OrthogonalPolynomial.java

package org.drip.numerical.quadrature;

/*
 * -*- 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>OrthogonalPolynomial</i> implements a Single Basis Orthogonal Polynomial used in the Construction of
 * the Quadrature. The References are:
 * 
 * <br><br>
 *  <ul>
 *      <li>
 *          Abramowitz, M., and I. A. Stegun (2007): <i>Handbook of Mathematics Functions</i> <b>Dover Book
 *              on Mathematics</b>
 *      </li>
 *      <li>
 *          Gil, A., J. Segura, and N. M. Temme (2007): <i>Numerical Methods for Special Functions</i>
 *              <b>Society for Industrial and Applied Mathematics</b> Philadelphia
 *      </li>
 *      <li>
 *          Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (2007): <i>Numerical Recipes:
 *              The Art of Scientific Computing 3rd Edition</i> <b>Cambridge University Press</b> New York
 *      </li>
 *      <li>
 *          Stoer, J., and R. Bulirsch (2002): <i>Introduction to Numerical Analysis 3rd Edition</i>
 *              <b>Springer</b>
 *      </li>
 *      <li>
 *          Wikipedia (2019): Gaussian Quadrature https://en.wikipedia.org/wiki/Gaussian_quadrature
 *      </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/NumericalAnalysisLibrary.md">Numerical Analysis Library</a></li>
 *      <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/numerical/README.md">Numerical Quadrature, Differentiation, Eigenization, Linear Algebra, and Utilities</a></li>
 *      <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/numerical/quadrature/README.md">R<sup>1</sup> Gaussian Integration Quadrature Schemes</a></li>
 *  </ul>
 *
 * @author Lakshmi Krishnamurthy
 */

public class OrthogonalPolynomial extends org.drip.numerical.estimation.R1ToR1Series
{

    /**
     * OrthogonalPolynomial Constructor
     * 
     * @param termWeightMap Error Term Weight Map
     * 
     * @throws java.lang.Exception Thrown if the Inputs are Invalid
     */

    public OrthogonalPolynomial (
        final java.util.TreeMap<java.lang.Integer, java.lang.Double> termWeightMap)
        throws java.lang.Exception
    {
        super (
            org.drip.numerical.estimation.R1ToR1SeriesTerm.Taylor(),
            false,
            termWeightMap
        );
    }

    /**
     * Retrieve the Degree of the Orthogonal Polynomial
     * 
     * @return Degree of the Orthogonal Polynomial
     */

    public int degree()
    {
        return termWeightMap().lastKey();
    }

    /**
     * Retrieve the Coefficient of the Degree of the Orthogonal Polynomial
     * 
     * @return Coefficient of the Degree of the Orthogonal Polynomial
     */

    public double degreeCoefficient()
    {
        return termWeightMap().lastEntry().getValue();
    }

    /**
     * Compute the Legendre (i.e., Unit Orthogonal Weight) Node Weight
     * 
     * @param xNode The Node X
     * 
     * @return The Legendre (i.e., Unit Orthogonal Weight) Node Weight
     * 
     * @throws java.lang.Exception Thrown if the Inputs are Invalid
     */

    public double legendreNodeWeight (
        final double xNode)
        throws java.lang.Exception
    {
        if (!org.drip.numerical.common.NumberUtil.IsValid (xNode))
        {
            throw new java.lang.Exception ("OrthogonalPolynomial::legendreNodeWeight => Invalid Inputs");
        }

        return 2. / (
            (1. - xNode * xNode) * derivative (
                xNode,
                1
            )
        );
    }

    /**
     * Compute the Lobatto (i.e., Unit Orthogonal Weight) Node Weight
     * 
     * @param xNode The Node X
     * 
     * @return The Lobatto (i.e., Unit Orthogonal Weight) Node Weight
     * 
     * @throws java.lang.Exception Thrown if the Inputs are Invalid
     */

    public double lobattoNodeWeight (
        final double xNode)
        throws java.lang.Exception
    {
        if (!org.drip.numerical.common.NumberUtil.IsValid (xNode))
        {
            throw new java.lang.Exception ("OrthogonalPolynomial::lobattoNodeWeight => Invalid Inputs");
        }

        int degree = termWeightMap().lastKey();

        double nodeValue = evaluate (xNode);

        return 2. / (degree * (degree + 1) * nodeValue * nodeValue);
    }
}