/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math.special;
import org.apache.commons.math.exception.MaxCountExceededException;
import org.apache.commons.math.util.ContinuedFraction;
import org.apache.commons.math.util.FastMath;
/**
* This is a utility class that provides computation methods related to the
* Gamma family of functions.
*
* @version $Id: Gamma.java 1131229 2011-06-03 20:49:25Z luc $
*/
public class Gamma {
/**
* <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a>
* @since 2.0
*/
public static final double GAMMA = 0.577215664901532860606512090082;
/** Maximum allowed numerical error. */
private static final double DEFAULT_EPSILON = 10e-15;
/** Lanczos coefficients */
private static final double[] LANCZOS = {
0.99999999999999709182,
57.156235665862923517,
-59.597960355475491248,
14.136097974741747174,
-0.49191381609762019978,
.33994649984811888699e-4,
.46523628927048575665e-4,
-.98374475304879564677e-4,
.15808870322491248884e-3,
-.21026444172410488319e-3,
.21743961811521264320e-3,
-.16431810653676389022e-3,
.84418223983852743293e-4,
-.26190838401581408670e-4,
.36899182659531622704e-5,
};
/** Avoid repeated computation of log of 2 PI in logGamma */
private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(2.0 * FastMath.PI);
// limits for switching algorithm in digamma
/** C limit. */
private static final double C_LIMIT = 49;
/** S limit. */
private static final double S_LIMIT = 1e-5;
/**
* Default constructor. Prohibit instantiation.
*/
private Gamma() {}
/**
* Returns the natural logarithm of the gamma function Γ(x).
*
* The implementation of this method is based on:
* <ul>
* <li><a href="http://mathworld.wolfram.com/GammaFunction.html">
* Gamma Function</a>, equation (28).</li>
* <li><a href="http://mathworld.wolfram.com/LanczosApproximation.html">
* Lanczos Approximation</a>, equations (1) through (5).</li>
* <li><a href="http://my.fit.edu/~gabdo/gamma.txt">Paul Godfrey, A note on
* the computation of the convergent Lanczos complex Gamma approximation
* </a></li>
* </ul>
*
* @param x Value.
* @return log(Γ(x))
*/
public static double logGamma(double x) {
double ret;
if (Double.isNaN(x) || (x <= 0.0)) {
ret = Double.NaN;
} else {
double g = 607.0 / 128.0;
double sum = 0.0;
for (int i = LANCZOS.length - 1; i > 0; --i) {
sum = sum + (LANCZOS[i] / (x + i));
}
sum = sum + LANCZOS[0];
double tmp = x + g + .5;
ret = ((x + .5) * FastMath.log(tmp)) - tmp +
HALF_LOG_2_PI + FastMath.log(sum / x);
}
return ret;
}
/**
* Returns the regularized gamma function P(a, x).
*
* @param a Parameter.
* @param x Value.
* @return the regularized gamma function P(a, x).
* @throws MaxCountExceededException if the algorithm fails to converge.
*/
public static double regularizedGammaP(double a, double x) {
return regularizedGammaP(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
}
/**
* Returns the regularized gamma function P(a, x).
*
* The implementation of this method is based on:
* <ul>
* <li>
* <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
* Regularized Gamma Function</a>, equation (1)
* </li>
* <li>
* <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">
* Incomplete Gamma Function</a>, equation (4).
* </li>
* <li>
* <a href="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html">
* Confluent Hypergeometric Function of the First Kind</a>, equation (1).
* </li>
* </ul>
*
* @param a the a parameter.
* @param x the value.
* @param epsilon When the absolute value of the nth item in the
* series is less than epsilon the approximation ceases to calculate
* further elements in the series.
* @param maxIterations Maximum number of "iterations" to complete.
* @return the regularized gamma function P(a, x)
* @throws MaxCountExceededException if the algorithm fails to converge.
*/
public static double regularizedGammaP(double a,
double x,
double epsilon,
int maxIterations) {
double ret;
if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
ret = Double.NaN;
} else if (x == 0.0) {
ret = 0.0;
} else if (x >= a + 1) {
// use regularizedGammaQ because it should converge faster in this
// case.
ret = 1.0 - regularizedGammaQ(a, x, epsilon, maxIterations);
} else {
// calculate series
double n = 0.0; // current element index
double an = 1.0 / a; // n-th element in the series
double sum = an; // partial sum
while (FastMath.abs(an/sum) > epsilon &&
n < maxIterations &&
sum < Double.POSITIVE_INFINITY) {
// compute next element in the series
n = n + 1.0;
an = an * (x / (a + n));
// update partial sum
sum = sum + an;
}
if (n >= maxIterations) {
throw new MaxCountExceededException(maxIterations);
} else if (Double.isInfinite(sum)) {
ret = 1.0;
} else {
ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * sum;
}
}
return ret;
}
/**
* Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
*
* @param a the a parameter.
