Gamma.java
/*
* 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
*
* https://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.
*/
/*
* This is not the original file distributed by the Apache Software Foundation
* It has been modified by the Hipparchus project
*/
package org.hipparchus.special;
import org.hipparchus.CalculusFieldElement;
import org.hipparchus.Field;
import org.hipparchus.exception.LocalizedCoreFormats;
import org.hipparchus.exception.MathIllegalArgumentException;
import org.hipparchus.exception.MathIllegalStateException;
import org.hipparchus.util.ContinuedFraction;
import org.hipparchus.util.FastMath;
import org.hipparchus.util.FieldContinuedFraction;
/**
* <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
* </p>
* <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>
* <p>
* 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 Hipparchus.
* </p>
*
*/
public class Gamma {
/**
* <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a>
*/
public static final double GAMMA = 0.577215664901532860606512090082; // NOPMD - the fact the function and the constant have the same name is intentional and comes from mathematics conventions
/**
* The value of the {@code g} constant in the Lanczos approximation, see
* {@link #lanczos(double)}.
*/
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-8;
/*
* 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;
}
/**
* <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.
* @param <T> Type of the field elements.
* @return the value of {@code log(Gamma(x))}, {@code Double.NaN} if
* {@code x <= 0.0}.
*/
public static <T extends CalculusFieldElement<T>> T logGamma(T x) {
final Field<T> field = x.getField();
T ret;
if (x.isNaN() || (x.getReal() <= 0.0)) {
ret = field.getOne().multiply(Double.NaN);
}
else if (x.getReal() < 0.5) {
return logGamma1p(x).subtract(x.log());
}
else if (x.getReal() <= 2.5) {
return logGamma1p(x.subtract(1));
}
else if (x.getReal() <= 8.0) {
final int n = (int) x.subtract(1.5).floor().getReal();
T prod = field.getOne();
for (int i = 1; i <= n; i++) {
prod = prod.multiply(x.subtract(i));
}
return logGamma1p(x.subtract(n + 1)).add(prod.log());
}
else {
T sum = lanczos(x);
T tmp = x.add(LANCZOS_G + .5);
ret = x.add(.5).multiply(tmp.log()).subtract(tmp).add(HALF_LOG_2_PI).add(sum.divide(x).log());
}
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 MathIllegalStateException 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).
*
* @param a Parameter.
* @param x Value.
* @param <T> Type of the field elements.
* @return the regularized gamma function P(a, x).
* @throws MathIllegalStateException if the algorithm fails to converge.
*/
public static <T extends CalculusFieldElement<T>> T regularizedGammaP(T a, T 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>
* <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 MathIllegalStateException 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 MathIllegalStateException(LocalizedCoreFormats.MAX_COUNT_EXCEEDED, 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 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>
* <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.
* @param <T> Type of the field elements.
* @return the regularized gamma function P(a, x)
* @throws MathIllegalStateException if the algorithm fails to converge.
*/
public static <T extends CalculusFieldElement<T>> T regularizedGammaP(T a,
T x,
double epsilon,
int maxIterations) {
final Field<T> field = x.getField();
final T zero = field.getZero();
final T one = field.getOne();
T ret;
if (a.isNaN() || x.isNaN() || (a.getReal() <= 0.0) || (x.getReal() < 0.0)) {
ret = one.multiply(Double.NaN);
}
else if (x.getReal() == 0.0) {
ret = zero;
}
else if (x.getReal() >= a.add(1).getReal()) {
// use regularizedGammaQ because it should converge faster in this
// case.
ret = one.subtract(regularizedGammaQ(a, x, epsilon, maxIterations));
}
else {
// calculate series
double n = 0.0; // current element index
T an = one.divide(a); // n-th element in the series
T sum = an; // partial sum
while (an.divide(sum).abs().getReal() > epsilon &&
n < maxIterations &&
sum.getReal() < Double.POSITIVE_INFINITY) {
// compute next element in the series
n += 1.0;
an = an.multiply(x.divide(a.add(n)));
// update partial sum
sum = sum.add(an);
}
if (n >= maxIterations) {
throw new MathIllegalStateException(LocalizedCoreFormats.MAX_COUNT_EXCEEDED, maxIterations);
}
else if (sum.isInfinite()) {
ret = one;
}
else {
ret = a.multiply(x.log()).subtract(logGamma(a)).subtract(x).exp().multiply(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 MathIllegalStateException 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).
*
* @param a the a parameter.
* @param x the value.
* @param <T> Type of the field elements.
* @return the regularized gamma function Q(a, x)
* @throws MathIllegalStateException if the algorithm fails to converge.
*/
public static <T extends CalculusFieldElement<T>> T regularizedGammaQ(T a, T 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>
* <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 MathIllegalStateException 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;
}
/**
* 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>
* <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.
* @param <T> Type fo the field elements.
