SaddlePoints.java

  1. package org.drip.specialfunction.digamma;

  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>SaddlePoints</i> contains the Hermite Based Saddle Point Roots of the Digamma Function. The References
  79.  * are:
  80.  * 
  81.  * <br><br>
  82.  *  <ul>
  83.  *      <li>
  84.  *          Abramowitz, M., and I. A. Stegun (2007): Handbook of Mathematics Functions <b>Dover Book on
  85.  *              Mathematics</b>
  86.  *      </li>
  87.  *      <li>
  88.  *          Blagouchine, I. V. (2018): Three Notes on Ser's and Hasse's Representations for the
  89.  *              Zeta-Functions https://arxiv.org/abs/1606.02044 <b>arXiv</b>
  90.  *      </li>
  91.  *      <li>
  92.  *          Mezo, I., and M. E. Hoffman (2017): Zeros of the Digamma Function and its Barnes G-function
  93.  *              Analogue <i>Integral Transforms and Special Functions</i> <b>28 (28)</b> 846-858
  94.  *      </li>
  95.  *      <li>
  96.  *          Whitaker, E. T., and G. N. Watson (1996): <i>A Course on Modern Analysis</i> <b>Cambridge
  97.  *              University Press</b> New York
  98.  *      </li>
  99.  *      <li>
  100.  *          Wikipedia (2019): Digamma Function https://en.wikipedia.org/wiki/Digamma_function
  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/FunctionAnalysisLibrary.md">Function Analysis Library</a></li>
  108.  *      <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>
  109.  *      <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/specialfunction/digamma/README.md">Estimation Techniques for Digamma Function</a></li>
  110.  *  </ul>
  111.  *
  112.  * @author Lakshmi Krishnamurthy
  113.  */

  115. public class SaddlePoints
  116. {

  118.     /**
  119.      * Generate the Set of Leading Digamma Saddle Points
  120.      * 
  121.      * @return Set of Leading Digamma Saddle Points
  122.      */

  124.     public static final java.util.TreeSet<java.lang.Double> LeadingZeros()
  125.     {
  126.         java.util.TreeSet<java.lang.Double> zeroSet = new java.util.TreeSet<java.lang.Double>();

  128.         zeroSet.add ( 1.461632144968);

  130.         zeroSet.add (-0.504083008000);

  132.         zeroSet.add (-1.573498473000);

  134.         zeroSet.add (-2.610720868000);

  136.         zeroSet.add (-3.635293366000);

  138.         return zeroSet;
  139.     }

  141.     /**
  142.      * Construct the R<sup>1</sup> to R<sup>1</sup> Hermite Digamma Root Function
  143.      * 
  144.      * @return The R<sup>1</sup> to R<sup>1</sup> Hermite Digamma Root Function
  145.      */

  147.     public static final org.drip.function.definition.R1ToR1 Hermite()
  148.     {
  149.         try
  150.         {
  151.             return new org.drip.function.definition.R1ToR1 (null)
  152.             {
  153.                 @Override public double evaluate (
  154.                     final double z)
  155.                     throws java.lang.Exception
  156.                 {
  157.                     if (1. > z)
  158.                     {
  159.                         throw new java.lang.Exception ("SaddlePoints::Hermite::evaluate => Invalid Inputs");
  160.                     }

  162.                     return 1. / java.lang.Math.log (z) - z;
  163.                 }
  164.             };
  165.         }
  166.         catch (java.lang.Exception e)
  167.         {
  168.             e.printStackTrace();
  169.         }

  171.         return null;
  172.     }

  174.     /**
  175.      * Construct the R<sup>1</sup> to R<sup>1</sup> Hermite Extension Digamma Root Function
  176.      * 
  177.      * @return The R<sup>1</sup> to R<sup>1</sup> Hermite Extension Digamma Root Function
  178.      */

  180.     public static final org.drip.function.definition.R1ToR1 HermiteExtension()
  181.     {
  182.         try
  183.         {
  184.             return new org.drip.function.definition.R1ToR1 (null)
  185.             {
  186.                 @Override public double evaluate (
  187.                     final double z)
  188.                     throws java.lang.Exception
  189.                 {
  190.                     if (1. > z)
  191.                     {
  192.                         throw new java.lang.Exception
  193.                             ("SaddlePoints::HermiteExtension::evaluate => Invalid Inputs");
  194.                     }

