MonodromyTransform2F1.java
package org.drip.specialfunction.group;
/*
* -*- 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>MonodromyTransform2F1</i> builds out the Monodromy Loop Solution Transformation Matrices for Paths
* around the Singular Points. The References are:
*
* <br><br>
* <ul>
* <li>
* Gessel, I., and D. Stanton (1982): Strange Evaluations of Hyper-geometric Series <i>SIAM Journal
* on Mathematical Analysis</i> <b>13 (2)</b> 295-308
* </li>
* <li>
* Koepf, W (1995): Algorithms for m-fold Hyper-geometric Summation <i>Journal of Symbolic
* Computation</i> <b>20 (4)</b> 399-417
* </li>
* <li>
* Lavoie, J. L., F. Grondin, and A. K. Rathie (1996): Generalization of Whipple’s Theorem on the
* Sum of a (_2^3)F(a,b;c;z) <i>Journal of Computational and Applied Mathematics</i> <b>72</b>
* 293-300
* </li>
* <li>
* National Institute of Standards and Technology (2019): Hyper-geometric Function
* https://dlmf.nist.gov/15
* </li>
* <li>
* Wikipedia (2019): Hyper-geometric Function https://en.wikipedia.org/wiki/Hypergeometric_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/group/README.md">Special Function Singularity Solution Group</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class MonodromyTransform2F1
{
/**
* Generate the Monodromy Group Matrix G0 around the '0' Singularity
*
* @param pathExponent1 Path Monodromy Exponents of the Fundamental Group #1
* @param pathExponent2 Path Monodromy Exponents of the Fundamental Group #2
*
* @return The Monodromy Group Matrix G0 around the '0' Singularity
*/
public static final org.drip.function.definition.CartesianComplexNumber[][] G0 (
final org.drip.specialfunction.group.FundamentalGroupPathExponent2F1 pathExponent1,
final org.drip.specialfunction.group.FundamentalGroupPathExponent2F1 pathExponent2)
{
if (null == pathExponent1 || null == pathExponent2)
{
return null;
}
org.drip.function.definition.CartesianComplexNumber[][] g0 = new org.drip.function.definition.CartesianComplexNumber[2][2];
double theta1 = 2. * java.lang.Math.PI * pathExponent1.alpha();
double theta2 = 2. * java.lang.Math.PI * pathExponent2.alpha();
try
{
g0[0][0] = new org.drip.function.definition.CartesianComplexNumber (
java.lang.Math.cos (theta1),
java.lang.Math.sin (theta1)
);
g0[0][1] = new org.drip.function.definition.CartesianComplexNumber (
0.,
0.
);
g0[1][0] = new org.drip.function.definition.CartesianComplexNumber (
0.,
0.
);
g0[1][1] = new org.drip.function.definition.CartesianComplexNumber (
java.lang.Math.cos (theta2),
java.lang.Math.sin (theta2)
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return g0;
}
/**
* Compute the "Mu" Intermediate for the G1 Monodromy Matrix
*
* @param pathExponent1 Path Monodromy Exponents of the Fundamental Group #1
* @param pathExponent2 Path Monodromy Exponents of the Fundamental Group #2
*
* @return The "Mu" Intermediate for the G1 Monodromy Matrix
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public static final double G1Mu (
final org.drip.specialfunction.group.FundamentalGroupPathExponent2F1 pathExponent1,
final org.drip.specialfunction.group.FundamentalGroupPathExponent2F1 pathExponent2)
throws java.lang.Exception
{
if (null == pathExponent1 || null == pathExponent2)
{
throw new java.lang.Exception ("MonodromyTransform2F1::G1Mu => Invalid Inputs");
}
double beta1 = pathExponent1.beta();
double beta2 = pathExponent2.beta();
double alpha1 = pathExponent1.alpha();
double alpha2 = pathExponent2.alpha();
double gamma2 = pathExponent2.gamma();
return java.lang.Math.sin (java.lang.Math.PI * (alpha1 + beta2 + gamma2)) *
java.lang.Math.sin (java.lang.Math.PI * (alpha2 + beta1 + gamma2)) /
java.lang.Math.sin (java.lang.Math.PI * (alpha2 + beta2 + gamma2)) /
java.lang.Math.sin (java.lang.Math.PI * (alpha1 + beta1 + gamma2));
}
/**
* Generate the Monodromy Group Matrix G1 around the '1' Singularity
*
* @param pathExponent1 Path Monodromy Exponents of the Fundamental Group #1
* @param pathExponent2 Path Monodromy Exponents of the Fundamental Group #2
*
* @return The Monodromy Group Matrix G1 around the '1' Singularity
*/
public static final org.drip.function.definition.CartesianComplexNumber[][] G1 (
final org.drip.specialfunction.group.FundamentalGroupPathExponent2F1 pathExponent1,
final org.drip.specialfunction.group.FundamentalGroupPathExponent2F1 pathExponent2)
{
if (null == pathExponent1 || null == pathExponent2)
{
return null;
}
org.drip.function.definition.CartesianComplexNumber[][] g0 = new org.drip.function.definition.CartesianComplexNumber[2][2];
double theta1 = 2. * java.lang.Math.PI * pathExponent1.beta();
double theta2 = 2. * java.lang.Math.PI * pathExponent2.beta();
try
{
double g1Mu = G1Mu (
pathExponent1,
pathExponent2
);
double muMinus1 = g1Mu - 1.;
double muMinus1Squared = muMinus1 * muMinus1;
g0[0][0] = new org.drip.function.definition.CartesianComplexNumber (
(g1Mu * java.lang.Math.cos (theta1) - java.lang.Math.cos (theta2)) / muMinus1,
(g1Mu * java.lang.Math.sin (theta1) - java.lang.Math.sin (theta2)) / muMinus1
);
g0[0][1] = new org.drip.function.definition.CartesianComplexNumber (
java.lang.Math.cos (theta2) - java.lang.Math.cos (theta1),
java.lang.Math.sin (theta2) - java.lang.Math.sin (theta1)
);
g0[1][0] = new org.drip.function.definition.CartesianComplexNumber (
g1Mu * (java.lang.Math.cos (theta2) - java.lang.Math.cos (theta1)) /
(muMinus1Squared * muMinus1Squared),
g1Mu * (java.lang.Math.sin (theta2) - java.lang.Math.sin (theta1)) /
(muMinus1Squared * muMinus1Squared)
);
g0[1][1] = new org.drip.function.definition.CartesianComplexNumber (
(g1Mu * java.lang.Math.cos (theta2) - java.lang.Math.cos (theta1)) / muMinus1,
(g1Mu * java.lang.Math.sin (theta2) - java.lang.Math.sin (theta1)) / muMinus1
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return g0;
}
}