/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
#pragma weak __jn = jn
#pragma weak __yn = yn
/*
* floating point Bessel's function of the 1st and 2nd kind
* of order n: jn(n,x),yn(n,x);
*
* 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 "libm.h"
#include <float.h> /* DBL_MIN */
#include <values.h> /* X_TLOSS */
#include "xpg6.h" /* __xpg6 */
#define GENERIC double
static const GENERIC
invsqrtpi = 5.641895835477562869480794515607725858441e-0001,
two = 2.0,
zero = 0.0,
one = 1.0;
GENERIC
jn(int n, GENERIC x) {
int i, sgn;
GENERIC a, b, temp = 0;
GENERIC z, w, ox, on;
/*
* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
* Thus, J(-n,x) = J(n,-x)
*/
ox = x; on = (GENERIC)n;
if (n < 0) {
n = -n;
x = -x;
}
if (isnan(x))
return (x*x); /* + -> * for Cheetah */
if (!((int) _lib_version == libm_ieee ||
(__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
if (fabs(x) > X_TLOSS)
return (_SVID_libm_err(on, ox, 38));
}
if (n == 0)
return (j0(x));
if (n == 1)
return (j1(x));
if ((n&1) == 0)
sgn = 0; /* even n */
else
sgn = signbit(x); /* old n */
x = fabs(x);
if (x == zero||!finite(x)) b = zero;
else if ((GENERIC)n <= x) {
/*
* Safe to use
* J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
*/
if (x > 1.0e91) {
/*
* 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*((GENERIC)(i+i)/x) - a; /* avoid underflow */
a = temp;
}
}
} else {
if (x < 1e-9) { /* use J(n,x) = 1/n!*(x/2)^n */
b = pow(0.5*x, (GENERIC) n);
if (b != zero) {
for (a = one, i = 1; i <= n; i++) a *= (GENERIC)i;
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 quaduple
*/
/* determin k */
GENERIC 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 is
* likely underflow to zero
*/
tmp = n;
v = two/x;
tmp = tmp*log(fabs(v*tmp));
if (tmp < 7.09782712893383973096e+02) {
for (i = n-1; i > 0; i--) {
temp = b;
b = ((i+i)/x)*b - a;
a = temp;
}
} else {
for (i = n-1; i > 0; i--) {
temp = b;
b = ((i+i)/x)*b - a;
a = temp;
if (b > 1e100) {
a /= b;
t /= b;
b = 1.0;
}
}
}
b = (t*j0(x)/b);
}
}
if (sgn == 1)
return (-b);
else
return (b);
}
GENERIC
yn(int n, GENERIC x) {
int i;
int sign;
GENERIC a, b, temp = 0, ox, on;
ox = x; on = (GENERIC)n;
if (isnan(x))
return (x*x); /* + -> * for Cheetah */
if (x <= zero) {
if (x == zero) {
/* return -one/zero; */
return (_SVID_libm_err((GENERIC)n, x, 12));
} else {
/* return zero/zero; */
return (_SVID_libm_err((GENERIC)n, x, 13));
}
}
if (!((int) _lib_version == libm_ieee ||
(__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
if (x > X_TLOSS)
return (_SVID_libm_err(on, ox, 39));
}
sign = 1;
if (n < 0) {
n = -n;
if ((n&1) == 1) sign = -1;
}
if (n == 0)
return (y0(x));
if (n == 1)
return (sign*y1(x));
if (!finite(x))
return (zero);
if (x > 1.0e91) {
/*
* 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);
/*
* fix 1262058 and take care of non-default rounding
*/
for (i = 1; i < n; i++) {
temp = b;
b *= (GENERIC) (i + i) / x;
if (b <= -DBL_MAX)
break;
b -= a;
a = temp;
}
}
if (sign > 0)
return (b);
else
return (-b);
}