GammaEqualityLemma.java

  1. package org.drip.specialfunction.property;

  2. /*
  3.  * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
  4.  */

  5. /*!
  6.  * Copyright (C) 2020 Lakshmi Krishnamurthy
  7.  * Copyright (C) 2019 Lakshmi Krishnamurthy
  8.  *
  9.  *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
  10.  *      asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
  11.  *      analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
  12.  *      equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
  13.  *      numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
  14.  *      and computational support.
  15.  *  
  16.  *      https://lakshmidrip.github.io/DROP/
  17.  *  
  18.  *  DROP is composed of three modules:
  19.  *  
  20.  *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
  21.  *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
  22.  *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
  23.  *
  24.  *  DROP Product Core implements libraries for the following:
  25.  *  - Fixed Income Analytics
  26.  *  - Loan Analytics
  27.  *  - Transaction Cost Analytics
  28.  *
  29.  *  DROP Portfolio Core implements libraries for the following:
  30.  *  - Asset Allocation Analytics
  31.  *  - Asset Liability Management Analytics
  32.  *  - Capital Estimation Analytics
  33.  *  - Exposure Analytics
  34.  *  - Margin Analytics
  35.  *  - XVA Analytics
  36.  *
  37.  *  DROP Computational Core implements libraries for the following:
  38.  *  - Algorithm Support
  39.  *  - Computation Support
  40.  *  - Function Analysis
  41.  *  - Model Validation
  42.  *  - Numerical Analysis
  43.  *  - Numerical Optimizer
  44.  *  - Spline Builder
  45.  *  - Statistical Learning
  46.  *
  47.  *  Documentation for DROP is Spread Over:
  48.  *
  49.  *  - Main                     => https://lakshmidrip.github.io/DROP/
  50.  *  - Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
  51.  *  - GitHub                   => https://github.com/lakshmiDRIP/DROP
  52.  *  - Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
  53.  *  - Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
  54.  *  - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
  55.  *  - Release Versions         => https://lakshmidrip.github.io/DROP/version.html
  56.  *  - Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
  57.  *  - Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
  58.  *  - JUnit                    => https://lakshmidrip.github.io/DROP/junit/index.html
  59.  *  - Jacoco                   => https://lakshmidrip.github.io/DROP/jacoco/index.html
  60.  *
  61.  *  Licensed under the Apache License, Version 2.0 (the "License");
  62.  *      you may not use this file except in compliance with the License.
  63.  *  
  64.  *  You may obtain a copy of the License at
  65.  *      http://www.apache.org/licenses/LICENSE-2.0
  66.  *  
  67.  *  Unless required by applicable law or agreed to in writing, software
  68.  *      distributed under the License is distributed on an "AS IS" BASIS,
  69.  *      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  70.  *  
  71.  *  See the License for the specific language governing permissions and
  72.  *      limitations under the License.
  73.  */

  74. /**
  75.  * <i>GammaEqualityLemma</i> contains the Verifiable Equality Lemmas of the Gamma Function. The References
  76.  * are:
  77.  *
  78.  * <br><br>
  79.  *  <ul>
  80.  *      <li>
  81.  *          Blagouchine, I. V. (2014): Re-discovery of Malmsten's Integrals, their Evaluation by Contour
  82.  *              Integration Methods, and some Related Results <i>Ramanujan Journal</i> <b>35 (1)</b> 21-110
  83.  *      </li>
  84.  *      <li>
  85.  *          Borwein, J. M., and R. M. Corless (2017): Gamma Function and the Factorial in the Monthly
  86.  *              https://arxiv.org/abs/1703.05349 <b>arXiv</b>
  87.  *      </li>
  88.  *      <li>
  89.  *          Davis, P. J. (1959): Leonhard Euler's Integral: A Historical Profile of the Gamma Function
  90.  *              <i>American Mathematical Monthly</i> <b>66 (10)</b> 849-869
  91.  *      </li>
  92.  *      <li>
  93.  *          Whitaker, E. T., and G. N. Watson (1996): <i>A Course on Modern Analysis</i> <b>Cambridge
  94.  *              University Press</b> New York
  95.  *      </li>
  96.  *      <li>
  97.  *          Wikipedia (2019): Gamma Function https://en.wikipedia.org/wiki/Gamma_function
  98.  *      </li>
  99.  *  </ul>
  100.  *
  101.  *  <br><br>
  102.  *  <ul>
  103.  *      <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
  104.  *      <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/FunctionAnalysisLibrary.md">Function Analysis Library</a></li>
  105.  *      <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>
  106.  *      <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>
  107.  *  </ul>
  108.  *
  109.  * @author Lakshmi Krishnamurthy
  110.  */

  111. public class GammaEqualityLemma
  112. {

  113.     /**
  114.      * Construct the Reflection Formula Verifier
  115.      *
  116.      * @return The Reflection Formula Verifier
  117.      */

  118.     public static final org.drip.function.definition.R1ToR1Property ReflectionFormula()
  119.     {
  120.         try
  121.         {
  122.             return new org.drip.function.definition.R1ToR1Property (
  123.                 org.drip.function.definition.R1ToR1Property.EQ,
  124.                 new org.drip.function.definition.R1ToR1 (null)
  125.                 {
  126.                     @Override public double evaluate (
  127.                         final double s)
  128.                         throws java.lang.Exception
  129.                     {
  130.                         if (!org.drip.numerical.common.NumberUtil.IsValid (s))
  131.                         {
  132.                             throw new java.lang.Exception
  133.                                 ("GammaEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
  134.                         }

  135.                         org.drip.specialfunction.loggamma.InfiniteSumEstimator weierstrassInfiniteProduct =
  136.                             org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);

