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Posted to commits@commons.apache.org by er...@apache.org on 2017/05/07 23:57:35 UTC
[2/2] commons-numbers git commit: NUMBERS-33: "Gamma" family of
functions (port from "Commons Math").
NUMBERS-33: "Gamma" family of functions (port from "Commons Math").
Code in module "commons-numbers-gamma" is a refactoring of the utilities
contained in the "o.a.c.math4.special.Gamma" class.
Project: http://git-wip-us.apache.org/repos/asf/commons-numbers/repo
Commit: http://git-wip-us.apache.org/repos/asf/commons-numbers/commit/0c9f1cde
Tree: http://git-wip-us.apache.org/repos/asf/commons-numbers/tree/0c9f1cde
Diff: http://git-wip-us.apache.org/repos/asf/commons-numbers/diff/0c9f1cde
Branch: refs/heads/task_NUMBERS-33__Gamma
Commit: 0c9f1cde2f639d537306e8cbe6d8dd54d0374720
Parents: 3540a23
Author: Gilles Sadowski <gi...@harfang.homelinux.org>
Authored: Mon May 8 01:48:52 2017 +0200
Committer: Gilles Sadowski <gi...@harfang.homelinux.org>
Committed: Mon May 8 01:48:52 2017 +0200
----------------------------------------------------------------------
commons-numbers-gamma/LICENSE.txt | 201 +++++++
commons-numbers-gamma/NOTICE.txt | 6 +
commons-numbers-gamma/README.md | 98 ++++
commons-numbers-gamma/pom.xml | 69 +++
.../apache/commons/numbers/gamma/Digamma.java | 90 +++
.../org/apache/commons/numbers/gamma/Gamma.java | 107 ++++
.../commons/numbers/gamma/GammaException.java | 52 ++
.../commons/numbers/gamma/InvGamma1pm1.java | 187 +++++++
.../numbers/gamma/LanczosApproximation.java | 69 +++
.../apache/commons/numbers/gamma/LogGamma.java | 77 +++
.../commons/numbers/gamma/LogGamma1p.java | 47 ++
.../commons/numbers/gamma/RegularizedGamma.java | 214 +++++++
.../apache/commons/numbers/gamma/Trigamma.java | 66 +++
.../commons/numbers/gamma/package-info.java | 39 ++
commons-numbers-gamma/src/site/site.xml | 35 ++
commons-numbers-gamma/src/site/xdoc/index.xml | 40 ++
.../commons/numbers/gamma/DigammaTest.java | 72 +++
.../apache/commons/numbers/gamma/GammaTest.java | 557 +++++++++++++++++++
.../commons/numbers/gamma/InvGamma1pm1Test.java | 91 +++
.../commons/numbers/gamma/LogGamma1pTest.java | 70 +++
.../commons/numbers/gamma/LogGammaTest.java | 232 ++++++++
.../numbers/gamma/RegularizedGammaTest.java | 67 +++
.../commons/numbers/gamma/TrigammaTest.java | 59 ++
pom.xml | 2 +-
24 files changed, 2546 insertions(+), 1 deletion(-)
----------------------------------------------------------------------
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/LICENSE.txt
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/LICENSE.txt b/commons-numbers-gamma/LICENSE.txt
new file mode 100644
index 0000000..261eeb9
--- /dev/null
+++ b/commons-numbers-gamma/LICENSE.txt
@@ -0,0 +1,201 @@
+ Apache License
+ Version 2.0, January 2004
+ http://www.apache.org/licenses/
+
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+ 1. Definitions.
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+ 5. Submission of Contributions. Unless You explicitly state otherwise,
+ any Contribution intentionally submitted for inclusion in the Work
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+ 6. Trademarks. This License does not grant permission to use the trade
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+ 7. Disclaimer of Warranty. Unless required by applicable law or
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http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/NOTICE.txt
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/NOTICE.txt b/commons-numbers-gamma/NOTICE.txt
new file mode 100644
index 0000000..9091baa
--- /dev/null
+++ b/commons-numbers-gamma/NOTICE.txt
@@ -0,0 +1,6 @@
+Apache Commons Numbers
+Copyright 2001-2017 The Apache Software Foundation
+
+This product includes software developed at
+The Apache Software Foundation (http://www.apache.org/).
+
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/README.md
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/README.md b/commons-numbers-gamma/README.md
new file mode 100644
index 0000000..0701bed
--- /dev/null
+++ b/commons-numbers-gamma/README.md
@@ -0,0 +1,98 @@
+<!---
+ 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.
