GammaEqualityLemma.java
- package org.drip.specialfunction.property;
- /*
- * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
- */
- /*!
- * Copyright (C) 2020 Lakshmi Krishnamurthy
- * Copyright (C) 2019 Lakshmi Krishnamurthy
- *
- * This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
- * asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
- * analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
- * equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
- * numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
- * and computational support.
- *
- * https://lakshmidrip.github.io/DROP/
- *
- * DROP is composed of three modules:
- *
- * - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
- * - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
- * - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
- *
- * DROP Product Core implements libraries for the following:
- * - Fixed Income Analytics
- * - Loan Analytics
- * - Transaction Cost Analytics
- *
- * DROP Portfolio Core implements libraries for the following:
- * - Asset Allocation Analytics
- * - Asset Liability Management Analytics
- * - Capital Estimation Analytics
- * - Exposure Analytics
- * - Margin Analytics
- * - XVA Analytics
- *
- * DROP Computational Core implements libraries for the following:
- * - Algorithm Support
- * - Computation Support
- * - Function Analysis
- * - Model Validation
- * - Numerical Analysis
- * - Numerical Optimizer
- * - Spline Builder
- * - Statistical Learning
- *
- * Documentation for DROP is Spread Over:
- *
- * - Main => https://lakshmidrip.github.io/DROP/
- * - Wiki => https://github.com/lakshmiDRIP/DROP/wiki
- * - GitHub => https://github.com/lakshmiDRIP/DROP
- * - Repo Layout Taxonomy => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
- * - Javadoc => https://lakshmidrip.github.io/DROP/Javadoc/index.html
- * - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
- * - Release Versions => https://lakshmidrip.github.io/DROP/version.html
- * - Community Credits => https://lakshmidrip.github.io/DROP/credits.html
- * - Issues Catalog => https://github.com/lakshmiDRIP/DROP/issues
- * - JUnit => https://lakshmidrip.github.io/DROP/junit/index.html
- * - Jacoco => https://lakshmidrip.github.io/DROP/jacoco/index.html
- *
- * Licensed 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.
- */
- /**
- * <i>GammaEqualityLemma</i> contains the Verifiable Equality Lemmas of the Gamma Function. The References
- * are:
- *
- * <br><br>
- * <ul>
- * <li>
- * Blagouchine, I. V. (2014): Re-discovery of Malmsten's Integrals, their Evaluation by Contour
- * Integration Methods, and some Related Results <i>Ramanujan Journal</i> <b>35 (1)</b> 21-110
- * </li>
- * <li>
- * Borwein, J. M., and R. M. Corless (2017): Gamma Function and the Factorial in the Monthly
- * https://arxiv.org/abs/1703.05349 <b>arXiv</b>
- * </li>
- * <li>
- * Davis, P. J. (1959): Leonhard Euler's Integral: A Historical Profile of the Gamma Function
- * <i>American Mathematical Monthly</i> <b>66 (10)</b> 849-869
- * </li>
- * <li>
- * Whitaker, E. T., and G. N. Watson (1996): <i>A Course on Modern Analysis</i> <b>Cambridge
- * University Press</b> New York
- * </li>
- * <li>
- * Wikipedia (2019): Gamma Function https://en.wikipedia.org/wiki/Gamma_function
- * </li>
- * </ul>
- *
- * <br><br>
- * <ul>
- * <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
- * <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/FunctionAnalysisLibrary.md">Function Analysis Library</a></li>
- * <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/specialfunction/README.md">Special Function Implementation Analysis</a></li>
- * <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/specialfunction/property/README.md">Special Function Property Lemma Verifiers</a></li>
- * </ul>
- *
- * @author Lakshmi Krishnamurthy
- */
- public class GammaEqualityLemma
- {
- /**
- * Construct the Reflection Formula Verifier
- *
- * @return The Reflection Formula Verifier
- */
- public static final org.drip.function.definition.R1ToR1Property ReflectionFormula()
- {
- try
- {
- return new org.drip.function.definition.R1ToR1Property (
- org.drip.function.definition.R1ToR1Property.EQ,
- new org.drip.function.definition.R1ToR1 (null)
- {
- @Override public double evaluate (
