| /* |
| * 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; |
| |
| /** |
| * <a href="http://en.wikipedia.org/wiki/Digamma_function">Digamma function</a>. |
| * <p> |
| * It is defined as the logarithmic derivative of the \( \Gamma \) |
| * ({@link Gamma}) function: |
| * \( \frac{d}{dx}(\ln \Gamma(x)) = \frac{\Gamma^\prime(x)}{\Gamma(x)} \). |
| * </p> |
| * |
| * @see Gamma |
| */ |
| public final class Digamma { |
| /** <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a>. */ |
| private static final double GAMMA = 0.577215664901532860606512090082; |
| |
| /** C limit. */ |
| private static final double C_LIMIT = 49; |
| /** S limit. */ |
| private static final double S_LIMIT = 1e-5; |
| /** Fraction. */ |
| private static final double F_M1_12 = -1d / 12; |
| /** Fraction. */ |
| private static final double F_1_120 = 1d / 120; |
| /** Fraction. */ |
| private static final double F_M1_252 = -1d / 252; |
| |
| /** Private constructor. */ |
| private Digamma() { |
| // intentional empty. |
| } |
| |
| /** |
| * Computes the digamma function. |
| * |
| * This is an independently written implementation of the algorithm described in |
| * <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf">Jose Bernardo, |
| * Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976</a>. |
| * A <a href="https://en.wikipedia.org/wiki/Digamma_function#Reflection_formula"> |
| * reflection formula</a> is incorporated to improve performance on negative values. |
| * |
| * Some of the constants have been changed to increase accuracy at the moderate |
| * expense of run-time. The result should be accurate to within {@code 1e-8}. |
| * relative tolerance for {@code 0 < x < 1e-5} and within {@code 1e-8} absolute |
| * tolerance otherwise. |
| * |
| * @param x Argument. |
| * @return digamma(x) to within {@code 1e-8} relative or absolute error whichever |
| * is larger. |
| */ |
| public static double value(double x) { |
| if (Double.isNaN(x) || Double.isInfinite(x)) { |
| return x; |
| } |
| |
| double digamma = 0; |
| if (x < 0) { |
| // Use reflection formula to fall back into positive values. |
| digamma -= Math.PI / Math.tan(Math.PI * x); |
| x = 1 - x; |
| } |
| |
| if (x > 0 && x <= S_LIMIT) { |
| // Use method 5 from Bernardo AS103, accurate to O(x). |
| return digamma - GAMMA - 1 / x; |
| } |
| |
| while (x < C_LIMIT) { |
| digamma -= 1 / x; |
| x += 1; |
| } |
| |
| // Use method 4, accurate to O(1/x^8) |
| final double inv = 1 / (x * x); |
| // 1 1 1 1 |
| // log(x) - --- - ------ + ------- - ------- |
| // 2 x 12 x^2 120 x^4 252 x^6 |
| digamma += Math.log(x) - 0.5 / x + inv * (F_M1_12 + inv * (F_1_120 + F_M1_252 * inv)); |
| |
| return digamma; |
| } |
| } |