+-->
+<!---
+ +======================================================================+
+ |**** ****|
+ |**** THIS FILE IS GENERATED BY THE COMMONS BUILD PLUGIN ****|
+ |**** DO NOT EDIT DIRECTLY ****|
+ |**** ****|
+ +======================================================================+
+ | TEMPLATE FILE: readme-md-template.md |
+ | commons-build-plugin/trunk/src/main/resources/commons-xdoc-templates |
+ +======================================================================+
+ | |
+ | 1) Re-generate using: mvn commons:readme-md |
+ | |
+ | 2) Set the following properties in the component's pom: |
+ | - commons.componentid (required, alphabetic, lower case) |
+ | - commons.release.version (required) |
+ | |
+ | 3) Example Properties |
+ | |
+ | <properties> |
+ | <commons.componentid>math</commons.componentid> |
+ | <commons.release.version>1.2</commons.release.version> |
+ | </properties> |
+ | |
+ +======================================================================+
+--->
+Apache Commons Numbers Core
+===================
+
+Basic utilities.
+
+Documentation
+-------------
+
+More information can be found on the [homepage](https://commons.apache.org/proper/commons-numbers).
+The [JavaDoc](https://commons.apache.org/proper/commons-numbers/javadocs/api-release) can be browsed.
+Questions related to the usage of Apache Commons Numbers Core should be posted to the [user mailing list][ml].
+
+Where can I get the latest release?
+-----------------------------------
+You can download source and binaries from our [download page](https://commons.apache.org/proper/commons-numbers/download_numbers.cgi).
+
+Alternatively you can pull it from the central Maven repositories:
+
+```xml
+<dependency>
+ <groupId>org.apache.commons</groupId>
+ <artifactId>commons-numbers-core</artifactId>
+ <version>1.0</version>
+</dependency>
+```
+
+Contributing
+------------
+
+We accept PRs via github. The [developer mailing list][ml] is the main channel of communication for contributors.
+There are some guidelines which will make applying PRs easier for us:
++ No tabs! Please use spaces for indentation.
++ Respect the code style.
++ Create minimal diffs - disable on save actions like reformat source code or organize imports. If you feel the source code should be reformatted create a separate PR for this change.
++ Provide JUnit tests for your changes and make sure your changes don't break any existing tests by running ```mvn clean test```.
+
+If you plan to contribute on a regular basis, please consider filing a [contributor license agreement](https://www.apache.org/licenses/#clas).
+You can learn more about contributing via GitHub in our [contribution guidelines](CONTRIBUTING.md).
+
+License
+-------
+Code is under the [Apache Licence v2](https://www.apache.org/licenses/LICENSE-2.0.txt).
+
+Donations
+---------
+You like Apache Commons Numbers Core? Then [donate back to the ASF](https://www.apache.org/foundation/contributing.html) to support the development.
+
+Additional Resources
+--------------------
+
++ [Apache Commons Homepage](https://commons.apache.org/)
++ [Apache Bugtracker (JIRA)](https://issues.apache.org/jira/)
++ [Apache Commons Twitter Account](https://twitter.com/ApacheCommons)
++ #apachecommons IRC channel on freenode.org
+
+[ml]:https://commons.apache.org/mail-lists.html
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/pom.xml
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/pom.xml b/commons-numbers-gamma/pom.xml
new file mode 100644
index 0000000..af530fe
--- /dev/null
+++ b/commons-numbers-gamma/pom.xml
@@ -0,0 +1,69 @@
+<?xml version="1.0"?>
+<!--
+ 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.
+-->
+<project xsi:schemaLocation="http://maven.apache.org/POM/4.0.0 http://maven.apache.org/xsd/maven-4.0.0.xsd"
+ xmlns="http://maven.apache.org/POM/4.0.0"
+ xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
+ <modelVersion>4.0.0</modelVersion>
+
+ <parent>
+ <groupId>org.apache.commons</groupId>
+ <artifactId>commons-numbers-parent</artifactId>
+ <version>1.0-SNAPSHOT</version>
+ </parent>
+
+ <groupId>org.apache.commons</groupId>
+ <artifactId>commons-numbers-gamma</artifactId>
+ <version>1.0-SNAPSHOT</version>
+ <name>Apache Commons Numbers Gamma</name>
+
+ <description>Gamma family of functions.</description>
+
+ <properties>
+ <!-- This value must reflect the current name of the base package. -->
+ <commons.osgi.symbolicName>org.apache.commons.numbers.gamma</commons.osgi.symbolicName>
+ <!-- OSGi -->
+ <commons.osgi.export>org.apache.commons.numbers.gamma</commons.osgi.export>
+ <!-- Workaround to avoid duplicating config files. -->
+ <numbers.parent.dir>${basedir}/..</numbers.parent.dir>
+ </properties>
+
+ <build>
+ <plugins>
+ <plugin>
+ <groupId>org.apache.maven.plugins</groupId>
+ <artifactId>maven-jar-plugin</artifactId>
+ <executions>
+ <execution>
+ <goals>
+ <goal>test-jar</goal>
+ </goals>
+ </execution>
+ </executions>
+ </plugin>
+ </plugins>
+ </build>
+
+ <dependencies>
+ <dependency>
+ <groupId>org.apache.commons</groupId>
+ <artifactId>commons-numbers-fraction</artifactId>
+ <version>1.0-SNAPSHOT</version>
+ </dependency>
+ </dependencies>
+
+</project>
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Digamma.java
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Digamma.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Digamma.java
new file mode 100644
index 0000000..8b096d5
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Digamma.java
@@ -0,0 +1,90 @@
+/*
+ * 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.numbers.gamma;
+
+/**
+ * <a href="http://en.wikipedia.org/wiki/Digamma_function">Digamma function</a>.
