mirror of https://github.com/jlizier/jidt
750 lines
28 KiB
Java
750 lines
28 KiB
Java
/*
|
|
* Java Information Dynamics Toolkit (JIDT)
|
|
* Copyright (C) 2017, Joseph T. Lizier
|
|
*
|
|
* This program is free software: you can redistribute it and/or modify
|
|
* it under the terms of the GNU General Public License as published by
|
|
* the Free Software Foundation, either version 3 of the License, or
|
|
* (at your option) any later version.
|
|
*
|
|
* This program is distributed in the hope that it will be useful,
|
|
* but WITHOUT ANY WARRANTY; without even the implied warranty of
|
|
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
|
* GNU General Public License for more details.
|
|
*
|
|
* You should have received a copy of the GNU General Public License
|
|
* along with this program. If not, see <http://www.gnu.org/licenses/>.
|
|
*/
|
|
|
|
/*
|
|
* This class was originally distributed as part of the Apache Commons
|
|
* Math3 library (3.6.1), under the Apache License Version 2.0, which is
|
|
* copied below. This Apache 2 software is now included as a derivative
|
|
* work in the GPLv3 licensed JIDT project, as per:
|
|
* http://www.apache.org/licenses/GPL-compatibility.html
|
|
*
|
|
* The original Apache source code has been modified as follows:
|
|
* -- We have modified package names to sit inside the JIDT structure.
|
|
*/
|
|
|
|
/*
|
|
* 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 infodynamics.utils.commonsmath3.special;
|
|
|
|
import infodynamics.utils.commonsmath3.exception.MaxCountExceededException;
|
|
import infodynamics.utils.commonsmath3.exception.NumberIsTooLargeException;
|
|
import infodynamics.utils.commonsmath3.exception.NumberIsTooSmallException;
|
|
import infodynamics.utils.commonsmath3.util.ContinuedFraction;
|
|
import infodynamics.utils.commonsmath3.util.FastMath;
|
|
|
|
/**
|
|
* <p>
|
|
* This is a utility class that provides computation methods related to the
|
|
* Γ (Gamma) family of functions.
|
|
* </p>
|
|
* <p>
|
|
* Implementation of {@link #invGamma1pm1(double)} and
|
|
* {@link #logGamma1p(double)} is based on the algorithms described in
|
|
* <ul>
|
|
* <li><a href="http://dx.doi.org/10.1145/22721.23109">Didonato and Morris
|
|
* (1986)</a>, <em>Computation of the Incomplete Gamma Function Ratios and
|
|
* their Inverse</em>, TOMS 12(4), 377-393,</li>
|
|
* <li><a href="http://dx.doi.org/10.1145/131766.131776">Didonato and Morris
|
|
* (1992)</a>, <em>Algorithm 708: Significant Digit Computation of the
|
|
* Incomplete Beta Function Ratios</em>, TOMS 18(3), 360-373,</li>
|
|
* </ul>
|
|
* and implemented in the
|
|
* <a href="http://www.dtic.mil/docs/citations/ADA476840">NSWC Library of Mathematical Functions</a>,
|
|
* available
|
|
* <a href="http://www.ualberta.ca/CNS/RESEARCH/Software/NumericalNSWC/site.html">here</a>.
|
|
* This library is "approved for public release", and the
|
|
* <a href="http://www.dtic.mil/dtic/pdf/announcements/CopyrightGuidance.pdf">Copyright guidance</a>
|
|
* indicates that unless otherwise stated in the code, all FORTRAN functions in
|
|
* this library are license free. Since no such notice appears in the code these
|
|
* functions can safely be ported to Commons-Math.
|
|
* </p>
|
|
*
|
|
*/
|
|
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;
|
|
|
|
/**
|
|
* The value of the {@code g} constant in the Lanczos approximation, see
|
|
* {@link #lanczos(double)}.
|
|
* @since 3.1
|
|
*/
|
|
public static final double LANCZOS_G = 607.0 / 128.0;
|
|
|
|
/** 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);
|
|
|
|
/** The constant value of √(2π). */
|
|
private static final double SQRT_TWO_PI = 2.506628274631000502;
|
|
|
|
// 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;
|
|
|
|
/*
|
|
* Constants for the computation of double invGamma1pm1(double).
