Author: celestin
Date: Wed Aug 29 06:20:21 2012
New Revision: 1378450
URL: http://svn.apache.org/viewvc?rev=1378450&view=rev
Log:
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.
Modified:
commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java
commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java
Modified:
commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java
URL:
http://svn.apache.org/viewvc/commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java?rev=1378450&r1=1378449&r2=1378450&view=diff
==============================================================================
---
commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java
(original)
+++
commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java
Wed Aug 29 06:20:21 2012
@@ -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.
+ * 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 $Id$
*/
@@ -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 Γ(x).
- *
- * The implementation of this method is based on:
+ * <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/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(Γ(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 / 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 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 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 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));
+ }
}
Modified:
commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java
URL:
http://svn.apache.org/viewvc/commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java?rev=1378450&r1=1378449&r2=1378450&view=diff
==============================================================================
---
commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java
(original)
+++
commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java
Wed Aug 29 06:20:21 2012
@@ -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];