+ *
+ * It is defined as the logarithmic derivative of the \( \Gamma \) function:
+ * \( \frac{d}{dx}(\ln \Gamma(x)) = \frac{\Gamma^\prime(x)}{\Gamma(x)} \).
+ */
+public class Digamma {
+ /** <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a>. */
+ private static final double GAMMA = 0.577215664901532860606512090082;
+ /** C limit. */
+ private static final double C_LIMIT = 49;
+ /** S limit. */
+ private static final double S_LIMIT = 1e-5;
+ /** Fraction. */
+ private static final double F_1_12 = 1d / 12;
+ /** Fraction. */
+ private static final double F_1_120 = 1d / 120;
+ /** Fraction. */
+ private static final double F_1_252 = 1d / 252;
+
+ /** Class only contains a static method. */
+ private Digamma() {}
+
+ /**
+ * Computes the digamma function.
+ *
+ * This is an independently written implementation of the algorithm described in
+ * <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf">Jose Bernardo,
+ * Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976</a>.
+ * A <a href="https://en.wikipedia.org/wiki/Digamma_function#Reflection_formula">
+ * reflection formula</a> is incorporated to improve performance on negative values.
+ *
+ * Some of the constants have been changed to increase accuracy at the moderate
+ * expense of run-time. The result should be accurate to within \( 10^{-8} \)
+ * relative tolerance for \( 0 < x < 10^{-5} \) and within \( 10^{-8} \) absolute
+ * tolerance otherwise.
+ *
+ * @param x Argument.
+ * @return digamma(x) to within \( 10^{-8} \) relative or absolute error whichever is larger.
+ */
+ public static double value(double x) {
+ if (Double.isNaN(x) || Double.isInfinite(x)) {
+ return x;
+ }
+
+ double digamma = 0;
+ if (x < 0) {
+ // Use reflection formula to fall back into positive values.
+ digamma -= Math.PI / Math.tan(Math.PI * x);
+ x = 1 - x;
+ }
+
+ if (x > 0 && x <= S_LIMIT) {
+ // Use method 5 from Bernardo AS103, accurate to O(x).
+ return digamma - GAMMA - 1 / x;
+ }
+
+ while (x < C_LIMIT) {
+ digamma -= 1 / x;
+ x += 1;
+ }
+
+ // Use method 4, accurate to O(1/x^8)
+ final double inv = 1 / (x * x);
+ // 1 1 1 1
+ // log(x) - --- - ------ + ------- - -------
+ // 2 x 12 x^2 120 x^4 252 x^6
+ digamma += Math.log(x) - 0.5 / x - inv * (F_1_12 + inv * (F_1_120 - F_1_252 * inv));
+
+ return digamma;
+ }
+}
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Gamma.java
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Gamma.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Gamma.java
new file mode 100644
index 0000000..797c93a
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Gamma.java
@@ -0,0 +1,107 @@
+/*
+ * 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.numbers.gamma;
+
+
+/**
+ * <a href="http://mathworld.wolfram.com/GammaFunction.html">Gamma function</a>.
+ *
+ * Class is immutable.
+ */
+public class Gamma {
+ /** \( g = \frac{607}{128} \). */
+ private static final double LANCZOS_G = 607d / 128d;
+ /** Helper. */
+ private static final LanczosApproximation LANCZOS_APPROXIMATION = LanczosApproximation.instance;
+ /** Helper. */
+ private static final InvGamma1pm1 INV_GAMMA_1P_M1 = InvGamma1pm1.instance;
+ /** √(2π). */
+ private static final double SQRT_TWO_PI = 2.506628274631000502;
+
+ /**
+ * Computes the value of \( \Gamma(x) \).