|
|
* Copied from DGAM1 in the NSWC library.
|
|
*/
|
|
|
|
/** The constant {@code A0} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_A0 = .611609510448141581788E-08;
|
|
|
|
/** The constant {@code A1} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_A1 = .624730830116465516210E-08;
|
|
|
|
/** The constant {@code B1} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_B1 = .203610414066806987300E+00;
|
|
|
|
/** The constant {@code B2} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_B2 = .266205348428949217746E-01;
|
|
|
|
/** The constant {@code B3} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_B3 = .493944979382446875238E-03;
|
|
|
|
/** The constant {@code B4} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_B4 = -.851419432440314906588E-05;
|
|
|
|
/** The constant {@code B5} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_B5 = -.643045481779353022248E-05;
|
|
|
|
/** The constant {@code B6} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_B6 = .992641840672773722196E-06;
|
|
|
|
/** The constant {@code B7} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_B7 = -.607761895722825260739E-07;
|
|
|
|
/** The constant {@code B8} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_B8 = .195755836614639731882E-09;
|
|
|
|
/** The constant {@code P0} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_P0 = .6116095104481415817861E-08;
|
|
|
|
/** The constant {@code P1} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_P1 = .6871674113067198736152E-08;
|
|
|
|
/** The constant {@code P2} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_P2 = .6820161668496170657918E-09;
|
|
|
|
/** The constant {@code P3} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_P3 = .4686843322948848031080E-10;
|
|
|
|
/** The constant {@code P4} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_P4 = .1572833027710446286995E-11;
|
|
|
|
/** The constant {@code P5} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_P5 = -.1249441572276366213222E-12;
|
|
|
|
/** The constant {@code P6} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_P6 = .4343529937408594255178E-14;
|
|
|
|
/** The constant {@code Q1} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_Q1 = .3056961078365221025009E+00;
|
|
|
|
/** The constant {@code Q2} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_Q2 = .5464213086042296536016E-01;
|
|
|
|
/** The constant {@code Q3} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_Q3 = .4956830093825887312020E-02;
|
|
|
|
/** The constant {@code Q4} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_Q4 = .2692369466186361192876E-03;
|
|
|
|
/** The constant {@code C} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C = -.422784335098467139393487909917598E+00;
|
|
|
|
/** The constant {@code C0} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C0 = .577215664901532860606512090082402E+00;
|
|
|
|
/** The constant {@code C1} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C1 = -.655878071520253881077019515145390E+00;
|
|
|
|
/** The constant {@code C2} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C2 = -.420026350340952355290039348754298E-01;
|
|
|
|
/** The constant {@code C3} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C3 = .166538611382291489501700795102105E+00;
|
|
|
|
/** The constant {@code C4} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C4 = -.421977345555443367482083012891874E-01;
|
|
|
|
/** The constant {@code C5} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C5 = -.962197152787697356211492167234820E-02;
|
|
|
|
/** The constant {@code C6} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C6 = .721894324666309954239501034044657E-02;
|
|
|
|
/** The constant {@code C7} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C7 = -.116516759185906511211397108401839E-02;
|
|
|
|
/** The constant {@code C8} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C8 = -.215241674114950972815729963053648E-03;
|
|
|
|
/** The constant {@code C9} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C9 = .128050282388116186153198626328164E-03;
|
|
|
|
/** The constant {@code C10} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C10 = -.201348547807882386556893914210218E-04;
|
|
|
|
/** The constant {@code C11} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C11 = -.125049348214267065734535947383309E-05;
|
|
|
|
/** The constant {@code C12} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C12 = .113302723198169588237412962033074E-05;
|
|
|
|
/** The constant {@code C13} defined in {@code DGAM1}. */
|
|
private static final double INV_GAMMA1P_M1_C13 = -.205633841697760710345015413002057E-06;
|
|
|
|
/**
|
|
* Default constructor. Prohibit instantiation.
