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MATH-849: new implementation of double Gamma.logGamma(double x) for x < 8.0. 

This greatly improves the accurarcy, from more than 130 ulps down to 3 ulps.
Unit tests updated accordingly.



git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1378450 13f79535-47bb-0310-9956-ffa450edef68
This commit is contained in:
Sebastien Brisard 2012-08-29 06:20:21 +00:00
parent 6d0dcae172
commit b6bf913d14
2 changed files with 282 additions and 12 deletions
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292
src/main/java/org/apache/commons/math3/special/Gamma.java
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@ -17,12 +17,37 @@
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
* Gamma family of functions.
* &Gamma; (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 $Id$
*/
@ -67,33 +92,163 @@ public class Gamma {
/** 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 natural logarithm of the gamma function &#915;(x).
*
* The implementation of this method is based on:
* <p>
* Returns the value of log&nbsp;&Gamma;(x) for x&nbsp;&gt;&nbsp;0.
* </p>
* <p>
* For x &lt; 8, the implementation is based on the double precision
* implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
* {@code DGAMLN}. For x &ge; 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/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>
* 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>
* the computation of the convergent Lanczos complex Gamma
* approximation</a></li>
* </ul>
*
* @param x Value.
* @return log(&#915;(x))
* @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;
@ -352,4 +507,119 @@ public class Gamma {
}
return sum + LANCZOS[0];
}
/**
* Returns the value of 1 / &Gamma;(1 + x) - 1 for -0&#46;5 &le; x &le;
* 1&#46;5. This implementation is based on the double precision
* implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
* {@code DGAM1}.
*
* @param x the 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}
*/
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 &Gamma;(1 + x) for -0&#46;5 &le; x &le; 1&#46;5.
* This implementation is based on the double precision implementation in
* the <em>NSWC Library of Mathematics Subroutines</em>, {@code DGMLN1}.
*
* @param x the argument
* @return the value of {@code log(Gamma(1 + x))}
* @throws NumberIsTooSmallException if {@code x < -0.5}
* @throws NumberIsTooLargeException if {@code x > 1.5}
*/
public static double logGamma1p(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);
}
return -FastMath.log1p(invGamma1pm1(x));
}
}

2
src/test/java/org/apache/commons/math3/special/GammaTest.java
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@ -262,7 +262,7 @@ public class GammaTest {
@Test
public void testLogGamma() {
final int ulps = 130;
final int ulps = 3;
for (int i = 0; i < LOG_GAMMA_REF.length; i++) {
final double[] data = LOG_GAMMA_REF[i];
final double x = data[0];