+ *
+ * Based on the <em>NSWC Library of Mathematics Subroutines</em> double
+ * precision implementation, {@code DGAMMA}.
+ *
+ * @param x Argument.
+ * @return \( \Gamma(x) \)
+ */
+ public double value(final double x) {
+
+ if ((x == Math.rint(x)) && (x <= 0.0)) {
+ return Double.NaN;
+ }
+
+ final double ret;
+ final double absX = Math.abs(x);
+ if (absX <= 20) {
+ if (x >= 1) {
+ /*
+ * From the recurrence relation
+ * Gamma(x) = (x - 1) * ... * (x - n) * Gamma(x - n),
+ * then
+ * Gamma(t) = 1 / [1 + InvGamma1pm1.value(t - 1)],
+ * where t = x - n. This means that t must satisfy
+ * -0.5 <= t - 1 <= 1.5.
+ */
+ double prod = 1;
+ double t = x;
+ while (t > 2.5) {
+ t -= 1;
+ prod *= t;
+ }
+ ret = prod / (1 + INV_GAMMA_1P_M1.value(t - 1));
+ } else {
+ /*
+ * From the recurrence relation
+ * Gamma(x) = Gamma(x + n + 1) / [x * (x + 1) * ... * (x + n)]
+ * then
+ * Gamma(x + n + 1) = 1 / [1 + InvGamma1pm1.value(x + n)],
+ * which requires -0.5 <= x + n <= 1.5.
+ */
+ double prod = x;
+ double t = x;
+ while (t < -0.5) {
+ t += 1;
+ prod *= t;
+ }
+ ret = 1 / (prod * (1 + INV_GAMMA_1P_M1.value(t)));
+ }
+ } else {
+ final double y = absX + LANCZOS_G + 0.5;
+ final double gammaAbs = SQRT_TWO_PI / absX *
+ Math.pow(y, absX + 0.5) *
+ Math.exp(-y) * LANCZOS_APPROXIMATION.value(absX);
+ if (x > 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 = -Math.PI / (x * Math.sin(Math.PI * x) * gammaAbs);
+ }
+ }
+
+ return ret;
+ }
+}
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/GammaException.java
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/GammaException.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/GammaException.java
new file mode 100644
index 0000000..7122f2e
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/GammaException.java
@@ -0,0 +1,52 @@
+/*
+ * 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.numbers.gamma;
+
+import java.text.MessageFormat;
+
+/**
+ * Package private exception class with constants for frequently used messages.
+ */
+class GammaException extends IllegalArgumentException {
+ /** Error message for "out of range" condition. */
+ static final String OUT_OF_RANGE = "Number {0} is out of range [{1}, {2}]";
+ /** Error message for convergence failure. */
+ static final String CONVERGENCE = "Failed to converge within {0} iterations";
+
+ /** Serializable version identifier. */
+ private static final long serialVersionUID = 20170505L;
+
+ /** Arguments for formatting the message. */
+ protected Object[] formatArguments;
+
+ /**
+ * Create an exception where the message is constructed by applying
+ * the {@code format()} method from {@code java.text.MessageFormat}.
+ *
+ * @param message the exception message with replaceable parameters
+ * @param formatArguments the arguments for formatting the message
+ */
+ GammaException(String message, Object... formatArguments) {
+ super(message);
+ this.formatArguments = formatArguments;
+ }
+
+ @Override
+ public String getMessage() {
+ return MessageFormat.format(super.getMessage(), formatArguments);
+ }
+}
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/InvGamma1pm1.java
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/InvGamma1pm1.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/InvGamma1pm1.java
new file mode 100644
index 0000000..77203a0
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/InvGamma1pm1.java
@@ -0,0 +1,187 @@
+/*
+ * 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.numbers.gamma;
+
+/**
+ * Function \( \frac{1}{\Gamma(1 + x)} - 1 \).
+ *
+ * Class is immutable.
+ */
+class InvGamma1pm1 {
+ /** Singleton. */
+ static final InvGamma1pm1 instance = new InvGamma1pm1();
+ /*
+ * Constants 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;
+
+ /**
+ * Computes the function \( \frac{1}{\Gamma(1 + x)} - 1 \) for \( -0.5 \leq x \leq 1.5 \).
+ *
+ * This implementation is based on the double precision implementation in
+ * the <em>NSWC Library of Mathematics Subroutines</em>, {@code DGAM1}.
+ *
+ * @param x Argument.
