BesselSecondEqualityLemma.java

  1. package org.drip.specialfunction.property;

  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>BesselSecondEqualityLemma</i> implements the implements the Equality Lemmas for the Cylindrical Bessel
  79.  * Function of the Second Kind. 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.  *          Arfken, G. B., and H. J. Weber (2005): <i>Mathematical Methods for Physicists 6<sup>th</sup>
  89.  *              Edition</i> <b>Harcourt</b> San Diego
  90.  *      </li>
  91.  *      <li>
  92.  *          Temme N. M. (1996): <i>Special Functions: An Introduction to the Classical Functions of
  93.  *              Mathematical Physics 2<sup>nd</sup> Edition</i> <b>Wiley</b> New York
  94.  *      </li>
  95.  *      <li>
  96.  *          Watson, G. N. (1995): <i>A Treatise on the Theory of Bessel Functions</i> <b>Cambridge University
  97.  *              Press</b>
  98.  *      </li>
  99.  *      <li>
  100.  *          Wikipedia (2019): Bessel Function https://en.wikipedia.org/wiki/Bessel_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/property/README.md">Special Function Property Lemma Verifiers</a></li>
  110.  *  </ul>
  111.  *
  112.  * @author Lakshmi Krishnamurthy
  113.  */

  115. public class BesselSecondEqualityLemma
  116. {

  118.     /**
  119.      * Construct the Bessel Second Kind Mirror Identity Verifier
  120.      * 
  121.      * @return The Bessel Second Kind Mirror Identity Verifier
  122.      */

  124.     public static final org.drip.function.definition.R2ToR1Property MirrorIdentity()
  125.     {
  126.         try
  127.         {
  128.             org.drip.function.definition.R1ToR1 gammaEstimator = new
  129.                 org.drip.specialfunction.gamma.EulerIntegralSecondKind (null);

  131.             final org.drip.specialfunction.definition.BesselSecondKindEstimator besselSecondKindEstimator =
  132.                 org.drip.specialfunction.bessel.SecondNISTSeriesEstimator.Standard (
  133.                     new org.drip.specialfunction.digamma.BinetFirstIntegral (null),
  134.                     gammaEstimator,
  135.                     org.drip.specialfunction.bessel.FirstFrobeniusSeriesEstimator.Standard (
  136.                         gammaEstimator,
  137.                         40
  138.                     ),
  139.                     40
  140.                 );

  142.             return new org.drip.function.definition.R2ToR1Property (
  143.                 org.drip.function.definition.RxToR1Property.EQ,
  144.                 new org.drip.function.definition.R2ToR1()
  145.                 {
  146.                     @Override public double evaluate (
  147.                         final double alpha,
  148.                         final double z)
  149.                         throws java.lang.Exception
  150.                     {
  151.                         if (!org.drip.numerical.common.NumberUtil.IsInteger (alpha))
  152.                         {
  153.                             throw new java.lang.Exception
  154.                                 ("BesselFirstEqualityLemma::MirrorIdentity => Invalid Inputs");
  155.                         }

  157.                         return besselSecondKindEstimator.bigY (
  158.                             -1. * alpha,
  159.                             z
  160.                         );
  161.                     }
  162.                 },
  163.                 new org.drip.function.definition.R2ToR1()
  164.                 {
  165.                     @Override public double evaluate (
  166.                         final double alpha,
  167.                         final double z)
  168.                         throws java.lang.Exception
  169.                     {
  170.                         if (!org.drip.numerical.common.NumberUtil.IsInteger (alpha))
  171.                         {
  172.                             throw new java.lang.Exception
  173.                                 ("BesselFirstEqualityLemma::MirrorIdentity => Invalid Inputs");
  174.                         }

  176.                         return (0 == ((int) z) % 2 ? 1. : -1.) * besselSecondKindEstimator.bigY (
  177.                             alpha,
  178.                             z
  179.                         );
  180.                     }
  181.                 },
  182.                 org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
  183.             );
  184.         }
  185.         catch (java.lang.Exception e)
  186.         {
  187.             e.printStackTrace();
  188.         }

  190.         return null;
  191.     }

  193.     /**
  194.      * Construct the Bessel Second Kind Half-Integer Identity Verifier
  195.      * 
  196.      * @return The Bessel Second Kind Half-Integer Identity Verifier
  197.      */

  199.     public static final org.drip.function.definition.R2ToR1Property HalfIntegerIdentity()
  200.     {
  201.         org.drip.function.definition.R1ToR1 gammaEstimator = new
  202.             org.drip.specialfunction.gamma.EulerIntegralSecondKind (null);

  204.         final org.drip.specialfunction.definition.BesselFirstKindEstimator besselFirstKindEstimator =
  205.             org.drip.specialfunction.bessel.FirstFrobeniusSeriesEstimator.Standard (
  206.                 gammaEstimator,
  207.                 50
  208.             );

  210.         try
  211.         {
  212.             final org.drip.specialfunction.definition.BesselSecondKindEstimator besselSecondKindEstimator =
  213.                 org.drip.specialfunction.bessel.SecondNISTSeriesEstimator.Standard (
  214.                     new org.drip.specialfunction.digamma.BinetFirstIntegral (null),
  215.                     gammaEstimator,
  216.                     org.drip.specialfunction.bessel.FirstFrobeniusSeriesEstimator.Standard (
  217.                         gammaEstimator,
  218.                         40
  219.                     ),
  220.                     40
  221.                 );

  223.             return new org.drip.function.definition.R2ToR1Property (
  224.                 org.drip.function.definition.RxToR1Property.EQ,
  225.                 new org.drip.function.definition.R2ToR1()
  226.                 {
  227.                     @Override public double evaluate (
  228.                         final double alpha,
  229.                         final double z)
  230.                         throws java.lang.Exception
  231.                     {
  232.                         if (!org.drip.numerical.common.NumberUtil.IsInteger (alpha))
  233.                         {
  234.                             throw new java.lang.Exception
  235.                                 ("BesselSecondEqualityLemma::HalfIntegerIdentity => Invalid Inputs");
  236.                         }

  238.                         return besselSecondKindEstimator.bigY (
  239.                             -1. * (alpha + 0.5),
  240.                             z
  241.                         );
  242.                     }
  243.                 },
  244.                 new org.drip.function.definition.R2ToR1()
  245.                 {
  246.                     @Override public double evaluate (
  247.                         final double alpha,
  248.                         final double z)
  249.                         throws java.lang.Exception
  250.                     {
  251.                         if (!org.drip.numerical.common.NumberUtil.IsInteger (alpha))
  252.                         {
  253.                             throw new java.lang.Exception
  254.                                 ("BesselSecondEqualityLemma::HalfIntegerIdentity => Invalid Inputs");
  255.                         }

  257.                         return (0 == ((int) (alpha)) % 2 ? 1. : -1.) * besselFirstKindEstimator.bigJ (
  258.                             alpha + 0.5,
  259.                             z
  260.                         );
  261.                     }
  262.                 },
  263.                 org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
  264.             );
  265.         }
  266.         catch (java.lang.Exception e)
  267.         {
  268.             e.printStackTrace();
  269.         }

  271.         return null;
  272.     }
  273. }
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