Author: celestin
Date: Sun Nov 25 16:00:43 2012
New Revision: 1413366
URL: http://svn.apache.org/viewvc?rev=1413366&view=rev
Log:
MATH-738: in class o.a.c.m3.special.Gamma, implemented function
logGammaMinusLogGammaSum(double, double),
which computes accurately log(Gamma(b)) - log(Gamma(a + b))
for a >= 0 and b >= 8. Based on the NSWC library.
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=1413366&r1=1413365&r2=1413366&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
Sun Nov 25 16:00:43 2012
@@ -215,6 +215,68 @@ public class Gamma {
private static final double INV_GAMMA1P_M1_C13 =
-.205633841697760710345015413002057E-06;
/**
+ * <p>
+ * The d<sub>0</sub> coefficient of the minimax approximation of the Î
+ * function. This function is defined as follows
+ * </p>
+ * <center>Î(x) = log Î(x) - (x - 0.5) log a + a - 0.5 log 2Ï,</center>
+ * <p>
+ * The minimax approximation is defined by the following sum
+ * </p>
+ * <pre>
+ * 5
+ * ====
+ * \ - 2 n - 1
+ * Î(x) = > d x
+ * / n
+ * ====
+ * n = 0
+ * <pre>
+ * <p>
+ * see equationS (23) and (25) in Didonato and Morris (1992).
+ * </p>
+ */
+ private static final double D0 = .833333333333333E-01;
+
+ /**
+ * The d<sub>1</sub> coefficent of the minimax approximation of the Î
+ * function (see {@link #D0}).
+ */
+ private static final double D1 = -.277777777760991E-02;
+
+ /**
+ * The d<sub>2</sub> coefficent of the minimax approximation of the Î
+ * function (see {@link #D0}).
+ */
+ private static final double D2 = .793650666825390E-03;
+
+ /**
+ * The d<sub>3</sub> coefficent of the minimax approximation of the Î
+ * function (see {@link #D0}).
+ */
+ private static final double D3 = -.595202931351870E-03;
+
+ /**
+ * The d<sub>4</sub> coefficent of the minimax approximation of the Î
+ * function (see {@link #D0}).
+ */
+ private static final double D4 = .837308034031215E-03;
+
+ /**
+ * The d<sub>5</sub> coefficent of the minimax approximation of the Î
+ * function (see {@link #D0}).
+ */
+ private static final double D5 = -.165322962780713E-02;
+ /*
+ * NOTA: the value of d0 published in Didonato and Morris (1992), eq. (25)
+ * and the value implemented in the NSWC library are NOT EQUAL
+ * - in Didonato and Morris (1992) D5 = -.125322962780713E-02,
+ * - while in the NSWC library D5 = -.165322962780713E-02.
+ * Checking the value of algdiv(1.0, 8.0)}, it seems that the second value
+ * leads to the smallest error. This is the one which is implemented here.
+ */
+
+ /**
* Default constructor. Prohibit instantiation.
*/
private Gamma() {}
@@ -734,4 +796,70 @@ public class Gamma {
return Gamma.logGamma1p(x - 1.0) + FastMath.log(x * (1.0 + x));
}
}
+
+ /**
+ * Returns the value of log[Î(b) / Î(a + b)] for a ⥠0 and b ⥠8. The
+ * present implementation is based on the double precision implementation
in
+ * the <em>NSWC Library of Mathematics Subroutines</em>, {@code ALGDIV}.
+ *
+ * @param a First argument.
+ * @param b Second argument.
+ * @return the value of {@code log(Gamma(b) / Gamma(a + b))}.
+ * @throws NumberIsTooSmallException if {@code a < 0.0} or {@code b < 8.0}.