+ * @return \( \frac{1}{\Gamma(1 + x)} - 1 \)
+ * @throws IllegalArgumentException if {@code x < -0.5} or {@code x > 1.5}
+ */
+ public double value(final double x) {
+ if (x < -0.5 || x > 1.5) {
+ throw new GammaException(GammaException.OUT_OF_RANGE, x, -0.5, 1.5);
+ }
+
+ final double t = x <= 0.5 ? x : (x - 0.5) - 0.5;
+ if (t < 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) {
+ return t * c / x;
+ } else {
+ return 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) {
+ return (t / x) * ((c - 0.5) - 0.5);
+ } else {
+ return x * c;
+ }
+ }
+ }
+}
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LanczosApproximation.java
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LanczosApproximation.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LanczosApproximation.java
new file mode 100644
index 0000000..e21bc54
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LanczosApproximation.java
@@ -0,0 +1,69 @@
+/*
+ * 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.numbers.gamma;
+
+/**
+ * <a href="http://mathworld.wolfram.com/LanczosApproximation.html">
+ * Lanczos approximation</a> to the Gamma function.
+ *
+ * It is related to the Gamma function by the following equation
+ * \[
+ * \Gamma(x) = \sqrt{2\pi} \, \frac{(g + x + \frac{1}{2})^{x + \frac{1}{2}} \, e^{-(g + x + \frac{1}{2})} \, \mathrm{lanczos}(x)}
+ * {x}
+ * \]
+ * where \(g\) is the Lanczos constant.
+ *
+ * @see 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>.
+ */
+class LanczosApproximation {
+ /** Singleton. */
+ static final LanczosApproximation instance = new LanczosApproximation();
+ /** 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,
+ };
+
+ /**
+ * Computes the Lanczos approximation.
+ *
+ * @param x Argument.
+ * @return the Lanczos approximation.
+ */
+ public double value(final double x) {
+ double sum = 0;
+ for (int i = LANCZOS.length - 1; i > 0; i--) {
+ sum += LANCZOS[i] / (x + i);
+ }
+ return sum + LANCZOS[0];
+ }
+}
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LogGamma.java
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LogGamma.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LogGamma.java
new file mode 100644
index 0000000..57146a2
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LogGamma.java
@@ -0,0 +1,77 @@
+/*
+ * 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.numbers.gamma;
+
+/**
+ * Function \( \ln \Gamma(x) \).
+ *
+ * Class is immutable.
+ */
+public class LogGamma {
+ /** Singleton. */
+ static final LogGamma instance = new LogGamma();
+ /** \( g = \frac{607}{128} \). */
+ private static final double LANCZOS_G = 607d / 128d;
+ /** Performance. */
+ private static final double HALF_LOG_2_PI = 0.5 * Math.log(2.0 * Math.PI);
+ /** Helper. */
+ private static final LanczosApproximation LANCZOS_APPROXIMATION = LanczosApproximation.instance;
+ /** Helper. */
+ private static final LogGamma1p LOG_GAMMA_1P = LogGamma1p.instance;
+
+ /**
+ * Computes the function \( \ln \Gamma(x) \) for \( x > 0 \).
+ *
+ * For \( x \leq 8 \), the implementation is based on the double precision
+ * implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
+ * {@code DGAMLN}. For \( x \geq 8 \), the implementation is based on
+ *
+ * <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 \( \ln \Gamma(x) \), or {@code NaN} if {@code x <= 0}.
+ */
+ public double value(double x) {
+ if (Double.isNaN(x) || (x <= 0.0)) {
+ return Double.NaN;
+ } else if (x < 0.5) {
+ return LOG_GAMMA_1P.value(x) - Math.log(x);
+ } else if (x <= 2.5) {
+ return LOG_GAMMA_1P.value((x - 0.5) - 0.5);
+ } else if (x <= 8.0) {
+ final int n = (int) Math.floor(x - 1.5);
+ double prod = 1.0;
+ for (int i = 1; i <= n; i++) {
+ prod *= x - i;
+ }
+ return LOG_GAMMA_1P.value(x - (n + 1)) + Math.log(prod);
+ } else {
+ final double sum = LANCZOS_APPROXIMATION.value(x);
+ final double tmp = x + LANCZOS_G + .5;
+ return ((x + .5) * Math.log(tmp)) - tmp +
+ HALF_LOG_2_PI + Math.log(sum / x);
+ }
+ }
+}
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LogGamma1p.java
----------------------------------------------------------------------
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LogGamma1p.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LogGamma1p.java
new file mode 100644
index 0000000..2f26fcc
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/LogGamma1p.java
@@ -0,0 +1,47 @@
+/*
+ * 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.numbers.gamma;
+
+/**
+ * Function \( \ln \Gamma(1 + x) \).
