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-/*-
- * 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);
-}
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