SaddlePoints.java
package org.drip.specialfunction.digamma;
/*
* -*- 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>SaddlePoints</i> contains the Hermite Based Saddle Point Roots of the Digamma Function. The References
* are:
*
* <br><br>
* <ul>
* <li>
* Abramowitz, M., and I. A. Stegun (2007): Handbook of Mathematics Functions <b>Dover Book on
* Mathematics</b>
* </li>
* <li>
* Blagouchine, I. V. (2018): Three Notes on Ser's and Hasse's Representations for the
* Zeta-Functions https://arxiv.org/abs/1606.02044 <b>arXiv</b>
* </li>
* <li>
* Mezo, I., and M. E. Hoffman (2017): Zeros of the Digamma Function and its Barnes G-function
* Analogue <i>Integral Transforms and Special Functions</i> <b>28 (28)</b> 846-858
* </li>
* <li>
* Whitaker, E. T., and G. N. Watson (1996): <i>A Course on Modern Analysis</i> <b>Cambridge
* University Press</b> New York
* </li>
* <li>
* Wikipedia (2019): Digamma Function https://en.wikipedia.org/wiki/Digamma_function
* </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/digamma/README.md">Estimation Techniques for Digamma Function</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class SaddlePoints
{
/**
* Generate the Set of Leading Digamma Saddle Points
*
* @return Set of Leading Digamma Saddle Points
*/
public static final java.util.TreeSet<java.lang.Double> LeadingZeros()
{
java.util.TreeSet<java.lang.Double> zeroSet = new java.util.TreeSet<java.lang.Double>();
zeroSet.add ( 1.461632144968);
zeroSet.add (-0.504083008000);
zeroSet.add (-1.573498473000);
zeroSet.add (-2.610720868000);
zeroSet.add (-3.635293366000);
return zeroSet;
}
/**
* Construct the R<sup>1</sup> to R<sup>1</sup> Hermite Digamma Root Function
*
* @return The R<sup>1</sup> to R<sup>1</sup> Hermite Digamma Root Function
*/
public static final org.drip.function.definition.R1ToR1 Hermite()
{
try
{
return new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (1. > z)
{
throw new java.lang.Exception ("SaddlePoints::Hermite::evaluate => Invalid Inputs");
}
return 1. / java.lang.Math.log (z) - z;
}
};
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the R<sup>1</sup> to R<sup>1</sup> Hermite Extension Digamma Root Function
*
* @return The R<sup>1</sup> to R<sup>1</sup> Hermite Extension Digamma Root Function
*/
public static final org.drip.function.definition.R1ToR1 HermiteExtension()
{
try
{
return new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (1. > z)
{
throw new java.lang.Exception
("SaddlePoints::HermiteExtension::evaluate => Invalid Inputs");
}
return java.lang.Math.atan (java.lang.Math.PI / java.lang.Math.log (z)) /
java.lang.Math.PI - z;
}
};
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the R<sup>1</sup> to R<sup>1</sup> Hermite Enhancement Digamma Root Function
*
* @return The R<sup>1</sup> to R<sup>1</sup> Hermite Enhancement Digamma Root Function
*/
public static final org.drip.function.definition.R1ToR1 HermiteEnhancement()
{
try
{
return new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (1. > z)
{
throw new java.lang.Exception
("SaddlePoints::HermiteEnhancement::evaluate => Invalid Inputs");
}
return java.lang.Math.atan (java.lang.Math.PI / (java.lang.Math.log (z) + (0.125 / z))) /
java.lang.Math.PI - z;
}
};
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the R<sup>1</sup> to R<sup>1</sup> Mezo-Hoffman (2017) Digamma Root Function
*
* @return The R<sup>1</sup> to R<sup>1</sup> Mezo-Hoffman (2017) Digamma Root Function
*/
public static final org.drip.function.definition.R1ToR1 MezoHoffman2017()
{
try
{
return new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (1. > z)
{
throw new java.lang.Exception
("SaddlePoints::MezoHoffman2017::evaluate => Invalid Inputs");
}
double logZReciprocal = 1. / java.lang.Math.log (z);
return logZReciprocal - z - 0.5 * logZReciprocal * logZReciprocal / z;
}
};
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Array of Leading Roots
*
* @param rootFunction The Root Generation Function
* @param rootCount The Root Count
*
* @return The Array of Leading Roots
*/
public static final double[] LeadingRoots (
final org.drip.function.definition.R1ToR1 rootFunction,
final int rootCount)
{
if (null == rootFunction ||
0 >= rootCount)
{
return null;
}
double[] leadingRootArray = new double[rootCount + 1];
leadingRootArray[0] = org.drip.specialfunction.gamma.Definitions.MINIMUM_VARIATE_LOCATION;
for (int rootIndex = 1; rootIndex <= rootCount; ++rootIndex)
{
try
{
leadingRootArray[rootIndex] = rootFunction.evaluate (rootIndex);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
}
return leadingRootArray;
}
}