+ *
+ * Class is immutable.
+ */
+class LogGamma1p {
+ /** Singleton. */
+ static final LogGamma1p instance = new LogGamma1p();
+ /** Helper. */
+ private static final InvGamma1pm1 invGamma1pm1 = InvGamma1pm1.instance;
+
+ /**
+ * Computes the function \( \ln \Gamma(1 + x) \) for \( -0.5 \leq x \leq 1.5 \).
+ *
+ * This implementation is based on the double precision implementation in
+ * the <em>NSWC Library of Mathematics Subroutines</em>, {@code DGMLN1}.
+ *
+ * @param x Argument.
+ * @return \( \ln \Gamma(1 + x) \)
+ * @throws IllegalArgumentException if {@code x < -0.5} or {@code x > 1.5}.
+ */
+ public double value(final double x) {
+ if (x < -0.5 || x > 1.5) {
+ throw new GammaException(GammaException.OUT_OF_RANGE, x, -0.5, 1.5);
+ }
+
+ return -Math.log1p(invGamma1pm1.value(x));
+ }
+}
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/RegularizedGamma.java
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diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/RegularizedGamma.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/RegularizedGamma.java
new file mode 100644
index 0000000..7f39690
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/RegularizedGamma.java
@@ -0,0 +1,214 @@
+/*
+ * 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.numbers.gamma;
+
+import org.apache.commons.numbers.fraction.ContinuedFraction;
+
+/**
+ * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
+ * Regularized Gamma functions</a>.
+ *
+ * Class is immutable.
+ */
+public abstract class RegularizedGamma {
+ /** Helper. */
+ private static final LogGamma LOG_GAMMA = LogGamma.instance;
+ /** Maximum allowed numerical error. */
+ private static final double DEFAULT_EPSILON = 10e-15; // XXX
+
+ /**
+ * @param a Parameter.
+ * @param x Argument.
+ * @param epsilon When the absolute value of the n-th element in the
+ * series is less than epsilon the approximation ceases to calculate
+ * further elements in the series.
+ * @param maxIterations Maximum number of iterations.
+ * @return the value of the function.
+ * @throws IllegalArgumentException if the algorithm fails to converge.
+ */
+ public abstract double value(double a,
+ double x,
+ double epsilon,
+ int maxIterations);
+
+ /**
+ * @param a Parameter.
+ * @param x Argument.
+ * @return the value of the function.
+ * @throws IllegalArgumentException if the algorithm fails to converge.
+ */
+ public abstract double value(double a,
+ double x);
+
+ /**
+ * \( P(a, x) \) <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
+ * regularized Gamma function</a>.
+ *
+ * Class is immutable.
+ */
+ public static class P {
+ /**
+ * Computes the regularized gamma function \( P(a, x) \).
+ *
+ * {@inheritDoc}
+ */
+ public double value(double a,
+ double x) {
+ return value(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
+ }
+
+ /**
+ * Computes the regularized gamma function \( P(a, x) \).
+ *
+ * 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>
+ *
+ * {@inheritDoc}
+ */
+ public double value(double a,
+ double x,
+ double epsilon,
+ int maxIterations) {
+ double ret;
+
+ if (Double.isNaN(a) ||
+ Double.isNaN(x) ||
+ a <= 0 ||
+ x < 0) {
+ ret = Double.NaN;
+ } else if (x == 0) {
+ ret = 0;
+ } else if (x >= a + 1) {
+ // Q should converge faster in this case.
+ final RegularizedGamma.Q q = new RegularizedGamma.Q();
+ ret = 1 - q.value(a, x, epsilon, maxIterations);
+ } else {
+ // Series.
+ double n = 0; // current element index
+ double an = 1 / a; // n-th element in the series
+ double sum = an; // partial sum
+ while (Math.abs(an / sum) > epsilon &&
+ n < maxIterations &&
+ sum < Double.POSITIVE_INFINITY) {
+ // compute next element in the series
+ n += 1;
+ an *= x / (a + n);
+
+ // update partial sum
+ sum += an;
+ }
+ if (n >= maxIterations) {
+ throw new GammaException(GammaException.CONVERGENCE, maxIterations);
+ } else if (Double.isInfinite(sum)) {
+ ret = 1;
+ } else {
+ ret = Math.exp(-x + (a * Math.log(x)) - LOG_GAMMA.value(a)) * sum;
+ }
+ }
+
+ return ret;
+ }
+ }
+
+ /**
+ * Creates the \( Q(a, x) \equiv 1 - P(a, x) \) \) <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
+ * regularized Gamma function</a>.
