NemesAnalytic.java

package org.drip.specialfunction.gamma;

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

/*!
 * Copyright (C) 2020 Lakshmi Krishnamurthy
 * Copyright (C) 2019 Lakshmi Krishnamurthy
 * 
 *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
 *      asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
 *      analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
 *      equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
 *      numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
 *      and computational support.
 *  
 *      https://lakshmidrip.github.io/DROP/
 *  
 *  DROP is composed of three modules:
 *  
 *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
 *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
 *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
 * 
 *  DROP Product Core implements libraries for the following:
 *  - Fixed Income Analytics
 *  - Loan Analytics
 *  - Transaction Cost Analytics
 * 
 *  DROP Portfolio Core implements libraries for the following:
 *  - Asset Allocation Analytics
 *  - Asset Liability Management Analytics
 *  - Capital Estimation Analytics
 *  - Exposure Analytics
 *  - Margin Analytics
 *  - XVA Analytics
 * 
 *  DROP Computational Core implements libraries for the following:
 *  - Algorithm Support
 *  - Computation Support
 *  - Function Analysis
 *  - Model Validation
 *  - Numerical Analysis
 *  - Numerical Optimizer
 *  - Spline Builder
 *  - Statistical Learning
 * 
 *  Documentation for DROP is Spread Over:
 * 
 *  - Main                     => https://lakshmidrip.github.io/DROP/
 *  - Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
 *  - GitHub                   => https://github.com/lakshmiDRIP/DROP
 *  - Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
 *  - Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
 *  - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
 *  - Release Versions         => https://lakshmidrip.github.io/DROP/version.html
 *  - Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
 *  - Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
 *  - JUnit                    => https://lakshmidrip.github.io/DROP/junit/index.html
 *  - Jacoco                   => https://lakshmidrip.github.io/DROP/jacoco/index.html
 * 
 *  Licensed under the Apache License, Version 2.0 (the "License");
 *      you may not use this file except in compliance with the License.
 *   
 *  You may obtain a copy of the License at
 *      http://www.apache.org/licenses/LICENSE-2.0
 *  
 *  Unless required by applicable law or agreed to in writing, software
 *      distributed under the License is distributed on an "AS IS" BASIS,
 *      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  
 *  See the License for the specific language governing permissions and
 *      limitations under the License.
 */

/**
 * <i>NemesAnalytic</i> implements the Nemes Analytic Estimate of the Gamma Function. The References are:
 * 
 * <br><br>
 *  <ul>
 *      <li>
 *          Mortici, C. (2011): Improved Asymptotic Formulas for the Gamma Function <i>Computers and
 *              Mathematics with Applications</i> <b>61 (11)</b> 3364-3369
 *      </li>
 *      <li>
 *          National Institute of Standards and Technology (2018): NIST Digital Library of Mathematical
 *              Functions https://dlmf.nist.gov/5.11
 *      </li>
 *      <li>
 *          Nemes, G. (2010): On the Coefficients of the Asymptotic Expansion of n!
 *              https://arxiv.org/abs/1003.2907 <b>arXiv</b>
 *      </li>
 *      <li>
 *          Toth V. T. (2016): Programmable Calculators – The Gamma Function
 *              http://www.rskey.org/CMS/index.php/the-library/11
 *      </li>
 *      <li>
 *          Wikipedia (2019): Stirling's Approximation
 *              https://en.wikipedia.org/wiki/Stirling%27s_approximation
 *      </li>
 *  </ul>
 *
 *  <br><br>
 *  <ul>
 *      <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
 *      <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/FunctionAnalysisLibrary.md">Function Analysis Library</a></li>
 *      <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>
 *      <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>
 *  </ul>
 *
 * @author Lakshmi Krishnamurthy
 */

public class NemesAnalytic extends org.drip.numerical.estimation.R1ToR1Estimator
{

    /**
     * NemesAnalytic Constructor
     * 
     * @param dc The Derivative Control
     */

    public NemesAnalytic (
        final org.drip.numerical.differentiation.DerivativeControl dc)
    {
        super (dc);
    }

    @Override public double evaluate (
        final double x)
        throws java.lang.Exception
    {
        if (!org.drip.numerical.common.NumberUtil.IsValid (x) || 0. > x)
        {
            throw new java.lang.Exception ("NemesAnalytic::evaluate => Invalid Inputs");
        }

        return java.lang.Math.sqrt (2. * java.lang.Math.PI / x) * java.lang.Math.pow (
            (x + (1. / (12. * x - (0.1 / x)))) / java.lang.Math.E,
            x
        );
    }
}