+ */
+ public static final double logGammaMinusLogGammaSum(final double a,
+ final double b)
+ throws NumberIsTooSmallException {
+
+ if (a < 0.0) {
+ throw new NumberIsTooSmallException(a, 0.0, true);
+ }
+ if (b < 8.0) {
+ throw new NumberIsTooSmallException(b, 8.0, true);
+ }
+
+ /*
+ * p = a / (a + b), q = b / (a + b), d = a + b - 0.5
+ */
+ final double p;
+ final double q;
+ final double d;
+ if (a <= b) {
+ final double h = a / b;
+ p = h / (1.0 + h);
+ q = 1.0 / (1.0 + h);
+ d = b + (a - 0.5);
+ } else {
+ final double h = b / a;
+ p = 1.0 / (1.0 + h);
+ q = h / (1.0 + h);
+ d = a + (b - 0.5);
+ }
+ /*
+ * s_n = 1 + q + ... + q^(n - 1)
+ */
+ final double q2 = q * q;
+ final double s3 = 1.0 + (q + q2);
+ final double s5 = 1.0 + (q + q2 * s3);
+ final double s7 = 1.0 + (q + q2 * s5);
+ final double s9 = 1.0 + (q + q2 * s7);
+ final double s11 = 1.0 + (q + q2 * s9);
+ /*
+ * w = Î(b) - Î(a + b)
+ */
+ final double tmp = 1.0 / b;
+ final double t = tmp * tmp;
+ double w = D5 * s11;
+ w = D4 * s9 + t * w;
+ w = D3 * s7 + t * w;
+ w = D2 * s5 + t * w;
+ w = D1 * s3 + t * w;
+ w = D0 + t * w;
+ w *= p / b;
+
+ final double u = d * FastMath.log1p(a / b);
+ final double v = a * (FastMath.log(b) - 1.0);
+
+ return u <= v ? (w - u) - v : (w - v) - u;
+ }
}
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=1413366&r1=1413365&r2=1413366&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
Sun Nov 25 16:00:43 2012
@@ -1075,6 +1075,156 @@ public class GammaTest {
}
}
+ private static final double[][] LOG_GAMMA_MINUS_LOG_GAMMA_SUM_REF = {
+ { 0.0 , 8.0 , 0.0 },
+ { 0.0 , 9.0 , 0.0 },
+ { 0.0 , 10.0 , 0.0 },
+ { 0.0 , 11.0 , 0.0 },
+ { 0.0 , 12.0 , 0.0 },
+ { 0.0 , 13.0 , 0.0 },
+ { 0.0 , 14.0 , 0.0 },
+ { 0.0 , 15.0 , 0.0 },
+ { 0.0 , 16.0 , 0.0 },
+ { 0.0 , 17.0 , 0.0 },
+ { 0.0 , 18.0 , 0.0 },
+ { 1.0 , 8.0 , - 2.079441541679836 },
+ { 1.0 , 9.0 , - 2.19722457733622 },
+ { 1.0 , 10.0 , - 2.302585092994046 },
+ { 1.0 , 11.0 , - 2.397895272798371 },
+ { 1.0 , 12.0 , - 2.484906649788 },
+ { 1.0 , 13.0 , - 2.564949357461537 },
+ { 1.0 , 14.0 , - 2.639057329615258 },
+ { 1.0 , 15.0 , - 2.70805020110221 },
+ { 1.0 , 16.0 , - 2.772588722239781 },
+ { 1.0 , 17.0 , - 2.833213344056216 },
+ { 1.0 , 18.0 , - 2.890371757896165 },
+ { 2.0 , 8.0 , - 4.276666119016055 },
+ { 2.0 , 9.0 , - 4.499809670330265 },
+ { 2.0 , 10.0 , - 4.700480365792417 },
+ { 2.0 , 11.0 , - 4.882801922586371 },
+ { 2.0 , 12.0 , - 5.049856007249537 },
+ { 2.0 , 13.0 , - 5.204006687076795 },
+ { 2.0 , 14.0 , - 5.347107530717468 },
+ { 2.0 , 15.0 , - 5.480638923341991 },
+ { 2.0 , 16.0 , - 5.605802066295998 },
+ { 2.0 , 17.0 , - 5.723585101952381 },
+ { 2.0 , 18.0 , - 5.834810737062605 },
+ { 3.0 , 8.0 , - 6.579251212010101 },
+ { 3.0 , 9.0 , - 6.897704943128636 },
+ { 3.0 , 10.0 , - 7.185387015580416 },
+ { 3.0 , 11.0 , - 7.447751280047908 },
+ { 3.0 , 12.0 , - 7.688913336864796 },
+ { 3.0 , 13.0 , - 7.912056888179006 },
+ { 3.0 , 14.0 , - 8.11969625295725 },
+ { 3.0 , 15.0 , - 8.313852267398207 },
+ { 3.0 , 16.0 , - 8.496173824192162 },
+ { 3.0 , 17.0 , - 8.668024081118821 },
+ { 3.0 , 18.0 , - 8.830543010616596 },
+ { 4.0 , 8.0 , - 8.977146484808472 },
+ { 4.0 , 9.0 , - 9.382611592916636 },
+ { 4.0 , 10.0 , - 9.750336373041954 },
+ { 4.0 , 11.0 , - 10.08680860966317 },
+ { 4.0 , 12.0 , - 10.39696353796701 },
+ { 4.0 , 13.0 , - 10.68464561041879 },
+ { 4.0 , 14.0 , - 10.95290959701347 },
+ { 4.0 , 15.0 , - 11.20422402529437 },
+ { 4.0 , 16.0 , - 11.4406128033586 },
+ { 4.0 , 17.0 , - 11.66375635467281 },
+ { 4.0 , 18.0 , - 11.87506544834002 },
+ { 5.0 , 8.0 , - 11.46205313459647 },
+ { 5.0 , 9.0 , - 11.94756095037817 },