+ *
+ * Class is immutable.
+ */
+ public static class Q {
+ /**
+ * Computes the regularized gamma function \( Q(a, x) = 1 - P(a, x) \).
+ *
+ * {@inheritDoc}
+ */
+ public double value(double a,
+ double x) {
+ return value(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
+ }
+
+ /**
+ * Computes the regularized gamma function \( Q(a, x) = 1 - P(a, x) \).
+ *
+ * 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>
+ *
+ * {@inheritDoc}
+ */
+ public double value(final double a,
+ double x,
+ double epsilon,
+ int maxIterations) {
+ double ret;
+
+ if (Double.isNaN(a) ||
+ Double.isNaN(x) ||
+ a <= 0 ||
+ x < 0) {
+ ret = Double.NaN;
+ } else if (x == 0) {
+ ret = 1;
+ } else if (x < a + 1) {
+ // P should converge faster in this case.
+ final RegularizedGamma.P p = new RegularizedGamma.P();
+ ret = 1 - p.value(a, x, epsilon, maxIterations);
+ } else {
+ final ContinuedFraction cf = new ContinuedFraction() {
+ /** {@inheritDoc} */
+ @Override
+ protected double getA(int n, double x) {
+ return ((2 * n) + 1) - a + x;
+ }
+
+ /** {@inheritDoc} */
+ @Override
+ protected double getB(int n, double x) {
+ return n * (a - n);
+ }
+ };
+
+ ret = 1 / cf.evaluate(x, epsilon, maxIterations);
+ ret = Math.exp(-x + (a * Math.log(x)) - LOG_GAMMA.value(a)) * ret;
+ }
+
+ return ret;
+ }
+ }
+}
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Trigamma.java
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diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Trigamma.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Trigamma.java
new file mode 100644
index 0000000..9e45601
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Trigamma.java
@@ -0,0 +1,66 @@
+/*
+ * 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.numbers.gamma;
+
+/**
+ * <a href="http://en.wikipedia.org/wiki/Trigamma_function">Trigamma function</a>.
+ *
+ * It is the derivative of the {@link Digamma digamma function}:
+ * \( \psi_1(x) = \frac{d^2}{dx^2} (\ln \Gamma(x)) \).
+ */
+public class Trigamma {
+ /** C limit. */
+ private static final double C_LIMIT = 49;
+ /** S limit. */
+ private static final double S_LIMIT = 1e-5;
+ /** Fraction. */
+ private static final double F_1_6 = 1d / 6;
+ /** Fraction. */
+ private static final double F_1_30 = 1d / 30;
+ /** Fraction. */
+ private static final double F_1_42 = 1d / 42;
+
+ /** Class only contains a static method. */
+ private Trigamma() {}
+
+ /**
+ * Computes the trigamma function.
+ *
+ * @param x Argument.
+ * @return trigamma(x) to within \( 10^{-8} \) relative or absolute error whichever is larger.
+ */
+ public static double value(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) {
+ final 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 * (F_1_6 - inv * (F_1_30 + F_1_42 * inv));
+ }
+
+ return value(x + 1) + 1 / (x * x);
+ }
+}
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/package-info.java
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diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/package-info.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/package-info.java
new file mode 100644
index 0000000..b44caa1
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/package-info.java
@@ -0,0 +1,39 @@
+/*
+ * 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.
+ */
+
+/**
+ * Γ (Gamma) family of functions.
+ *
+ * Implementation of {@link org.apache.commons.numbers.gamma.InvGamma1pm1 InvGamma1pm1}
+ * and {@link org.apache.commons.numbers.gamma.LogGamma1p LogGamma1p} 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>.
+ */
+package org.apache.commons.numbers.gamma;
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/site/site.xml
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diff --git a/commons-numbers-gamma/src/site/site.xml b/commons-numbers-gamma/src/site/site.xml
new file mode 100644
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+++ b/commons-numbers-gamma/src/site/site.xml
@@ -0,0 +1,35 @@
+<?xml version="1.0" encoding="ISO-8859-1"?>
+<!--
+ 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.