|
|
*/
|
|
private Gamma() {}
|
|
|
|
/**
|
|
* <p>
|
|
* Returns the value of log Γ(x) for x > 0.
|
|
* </p>
|
|
* <p>
|
|
* For x ≤ 8, the implementation is based on the double precision
|
|
* implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
|
|
* {@code DGAMLN}. For x > 8, the implementation is based on
|
|
* </p>
|
|
* <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 Argument.
|
|
* @return the value of {@code log(Gamma(x))}, {@code Double.NaN} if
|
|
* {@code x <= 0.0}.
|
|
*/
|
|
public static double logGamma(double x) {
|
|
double ret;
|
|
|
|
if (Double.isNaN(x) || (x <= 0.0)) {
|
|
ret = Double.NaN;
|
|
} else if (x < 0.5) {
|
|
return logGamma1p(x) - FastMath.log(x);
|
|
} else if (x <= 2.5) {
|
|
return logGamma1p((x - 0.5) - 0.5);
|
|
} else if (x <= 8.0) {
|
|
final int n = (int) FastMath.floor(x - 1.5);
|
|
double prod = 1.0;
|
|
for (int i = 1; i <= n; i++) {
|
|
prod *= x - i;
|
|
}
|
|
return logGamma1p(x - (n + 1)) + FastMath.log(prod);
|
|
} else {
|
|
double sum = lanczos(x);
|
|
double tmp = x + LANCZOS_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 += 1.0;
|
|
an *= x / (a + n);
|
|
|
|
// update partial 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() {
|
|
|
|
/** {@inheritDoc} */
|
|
@Override
|
|
protected double getA(int n, double x) {
|
|
return ((2.0 * n) + 1.0) - a + x;
|
|
}
|
|
|
|
/** {@inheritDoc} */
|
|
@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 (Double.isNaN(x) || Double.isInfinite(x)) {
|
|
return 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 (Double.isNaN(x) || Double.isInfinite(x)) {
|
|
return 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);
|
|
}
|
|
|
|
/**
|
|
* <p>
|
|
* Returns the Lanczos approximation used to compute the gamma function.
|
|
* The Lanczos approximation is related to the Gamma function by the
|
|
* following equation
|
|
* <center>
|
|
* {@code gamma(x) = sqrt(2 * pi) / x * (x + g + 0.5) ^ (x + 0.5)
|
|
* * exp(-x - g - 0.5) * lanczos(x)},
|
|
* </center>
|
|
* where {@code g} is the Lanczos constant.
|
|
* </p>
|
|
*
|
|
* @param x Argument.
|
|
* @return The Lanczos approximation.
|
|
* @see <a href="http://mathworld.wolfram.com/LanczosApproximation.html">Lanczos Approximation</a>
|
|
* equations (1) through (5), and Paul Godfrey's
|
|
* <a href="http://my.fit.edu/~gabdo/gamma.txt">Note on the computation
|
|
* of the convergent Lanczos complex Gamma approximation</a>
|
|
* @since 3.1
|
|
*/
|
|
public static double lanczos(final double x) {
|
|
double sum = 0.0;
|
|
for (int i = LANCZOS.length - 1; i > 0; --i) {
|
|
sum += LANCZOS[i] / (x + i);
|
|
}
|
|
return sum + LANCZOS[0];
|
|
}
|
|
|
|
/**
|
|
* Returns the value of 1 / Γ(1 + x) - 1 for -0.5 ≤ x ≤
|
|
* 1.5. This implementation is based on the double precision
|
|
* implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
|
|
* {@code DGAM1}.
|
|
*
|
|
* @param x Argument.
|
|
* @return The value of {@code 1.0 / Gamma(1.0 + x) - 1.0}.
