ArcTangentGeneralizedMidPoint.java

  1. package org.drip.sample.newtoncotes;

  3. import org.drip.numerical.common.FormatUtil;
  4. import org.drip.service.env.EnvManager;

  6. /*
  7.  * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
  8.  */

  10. /*!
  11.  * Copyright (C) 2019 Lakshmi Krishnamurthy
  12.  * 
  13.  *  This file is part of DROP, an open-source library targeting risk, transaction costs, exposure, margin
  14.  *      calculations, and portfolio construction within and across fixed income, credit, commodity, equity,
  15.  *      FX, and structured products.
  16.  *  
  17.  *      https://lakshmidrip.github.io/DROP/
  18.  *  
  19.  *  DROP is composed of three main modules:
  20.  *  
  21.  *  - DROP Analytics Core - https://lakshmidrip.github.io/DROP-Analytics-Core/
  22.  *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
  23.  *  - DROP Numerical Core - https://lakshmidrip.github.io/DROP-Numerical-Core/
  24.  * 
  25.  *  DROP Analytics Core implements libraries for the following:
  26.  *  - Fixed Income Analytics
  27.  *  - Asset Backed Analytics
  28.  *  - XVA Analytics
  29.  *  - Exposure and Margin Analytics
  30.  * 
  31.  *  DROP Portfolio Core implements libraries for the following:
  32.  *  - Asset Allocation Analytics
  33.  *  - Transaction Cost Analytics
  34.  * 
  35.  *  DROP Numerical Core implements libraries for the following:
  36.  *  - Statistical Learning Library
  37.  *  - Numerical Optimizer Library
  38.  *  - Machine Learning Library
  39.  *  - Spline Builder Library
  40.  * 
  41.  *  Documentation for DROP is Spread Over:
  42.  * 
  43.  *  - Main                     => https://lakshmidrip.github.io/DROP/
  44.  *  - Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
  45.  *  - GitHub                   => https://github.com/lakshmiDRIP/DROP
  46.  *  - Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
  47.  *  - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
  48.  *  - Release Versions         => https://lakshmidrip.github.io/DROP/version.html
  49.  *  - Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
  50.  *  - Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
  51.  *  - JUnit                    => https://lakshmidrip.github.io/DROP/junit/index.html
  52.  *  - Jacoco                   => https://lakshmidrip.github.io/DROP/jacoco/index.html
  53.  * 
  54.  *  Licensed under the Apache License, Version 2.0 (the "License");
  55.  *      you may not use this file except in compliance with the License.
  56.  *   
  57.  *  You may obtain a copy of the License at
  58.  *      http://www.apache.org/licenses/LICENSE-2.0
  59.  *  
  60.  *  Unless required by applicable law or agreed to in writing, software
  61.  *      distributed under the License is distributed on an "AS IS" BASIS,
  62.  *      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  63.  *  
  64.  *  See the License for the specific language governing permissions and
  65.  *      limitations under the License.
  66.  */

  68. /**
  69.  * <i>ArcTangentGeneralizedMidPoint</i> computes the R<sup>1</sup> Numerical Estimate of the tan<sup>-1</sup>
  70.  * using the Generalized Mid-Point Quadrature. The References are:
  71.  * 
  72.  * <br><br>
  73.  *  <ul>
  74.  *      <li>
  75.  *          Briol, F. X., C. J. Oates, M. Girolami, and M. A. Osborne (2015): <i>Frank-Wolfe Bayesian
  76.  *              Quadrature: Probabilistic Integration with Theoretical Guarantees</i> <b>arXiv</b>
  77.  *      </li>
  78.  *      <li>
  79.  *          Forsythe, G. E., M. A. Malcolm, and C. B. Moler (1977): <i>Computer Methods for Mathematical
  80.  *              Computation</i> <b>Prentice Hall</b> Englewood Cliffs NJ
  81.  *      </li>
  82.  *      <li>
  83.  *          Leader, J. J. (2004): <i>Numerical Analysis and Scientific Computation</i> <b>Addison Wesley</b>
  84.  *      </li>
  85.  *      <li>
  86.  *          Stoer, J., and R. Bulirsch (1980): <i>Introduction to Numerical Analysis</i>
  87.  *              <b>Springer-Verlag</b> New York
  88.  *      </li>
  89.  *      <li>
  90.  *          Wikipedia (2019): Numerical Integration https://en.wikipedia.org/wiki/Numerical_integration
  91.  *      </li>
  92.  *  </ul>
  93.  *
  94.  *  <br><br>
  95.  *  <ul>
  96.  *      <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/NumericalCore.md">Numerical Core Module</a></li>
  97.  *      <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/NumericalOptimizerLibrary.md">Numerical Optimizer</a></li>
  98.  *      <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/README.md">Sample</a></li>
  99.  *      <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/newtoncotes/README.md">R<sup>1</sup> Newton-Cotes Quadrature Schemes</a></li>
  100.  *  </ul>
  101.  *
  102.  * @author Lakshmi Krishnamurthy
  103.  */

  105. public class ArcTangentGeneralizedMidPoint
  106. {

  108.     private static final double ArcTangent (
  109.         final double z,
  110.         final int termCount)
  111.         throws Exception
  112.     {
  113.         double b = 1.;
  114.         double a = 2. / z;
  115.         double arcTangent = 0.;

  117.         for (int termIndex = 1; termIndex <= termCount; ++termIndex)
  118.         {
  119.             arcTangent = arcTangent + (a / ((a * a + b * b) * (2 * termIndex - 1)));
  120.             a = a * (1. - (4. / (z * z))) + 4. * (b / z);
  121.             b = b * (1. - (4. / (z * z))) - 4. * (a / z);
  122.         }

  124.         return 2. * arcTangent;
  125.     }

  127.     public static final void main (
  128.         final String[] argumentArray)
  129.         throws Exception
  130.     {
  131.         EnvManager.InitEnv ("");

  133.         int termCount = 10;
  134.         double[] rValueArray =
  135.         {
  136.              1. / 32.,
  137.              1. / 16.,
  138.              1. /  8.,
  139.              1. /  4.,
  140.              1. /  2.,
  141.              1.,
  142.              2.,
  143.              4.,
  144.              8.,
  145.             16.,
  146.             32.,
  147.             64.
  148.         };

  150.         System.out.println ("\t|-----------------------------------||");

  152.         System.out.println ("\t|   ARC TANGENT SERIES ESTIMATION   ||");

  154.         System.out.println ("\t|-----------------------------------||");

  156.         System.out.println ("\t|      L -> R:                      ||");

  158.         System.out.println ("\t|            - z                    ||");

  160.         System.out.println ("\t|            - System               ||");

  162.         System.out.println ("\t|            - Series               ||");

  164.         System.out.println ("\t|-----------------------------------||");

  166.         for (double rValue : rValueArray)
  167.         {
  168.             System.out.println (
  169.                 "\t|" + FormatUtil.FormatDouble (rValue, 2, 5, 1.) + " => " +
  170.                 FormatUtil.FormatDouble (Math.atan (rValue), 1, 6, 1.) + " | " +
  171.                 FormatUtil.FormatDouble (
  172.                     ArcTangent (
  173.                         rValue,
  174.                         termCount
  175.                     ), 1, 6, 1.
  176.                 ) + " ||"
  177.             );
  178.         }

  180.         System.out.println ("\t|-----------------------------------||");

  182.         EnvManager.TerminateEnv();
  183.     }
  184. }
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