/*
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 cern.jet.stat;
import cern.jet.math.Polynomial;
/**
* 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 extends cern.jet.math.Constants {
/**
* Makes this class non instantiable, but still let's others inherit from it.
*/
protected Gamma() {}
/**
* Returns the beta function of the arguments.
* <pre>
* - -
* | (a) | (b)
* beta( a, b ) = -----------.
* -
* | (a+b)
* </pre>
*/
static public double beta(double a, double b) throws ArithmeticException {
double y;
y = a + b;
y = gamma(y);
if( y == 0.0 ) return 1.0;
if( a > b ) {
y = gamma(a)/y;
y *= gamma(b);
}
else {
y = gamma(b)/y;
y *= gamma(a);
}
return(y);
}
/**
* Returns the Gamma function of the argument.
*/
static public double gamma(double x) throws ArithmeticException {
double P[] = {
1.60119522476751861407E-4,
1.19135147006586384913E-3,
1.04213797561761569935E-2,
4.76367800457137231464E-2,
2.07448227648435975150E-1,
4.94214826801497100753E-1,
9.99999999999999996796E-1
};
double Q[] = {
-2.31581873324120129819E-5,
5.39605580493303397842E-4,
-4.45641913851797240494E-3,
1.18139785222060435552E-2,
3.58236398605498653373E-2,
-2.34591795718243348568E-1,
7.14304917030273074085E-2,
1.00000000000000000320E0
};
//double MAXGAM = 171.624376956302725;
//double LOGPI = 1.14472988584940017414;
double p, z;
int i;
double q = Math.abs(x);
if( q > 33.0 ) {
if( x < 0.0 ) {
p = Math.floor(q);
if( p == q ) throw new ArithmeticException("gamma: overflow");
i = (int)p;
z = q - p;
if( z > 0.5 ) {
p += 1.0;
z = q - p;
}
z = q * Math.sin( Math.PI * z );
if( z == 0.0 ) throw new ArithmeticException("gamma: overflow");
z = Math.abs(z);
z = Math.PI/(z * stirlingFormula(q) );
return -z;
} else {
return stirlingFormula(x);
}
}
z = 1.0;
while( x >= 3.0 ) {
x -= 1.0;
z *= x;
}
while( x < 0.0 ) {
if( x == 0.0 ) {
throw new ArithmeticException("gamma: singular");
} else
if( x > -1.E-9 ) {
return( z/((1.0 + 0.5772156649015329 * x) * x) );
}
z /= x;
x += 1.0;
}
while( x < 2.0 ) {
if( x == 0.0 ) {
throw new ArithmeticException("gamma: singular");
} else
if( x < 1.e-9 ) {
return( z/((1.0 + 0.5772156649015329 * x) * x) );
}
z /= x;
x += 1.0;
}
if( (x == 2.0) || (x == 3.0) ) return z;
x -= 2.0;
p = Polynomial.polevl( x, P, 6 );
q = Polynomial.polevl( x, Q, 7 );
return z * p / q;
}
/**
* Returns the Incomplete Beta Function evaluated from zero to <tt>xx</tt>; formerly named <tt>ibeta</tt>.
*
* @param aa the alpha parameter of the beta distribution.
* @param bb the beta parameter of the beta distribution.
* @param xx the integration end point.