  137.                         return java.lang.Math.exp (weierstrassInfiniteProduct.evaluate (1. - s) +
  138.                             weierstrassInfiniteProduct.evaluate (s));
  139.                     }
  140.                 },
  141.                 new org.drip.function.definition.R1ToR1 (null)
  142.                 {
  143.                     @Override public double evaluate (
  144.                         final double s)
  145.                         throws java.lang.Exception
  146.                     {
  147.                         if (!org.drip.numerical.common.NumberUtil.IsValid (s))
  148.                         {
  149.                             throw new java.lang.Exception
  150.                                 ("GammaEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
  151.                         }

  152.                         return java.lang.Math.PI / java.lang.Math.sin (java.lang.Math.PI * s);
  153.                     }
  154.                 },
  155.                 org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
  156.             );
  157.         }
  158.         catch (java.lang.Exception e)
  159.         {
  160.             e.printStackTrace();
  161.         }

  162.         return null;
  163.     }

  164.     /**
  165.      * Construct the Duplication Formula Verifier
  166.      *
  167.      * @return The Duplication Formula Verifier
  168.      */

  169.     public static final org.drip.function.definition.R1ToR1Property DuplicationFormula()
  170.     {
  171.         try
  172.         {
  173.             return new org.drip.function.definition.R1ToR1Property (
  174.                 org.drip.function.definition.R1ToR1Property.EQ,
  175.                 new org.drip.function.definition.R1ToR1 (null)
  176.                 {
  177.                     @Override public double evaluate (
  178.                         final double s)
  179.                         throws java.lang.Exception
  180.                     {
  181.                         if (!org.drip.numerical.common.NumberUtil.IsValid (s))
  182.                         {
  183.                             throw new java.lang.Exception
  184.                                 ("GammaEqualityLemma::DuplicationFormula::evaluate => Invalid Inputs");
  185.                         }

  186.                         org.drip.specialfunction.loggamma.InfiniteSumEstimator weierstrassInfiniteProduct =
  187.                             org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);

  188.                         return weierstrassInfiniteProduct.evaluate (s) + weierstrassInfiniteProduct.evaluate
  189.                             (s + 0.5);
  190.                     }
  191.                 },
  192.                 new org.drip.function.definition.R1ToR1 (null)
  193.                 {
  194.                     @Override public double evaluate (
  195.                         final double s)
  196.                         throws java.lang.Exception
  197.                     {
  198.                         if (!org.drip.numerical.common.NumberUtil.IsValid (s))
  199.                         {
  200.                             throw new java.lang.Exception
  201.                                 ("GammaEqualityLemma::DuplicationFormula::evaluate => Invalid Inputs");
  202.                         }

  203.                         return (1. - 2. * s) * java.lang.Math.log (2.) +
  204.                             0.5 * java.lang.Math.log (java.lang.Math.PI)  +
  205.                             org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400).evaluate (2. * s);
  206.                     }
  207.                 },
  208.                 org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
  209.             );
  210.         }
  211.         catch (java.lang.Exception e)
  212.         {
  213.             e.printStackTrace();
  214.         }

  215.         return null;
  216.     }

  217.     /**
  218.      * Construct the Multiplication Formula Verifier
  219.      *
  220.      * @param m m
  221.      *
  222.      * @return The Multiplication Formula Verifier
  223.      */

  224.     public static final org.drip.function.definition.R1ToR1Property MultiplicationFormula (
  225.         final int m)
  226.     {
  227.         if (1 >= m)
  228.         {
  229.             return null;
  230.         }

  231.         try
  232.         {
  233.             return new org.drip.function.definition.R1ToR1Property (
  234.                 org.drip.function.definition.R1ToR1Property.EQ,
  235.                 new org.drip.function.definition.R1ToR1 (null)
  236.                 {
  237.                     @Override public double evaluate (
  238.                         final double s)
  239.                         throws java.lang.Exception
  240.                     {
  241.                         if (!org.drip.numerical.common.NumberUtil.IsValid (s))
  242.                         {
  243.                             throw new java.lang.Exception
  244.                                 ("GammaEqualityLemma::MultiplicationFormula::evaluate => Invalid Inputs");
  245.                         }

  246.                         double logGammaSum = 0.;

  247.                         org.drip.specialfunction.loggamma.InfiniteSumEstimator weierstrassInfiniteProduct =
  248.                             org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);

  249.                         for (double i = 0; i < m; ++i)
  250.                         {
  251.                             logGammaSum = logGammaSum + weierstrassInfiniteProduct.evaluate (s + (i / m));
  252.                         }

  253.                         return logGammaSum;
  254.                     }
  255.                 },
  256.                 new org.drip.function.definition.R1ToR1 (null)
  257.                 {
  258.                     @Override public double evaluate (
  259.                         final double s)
  260.                         throws java.lang.Exception
  261.                     {
  262.                         if (!org.drip.numerical.common.NumberUtil.IsValid (s))
  263.                         {
  264.                             throw new java.lang.Exception
  265.                                 ("GammaEqualityLemma::MultiplicationFormula::evaluate => Invalid Inputs");
  266.                         }

  267.                         return 0.5 * (m - 1.) * java.lang.Math.log (2. * java.lang.Math.PI) +
  268.                             (0.5 - m * s) * java.lang.Math.log (m)  +
  269.                             org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400).evaluate (m * s);
  270.                     }
  271.                 },
  272.                 org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
  273.             );
  274.         }
  275.         catch (java.lang.Exception e)
  276.         {
  277.             e.printStackTrace();
  278.         }

  279.         return null;
  280.     }
  281. }