GaussContinuedFraction.java

  1. package org.drip.specialfunction.incompletegamma;

  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>GaussContinuedFraction</i> implements the Gauss Continued Fraction Based Estimates for the Lower/Upper
  79.  * Incomplete Gamma Function. The References are:
  80.  * 
  81.  * <br><br>
  82.  *  <ul>
  83.  *      <li>
  84.  *          Geddes, K. O., M. L. Glasser, R. A. Moore, and T. C. Scott (1990): Evaluation of Classes of
  85.  *              Definite Integrals involving Elementary Functions via Differentiation of Special Functions
  86.  *              <i>Applicable Algebra in Engineering, Communications, and </i> <b>1 (2)</b> 149-165
  87.  *      </li>
  88.  *      <li>
  89.  *          Gradshteyn, I. S., I. M. Ryzhik, Y. V. Geronimus, M. Y. Tseytlin, and A. Jeffrey (2015):
  90.  *              <i>Tables of Integrals, Series, and Products</i> <b>Academic Press</b>
  91.  *      </li>
  92.  *      <li>
  93.  *          Mathar, R. J. (2010): Numerical Evaluation of the Oscillatory Integral over
  94.  *              e<sup>iÏ€x</sup> x<sup>(1/x)</sup> between 1 and ∞
  95.  *              https://arxiv.org/pdf/0912.3844.pdf <b>arXiV</b>
  96.  *      </li>
  97.  *      <li>
  98.  *          National Institute of Standards and Technology (2019): Incomplete Gamma and Related Functions
  99.  *              https://dlmf.nist.gov/8
  100.  *      </li>
  101.  *      <li>
  102.  *          Wikipedia (2019): Incomplete Gamma Function
  103.  *              https://en.wikipedia.org/wiki/Incomplete_gamma_function
  104.  *      </li>
  105.  *  </ul>
  106.  *
  107.  *  <br><br>
  108.  *  <ul>
  109.  *      <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
  110.  *      <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/FunctionAnalysisLibrary.md">Function Analysis Library</a></li>
  111.  *      <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>
  112.  *      <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/specialfunction/gammaincomplete/README.md">Upper/Lower Incomplete Gamma Functions</a></li>
  113.  *  </ul>
  114.  *
  115.  * @author Lakshmi Krishnamurthy
  116.  */

  118. public class GaussContinuedFraction
  119. {

  121.     private static final double Lower (
  122.         final double z,
  123.         final double s,
  124.         final int k,
  125.         final int n)
  126.     {
  127.         if (k == n)
  128.         {
  129.             return 0;
  130.         }

  132.         double bottom = s + 2 * k + 1 + ((k + 1) * z) / Lower (
  133.             z,
  134.             s,
  135.             k + 1,
  136.             n
  137.         );

  139.         return s + 2 * k - (((k + s) * z) / bottom);
  140.     }

  142.     private static final double UpperAbramowitzStegun2007 (
  143.         final double z,
  144.         final double s,
  145.         final int k,
  146.         final int n)
  147.     {
  148.         if (k == n)
  149.         {
  150.             return 0;
  151.         }

  153.         double bottom = 1 + (k + 1.) / UpperAbramowitzStegun2007 (
  154.             z,
  155.             s,
  156.             k + 1,
  157.             n
  158.         );

  160.         return z + (k + 1 - s) / bottom;
  161.     }

  163.     private static final double Upper (
  164.         final double z,
  165.         final double s,
  166.         final int k,
  167.         final int n)
  168.     {
  169.         if (k == n)
  170.         {
  171.             return 0;
  172.         }

  174.         return 1. + 2. * k + z - s + (k + 1.) * ((s - k - 1.) / Upper (
  175.             z,
  176.             s,
  177.             k + 1,
  178.             n
  179.         ));
  180.     }

  182.     /**
  183.      * Compute the Lower Incomplete Gamma Function using the Gauss Continued Fraction
  184.      * 
  185.      * @param z z
  186.      * @param s s
  187.      * @param n Count of the Number of Terms
  188.      * 
  189.      * @return The Lower Incomplete Gamma Function
  190.      * 
  191.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  192.      */

  194.     public static final double Lower (
  195.         final double z,
  196.         final double s,
  197.         final int n)
  198.         throws java.lang.Exception
  199.     {
  200.         if (!org.drip.numerical.common.NumberUtil.IsValid (z) ||
  201.             !org.drip.numerical.common.NumberUtil.IsValid (s) ||
  202.             0 == n)
  203.         {
  204.             throw new java.lang.Exception ("GaussContinuedFraction::Lower => Invalid Inputs");
  205.         }

  207.         return java.lang.Math.pow (
  208.             z,
  209.             s
  210.         ) * java.lang.Math.exp (-z) / Lower (
  211.             z,
  212.             s,
  213.             0,
  214.             n
  215.         );
  216.     }

  218.     /**
  219.      * Compute the Upper Incomplete Gamma Function using the Abramowitz-Stegun Gauss Continued Fraction
  220.      * 
  221.      * @param z z
  222.      * @param s s
  223.      * @param n Count of the Number of Terms
  224.      * 
  225.      * @return The Upper Incomplete Gamma Function using the Abramowitz-Stegun Gauss Continued Fraction
  226.      * 
  227.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  228.      */

  230.     public static final double UpperAbramowitzStegun2007 (
  231.         final double z,
  232.         final double s,
  233.         final int n)
  234.         throws java.lang.Exception
  235.     {
  236.         if (!org.drip.numerical.common.NumberUtil.IsValid (z) ||
  237.             !org.drip.numerical.common.NumberUtil.IsValid (s) ||
  238.             0 == n)
  239.         {
  240.             throw new java.lang.Exception
  241.                 ("GaussContinuedFraction::UpperAbramowitzStegun2007 => Invalid Inputs");
  242.         }

  244.         return java.lang.Math.pow (
  245.             z,
  246.             s
  247.         ) * java.lang.Math.exp (-z) / UpperAbramowitzStegun2007 (
  248.             z,
  249.             s,
  250.             0,
  251.             n
  252.         );
  253.     }

  255.     /**
  256.      * Compute the Upper Incomplete Gamma Function using Gauss Continued Fraction
  257.      * 
  258.      * @param z z
  259.      * @param s s
  260.      * @param n Count of the Number of Terms
  261.      * 
  262.      * @return The Upper Incomplete Gamma Function using Gauss Continued Fraction
  263.      * 
  264.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  265.      */

  267.     public static final double Upper (
  268.         final double z,
  269.         final double s,
  270.         final int n)
  271.         throws java.lang.Exception
  272.     {
  273.         if (!org.drip.numerical.common.NumberUtil.IsValid (z) ||
  274.             !org.drip.numerical.common.NumberUtil.IsValid (s) ||
  275.             0 == n)
  276.         {
  277.             throw new java.lang.Exception ("GaussContinuedFraction::Upper => Invalid Inputs");
  278.         }

  280.         return java.lang.Math.pow (
  281.             z,
  282.             s
  283.         ) * java.lang.Math.exp (-z) / Upper (
  284.             z,
  285.             s,
  286.             0,
  287.             n
  288.         );
  289.     }
  290. }
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