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diff --git a/src/main/java/org/apache/commons/math3/special/Gamma.java b/src/main/java/org/apache/commons/math3/special/Gamma.java new file mode 100644 index 0000000..8f7e248 --- /dev/null +++ b/src/main/java/org/apache/commons/math3/special/Gamma.java @@ -0,0 +1,692 @@ +/* + * 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; + +/** + * This is a utility class that provides computation methods related to the Γ (Gamma) family + * of functions. + * + * <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><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, + * </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. + */ +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() {} + + /** + * Returns the value of log Γ(x) for x > 0. + * + * <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 + * + * <ul> + * <li><a href="http://mathworld.wolfram.com/GammaFunction.html">Gamma Function</a>, equation + * (28). + * <li><a href="http://mathworld.wolfram.com/LanczosApproximation.html">Lanczos + * Approximation</a>, equations (1) through (5). + * <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> + * </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). + * + * <p>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><a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma + * Function</a>, equation (4). + * <li><a + * href="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html"> + * Confluent Hypergeometric Function of the First Kind</a>, equation (1). + * </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). + * + * <p>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><a href="http://functions.wolfram.com/GammaBetaErf/GammaRegularized/10/0003/"> + * Regularized incomplete gamma function: Continued fraction representations (formula + * 06.08.10.0003)</a> + * </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; + } + + /** + * Computes the digamma function of x. + * + * <p>This is an independently written implementation of the algorithm described in Jose + * Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976. + * + * <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>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. + * + * @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); + } + + /** + * 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. + * + * @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; + } +} |