BigPiEqualityLemma.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>BigPiEqualityLemma</i> verifies the Specified Property Lemmas of the Big Pi 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 BigPiEqualityLemma
{
/**
* Construct the Reflection Formula Verifier
*
* @return The Reflection Formula Verifier
*/
public static final org.drip.function.definition.R1ToR1Property ReflectionFormula()
{
final org.drip.specialfunction.derived.LogBigPi weierstrassBigPi = org.drip.specialfunction.derived.LogBigPi.Weierstrass
(1638400);
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 z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("BigPiEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
}
return java.lang.Math.exp (weierstrassBigPi.evaluate (-1. * z) +
weierstrassBigPi.evaluate (z));
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("BigPiEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
}
return 1. / new org.drip.function.r1tor1.Sinc (null).evaluate (z);
}
},
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;
}
final org.drip.specialfunction.derived.LogBigPi weierstrassBigPi = org.drip.specialfunction.derived.LogBigPi.Weierstrass
(1638400);
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 z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("BigPiEqualityLemma::MultiplicationFormula::evaluate => Invalid Inputs");
}
double logBigPiSum = 0.;
for (double i = 0; i < m; ++i)
{
logBigPiSum = logBigPiSum + weierstrassBigPi.evaluate ((z - i) / m);
}
return logBigPiSum;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("BigPiEqualityLemma::MultiplicationFormula::evaluate => Invalid Inputs");
}
return 0.5 * (m - 1.) * java.lang.Math.log (2. * java.lang.Math.PI) -
(0.5 + z) * java.lang.Math.log (m) + weierstrassBigPi.evaluate (z);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
}