| /* |
| * 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.numbers.gamma; |
| |
| import java.text.MessageFormat; |
| |
| import org.apache.commons.numbers.fraction.ContinuedFraction; |
| |
| /** |
| * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"> |
| * Regularized Gamma functions</a>. |
| * |
| * Class is immutable. |
| */ |
| public final class RegularizedGamma { |
| /** Maximum allowed numerical error. */ |
| private static final double DEFAULT_EPSILON = 1e-15; |
| |
| /** Private constructor. */ |
| private RegularizedGamma() { |
| // intentionally empty. |
| } |
| |
| /** |
| * \( P(a, x) \) <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"> |
| * regularized Gamma function</a>. |
| * |
| * Class is immutable. |
| */ |
| public static final class P { |
| /** Prevent instantiation. */ |
| private P() {} |
| |
| /** |
| * Computes the regularized gamma function \( P(a, x) \). |
| * |
| * @param a Argument. |
| * @param x Argument. |
| * @return \( P(a, x) \). |
| * @throws ArithmeticException if the continued fraction fails to converge. |
| */ |
| public static double value(double a, |
| double x) { |
| return value(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE); |
| } |
| |
| /** |
| * Computes 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 Argument. |
| * @param x Argument. |
| * @param epsilon Tolerance in continued fraction evaluation. |
| * @param maxIterations Maximum number of iterations in continued fraction evaluation. |
| * @return \( P(a, x) \). |
| * @throws ArithmeticException if the continued fraction fails to converge. |
| */ |
| public static double value(double a, |
| double x, |
| double epsilon, |
| int maxIterations) { |
| if (Double.isNaN(a) || |
| Double.isNaN(x) || |
| a <= 0 || |
| x < 0) { |
| return Double.NaN; |
| } else if (x == 0) { |
| return 0; |
| } else if (x >= a + 1) { |
| // Q should converge faster in this case. |
| return 1 - RegularizedGamma.Q.value(a, x, epsilon, maxIterations); |
| } else { |
| // Series. |
| double n = 0; // current element index |
| double an = 1 / a; // n-th element in the series |
| double sum = an; // partial sum |
| while (Math.abs(an / sum) > epsilon && |
| n < maxIterations && |
| sum < Double.POSITIVE_INFINITY) { |
| // compute next element in the series |
| n += 1; |
| an *= x / (a + n); |
| |
| // update partial sum |
| sum += an; |
| } |
| if (n >= maxIterations) { |
| throw new ArithmeticException( |
| MessageFormat.format("Failed to converge within {0} iterations", maxIterations)); |
| } else if (Double.isInfinite(sum)) { |
| return 1; |
| } else { |
| // Ensure result is in the range [0, 1] |
| final double result = Math.exp(-x + (a * Math.log(x)) - LogGamma.value(a)) * sum; |
| return result > 1.0 ? 1.0 : result; |
| } |
| } |
| } |
| } |
| |
| /** |
| * Creates the \( Q(a, x) \equiv 1 - P(a, x) \) <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"> |
| * regularized Gamma function</a>. |
| * |
| * Class is immutable. |
| */ |
| public static final class Q { |
| /** Prevent instantiation. */ |
| private Q() {} |
| |
| /** |
| * Computes the regularized gamma function \( Q(a, x) = 1 - P(a, x) \). |
| * |
| * @param a Argument. |
| * @param x Argument. |
| * @return \( Q(a, x) \). |
| * @throws ArithmeticException if the continued fraction fails to converge. |
| */ |
| public static double value(double a, |
| double x) { |
| return value(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE); |
| } |
| |
| /** |
| * Computes 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 Argument. |
| * @param x Argument. |
| * @param epsilon Tolerance in continued fraction evaluation. |
| * @param maxIterations Maximum number of iterations in continued fraction evaluation. |
| * @throws ArithmeticException if the continued fraction fails to converge. |
| * @return \( Q(a, x) \). |
| */ |
| public static double value(final double a, |
| double x, |
| double epsilon, |
| int maxIterations) { |
| if (Double.isNaN(a) || |
| Double.isNaN(x) || |
| a <= 0 || |
| x < 0) { |
| return Double.NaN; |
| } else if (x == 0) { |
| return 1; |
| } else if (x < a + 1) { |
| // P should converge faster in this case. |
| return 1 - RegularizedGamma.P.value(a, x, epsilon, maxIterations); |
| } else { |
| final ContinuedFraction cf = new ContinuedFraction() { |
| /** {@inheritDoc} */ |
| @Override |
| protected double getA(int n, double x) { |
| return n * (a - n); |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| protected double getB(int n, double x) { |
| return ((2 * n) + 1) - a + x; |
| } |
| }; |
| |
| return Math.exp(-x + (a * Math.log(x)) - LogGamma.value(a)) / |
| cf.evaluate(x, epsilon, maxIterations); |
| } |
| } |
| } |
| } |