org.apache.commons.numbers.gamma.Digamma Maven / Gradle / Ivy
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* 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
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*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
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package org.apache.commons.numbers.gamma;
/**
* Digamma function.
*
* It is defined as the logarithmic derivative of the \( \Gamma \)
* ({@link Gamma}) function:
* \( \frac{d}{dx}(\ln \Gamma(x)) = \frac{\Gamma^\prime(x)}{\Gamma(x)} \).
*
*
* @see Gamma
*/
public final class Digamma {
/** Euler-Mascheroni constant. */
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
* Jose Bernardo,
* Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.
* A
* reflection formula 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;
}
}