*/
public static double incompleteBeta( double aa, double bb, double xx ) throws ArithmeticException {
double a, b, t, x, xc, w, y;
boolean flag;
if( aa <= 0.0 || bb <= 0.0 ) throw new
ArithmeticException("ibeta: Domain error!");
if( (xx <= 0.0) || ( xx >= 1.0) ) {
if( xx == 0.0 ) return 0.0;
if( xx == 1.0 ) return 1.0;
throw new ArithmeticException("ibeta: Domain error!");
}
flag = false;
if( (bb * xx) <= 1.0 && xx <= 0.95) {
t = powerSeries(aa, bb, xx);
return t;
}
w = 1.0 - xx;
/* Reverse a and b if x is greater than the mean. */
if( xx > (aa/(aa+bb)) ) {
flag = true;
a = bb;
b = aa;
xc = xx;
x = w;
} else {
a = aa;
b = bb;
xc = w;
x = xx;
}
if( flag && (b * x) <= 1.0 && x <= 0.95) {
t = powerSeries(a, b, x);
if( t <= MACHEP ) t = 1.0 - MACHEP;
else t = 1.0 - t;
return t;
}
/* Choose expansion for better convergence. */
y = x * (a+b-2.0) - (a-1.0);
if( y < 0.0 )
w = incompleteBetaFraction1( a, b, x );
else
w = incompleteBetaFraction2( a, b, x ) / xc;
/* Multiply w by the factor
a b _ _ _
x (1-x) | (a+b) / ( a | (a) | (b) ) . */
y = a * Math.log(x);
t = b * Math.log(xc);
if( (a+b) < MAXGAM && Math.abs(y) < MAXLOG && Math.abs(t) < MAXLOG ) {
t = Math.pow(xc,b);
t *= Math.pow(x,a);
t /= a;
t *= w;
t *= gamma(a+b) / (gamma(a) * gamma(b));
if( flag ) {
if( t <= MACHEP ) t = 1.0 - MACHEP;
else t = 1.0 - t;
}
return t;
}
/* Resort to logarithms. */
y += t + logGamma(a+b) - logGamma(a) - logGamma(b);
y += Math.log(w/a);
if( y < MINLOG )
t = 0.0;
else
t = Math.exp(y);
if( flag ) {
if( t <= MACHEP ) t = 1.0 - MACHEP;
else t = 1.0 - t;
}
return t;
}
/**
* 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;
}
/**
* Power series for incomplete beta integral; formerly named <tt>pseries</tt>.
* Use when b*x is small and x not too close to 1.
*/
static double powerSeries( double a, double b, double x ) throws ArithmeticException {
double s, t, u, v, n, t1, z, ai;
ai = 1.0 / a;
u = (1.0 - b) * x;
v = u / (a + 1.0);
t1 = v;
t = u;
n = 2.0;
s = 0.0;
z = MACHEP * ai;
while( Math.abs(v) > z ) {
u = (n - b) * x / n;
t *= u;
v = t / (a + n);
s += v;
n += 1.0;
}
s += t1;
s += ai;
u = a * Math.log(x);
if( (a+b) < MAXGAM && Math.abs(u) < MAXLOG ) {
t = Gamma.gamma(a+b)/(Gamma.gamma(a)*Gamma.gamma(b));
s = s * t * Math.pow(x,a);
} else {
t = Gamma.logGamma(a+b) - Gamma.logGamma(a) - Gamma.logGamma(b) + u + Math.log(s);
if( t < MINLOG ) s = 0.0;
else s = Math.exp(t);
}
return s;
}
/**
* Returns the Gamma function computed by Stirling's formula; formerly named <tt>stirf</tt>.
* The polynomial STIR is valid for 33 <= x <= 172.
*/
static double stirlingFormula(double x) throws ArithmeticException {
double STIR[] = {
7.87311395793093628397E-4,
-2.29549961613378126380E-4,
-2.68132617805781232825E-3,
3.47222221605458667310E-3,
8.33333333333482257126E-2,
};
double MAXSTIR = 143.01608;
double w = 1.0/x;
double y = Math.exp(x);
w = 1.0 + w * Polynomial.polevl( w, STIR, 4 );
if( x > MAXSTIR ) {
/* Avoid overflow in Math.pow() */
double v = Math.pow( x, 0.5 * x - 0.25 );
y = v * (v / y);
} else {
y = Math.pow( x, x - 0.5 ) / y;
}
y = SQTPI * y * w;
return y;
}
}