* @return the regularized gamma function P(a, x)
* @throws MathIllegalStateException if the algorithm fails to converge.
*/
public static <T extends CalculusFieldElement<T>> T regularizedGammaQ(final T a,
T x,
double epsilon,
int maxIterations) {
final Field<T> field = x.getField();
final T one = field.getOne();
T ret;
if (a.isNaN() || x.isNaN() || a.getReal() <= 0.0 || x.getReal() < 0.0) {
ret = field.getOne().multiply(Double.NaN);
}
else if (x.getReal() == 0.0) {
ret = one;
}
else if (x.getReal() < a.add(1.0).getReal()) {
// use regularizedGammaP because it should converge faster in this
// case.
ret = one.subtract(regularizedGammaP(a, x, epsilon, maxIterations));
}
else {
// create continued fraction
FieldContinuedFraction cf = new FieldContinuedFraction() {
/** {@inheritDoc} */
@Override
@SuppressWarnings("unchecked")
public <C extends CalculusFieldElement<C>> C getA(final int n, final C x) {
return x.subtract((C) a).add((2.0 * n) + 1.0);
}
/** {@inheritDoc} */
@Override
@SuppressWarnings("unchecked")
public <C extends CalculusFieldElement<C>> C getB(final int n, final C x) {
return (C) a.subtract(n).multiply(n);
}
};
ret = one.divide(cf.evaluate(x, epsilon, maxIterations));
ret = a.multiply(x.log()).subtract(logGamma(a)).subtract(x).exp().multiply(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>
*/
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 8 (accurate to O(1/x^8))
double inv = 1 / (x * x);
// 1 1 1 1 1 5 691 1
// log(x) - --- - ------ + ------- - ------- + ------- - ------- + ---------- - -------
// 2 x 12 x^2 120 x^4 252 x^6 240 x^8 660 x^10 32760 x^12 12 x^14
return FastMath.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv * (1.0 / 252 + inv *
(1.0 / 240 - inv * (5.0 / 660 + inv * (691.0 / 32760 - inv / 12))))));
}
return digamma(x + 1) - 1 / x;
}
/**
* <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.
* @param <T> Type of the field elements.
* @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>
*/
public static <T extends CalculusFieldElement<T>> T digamma(T x) {
if (x.isNaN() || x.isInfinite()) {
return x;
}
if (x.getReal() > 0 && x.getReal() <= S_LIMIT) {
// use method 5 from Bernardo AS103
// accurate to O(x)
return x.pow(-1).negate().subtract(GAMMA);
}
if (x.getReal() >= C_LIMIT) {
// use method 8 (accurate to O(1/x^8))
T inv = x.square().reciprocal();
// 1 1 1 1 1 5 691 1
// log(x) - --- - ------ + ------- - ------- + ------- - ------- + ---------- - -------
// 2 x 12 x^2 120 x^4 252 x^6 240 x^8 660 x^10 32760 x^12 12 x^14
return x.log().subtract(x.pow(-1).multiply(0.5)).add(
inv.multiply(
inv.multiply(
inv.multiply(
inv.multiply(
inv.multiply(
inv.multiply(inv.divide(-12.)
.add(691. / 32760))
.subtract(5. / 660))
.add(1.0 / 240))
.subtract(1.0 / 252))
.add(1.0 / 120))
.subtract(1.0 / 12)));
}
return digamma(x.add(1.)).subtract(x.pow(-1));
}
/**
* 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)
*/
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 1 5 691 7
// - + ---- + ---- - ----- + ----- - ----- + ------- - -------- + ------
// x 2 3 5 7 9 11 13 15
// 2 x 6 x 30 x 42 x 30 x 66 x 2730 x 6 x
return 1 / x + inv * 0.5 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv * (1.0 / 42 - inv * (1.0 / 30 + inv *
(5.0 / 66 - inv * (691. / 2730 + inv * 7. / 15))))));
}
return trigamma(x + 1) + 1 / (x * x);
}
/**
* Computes the trigamma function of x.
* This function is derived by taking the derivative of the implementation
* of digamma.
*
* @param x Argument.
* @param <T> Type of the field elements.