|
|
* @throws NumberIsTooSmallException if {@code x < -0.5}
|
|
* @throws NumberIsTooLargeException if {@code x > 1.5}
|
|
* @since 3.1
|
|
*/
|
|
public static double invGamma1pm1(final double x) {
|
|
|
|
if (x < -0.5) {
|
|
throw new NumberIsTooSmallException(x, -0.5, true);
|
|
}
|
|
if (x > 1.5) {
|
|
throw new NumberIsTooLargeException(x, 1.5, true);
|
|
}
|
|
|
|
final double ret;
|
|
final double t = x <= 0.5 ? x : (x - 0.5) - 0.5;
|
|
if (t < 0.0) {
|
|
final double a = INV_GAMMA1P_M1_A0 + t * INV_GAMMA1P_M1_A1;
|
|
double b = INV_GAMMA1P_M1_B8;
|
|
b = INV_GAMMA1P_M1_B7 + t * b;
|
|
b = INV_GAMMA1P_M1_B6 + t * b;
|
|
b = INV_GAMMA1P_M1_B5 + t * b;
|
|
b = INV_GAMMA1P_M1_B4 + t * b;
|
|
b = INV_GAMMA1P_M1_B3 + t * b;
|
|
b = INV_GAMMA1P_M1_B2 + t * b;
|
|
b = INV_GAMMA1P_M1_B1 + t * b;
|
|
b = 1.0 + t * b;
|
|
|
|
double c = INV_GAMMA1P_M1_C13 + t * (a / b);
|
|
c = INV_GAMMA1P_M1_C12 + t * c;
|
|
c = INV_GAMMA1P_M1_C11 + t * c;
|
|
c = INV_GAMMA1P_M1_C10 + t * c;
|
|
c = INV_GAMMA1P_M1_C9 + t * c;
|
|
c = INV_GAMMA1P_M1_C8 + t * c;
|
|
c = INV_GAMMA1P_M1_C7 + t * c;
|
|
c = INV_GAMMA1P_M1_C6 + t * c;
|
|
c = INV_GAMMA1P_M1_C5 + t * c;
|
|
c = INV_GAMMA1P_M1_C4 + t * c;
|
|
c = INV_GAMMA1P_M1_C3 + t * c;
|
|
c = INV_GAMMA1P_M1_C2 + t * c;
|
|
c = INV_GAMMA1P_M1_C1 + t * c;
|
|
c = INV_GAMMA1P_M1_C + t * c;
|
|
if (x > 0.5) {
|
|
ret = t * c / x;
|
|
} else {
|
|
ret = x * ((c + 0.5) + 0.5);
|
|
}
|
|
} else {
|
|
double p = INV_GAMMA1P_M1_P6;
|
|
p = INV_GAMMA1P_M1_P5 + t * p;
|
|
p = INV_GAMMA1P_M1_P4 + t * p;
|
|
p = INV_GAMMA1P_M1_P3 + t * p;
|
|
p = INV_GAMMA1P_M1_P2 + t * p;
|
|
p = INV_GAMMA1P_M1_P1 + t * p;
|
|
p = INV_GAMMA1P_M1_P0 + t * p;
|
|
|
|
double q = INV_GAMMA1P_M1_Q4;
|
|
q = INV_GAMMA1P_M1_Q3 + t * q;
|
|
q = INV_GAMMA1P_M1_Q2 + t * q;
|
|
q = INV_GAMMA1P_M1_Q1 + t * q;
|
|
q = 1.0 + t * q;
|
|
|
|
double c = INV_GAMMA1P_M1_C13 + (p / q) * t;
|
|
c = INV_GAMMA1P_M1_C12 + t * c;
|
|
c = INV_GAMMA1P_M1_C11 + t * c;
|
|
c = INV_GAMMA1P_M1_C10 + t * c;
|
|
c = INV_GAMMA1P_M1_C9 + t * c;
|
|
c = INV_GAMMA1P_M1_C8 + t * c;
|
|
c = INV_GAMMA1P_M1_C7 + t * c;
|
|
c = INV_GAMMA1P_M1_C6 + t * c;
|
|
c = INV_GAMMA1P_M1_C5 + t * c;
|
|
c = INV_GAMMA1P_M1_C4 + t * c;
|
|
c = INV_GAMMA1P_M1_C3 + t * c;
|
|
c = INV_GAMMA1P_M1_C2 + t * c;
|
|
c = INV_GAMMA1P_M1_C1 + t * c;
|
|
c = INV_GAMMA1P_M1_C0 + t * c;
|
|
|
|
if (x > 0.5) {
|
|
ret = (t / x) * ((c - 0.5) - 0.5);
|
|
} else {
|
|
ret = x * c;
|
|
}
|
|
}
|
|
|
|
return ret;
|
|
}
|
|
|
|
/**
|
|
* Returns the value of log Γ(1 + x) for -0.5 ≤ x ≤ 1.5.
