BigPiEqualityLemma.java

  1. package org.drip.specialfunction.property;

  3. /*
  4.  * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
  5.  */

  7. /*!
  8.  * Copyright (C) 2020 Lakshmi Krishnamurthy
  9.  * Copyright (C) 2019 Lakshmi Krishnamurthy
  10.  * 
  11.  *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
  12.  *      asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
  13.  *      analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
  14.  *      equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
  15.  *      numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
  16.  *      and computational support.
  17.  *  
  18.  *      https://lakshmidrip.github.io/DROP/
  19.  *  
  20.  *  DROP is composed of three modules:
  21.  *  
  22.  *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
  23.  *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
  24.  *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
  25.  * 
  26.  *  DROP Product Core implements libraries for the following:
  27.  *  - Fixed Income Analytics
  28.  *  - Loan Analytics
  29.  *  - Transaction Cost Analytics
  30.  * 
  31.  *  DROP Portfolio Core implements libraries for the following:
  32.  *  - Asset Allocation Analytics
  33.  *  - Asset Liability Management Analytics
  34.  *  - Capital Estimation Analytics
  35.  *  - Exposure Analytics
  36.  *  - Margin Analytics
  37.  *  - XVA Analytics
  38.  * 
  39.  *  DROP Computational Core implements libraries for the following:
  40.  *  - Algorithm Support
  41.  *  - Computation Support
  42.  *  - Function Analysis
  43.  *  - Model Validation
  44.  *  - Numerical Analysis
  45.  *  - Numerical Optimizer
  46.  *  - Spline Builder
  47.  *  - Statistical Learning
  48.  * 
  49.  *  Documentation for DROP is Spread Over:
  50.  * 
  51.  *  - Main                     => https://lakshmidrip.github.io/DROP/
  52.  *  - Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
  53.  *  - GitHub                   => https://github.com/lakshmiDRIP/DROP
  54.  *  - Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
  55.  *  - Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
  56.  *  - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
  57.  *  - Release Versions         => https://lakshmidrip.github.io/DROP/version.html
  58.  *  - Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
  59.  *  - Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
  60.  *  - JUnit                    => https://lakshmidrip.github.io/DROP/junit/index.html
  61.  *  - Jacoco                   => https://lakshmidrip.github.io/DROP/jacoco/index.html
  62.  * 
  63.  *  Licensed under the Apache License, Version 2.0 (the "License");
  64.  *      you may not use this file except in compliance with the License.
  65.  *   
  66.  *  You may obtain a copy of the License at
  67.  *      http://www.apache.org/licenses/LICENSE-2.0
  68.  *  
  69.  *  Unless required by applicable law or agreed to in writing, software
  70.  *      distributed under the License is distributed on an "AS IS" BASIS,
  71.  *      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  72.  *  
  73.  *  See the License for the specific language governing permissions and
  74.  *      limitations under the License.
  75.  */

  77. /**
  78.  * <i>BigPiEqualityLemma</i> verifies the Specified Property Lemmas of the Big Pi Function. The References
  79.  * are:
  80.  * 
  81.  * <br><br>
  82.  *  <ul>
  83.  *      <li>
  84.  *          Blagouchine, I. V. (2014): Re-discovery of Malmsten's Integrals, their Evaluation by Contour
  85.  *              Integration Methods, and some Related Results <i>Ramanujan Journal</i> <b>35 (1)</b> 21-110
  86.  *      </li>
  87.  *      <li>
  88.  *          Borwein, J. M., and R. M. Corless (2017): Gamma Function and the Factorial in the Monthly
  89.  *              https://arxiv.org/abs/1703.05349 <b>arXiv</b>
  90.  *      </li>
  91.  *      <li>
  92.  *          Davis, P. J. (1959): Leonhard Euler's Integral: A Historical Profile of the Gamma Function
  93.  *              <i>American Mathematical Monthly</i> <b>66 (10)</b> 849-869
  94.  *      </li>
  95.  *      <li>
  96.  *          Whitaker, E. T., and G. N. Watson (1996): <i>A Course on Modern Analysis</i> <b>Cambridge
  97.  *              University Press</b> New York
  98.  *      </li>
  99.  *      <li>
  100.  *          Wikipedia (2019): Gamma Function https://en.wikipedia.org/wiki/Gamma_function
  101.  *      </li>
  102.  *  </ul>
  103.  *
  104.  *  <br><br>
  105.  *  <ul>
  106.  *      <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
  107.  *      <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/FunctionAnalysisLibrary.md">Function Analysis Library</a></li>
  108.  *      <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>
  109.  *      <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>
  110.  *  </ul>
  111.  *
  112.  * @author Lakshmi Krishnamurthy
  113.  */

