GammaPolynomialQuotientLemma.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
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 *  - Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
 *  - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
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 *  - 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
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 *  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>GammaPolynomialQuotientLemma</i> contains the Verifiable Gamma Polynomial Quotient Equality Lemma. 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 GammaPolynomialQuotientLemma
{
    private org.drip.function.r1tor1.MonicPolynomial _monicPolynomialP = null;
    private org.drip.function.r1tor1.MonicPolynomial _monicPolynomialQ = null;

    /**
     * Retrieve the Monic Polynomial "P"
     * 
     * @return The Monic Polynomial "P"
     */

    public org.drip.function.r1tor1.MonicPolynomial _monicPolynomialP()
    {
        return _monicPolynomialP;
    }

    /**
     * Retrieve the Monic Polynomial "Q"
     * 
     * @return The Monic Polynomial "Q"
     */

    public org.drip.function.r1tor1.MonicPolynomial _monicPolynomialQ()
    {
        return _monicPolynomialQ;
    }

    /**
     * Run the Equality Lemma over the specified a and b
     * 
     * @param a a
     * @param b b
     * 
     * @return Result of the Equality Lemma Run
     */

    public org.drip.function.definition.R1PropertyVerification equalityLemma (
        final int a,
        final int b)
    {
        if (a >= b)
        {
            return null;
        }

        org.drip.specialfunction.loggamma.InfiniteSumEstimator logGammaEstimator =
            org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);

        try
        {
            double lValue = 1.;

            for (int i = a; i <= b; ++i)
            {
                lValue = lValue * _monicPolynomialP.evaluate (i) / _monicPolynomialQ.evaluate (i);
            }

            double rValue = java.lang.Math.exp (
                _monicPolynomialP.logGammaProduct (
                    a,
                    b,
                    logGammaEstimator
                ) - _monicPolynomialQ.logGammaProduct (
                    a,
                    b,
                    logGammaEstimator
                )
            );

            double leftTolerance = java.lang.Math.abs (lValue *
                org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE);

            double rightTolerance = java.lang.Math.abs (rValue *
                org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE);

            double tolerance = leftTolerance < rightTolerance ? leftTolerance: rightTolerance;
            tolerance = tolerance < org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE ?
                org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE : tolerance;

            return new org.drip.function.definition.R1PropertyVerification (
                lValue,
                rValue,
                java.lang.Math.abs (lValue - rValue) < tolerance
            );
        }
        catch (java.lang.Exception e)
        {
            e.printStackTrace();
        }

        return null;
    }
}