|
|
* This implementation is based on the double precision implementation in
|
|
* the <em>NSWC Library of Mathematics Subroutines</em>, {@code DGMLN1}.
|
|
*
|
|
* @param x Argument.
|
|
* @return The value of {@code log(Gamma(1 + x))}.
|
|
* @throws NumberIsTooSmallException if {@code x < -0.5}.
|
|
* @throws NumberIsTooLargeException if {@code x > 1.5}.
|
|
* @since 3.1
|
|
*/
|
|
public static double logGamma1p(final double x)
|
|
throws NumberIsTooSmallException, NumberIsTooLargeException {
|
|
|
|
if (x < -0.5) {
|
|
throw new NumberIsTooSmallException(x, -0.5, true);
|
|
}
|
|
if (x > 1.5) {
|
|
throw new NumberIsTooLargeException(x, 1.5, true);
|
|
}
|
|
|
|
return -FastMath.log1p(invGamma1pm1(x));
|
|
}
|
|
|
|
|
|
/**
|
|
* Returns the value of Γ(x). Based on the <em>NSWC Library of
|
|
* Mathematics Subroutines</em> double precision implementation,
|
|
* {@code DGAMMA}.
|
|
*
|
|
* @param x Argument.
|
|
* @return the value of {@code Gamma(x)}.
|
|
* @since 3.1
|
|
*/
|
|
public static double gamma(final double x) {
|
|
|
|
if ((x == FastMath.rint(x)) && (x <= 0.0)) {
|
|
return Double.NaN;
|
|
}
|
|
|
|
final double ret;
|
|
final double absX = FastMath.abs(x);
|
|
if (absX <= 20.0) {
|
|
if (x >= 1.0) {
|
|
/*
|
|
* From the recurrence relation
|
|
* Gamma(x) = (x - 1) * ... * (x - n) * Gamma(x - n),
|
|
* then
|
|
* Gamma(t) = 1 / [1 + invGamma1pm1(t - 1)],
|
|
* where t = x - n. This means that t must satisfy
|
|
* -0.5 <= t - 1 <= 1.5.
|
|
*/
|
|
double prod = 1.0;
|
|
double t = x;
|
|
while (t > 2.5) {
|
|
t -= 1.0;
|
|
prod *= t;
|
|
}
|
|
ret = prod / (1.0 + invGamma1pm1(t - 1.0));
|
|
} else {
|
|
/*
|
|
* From the recurrence relation
|
|
* Gamma(x) = Gamma(x + n + 1) / [x * (x + 1) * ... * (x + n)]
|
|
* then
|
|
* Gamma(x + n + 1) = 1 / [1 + invGamma1pm1(x + n)],
|
|
* which requires -0.5 <= x + n <= 1.5.
|
|
*/
|
|
double prod = x;
|
|
double t = x;
|
|
while (t < -0.5) {
|
|
t += 1.0;
|
|
prod *= t;
|
|
}
|
|
ret = 1.0 / (prod * (1.0 + invGamma1pm1(t)));
|
|
}
|
|
} else {
|
|
final double y = absX + LANCZOS_G + 0.5;
|
|
final double gammaAbs = SQRT_TWO_PI / absX *
|
|
FastMath.pow(y, absX + 0.5) *
|
|
FastMath.exp(-y) * lanczos(absX);
|
|
if (x > 0.0) {
|
|
ret = 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)].
|
|
*/
|
|
ret = -FastMath.PI /
|
|
(x * FastMath.sin(FastMath.PI * x) * gammaAbs);
|
|
}
|
|
}
|
|
return ret;
|
|
}
|
|
}
|