* @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)
*/
public static <T extends CalculusFieldElement<T>> T trigamma(T x) {
if (x.isNaN() || x.isInfinite()) {
return x;
}
if (x.getReal() > 0 && x.getReal() <= S_LIMIT) {
// use method 5 from Bernardo AS103
// accurate to O(x)
return x.square().reciprocal();
}
if (x.getReal() >= C_LIMIT) {
// use method 4 (accurate to O(1/x^8)
T inv = x.square().reciprocal();
T invCub = inv.multiply(x.reciprocal());
// 1 1 1 1 1 1 5 691 7
// - + ---- + ---- - ----- + ----- + ----- + ------- - -------- + ------
// x 2 3 5 7 9 11 13 15
// 2 x 6 x 30 x 42 x 30 x 66 x 2730 x 6 x
return x.pow(-1).add(
inv.multiply(0.5)).add(
invCub.multiply(
inv.multiply(
inv.multiply(
inv.multiply(
inv.multiply(
inv.multiply(inv.multiply(7. / 6)
.subtract(691. / 2730))
.add(5. / 66))
.subtract(1.0 / 30))
.add(1.0 / 42))
.subtract(1.0 / 30))
.add(1.0 / 6)));
}
return trigamma(x.add(1.)).add(x.square().reciprocal());
}
/**
* <p>
* Returns the Lanczos approximation used to compute the gamma function.
* The Lanczos approximation is related to the Gamma function by the
* following equation
* \[
* \Gamma(x) = \frac{\sqrt{2\pi}}{x} \times (x + g + \frac{1}{2}) ^ (x + \frac{1}{2})
* \times e^{-x - g - 0.5} \times \mathrm{lanczos}(x)
* \]
* 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>
*/
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];
}
/**
* <p>
* Returns the Lanczos approximation used to compute the gamma function.
* The Lanczos approximation is related to the Gamma function by the
* following equation
* \[
* \Gamma(x) = \frac{\sqrt{2\pi}}{x} \times (x + g + \frac{1}{2}) ^ (x + \frac{1}{2})
* \times e^{-x - g - 0.5} \times \mathrm{lanczos}(x)
* \]
* where {@code g} is the Lanczos constant.
* </p>
*
* @param x Argument.
* @param <T> Type of the field elements.
* @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>
*/
public static <T extends CalculusFieldElement<T>> T lanczos(final T x) {
final Field<T> field = x.getField();
T sum = field.getZero();
for (int i = LANCZOS.length - 1; i > 0; --i) {
sum = sum.add(x.add(i).pow(-1.).multiply(LANCZOS[i]));
}
return sum.add(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 MathIllegalArgumentException if {@code x < -0.5}
* @throws MathIllegalArgumentException if {@code x > 1.5}
*/
public static double invGamma1pm1(final double x) {
if (x < -0.5) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL,
x, -0.5);
}
if (x > 1.5) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
x, 1.5);
}
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 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.
* @param <T> Type of the field elements.
* @return The value of {@code 1.0 / Gamma(1.0 + x) - 1.0}.
* @throws MathIllegalArgumentException if {@code x < -0.5}
* @throws MathIllegalArgumentException if {@code x > 1.5}
*/
public static <T extends CalculusFieldElement<T>> T invGamma1pm1(final T x) {
final T one = x.getField().getOne();
if (x.getReal() < -0.5) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL,
x, -0.5);
}
if (x.getReal() > 1.5) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
x, 1.5);
}
final T ret;
final T t = x.getReal() <= 0.5 ? x : x.subtract(1);
if (t.getReal() < 0.0) {
final T a = one.newInstance(INV_GAMMA1P_M1_A0).add(t.multiply(INV_GAMMA1P_M1_A1));
T b = one.newInstance(INV_GAMMA1P_M1_B8);
b = t.multiply(b).add(INV_GAMMA1P_M1_B7);
b = t.multiply(b).add(INV_GAMMA1P_M1_B6);
b = t.multiply(b).add(INV_GAMMA1P_M1_B5);
b = t.multiply(b).add(INV_GAMMA1P_M1_B4);
b = t.multiply(b).add(INV_GAMMA1P_M1_B3);
b = t.multiply(b).add(INV_GAMMA1P_M1_B2);
b = t.multiply(b).add(INV_GAMMA1P_M1_B1);
b = t.multiply(b).add(1.);
T c = one.newInstance(INV_GAMMA1P_M1_C13).add(t.multiply(a.divide(b)));
c = t.multiply(c).add(INV_GAMMA1P_M1_C12);
c = t.multiply(c).add(INV_GAMMA1P_M1_C11);
c = t.multiply(c).add(INV_GAMMA1P_M1_C10);
c = t.multiply(c).add(INV_GAMMA1P_M1_C9);
c = t.multiply(c).add(INV_GAMMA1P_M1_C8);
c = t.multiply(c).add(INV_GAMMA1P_M1_C7);
c = t.multiply(c).add(INV_GAMMA1P_M1_C6);
c = t.multiply(c).add(INV_GAMMA1P_M1_C5);