+-->
+<project name="Numbers">
+ <bannerRight>
+ <name>Apache Commons Numbers</name>
+ <src>/images/commons_numbers.small.png</src>
+ <href>/index.html</href>
+ </bannerRight>
+
+ <body>
+ <menu name="Numbers Gamma">
+ <item name="Overview" href="index.html"/>
+ <item name="Latest API docs (development)"
+ href="apidocs/index.html"/>
+ <!--item name="Javadoc (1.0 release)"
+ href="http://commons.apache.org/rng/commons-numbers-gamma/javadocs/api-1.0/index.html"/-->
+ </menu>
+
+ </body>
+</project>
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/site/xdoc/index.xml
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diff --git a/commons-numbers-gamma/src/site/xdoc/index.xml b/commons-numbers-gamma/src/site/xdoc/index.xml
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--- /dev/null
+++ b/commons-numbers-gamma/src/site/xdoc/index.xml
@@ -0,0 +1,40 @@
+<?xml version="1.0"?>
+
+<!--
+ 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.
+ -->
+
+<document>
+
+ <properties>
+ <title>Commons Numbers Core</title>
+ </properties>
+
+ <body>
+
+ <section name="Apache Commons Numbers: Number types" href="summary">
+ <p>
+ Commons Numbers provides utilities such as complex numbers and fractions.
+ </p>
+
+ <p>
+ The "gamma" module contains utilities related to the "Gamma" function.
+ </p>
+ </section>
+
+ </body>
+
+</document>
http://git-wip-us.apache.org/repos/asf/commons-numbers/blob/0c9f1cde/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/DigammaTest.java
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diff --git a/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/DigammaTest.java b/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/DigammaTest.java
new file mode 100644
index 0000000..17b459a
--- /dev/null
+++ b/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/DigammaTest.java
@@ -0,0 +1,72 @@
+/*
+ * 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.numbers.gamma;
+
+import org.junit.Assert;
+import org.junit.Test;
+
+/**
+ * Tests for {@link Digamma}.
+ */
+public class DigammaTest {
+ @Test
+ public void testDigammaLargeArgs() {
+ double eps = 1e-8;
+ Assert.assertEquals(4.6001618527380874002, Digamma.value(100), eps);
+ Assert.assertEquals(3.9019896734278921970, Digamma.value(50), eps);
+ Assert.assertEquals(2.9705239922421490509, Digamma.value(20), eps);
+ Assert.assertEquals(2.9958363947076465821, Digamma.value(20.5), eps);
+ Assert.assertEquals(2.2622143570941481605, Digamma.value(10.1), eps);
+ Assert.assertEquals(2.1168588189004379233, Digamma.value(8.8), eps);
+ Assert.assertEquals(1.8727843350984671394, Digamma.value(7), eps);
+ Assert.assertEquals(0.42278433509846713939, Digamma.value(2), eps);
+ Assert.assertEquals(-100.56088545786867450, Digamma.value(0.01), eps);
+ Assert.assertEquals(-4.0390398965921882955, Digamma.value(-0.8), eps);
+ Assert.assertEquals(4.2003210041401844726, Digamma.value(-6.3), eps);
+ Assert.assertEquals(-3.110625123035E-5, Digamma.value(1.4616), eps);
+ }
+
+ @Test
+ public void testDigammaSmallArgs() {
+ // values for negative powers of 10 from 1 to 30 as computed by webMathematica with 20 digits
+ // see functions.wolfram.com
+ double[] expected = {-10.423754940411076795, -100.56088545786867450, -1000.5755719318103005,
+ -10000.577051183514335, -100000.57719921568107, -1.0000005772140199687e6, -1.0000000577215500408e7,
+ -1.0000000057721564845e8, -1.0000000005772156633e9, -1.0000000000577215665e10, -1.0000000000057721566e11,
+ -1.0000000000005772157e12, -1.0000000000000577216e13, -1.0000000000000057722e14, -1.0000000000000005772e15, -1e+16,
+ -1e+17, -1e+18, -1e+19, -1e+20, -1e+21, -1e+22, -1e+23, -1e+24, -1e+25, -1e+26,
+ -1e+27, -1e+28, -1e+29, -1e+30};
+ for (double n = 1; n < 30; n++) {
+ checkRelativeError(String.format("Test %.0f: ", n), expected[(int) (n - 1)], Digamma.value(Math.pow(10.0, -n)), 1e-8);
+ }
+ }
+
+ @Test
+ public void testDigammaNonRealArgs() {
+ Assert.assertTrue(Double.isNaN(Digamma.value(Double.NaN)));
+ Assert.assertTrue(Double.isInfinite(Digamma.value(Double.POSITIVE_INFINITY)));
+ Assert.assertTrue(Double.isInfinite(Digamma.value(Double.NEGATIVE_INFINITY)));
+ }
+
+ private void checkRelativeError(String msg,
+ double expected,
+ double actual,
+ double tolerance) {
+ Assert.assertEquals(msg, expected, actual, Math.abs(tolerance * actual));
+ }
+}
+