StirlingSeries.java

  1. package org.drip.specialfunction.gamma;

  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>StirlingSeries</i> implements the Stirling's Series Approximation of the Gamma Functions. The
  79.  * References are:
  80.  * <br><br>
  81.  *  <ul>
  82.  *      <li>
  83.  *          Mortici, C. (2011): Improved Asymptotic Formulas for the Gamma Function <i>Computers and
  84.  *              Mathematics with Applications</i> <b>61 (11)</b> 3364-3369
  85.  *      </li>
  86.  *      <li>
  87.  *          National Institute of Standards and Technology (2018): NIST Digital Library of Mathematical
  88.  *              Functions https://dlmf.nist.gov/5.11
  89.  *      </li>
  90.  *      <li>
  91.  *          Nemes, G. (2010): On the Coefficients of the Asymptotic Expansion of n!
  92.  *              https://arxiv.org/abs/1003.2907 <b>arXiv</b>
  93.  *      </li>
  94.  *      <li>
  95.  *          Toth V. T. (2016): Programmable Calculators – The Gamma Function
  96.  *              http://www.rskey.org/CMS/index.php/the-library/11
  97.  *      </li>
  98.  *      <li>
  99.  *          Wikipedia (2019): Stirling's Approximation
  100.  *              https://en.wikipedia.org/wiki/Stirling%27s_approximation
  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/gamma/README.md">Analytic/Series/Integral Gamma Estimators</a></li>
  110.  *  </ul>
  111.  *
  112.  * @author Lakshmi Krishnamurthy
  113.  */

  115. public class StirlingSeries extends org.drip.numerical.estimation.R1ToR1Estimator
  116. {

  118.     /**
  119.      * StirlingSeries Constructor
  120.      * 
  121.      * @param dc The Derivative Control
  122.      */

  124.     public StirlingSeries (
  125.         final org.drip.numerical.differentiation.DerivativeControl dc)
  126.     {
  127.         super (dc);
  128.     }

  130.     /**
  131.      * Compute the de-Moivre Term
  132.      * 
  133.      * @param x X
  134.      * 
  135.      * @return The de-Moivre Term
  136.      * 
  137.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  138.      */

  140.     public double deMoivreTerm (
  141.         final double x)
  142.         throws java.lang.Exception
  143.     {
  144.         if (!org.drip.numerical.common.NumberUtil.IsValid (x) || 0. > x)
  145.         {
  146.             throw new java.lang.Exception ("StirlingSeries::deMoivreTerm => Invalid Inputs");
  147.         }

  149.         return java.lang.Math.exp (1. - x) * java.lang.Math.pow (
  150.             x - 1.,
  151.             x - 0.5
  152.         );
  153.     }

  155.     @Override public double evaluate (
  156.         final double x)
  157.         throws java.lang.Exception
  158.     {
  159.         return java.lang.Math.sqrt (2. * java.lang.Math.PI) * deMoivreTerm (x);
  160.     }

  162.     @Override public org.drip.numerical.estimation.R1Estimate boundedEstimate (
  163.         final double x)
  164.     {
  165.         try
  166.         {
  167.             double deMoivreTerm = deMoivreTerm (x);

  169.             double estimate = java.lang.Math.sqrt (2. * java.lang.Math.PI) * deMoivreTerm;

  171.             return new org.drip.numerical.estimation.R1Estimate (
  172.                 estimate,
  173.                 estimate,
  174.                 java.lang.Math.E * deMoivreTerm
  175.             );
  176.         }
  177.         catch (java.lang.Exception e)
  178.         {
  179.             e.printStackTrace();
  180.         }

  182.         return null;
  183.     }

  185.     /**
  186.      * Compute the Bounded Function Estimates along with the First Order Laplace Correction
  187.      * 
  188.      * @param x X
  189.      * 
  190.      * @return The Bounded Function Estimates along with the First Order Laplace Correction
  191.      */

  193.     public org.drip.numerical.estimation.R1Estimate laplaceCorrectionEstimate (
  194.         final double x)
  195.     {
  196.         java.util.TreeMap<java.lang.Integer, java.lang.Double> termWeightMap = new
  197.             java.util.TreeMap<java.lang.Integer, java.lang.Double>();

  199.         termWeightMap.put (
  200.             1,
  201.             1. / 12.
  202.         );

  204.         try
  205.         {
  206.             return seriesEstimate (
  207.                 x,
  208.                 termWeightMap,
  209.                 new org.drip.numerical.estimation.R1ToR1Series (
  210.                     org.drip.numerical.estimation.R1ToR1SeriesTerm.Asymptotic(),
  211.                     true,
  212.                     termWeightMap
  213.                 )
  214.             );
  215.         }
  216.         catch (java.lang.Exception e)
  217.         {
  218.             e.printStackTrace();
  219.         }

  221.         return null;
  222.     }

  224.     /**
  225.      * Compute the Bounded Function Estimates along with the Higher Order Nemes Correction
  226.      * 
  227.      * @param x X
  228.      * 
  229.      * @return The Bounded Function Estimates along with the Higher Order Nemes Correction
  230.      */

  232.     public org.drip.numerical.estimation.R1Estimate nemesCorrectionEstimate (
  233.         final double x)
  234.     {
  235.         java.util.TreeMap<java.lang.Integer, java.lang.Double> termWeightMap = new
  236.             java.util.TreeMap<java.lang.Integer, java.lang.Double>();

  238.         termWeightMap.put (
  239.             1,
  240.             1. / 12.
  241.         );

  243.         termWeightMap.put (
  244.             2,
  245.             1. / 288.
  246.         );

  248.         termWeightMap.put (
  249.             3,
  250.             -139. / 51840.
  251.         );

  253.         termWeightMap.put (
  254.             4,
  255.             -571. / 2488320.
  256.         );

  258.         try
  259.         {
  260.             return seriesEstimate (
  261.                 x,
  262.                 termWeightMap,
  263.                 new org.drip.numerical.estimation.R1ToR1Series (
  264.                     org.drip.numerical.estimation.R1ToR1SeriesTerm.Asymptotic(),
  265.                     true,
  266.                     termWeightMap
  267.                 )
  268.             );
  269.         }
  270.         catch (java.lang.Exception e)
  271.         {
  272.             e.printStackTrace();
  273.         }

  275.         return null;
  276.     }
  277. }
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