c = t.multiply(c).add(INV_GAMMA1P_M1_C4);
c = t.multiply(c).add(INV_GAMMA1P_M1_C3);
c = t.multiply(c).add(INV_GAMMA1P_M1_C2);
c = t.multiply(c).add(INV_GAMMA1P_M1_C1);
c = t.multiply(c).add(INV_GAMMA1P_M1_C);
if (x.getReal() > 0.5) {
ret = t.multiply(c).divide(x);
}
else {
ret = x.multiply(c.add(1));
}
}
else {
T p = one.newInstance(INV_GAMMA1P_M1_P6);
p = t.multiply(p).add(INV_GAMMA1P_M1_P5);
p = t.multiply(p).add(INV_GAMMA1P_M1_P4);
p = t.multiply(p).add(INV_GAMMA1P_M1_P3);
p = t.multiply(p).add(INV_GAMMA1P_M1_P2);
p = t.multiply(p).add(INV_GAMMA1P_M1_P1);
p = t.multiply(p).add(INV_GAMMA1P_M1_P0);
T q = one.newInstance(INV_GAMMA1P_M1_Q4);
q = t.multiply(q).add(INV_GAMMA1P_M1_Q3);
q = t.multiply(q).add(INV_GAMMA1P_M1_Q2);
q = t.multiply(q).add(INV_GAMMA1P_M1_Q1);
q = t.multiply(q).add(1.);
T c = one.newInstance(INV_GAMMA1P_M1_C13).add(t.multiply(p.divide(q)));
c = t.multiply(c).add(INV_GAMMA1P_M1_C12);
c = t.multiply(c).add(INV_GAMMA1P_M1_C11);
c = t.multiply(c).add(INV_GAMMA1P_M1_C10);
c = t.multiply(c).add(INV_GAMMA1P_M1_C9);
c = t.multiply(c).add(INV_GAMMA1P_M1_C8);
c = t.multiply(c).add(INV_GAMMA1P_M1_C7);
c = t.multiply(c).add(INV_GAMMA1P_M1_C6);
c = t.multiply(c).add(INV_GAMMA1P_M1_C5);
c = t.multiply(c).add(INV_GAMMA1P_M1_C4);
c = t.multiply(c).add(INV_GAMMA1P_M1_C3);
c = t.multiply(c).add(INV_GAMMA1P_M1_C2);
c = t.multiply(c).add(INV_GAMMA1P_M1_C1);
c = t.multiply(c).add(INV_GAMMA1P_M1_C0);
if (x.getReal() > 0.5) {
ret = t.divide(x).multiply(c.subtract(1));
}
else {
ret = x.multiply(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 MathIllegalArgumentException if {@code x < -0.5}.
* @throws MathIllegalArgumentException if {@code x > 1.5}.
*/
public static double logGamma1p(final double x)
throws MathIllegalArgumentException {
if (x < -0.5) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL,
x, -0.5);
}
if (x > 1.5) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
x, 1.5);
}
return -FastMath.log1p(invGamma1pm1(x));
}
/**
* 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.
* @param <T> Type of the field elements.
* @return The value of {@code log(Gamma(1 + x))}.
* @throws MathIllegalArgumentException if {@code x < -0.5}.
* @throws MathIllegalArgumentException if {@code x > 1.5}.
*/
public static <T extends CalculusFieldElement<T>> T logGamma1p(final T x)
throws MathIllegalArgumentException {
if (x.getReal() < -0.5) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL,
x, -0.5);
}
if (x.getReal() > 1.5) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
x, 1.5);
}
return invGamma1pm1(x).log1p().negate();
}
/**
* 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)}.
*/
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;
}
/**
* Returns the value of Γ(x). Based on the <em>NSWC Library of
* Mathematics Subroutines</em> double precision implementation,
* {@code DGAMMA}.
*
* @param x Argument.
* @param <T> Type of the field elements.
* @return the value of {@code Gamma(x)}.
*/
public static <T extends CalculusFieldElement<T>> T gamma(final T x) {
final T one = x.getField().getOne();
if ((x.getReal() == x.rint().getReal()) && (x.getReal() <= 0.0)) {
return one.multiply(Double.NaN);
}
final T ret;
final T absX = x.abs();
if (absX.getReal() <= 20.0) {
if (x.getReal() >= 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.
*/
T prod = one;
T t = x;
while (t.getReal() > 2.5) {
t = t.subtract(1.0);
prod = prod.multiply(t);
}
ret = prod.divide(invGamma1pm1(t.subtract(1.0)).add(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.
*/
T prod = x;
T t = x;
while (t.getReal() < -0.5) {
t = t.add(1.0);
prod = prod.multiply(t);
}
ret = prod.multiply(invGamma1pm1(t).add(1)).reciprocal();
}
}
else {
final T y = absX.add(LANCZOS_G + 0.5);
final T gammaAbs = absX.reciprocal().multiply(SQRT_TWO_PI).multiply(y.pow(absX.add(0.5)))
.multiply(y.negate().exp()).multiply(lanczos(absX));
if (x.getReal() > 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 = x.multiply(x.multiply(FastMath.PI).sin()).multiply(gammaAbs).reciprocal().multiply(-FastMath.PI);
}
}
return ret;
}
}