Java tutorial
/* * 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.math3.special; import org.apache.commons.math3.exception.MaxCountExceededException; import org.apache.commons.math3.exception.NumberIsTooLargeException; import org.apache.commons.math3.exception.NumberIsTooSmallException; import org.apache.commons.math3.util.ContinuedFraction; import org.apache.commons.math3.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> * * @version $Id: Gamma.java 1422313 2012-12-15 18:53:41Z psteitz $ */ 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 = n + 1.0; an = an * (x / (a + n)); // update partial sum 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() { @Override protected double getA(int n, double x) { return ((2.0 * n) + 1.0) - a + x; } @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 (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 (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 = 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 (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 = 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 = 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 / x * 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; } }