org.apache.commons.numbers.gamma.Gamma Maven / Gradle / Ivy
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* 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;
/**
* Gamma
* function.
*
* The gamma
* function can be seen to extend the factorial function to cover real and
* complex numbers, but with its argument shifted by {@code -1}. This
* implementation supports real numbers.
*
*
* This class is immutable.
*
*/
public final class Gamma {
/** The threshold value for choosing the Lanczos approximation. */
private static final double LANCZOS_THRESHOLD = 20;
/** √(2π). */
private static final double SQRT_TWO_PI = 2.506628274631000502;
/** Private constructor. */
private Gamma() {
// intentionally empty.
}
/**
* Computes the value of \( \Gamma(x) \).
*
* Based on the NSWC Library of Mathematics Subroutines double
* precision implementation, {@code DGAMMA}.
*
* @param x Argument.
* @return \( \Gamma(x) \)
*/
public static double value(final double x) {
if ((x == Math.rint(x)) && (x <= 0.0)) {
return Double.NaN;
}
final double absX = Math.abs(x);
if (absX <= LANCZOS_THRESHOLD) {
if (x >= 1) {
/*
* From the recurrence relation
* Gamma(x) = (x - 1) * ... * (x - n) * Gamma(x - n),
* then
* Gamma(t) = 1 / [1 + InvGamma1pm1.value(t - 1)],
* where t = x - n. This means that t must satisfy
* -0.5 <= t - 1 <= 1.5.
*/
double prod = 1;
double t = x;
while (t > 2.5) {
t -= 1;
prod *= t;
}
return prod / (1 + InvGamma1pm1.value(t - 1));
} else {
/*
* From the recurrence relation
* Gamma(x) = Gamma(x + n + 1) / [x * (x + 1) * ... * (x + n)]
* then
* Gamma(x + n + 1) = 1 / [1 + InvGamma1pm1.value(x + n)],
* which requires -0.5 <= x + n <= 1.5.
*/
double prod = x;
double t = x;
while (t < -0.5) {
t += 1;
prod *= t;
}
return 1 / (prod * (1 + InvGamma1pm1.value(t)));
}
} else {
final double y = absX + LanczosApproximation.g() + 0.5;
final double gammaAbs = SQRT_TWO_PI / absX *
Math.pow(y, absX + 0.5) *
Math.exp(-y) * LanczosApproximation.value(absX);
if (x > 0) {
return gammaAbs;
} else {
/*
* From the reflection formula
* Gamma(x) * Gamma(1 - x) * sin(pi * x) = pi,
* and the recurrence relation
* Gamma(1 - x) = -x * Gamma(-x),
* it is found
* Gamma(x) = -pi / [x * sin(pi * x) * Gamma(-x)].
*/
return -Math.PI / (x * Math.sin(Math.PI * x) * gammaAbs);
}
}
}
}