/*
Copyright © 1999 CERN - European Organization for Nuclear Research.
Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose
is hereby granted without fee, provided that the above copyright notice appear in all copies and
that both that copyright notice and this permission notice appear in supporting documentation.
CERN makes no representations about the suitability of this software for any purpose.
It is provided "as is" without expressed or implied warranty.
*/
package tr.gov.ulakbim.jDenetX.clusterers.clustree.util;
/**
* Gamma and Beta functions.
* <p/>
* <b>Implementation:</b>
* <dt>
* Some code taken and adapted from the <A HREF="http://www.sci.usq.edu.au/staff/leighb/graph/Top.html">Java 2D Graph Package 2.4</A>,
* which in turn is a port from the <A HREF="http://people.ne.mediaone.net/moshier/index.html#Cephes">Cephes 2.2</A> Math Library (C).
* Most Cephes code (missing from the 2D Graph Package) directly ported.
*
* @author wolfgang.hoschek@cern.ch
* @version 0.9, 22-Jun-99
*/
public class Gamma {
protected static final double MACHEP = 1.11022302462515654042E-16;
protected static final double MAXLOG = 7.09782712893383996732E2;
protected static final double MINLOG = -7.451332191019412076235E2;
protected static final double MAXGAM = 171.624376956302725;
protected static final double SQTPI = 2.50662827463100050242E0;
protected static final double SQRTH = 7.07106781186547524401E-1;
protected static final double LOGPI = 1.14472988584940017414;
protected static final double big = 4.503599627370496e15;
protected static final double biginv = 2.22044604925031308085e-16;
/**
* Makes this class non instantiable, but still let's others inherit from it.
*/
protected Gamma() {
}
/**
* Continued fraction expansion #1 for incomplete beta integral; formerly named <tt>incbcf</tt>.
*/
static double incompleteBetaFraction1(double a, double b, double x) throws ArithmeticException {
double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
double k1, k2, k3, k4, k5, k6, k7, k8;
double r, t, ans, thresh;
int n;
k1 = a;
k2 = a + b;
k3 = a;
k4 = a + 1.0;
k5 = 1.0;
k6 = b - 1.0;
k7 = k4;
k8 = a + 2.0;
pkm2 = 0.0;
qkm2 = 1.0;
pkm1 = 1.0;
qkm1 = 1.0;
ans = 1.0;
r = 1.0;
n = 0;
thresh = 3.0 * MACHEP;
do {
xk = -(x * k1 * k2) / (k3 * k4);
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
xk = (x * k5 * k6) / (k7 * k8);
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if (qk != 0) {
r = pk / qk;
}
if (r != 0) {
t = Math.abs((ans - r) / r);
ans = r;
} else {
t = 1.0;
}
if (t < thresh) {
return ans;
}
k1 += 1.0;
k2 += 1.0;
k3 += 2.0;
k4 += 2.0;
k5 += 1.0;
k6 -= 1.0;
k7 += 2.0;
k8 += 2.0;
if ((Math.abs(qk) + Math.abs(pk)) > big) {
pkm2 *= biginv;
pkm1 *= biginv;
qkm2 *= biginv;
qkm1 *= biginv;
}
if ((Math.abs(qk) < biginv) || (Math.abs(pk) < biginv)) {
pkm2 *= big;
pkm1 *= big;
qkm2 *= big;
qkm1 *= big;
}
} while (++n < 300);
return ans;
}
/**
* Continued fraction expansion #2 for incomplete beta integral; formerly named <tt>incbd</tt>.
*/
static double incompleteBetaFraction2(double a, double b, double x) throws ArithmeticException {
double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
double k1, k2, k3, k4, k5, k6, k7, k8;
double r, t, ans, z, thresh;
int n;
k1 = a;
k2 = b - 1.0;
k3 = a;
k4 = a + 1.0;
k5 = 1.0;
k6 = a + b;
k7 = a + 1.0;
k8 = a + 2.0;
pkm2 = 0.0;
qkm2 = 1.0;
pkm1 = 1.0;
qkm1 = 1.0;
z = x / (1.0 - x);
ans = 1.0;
r = 1.0;
n = 0;
thresh = 3.0 * MACHEP;
do {
xk = -(z * k1 * k2) / (k3 * k4);
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
xk = (z * k5 * k6) / (k7 * k8);
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if (qk != 0) {
r = pk / qk;
}
if (r != 0) {
t = Math.abs((ans - r) / r);
ans = r;
} else {
t = 1.0;
}
if (t < thresh) {
return ans;
}
k1 += 1.0;
k2 -= 1.0;
k3 += 2.0;
k4 += 2.0;
k5 += 1.0;
k6 += 1.0;
k7 += 2.0;
k8 += 2.0;
if ((Math.abs(qk) + Math.abs(pk)) > big) {
pkm2 *= biginv;
pkm1 *= biginv;
qkm2 *= biginv;
qkm1 *= biginv;
}
if ((Math.abs(qk) < biginv) || (Math.abs(pk) < biginv)) {
pkm2 *= big;
pkm1 *= big;
qkm2 *= big;
qkm1 *= big;
}
} while (++n < 300);
return ans;
}
/**
* Returns the Incomplete Gamma function; formerly named <tt>igamma</tt>.