  115. public class BigPiEqualityLemma
  116. {

  118.     /**
  119.      * Construct the Reflection Formula Verifier
  120.      * 
  121.      * @return The Reflection Formula Verifier
  122.      */

  124.     public static final org.drip.function.definition.R1ToR1Property ReflectionFormula()
  125.     {
  126.         final org.drip.specialfunction.derived.LogBigPi weierstrassBigPi = org.drip.specialfunction.derived.LogBigPi.Weierstrass
  127.             (1638400);

  129.         try
  130.         {
  131.             return new org.drip.function.definition.R1ToR1Property (
  132.                 org.drip.function.definition.R1ToR1Property.EQ,
  133.                 new org.drip.function.definition.R1ToR1 (null)
  134.                 {
  135.                     @Override public double evaluate (
  136.                         final double z)
  137.                         throws java.lang.Exception
  138.                     {
  139.                         if (!org.drip.numerical.common.NumberUtil.IsValid (z))
  140.                         {
  141.                             throw new java.lang.Exception
  142.                                 ("BigPiEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
  143.                         }

  145.                         return java.lang.Math.exp (weierstrassBigPi.evaluate (-1. * z) +
  146.                             weierstrassBigPi.evaluate (z));
  147.                     }
  148.                 },
  149.                 new org.drip.function.definition.R1ToR1 (null)
  150.                 {
  151.                     @Override public double evaluate (
  152.                         final double z)
  153.                         throws java.lang.Exception
  154.                     {
  155.                         if (!org.drip.numerical.common.NumberUtil.IsValid (z))
  156.                         {
  157.                             throw new java.lang.Exception
  158.                                 ("BigPiEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
  159.                         }

  161.                         return 1. / new org.drip.function.r1tor1.Sinc (null).evaluate (z);
  162.                     }
  163.                 },
  164.                 org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
  165.             );
  166.         }
  167.         catch (java.lang.Exception e)
  168.         {
  169.             e.printStackTrace();
  170.         }

  172.         return null;
  173.     }

  175.     /**
  176.      * Construct the Multiplication Formula Verifier
  177.      * 
  178.      * @param m m
  179.      * 
  180.      * @return The Multiplication Formula Verifier
  181.      */

  183.     public static final org.drip.function.definition.R1ToR1Property MultiplicationFormula (
  184.         final int m)
  185.     {
  186.         if (1 >= m)
  187.         {
  188.             return null;
  189.         }

  191.         final org.drip.specialfunction.derived.LogBigPi weierstrassBigPi = org.drip.specialfunction.derived.LogBigPi.Weierstrass
  192.             (1638400);

  194.         try
  195.         {
  196.             return new org.drip.function.definition.R1ToR1Property (
  197.                 org.drip.function.definition.R1ToR1Property.EQ,
  198.                 new org.drip.function.definition.R1ToR1 (null)
  199.                 {
  200.                     @Override public double evaluate (
  201.                         final double z)
  202.                         throws java.lang.Exception
  203.                     {
  204.                         if (!org.drip.numerical.common.NumberUtil.IsValid (z))
  205.                         {
  206.                             throw new java.lang.Exception
  207.                                 ("BigPiEqualityLemma::MultiplicationFormula::evaluate => Invalid Inputs");
  208.                         }

  210.                         double logBigPiSum = 0.;

  212.                         for (double i = 0; i < m; ++i)
  213.                         {
  214.                             logBigPiSum = logBigPiSum + weierstrassBigPi.evaluate ((z - i) / m);
  215.                         }

  217.                         return logBigPiSum;
  218.                     }
  219.                 },
  220.                 new org.drip.function.definition.R1ToR1 (null)
  221.                 {
  222.                     @Override public double evaluate (
  223.                         final double z)
  224.                         throws java.lang.Exception
  225.                     {
  226.                         if (!org.drip.numerical.common.NumberUtil.IsValid (z))
  227.                         {
  228.                             throw new java.lang.Exception
  229.                                 ("BigPiEqualityLemma::MultiplicationFormula::evaluate => Invalid Inputs");
  230.                         }

  232.                         return 0.5 * (m - 1.) * java.lang.Math.log (2. * java.lang.Math.PI) -
  233.                             (0.5 + z) * java.lang.Math.log (m) + weierstrassBigPi.evaluate (z);
  234.                     }
  235.                 },
  236.                 org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
  237.             );
  238.         }
  239.         catch (java.lang.Exception e)
  240.         {
  241.             e.printStackTrace();
  242.         }

  244.         return null;
  245.     }
  246. }
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