GolubWelsch.java

  1. package org.drip.numerical.quadrature;

  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>GolubWelsch</i> implements the Golub-Welsch Algorithm that extracts the Quadrature Nodes and Weights.
  79.  * The References are:
  80.  * 
  81.  * <br><br>
  82.  *  <ul>
  83.  *      <li>
  84.  *          Abramowitz, M., and I. A. Stegun (2007): <i>Handbook of Mathematics Functions</i> <b>Dover Book
  85.  *              on Mathematics</b>
  86.  *      </li>
  87.  *      <li>
  88.  *          Gil, A., J. Segura, and N. M. Temme (2007): <i>Numerical Methods for Special Functions</i>
  89.  *              <b>Society for Industrial and Applied Mathematics</b> Philadelphia
  90.  *      </li>
  91.  *      <li>
  92.  *          Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (2007): <i>Numerical Recipes:
  93.  *              The Art of Scientific Computing 3rd Edition</i> <b>Cambridge University Press</b> New York
  94.  *      </li>
  95.  *      <li>
  96.  *          Stoer, J., and R. Bulirsch (2002): <i>Introduction to Numerical Analysis 3rd Edition</i>
  97.  *              <b>Springer</b>
  98.  *      </li>
  99.  *      <li>
  100.  *          Wikipedia (2019): Gaussian Quadrature https://en.wikipedia.org/wiki/Gaussian_quadrature
  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/NumericalAnalysisLibrary.md">Numerical Analysis Library</a></li>
  108.  *      <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>
  109.  *      <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>
  110.  *  </ul>
  111.  *
  112.  * @author Lakshmi Krishnamurthy
  113.  */

  115. public class GolubWelsch
  116. {
  117.     private double[][] _recurrenceJ = null;

  119.     /**
  120.      * GolubWelsch Constructor
  121.      * 
  122.      * @param recurrenceJ The J Matrix derived from Orthogonal Polynomial Recursion
  123.      * 
  124.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  125.      */

  127.     public GolubWelsch (
  128.         final double[][] recurrenceJ)
  129.         throws java.lang.Exception
  130.     {
  131.         if (null == (_recurrenceJ = recurrenceJ))
  132.         {
  133.             throw new java.lang.Exception ("GolubWelsch Constructor => Invalid Inputs");
  134.         }

  136.         int size = _recurrenceJ.length;

  138.         if (0 == size)
  139.         {
  140.             throw new java.lang.Exception ("GolubWelsch Constructor => Invalid Inputs");
  141.         }

  143.         for (int column = 0; column < size; ++column)
  144.         {
  145.             if (null == _recurrenceJ || size != _recurrenceJ[column].length ||
  146.                 !org.drip.numerical.common.NumberUtil.IsValid (_recurrenceJ[column]))
  147.             {
  148.                 throw new java.lang.Exception ("GolubWelsch Constructor => Invalid Inputs");
  149.             }
  150.         }
  151.     }

  153.     /**
  154.      * Retrieve the Recurrence Matrix J
  155.      * 
  156.      * @return The Recurrence Matrix J
  157.      */

  159.     public double[][] recurrenceJ()
  160.     {
  161.         return _recurrenceJ;
  162.     }

  164.     /**
  165.      * Generate the Symmetric Tri-diagonal Matrix from the Recurrence J Matrix
  166.      * 
  167.      * @return The Symmetric Tri-diagonal Matrix from the Recurrence J Matrix
  168.      */

  170.     public double[][] symmetricTridiagonal()
  171.     {
  172.         int size = _recurrenceJ.length;
  173.         double[][] symmetricTridiagonal = new double[size][size];

  175.         for (int row = 0; row < size; ++row)
  176.         {
  177.             for (int column = 0; column < size; ++column)
  178.             {
  179.                 symmetricTridiagonal[row][column] = 0.;
  180.             }
  181.         }

  183.         for (int row = 0; row < size; ++row)
  184.         {
  185.             int column = row + 1;

  187.             if (column < size)
  188.             {
  189.                 double sqrtRecurrenceB = java.lang.Math.sqrt (_recurrenceJ[row][column]);

  191.                 symmetricTridiagonal[row][column] = sqrtRecurrenceB;
  192.                 symmetricTridiagonal[column][row] = sqrtRecurrenceB;
  193.             }

  195.             symmetricTridiagonal[row][row] = _recurrenceJ[row][row];
  196.         }

  198.         return symmetricTridiagonal;
  199.     }

  201.     /**
  202.      * Generate the Quadrature Nodes and Unscaled Weights
  203.      * 
  204.      * @return The Quadrature Nodes and Unscaled Weights
  205.      */

  207.     public org.drip.numerical.common.Array2D nodesAndUnscaledWeights()
  208.     {
  209.         int size = _recurrenceJ.length;
  210.         double[] nodeArray = new double[size];
  211.         double[] unscaledWeightArray = new double[size];

  213.         try
  214.         {
  215.             org.drip.numerical.eigen.EigenComponent[] orderedEigenComponentArray = new
  216.                 org.drip.numerical.eigen.QREigenComponentExtractor (
  217.                     100,
  218.                     1.e-06
  219.                 ).orderedEigenComponentArray (symmetricTridiagonal());

  221.             if (null == orderedEigenComponentArray || 0 == orderedEigenComponentArray.length)
  222.             {
  223.                 return null;
  224.             }

  226.             for (int componentIndex = 0;
  227.                 componentIndex < orderedEigenComponentArray.length;
  228.                 ++componentIndex)
  229.             {
  230.                 nodeArray[componentIndex] = orderedEigenComponentArray[componentIndex].eigenValue();

  232.                 double leadingEigenComponentCoefficient =
  233.                     orderedEigenComponentArray[componentIndex].eigenVector()[0];

  235.                 unscaledWeightArray[componentIndex] = leadingEigenComponentCoefficient *
  236.                     leadingEigenComponentCoefficient;
  237.             }

  239.             return org.drip.numerical.common.Array2D.FromArray (
  240.                 nodeArray,
  241.                 unscaledWeightArray
  242.             );
  243.         }
  244.         catch (java.lang.Exception e)
  245.         {
  246.             e.printStackTrace();
  247.         }

  249.         return null;
  250.     }
  251. }
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