ArcTangentGeneralizedMidPoint.java
package org.drip.sample.newtoncotes;
import org.drip.numerical.common.FormatUtil;
import org.drip.service.env.EnvManager;
/*
* -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
*/
/*!
* Copyright (C) 2019 Lakshmi Krishnamurthy
*
* This file is part of DROP, an open-source library targeting risk, transaction costs, exposure, margin
* calculations, and portfolio construction within and across fixed income, credit, commodity, equity,
* FX, and structured products.
*
* https://lakshmidrip.github.io/DROP/
*
* DROP is composed of three main modules:
*
* - DROP Analytics Core - https://lakshmidrip.github.io/DROP-Analytics-Core/
* - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
* - DROP Numerical Core - https://lakshmidrip.github.io/DROP-Numerical-Core/
*
* DROP Analytics Core implements libraries for the following:
* - Fixed Income Analytics
* - Asset Backed Analytics
* - XVA Analytics
* - Exposure and Margin Analytics
*
* DROP Portfolio Core implements libraries for the following:
* - Asset Allocation Analytics
* - Transaction Cost Analytics
*
* DROP Numerical Core implements libraries for the following:
* - Statistical Learning Library
* - Numerical Optimizer Library
* - Machine Learning Library
* - Spline Builder Library
*
* 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
* - 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>ArcTangentGeneralizedMidPoint</i> computes the R<sup>1</sup> Numerical Estimate of the tan<sup>-1</sup>
* using the Generalized Mid-Point Quadrature. The References are:
*
* <br><br>
* <ul>
* <li>
* Briol, F. X., C. J. Oates, M. Girolami, and M. A. Osborne (2015): <i>Frank-Wolfe Bayesian
* Quadrature: Probabilistic Integration with Theoretical Guarantees</i> <b>arXiv</b>
* </li>
* <li>
* Forsythe, G. E., M. A. Malcolm, and C. B. Moler (1977): <i>Computer Methods for Mathematical
* Computation</i> <b>Prentice Hall</b> Englewood Cliffs NJ
* </li>
* <li>
* Leader, J. J. (2004): <i>Numerical Analysis and Scientific Computation</i> <b>Addison Wesley</b>
* </li>
* <li>
* Stoer, J., and R. Bulirsch (1980): <i>Introduction to Numerical Analysis</i>
* <b>Springer-Verlag</b> New York
* </li>
* <li>
* Wikipedia (2019): Numerical Integration https://en.wikipedia.org/wiki/Numerical_integration
* </li>
* </ul>
*
* <br><br>
* <ul>
* <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/NumericalCore.md">Numerical Core Module</a></li>
* <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/NumericalOptimizerLibrary.md">Numerical Optimizer</a></li>
* <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/README.md">Sample</a></li>
* <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/newtoncotes/README.md">R<sup>1</sup> Newton-Cotes Quadrature Schemes</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class ArcTangentGeneralizedMidPoint
{
private static final double ArcTangent (
final double z,
final int termCount)
throws Exception
{
double b = 1.;
double a = 2. / z;
double arcTangent = 0.;
for (int termIndex = 1; termIndex <= termCount; ++termIndex)
{
arcTangent = arcTangent + (a / ((a * a + b * b) * (2 * termIndex - 1)));
a = a * (1. - (4. / (z * z))) + 4. * (b / z);
b = b * (1. - (4. / (z * z))) - 4. * (a / z);
}
return 2. * arcTangent;
}
public static final void main (
final String[] argumentArray)
throws Exception
{
EnvManager.InitEnv ("");
int termCount = 10;
double[] rValueArray =
{
1. / 32.,
1. / 16.,
1. / 8.,
1. / 4.,
1. / 2.,
1.,
2.,
4.,
8.,
16.,
32.,
64.
};
System.out.println ("\t|-----------------------------------||");
System.out.println ("\t| ARC TANGENT SERIES ESTIMATION ||");
System.out.println ("\t|-----------------------------------||");
System.out.println ("\t| L -> R: ||");
System.out.println ("\t| - z ||");
System.out.println ("\t| - System ||");
System.out.println ("\t| - Series ||");
System.out.println ("\t|-----------------------------------||");
for (double rValue : rValueArray)
{
System.out.println (
"\t|" + FormatUtil.FormatDouble (rValue, 2, 5, 1.) + " => " +
FormatUtil.FormatDouble (Math.atan (rValue), 1, 6, 1.) + " | " +
FormatUtil.FormatDouble (
ArcTangent (
rValue,
termCount
), 1, 6, 1.
) + " ||"
);
}
System.out.println ("\t|-----------------------------------||");
EnvManager.TerminateEnv();
}
}