|
if (n<0){
|
|
n = -n;
|
|
x = -x;
|
|
}
|
|
if (n==0) return(j0(x));
|
|
if (n==1) return(j1(x));
|
|
sgn = (n&1)&(x < zero); /* even n -- 0, odd n -- sign(x) */
|
|
x = fabs(x);
|
|
if (x == 0 || !finite (x)) /* 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 (_IEEE && x >= 8.148143905337944345e+090) {
|
|
/* 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 = j0(x);
|
|
b = j1(x);
|
|
for(i=1;i<n;i++){
|
|
temp = b;
|
|
b = b*((double)(i+i)/x) - a; /* avoid underflow */
|
|
a = temp;
|
|
}
|
|
}
|
|
} else {
|
|
if (x < 1.86264514923095703125e-009) { /* 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; int 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 will
|
|
* likely underflow to zero
|
|
*/
|
|
tmp = n;
|
|
v = two/x;
|
|
tmp = tmp*log(fabs(v*tmp));
|
|
for (i=n-1;i>0;i--){
|
|
temp = b;
|
|
b = ((i+i)/x)*b - a;
|
|
a = temp;
|
|
/* scale b to avoid spurious overflow */
|
|
# if defined(vax) || defined(tahoe)
|
|
# define BMAX 1e13
|
|
# else
|
|
# define BMAX 1e100
|
|
# endif /* defined(vax) || defined(tahoe) */
|
|
if (b > BMAX) {
|
|
a /= b;
|
|
t /= b;
|
|
b = one;
|
|
}
|
|
}
|
|
b = (t*j0(x)/b);
|
|
}
|
|
}
|
|
return ((sgn == 1) ? -b : b);
|
|
}
|
|
double yn(n,x)
|
|
int n; double x;
|
|
{
|
|
int i, sign;
|
|
double a, b, temp;
|
|
|
|
/* Y(n,NaN), Y(n, x < 0) is NaN */
|
|
if (x <= 0 || (_IEEE && x != x))
|
|
if (_IEEE && x < 0) return zero/zero;
|
|
else if (x < 0) return (infnan(EDOM));
|
|
else if (_IEEE) return -one/zero;
|
|
else return(infnan(-ERANGE));
|
|
else if (!finite(x)) return(0);
|
|
sign = 1;
|
|
if (n<0){
|
|
n = -n;
|
|
sign = 1 - ((n&1)<<2);
|
|
}
|
|
if (n == 0) return(y0(x));
|
|
if (n == 1) return(sign*y1(x));
|
|
if(_IEEE && x >= 8.148143905337944345e+090) { /* 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 {
|
|
a = y0(x);
|
|
b = y1(x);
|
|
/* quit if b is -inf */
|
|
for (i = 1; i < n && !finite(b); i++){
|
|
temp = b;
|
|
b = ((double)(i+i)/x)*b - a;
|
|
a = temp;
|
|
}
|
|
}
|
|
if (!_IEEE && !finite(b))
|
|
return (infnan(-sign * ERANGE));
|
|
return ((sign > 0) ? b : -b);
|
|
}
|