  196.                     return java.lang.Math.atan (java.lang.Math.PI / java.lang.Math.log (z)) /
  197.                         java.lang.Math.PI - z;
  198.                 }
  199.             };
  200.         }
  201.         catch (java.lang.Exception e)
  202.         {
  203.             e.printStackTrace();
  204.         }

  206.         return null;
  207.     }

  209.     /**
  210.      * Construct the R<sup>1</sup> to R<sup>1</sup> Hermite Enhancement Digamma Root Function
  211.      * 
  212.      * @return The R<sup>1</sup> to R<sup>1</sup> Hermite Enhancement Digamma Root Function
  213.      */

  215.     public static final org.drip.function.definition.R1ToR1 HermiteEnhancement()
  216.     {
  217.         try
  218.         {
  219.             return new org.drip.function.definition.R1ToR1 (null)
  220.             {
  221.                 @Override public double evaluate (
  222.                     final double z)
  223.                     throws java.lang.Exception
  224.                 {
  225.                     if (1. > z)
  226.                     {
  227.                         throw new java.lang.Exception
  228.                             ("SaddlePoints::HermiteEnhancement::evaluate => Invalid Inputs");
  229.                     }

  231.                     return java.lang.Math.atan (java.lang.Math.PI / (java.lang.Math.log (z) + (0.125 / z))) /
  232.                         java.lang.Math.PI - z;
  233.                 }
  234.             };
  235.         }
  236.         catch (java.lang.Exception e)
  237.         {
  238.             e.printStackTrace();
  239.         }

  241.         return null;
  242.     }

  244.     /**
  245.      * Construct the R<sup>1</sup> to R<sup>1</sup> Mezo-Hoffman (2017) Digamma Root Function
  246.      * 
  247.      * @return The R<sup>1</sup> to R<sup>1</sup> Mezo-Hoffman (2017) Digamma Root Function
  248.      */

  250.     public static final org.drip.function.definition.R1ToR1 MezoHoffman2017()
  251.     {
  252.         try
  253.         {
  254.             return new org.drip.function.definition.R1ToR1 (null)
  255.             {
  256.                 @Override public double evaluate (
  257.                     final double z)
  258.                     throws java.lang.Exception
  259.                 {
  260.                     if (1. > z)
  261.                     {
  262.                         throw new java.lang.Exception
  263.                             ("SaddlePoints::MezoHoffman2017::evaluate => Invalid Inputs");
  264.                     }

  266.                     double logZReciprocal = 1. / java.lang.Math.log (z);

  268.                     return logZReciprocal - z - 0.5 * logZReciprocal * logZReciprocal / z;
  269.                 }
  270.             };
  271.         }
  272.         catch (java.lang.Exception e)
  273.         {
  274.             e.printStackTrace();
  275.         }

  277.         return null;
  278.     }

  280.     /**
  281.      * Generate the Array of Leading Roots
  282.      * 
  283.      * @param rootFunction The Root Generation Function
  284.      * @param rootCount The Root Count
  285.      * 
  286.      * @return The Array of Leading Roots
  287.      */

  289.     public static final double[] LeadingRoots (
  290.         final org.drip.function.definition.R1ToR1 rootFunction,
  291.         final int rootCount)
  292.     {
  293.         if (null == rootFunction ||
  294.             0 >= rootCount)
  295.         {
  296.             return null;
  297.         }

  299.         double[] leadingRootArray = new double[rootCount + 1];
  300.         leadingRootArray[0] = org.drip.specialfunction.gamma.Definitions.MINIMUM_VARIATE_LOCATION;

  302.         for (int rootIndex = 1; rootIndex <= rootCount; ++rootIndex)
  303.         {
  304.             try
  305.             {
  306.                 leadingRootArray[rootIndex] = rootFunction.evaluate (rootIndex);
  307.             }
  308.             catch (java.lang.Exception e)
  309.             {
  310.                 e.printStackTrace();
  311.             }
  312.         }

  314.         return leadingRootArray;
  315.     }
  316. }
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