*
* @param a the parameter of the gamma distribution.
* @param x the integration end point.
*/
static public double incompleteGamma(double a, double x)
throws ArithmeticException {
double ans, ax, c, r;
if (x <= 0 || a <= 0) {
return 0.0;
}
if (x > 1.0 && x > a) {
return 1.0 - incompleteGammaComplement(a, x);
}
/* Compute x**a * exp(-x) / gamma(a) */
ax = a * Math.log(x) - x - logGamma(a);
if (ax < -MAXLOG) {
return (0.0);
}
ax = Math.exp(ax);
/* power series */
r = a;
c = 1.0;
ans = 1.0;
do {
r += 1.0;
c *= x / r;
ans += c;
} while (c / ans > MACHEP);
return (ans * ax / a);
}
/**
* Returns the Complemented Incomplete Gamma function; formerly named <tt>igamc</tt>.
*
* @param a the parameter of the gamma distribution.
* @param x the integration start point.
*/
static public double incompleteGammaComplement(double a, double x) throws ArithmeticException {
double ans, ax, c, yc, r, t, y, z;
double pk, pkm1, pkm2, qk, qkm1, qkm2;
if (x <= 0 || a <= 0) {
return 1.0;
}
if (x < 1.0 || x < a) {
return 1.0 - incompleteGamma(a, x);
}
ax = a * Math.log(x) - x - logGamma(a);
if (ax < -MAXLOG) {
return 0.0;
}
ax = Math.exp(ax);
/* continued fraction */
y = 1.0 - a;
z = x + y + 1.0;
c = 0.0;
pkm2 = 1.0;
qkm2 = x;
pkm1 = x + 1.0;
qkm1 = z * x;
ans = pkm1 / qkm1;
do {
c += 1.0;
y += 1.0;
z += 2.0;
yc = y * c;
pk = pkm1 * z - pkm2 * yc;
qk = qkm1 * z - qkm2 * yc;
if (qk != 0) {
r = pk / qk;
t = Math.abs((ans - r) / r);
ans = r;
} else {
t = 1.0;
}
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if (Math.abs(pk) > big) {
pkm2 *= biginv;
pkm1 *= biginv;
qkm2 *= biginv;
qkm1 *= biginv;
}
} while (t > MACHEP);
return ans * ax;
}
/**
* Returns the natural logarithm of the gamma function; formerly named <tt>lgamma</tt>.
*/
public static double logGamma(double x) throws ArithmeticException {
double p, q, w, z;
double A[] = {
8.11614167470508450300E-4,
-5.95061904284301438324E-4,
7.93650340457716943945E-4,
-2.77777777730099687205E-3,
8.33333333333331927722E-2
};
double B[] = {
-1.37825152569120859100E3,
-3.88016315134637840924E4,
-3.31612992738871184744E5,
-1.16237097492762307383E6,
-1.72173700820839662146E6,
-8.53555664245765465627E5
};
double C[] = {
/* 1.00000000000000000000E0, */
-3.51815701436523470549E2,
-1.70642106651881159223E4,
-2.20528590553854454839E5,
-1.13933444367982507207E6,
-2.53252307177582951285E6,
-2.01889141433532773231E6
};
if (x < -34.0) {
q = -x;
w = logGamma(q);
p = Math.floor(q);
if (p == q) {
throw new ArithmeticException("lgam: Overflow");
}
z = q - p;
if (z > 0.5) {
p += 1.0;
z = p - q;
}
z = q * Math.sin(Math.PI * z);
if (z == 0.0) {
throw new ArithmeticException("lgamma: Overflow");
}
z = LOGPI - Math.log(z) - w;
return z;
}
if (x < 13.0) {
z = 1.0;
while (x >= 3.0) {
x -= 1.0;
z *= x;
}
while (x < 2.0) {
if (x == 0.0) {
throw new ArithmeticException("lgamma: Overflow");
}
z /= x;
x += 1.0;
}
if (z < 0.0) {
z = -z;
}
if (x == 2.0) {
return Math.log(z);
}
x -= 2.0;
p = x * Polynomial.polevl(x, B, 5) / Polynomial.p1evl(x, C, 6);
return (Math.log(z) + p);
}
if (x > 2.556348e305) {
throw new ArithmeticException("lgamma: Overflow");
}
q = (x - 0.5) * Math.log(x) - x + 0.91893853320467274178;
//if( x > 1.0e8 ) return( q );
if (x > 1.0e8) {
return (q);
}
p = 1.0 / (x * x);
if (x >= 1000.0) {
q += ((7.9365079365079365079365e-4 * p
- 2.7777777777777777777778e-3) * p
+ 0.0833333333333333333333) / x;
} else {
q += Polynomial.polevl(p, A, 4) / x;
}
return q;
}
}