+ { 5.0 , 10.0 , - 12.38939370265721 },
+ { 5.0 , 11.0 , - 12.79485881076538 },
+ { 5.0 , 12.0 , - 13.16955226020679 },
+ { 5.0 , 13.0 , - 13.517858954475 },
+ { 5.0 , 14.0 , - 13.84328135490963 },
+ { 5.0 , 15.0 , - 14.14866300446081 },
+ { 5.0 , 16.0 , - 14.43634507691259 },
+ { 5.0 , 17.0 , - 14.70827879239624 },
+ { 5.0 , 18.0 , - 14.96610790169833 },
+ { 6.0 , 8.0 , - 14.02700249205801 },
+ { 6.0 , 9.0 , - 14.58661827999343 },
+ { 6.0 , 10.0 , - 15.09744390375942 },
+ { 6.0 , 11.0 , - 15.56744753300516 },
+ { 6.0 , 12.0 , - 16.002765604263 },
+ { 6.0 , 13.0 , - 16.40823071237117 },
+ { 6.0 , 14.0 , - 16.78772033407607 },
+ { 6.0 , 15.0 , - 17.14439527801481 },
+ { 6.0 , 16.0 , - 17.48086751463602 },
+ { 6.0 , 17.0 , - 17.79932124575455 },
+ { 6.0 , 18.0 , - 18.10160211762749 },
+ { 7.0 , 8.0 , - 16.66605982167327 },
+ { 7.0 , 9.0 , - 17.29466848109564 },
+ { 7.0 , 10.0 , - 17.8700326259992 },
+ { 7.0 , 11.0 , - 18.40066087706137 },
+ { 7.0 , 12.0 , - 18.89313736215917 },
+ { 7.0 , 13.0 , - 19.35266969153761 },
+ { 7.0 , 14.0 , - 19.78345260763006 },
+ { 7.0 , 15.0 , - 20.18891771573823 },
+ { 7.0 , 16.0 , - 20.57190996799433 },
+ { 7.0 , 17.0 , - 20.9348154616837 },
+ { 7.0 , 18.0 , - 21.27965594797543 },
+ { 8.0 , 8.0 , - 19.37411002277548 },
+ { 8.0 , 9.0 , - 20.06725720333542 },
+ { 8.0 , 10.0 , - 20.70324597005542 },
+ { 8.0 , 11.0 , - 21.29103263495754 },
+ { 8.0 , 12.0 , - 21.83757634132561 },
+ { 8.0 , 13.0 , - 22.3484019650916 },
+ { 8.0 , 14.0 , - 22.82797504535349 },
+ { 8.0 , 15.0 , - 23.27996016909654 },
+ { 8.0 , 16.0 , - 23.70740418392348 },
+ { 8.0 , 17.0 , - 24.11286929203165 },
+ { 8.0 , 18.0 , - 24.49853177284363 },
+ { 9.0 , 8.0 , - 22.14669874501526 },
+ { 9.0 , 9.0 , - 22.90047054739164 },
+ { 9.0 , 10.0 , - 23.59361772795159 },
+ { 9.0 , 11.0 , - 24.23547161412398 },
+ { 9.0 , 12.0 , - 24.8333086148796 },
+ { 9.0 , 13.0 , - 25.39292440281502 },
+ { 9.0 , 14.0 , - 25.9190174987118 },
+ { 9.0 , 15.0 , - 26.41545438502569 },
+ { 9.0 , 16.0 , - 26.88545801427143 },
+ { 9.0 , 17.0 , - 27.33174511689985 },
+ { 9.0 , 18.0 , - 27.75662831086511 },
+ { 10.0 , 8.0 , - 24.97991208907148 },
+ { 10.0 , 9.0 , - 25.7908423052878 },
+ { 10.0 , 10.0 , - 26.53805670711802 },
+ { 10.0 , 11.0 , - 27.23120388767797 },
+ { 10.0 , 12.0 , - 27.87783105260302 },
+ { 10.0 , 13.0 , - 28.48396685617334 },
+ { 10.0 , 14.0 , - 29.05451171464095 },
+ { 10.0 , 15.0 , - 29.59350821537364 },
+ { 10.0 , 16.0 , - 30.10433383913963 },
+ { 10.0 , 17.0 , - 30.58984165492133 },
+ { 10.0 , 18.0 , - 31.05246517686944 },
+ };
+
+ @Test
+ public void testLogGammaMinusLogGammaSum() {
+ final int ulps = 4;
+ for (int i = 0; i < LOG_GAMMA_MINUS_LOG_GAMMA_SUM_REF.length; i++) {
+ final double[] ref = LOG_GAMMA_MINUS_LOG_GAMMA_SUM_REF[i];
+ final double a = ref[0];
+ final double b = ref[1];
+ final double expected = ref[2];
+ final double actual = Gamma.logGammaMinusLogGammaSum(a, b);
+ final double tol = ulps * FastMath.ulp(expected);
+ final StringBuilder builder = new StringBuilder();
+ builder.append(a).append(", ").append(b);
+ Assert.assertEquals(builder.toString(), expected, actual, tol);
+ }
+ }
+
+ @Test(expected = NumberIsTooSmallException.class)
+ public void testLogGammaMinusLogGammaSumPrecondition1() {
+ Gamma.logGammaMinusLogGammaSum(-1.0, 8.0);
+ }
+
+ @Test(expected = NumberIsTooSmallException.class)
+ public void testLogGammaMinusLogGammaSumPrecondition2() {
+ Gamma.logGammaMinusLogGammaSum(1.0, 7.0);
+ }
+
private void checkRelativeError(String msg, double expected, double
actual, double tolerance) {
Assert.assertEquals(msg, expected, actual, FastMath.abs(tolerance *
actual));
}