* @param x the value.
* @return the regularized gamma function Q(a, x)
* @throws MaxCountExceededException if the algorithm fails to converge.
*/
public static double regularizedGammaQ(double a, double x) {
return regularizedGammaQ(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
}
/**
* Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
*
* The implementation of this method is based on:
* <ul>
* <li>
* <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
* Regularized Gamma Function</a>, equation (1).
* </li>
* <li>
* <a href="http://functions.wolfram.com/GammaBetaErf/GammaRegularized/10/0003/">
* Regularized incomplete gamma function: Continued fraction representations
* (formula 06.08.10.0003)</a>
* </li>
* </ul>
*
* @param a the a parameter.
* @param x the value.
* @param epsilon When the absolute value of the nth item in the
* series is less than epsilon the approximation ceases to calculate
* further elements in the series.
* @param maxIterations Maximum number of "iterations" to complete.
* @return the regularized gamma function P(a, x)
* @throws MaxCountExceededException if the algorithm fails to converge.
*/
public static double regularizedGammaQ(final double a,
double x,
double epsilon,
int maxIterations) {
double ret;
if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
ret = Double.NaN;
} else if (x == 0.0) {
ret = 1.0;
} else if (x < a + 1.0) {
// use regularizedGammaP because it should converge faster in this
// case.
ret = 1.0 - regularizedGammaP(a, x, epsilon, maxIterations);
} else {
// create continued fraction
ContinuedFraction cf = new ContinuedFraction() {
@Override
protected double getA(int n, double x) {
return ((2.0 * n) + 1.0) - a + x;
}
@Override
protected double getB(int n, double x) {
return n * (a - n);
}
};
ret = 1.0 / cf.evaluate(x, epsilon, maxIterations);
ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * ret;
}
return ret;
}
/**
* <p>Computes the digamma function of x.</p>
*
* <p>This is an independently written implementation of the algorithm described in
* Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.</p>
*
* <p>Some of the constants have been changed to increase accuracy at the moderate expense
* of run-time. The result should be accurate to within 10^-8 absolute tolerance for
* x >= 10^-5 and within 10^-8 relative tolerance for x > 0.</p>
*
* <p>Performance for large negative values of x will be quite expensive (proportional to
* |x|). Accuracy for negative values of x should be about 10^-8 absolute for results
* less than 10^5 and 10^-8 relative for results larger than that.</p>
*
* @param x Argument.
* @return digamma(x) to within 10-8 relative or absolute error whichever is smaller.
* @see <a href="http://en.wikipedia.org/wiki/Digamma_function">Digamma</a>
* @see <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf">Bernardo's original article </a>
* @since 2.0
*/
public static double digamma(double x) {
if (x > 0 && x <= S_LIMIT) {
// use method 5 from Bernardo AS103
// accurate to O(x)
return -GAMMA - 1 / x;
}
if (x >= C_LIMIT) {
// use method 4 (accurate to O(1/x^8)
double inv = 1 / (x * x);
// 1 1 1 1
// log(x) - --- - ------ + ------- - -------
// 2 x 12 x^2 120 x^4 252 x^6
return FastMath.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv / 252));
}
return digamma(x + 1) - 1 / x;
}
/**
* Computes the trigamma function of x.
* This function is derived by taking the derivative of the implementation
* of digamma.
*
* @param x Argument.
* @return trigamma(x) to within 10-8 relative or absolute error whichever is smaller
* @see <a href="http://en.wikipedia.org/wiki/Trigamma_function">Trigamma</a>
* @see Gamma#digamma(double)
* @since 2.0
*/
public static double trigamma(double x) {
if (x > 0 && x <= S_LIMIT) {
return 1 / (x * x);
}
if (x >= C_LIMIT) {
double inv = 1 / (x * x);
// 1 1 1 1 1
// - + ---- + ---- - ----- + -----
// x 2 3 5 7
// 2 x 6 x 30 x 42 x
return 1 / x + inv / 2 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv / 42));
}
return trigamma(x + 1) + 1 / (x * x);
}
}