- final double s)
- throws java.lang.Exception
- {
- if (!org.drip.numerical.common.NumberUtil.IsValid (s))
- {
- throw new java.lang.Exception
- ("GammaEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
- }
- org.drip.specialfunction.loggamma.InfiniteSumEstimator weierstrassInfiniteProduct =
- org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);
- return java.lang.Math.exp (weierstrassInfiniteProduct.evaluate (1. - s) +
- weierstrassInfiniteProduct.evaluate (s));
- }
- },
- new org.drip.function.definition.R1ToR1 (null)
- {
- @Override public double evaluate (
- final double s)
- throws java.lang.Exception
- {
- if (!org.drip.numerical.common.NumberUtil.IsValid (s))
- {
- throw new java.lang.Exception
- ("GammaEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
- }
- return java.lang.Math.PI / java.lang.Math.sin (java.lang.Math.PI * s);
- }
- },
- org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
- );
- }
- catch (java.lang.Exception e)
- {
- e.printStackTrace();
- }
- return null;
- }
- /**
- * Construct the Duplication Formula Verifier
- *
- * @return The Duplication Formula Verifier
- */
- public static final org.drip.function.definition.R1ToR1Property DuplicationFormula()
- {
- try
- {
- return new org.drip.function.definition.R1ToR1Property (
- org.drip.function.definition.R1ToR1Property.EQ,
- new org.drip.function.definition.R1ToR1 (null)
- {
- @Override public double evaluate (
- final double s)
- throws java.lang.Exception
- {
- if (!org.drip.numerical.common.NumberUtil.IsValid (s))
- {
- throw new java.lang.Exception
- ("GammaEqualityLemma::DuplicationFormula::evaluate => Invalid Inputs");
- }
- org.drip.specialfunction.loggamma.InfiniteSumEstimator weierstrassInfiniteProduct =
- org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);
- return weierstrassInfiniteProduct.evaluate (s) + weierstrassInfiniteProduct.evaluate
- (s + 0.5);
- }
- },
- new org.drip.function.definition.R1ToR1 (null)
- {
- @Override public double evaluate (
- final double s)
- throws java.lang.Exception
- {
- if (!org.drip.numerical.common.NumberUtil.IsValid (s))
- {
- throw new java.lang.Exception
- ("GammaEqualityLemma::DuplicationFormula::evaluate => Invalid Inputs");
- }
- return (1. - 2. * s) * java.lang.Math.log (2.) +
- 0.5 * java.lang.Math.log (java.lang.Math.PI) +
- org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400).evaluate (2. * s);
- }
- },
- org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
- );
- }
- catch (java.lang.Exception e)
- {
- e.printStackTrace();
- }
- return null;
- }
- /**
- * Construct the Multiplication Formula Verifier
- *
- * @param m m
- *
- * @return The Multiplication Formula Verifier
- */
- public static final org.drip.function.definition.R1ToR1Property MultiplicationFormula (
- final int m)
- {
- if (1 >= m)
- {
- return null;
- }
- try
- {
- return new org.drip.function.definition.R1ToR1Property (
- org.drip.function.definition.R1ToR1Property.EQ,
- new org.drip.function.definition.R1ToR1 (null)
- {
- @Override public double evaluate (
- final double s)
- throws java.lang.Exception
- {
- if (!org.drip.numerical.common.NumberUtil.IsValid (s))
- {
- throw new java.lang.Exception
- ("GammaEqualityLemma::MultiplicationFormula::evaluate => Invalid Inputs");
- }
- double logGammaSum = 0.;
- org.drip.specialfunction.loggamma.InfiniteSumEstimator weierstrassInfiniteProduct =
- org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);
- for (double i = 0; i < m; ++i)
- {
- logGammaSum = logGammaSum + weierstrassInfiniteProduct.evaluate (s + (i / m));
- }
- return logGammaSum;
- }
- },
- new org.drip.function.definition.R1ToR1 (null)
- {
- @Override public double evaluate (
- final double s)
- throws java.lang.Exception
- {
- if (!org.drip.numerical.common.NumberUtil.IsValid (s))
- {
- throw new java.lang.Exception
- ("GammaEqualityLemma::MultiplicationFormula::evaluate => Invalid Inputs");
- }
- return 0.5 * (m - 1.) * java.lang.Math.log (2. * java.lang.Math.PI) +
- (0.5 - m * s) * java.lang.Math.log (m) +
- org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400).evaluate (m * s);
- }
- },
- org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
- );
- }
- catch (java.lang.Exception e)
- {
- e.printStackTrace();
- }
- return null;
- }
- }