GaussContiguousEqualityLemma.java
package org.drip.specialfunction.property;
/*
* -*- 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>GaussContiguousEqualityLemma</i> verifies the Hyper-geometric Gauss Contiguous Equality Lemma
* Properties. 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/property/README.md">Special Function Property Lemma Verifiers</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class GaussContiguousEqualityLemma
{
/**
* Construct the Hyper-geometric Gauss Contiguous Identity #2 Verifier
*
* @param a A
* @param b B
* @param c C
*
* @return The Hyper-geometric Gauss Contiguous Identity #2 Verifier
*/
public static final org.drip.function.definition.R1ToR1Property RelationIdentity2 (
final double a,
final double b,
final double c)
{
org.drip.function.definition.R2ToR1 logBetaEstimator =
org.drip.specialfunction.beta.LogGammaEstimator.Weierstrass (1000);
if (null == logBetaEstimator)
{
return null;
}
try
{
final org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator hypergeometricEstimator =
new org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator (
new org.drip.specialfunction.definition.HypergeometricParameters (
a,
b,
c
),
logBetaEstimator,
10000
);
final org.drip.specialfunction.definition.RegularHypergeometricEstimator
aPlusRegularHypergeometricEstimator = new
org.drip.specialfunction.hypergeometric.GaussContiguousRelations
(hypergeometricEstimator).aPlus();
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.EQ,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return z * hypergeometricEstimator.derivative (
z,
1
);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return a * (
aPlusRegularHypergeometricEstimator.regularHypergeometric (z) -
hypergeometricEstimator.regularHypergeometric (z)
);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Hyper-geometric Gauss Contiguous Identity #3 Verifier
*
* @param a A
* @param b B
* @param c C
*
* @return The Hyper-geometric Gauss Contiguous Identity #3 Verifier
*/
public static final org.drip.function.definition.R1ToR1Property RelationIdentity3 (
final double a,
final double b,
final double c)
{
org.drip.function.definition.R2ToR1 logBetaEstimator =
org.drip.specialfunction.beta.LogGammaEstimator.Weierstrass (1000);
if (null == logBetaEstimator)
{
return null;
}
try
{
final org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator hypergeometricEstimator =
new org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator (
new org.drip.specialfunction.definition.HypergeometricParameters (
a,
b,
c
),
logBetaEstimator,
10000
);
final org.drip.specialfunction.definition.RegularHypergeometricEstimator
bPlusHypergeometricEstimator = new
org.drip.specialfunction.hypergeometric.GaussContiguousRelations
(hypergeometricEstimator).bPlus();
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.EQ,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return z * hypergeometricEstimator.derivative (
z,
1
);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return b * (
bPlusHypergeometricEstimator.regularHypergeometric (z) -
hypergeometricEstimator.regularHypergeometric (z)
);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Hyper-geometric Gauss Contiguous Identity #4 Verifier
*
* @param a A
* @param b B
* @param c C
*
* @return The Hyper-geometric Gauss Contiguous Identity #4 Verifier
*/
public static final org.drip.function.definition.R1ToR1Property RelationIdentity4 (
final double a,
final double b,
final double c)
{
org.drip.function.definition.R2ToR1 logBetaEstimator =
org.drip.specialfunction.beta.LogGammaEstimator.Weierstrass (10000);
if (null == logBetaEstimator)
{
return null;
}
try
{
final org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator hypergeometricEstimator =
new org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator (
new org.drip.specialfunction.definition.HypergeometricParameters (
a,
b,
c
),
logBetaEstimator,
100000
);
final org.drip.specialfunction.definition.RegularHypergeometricEstimator
cMinusHypergeometricEstimator = new
org.drip.specialfunction.hypergeometric.GaussContiguousRelations
(hypergeometricEstimator).cMinus();
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.EQ,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return z * hypergeometricEstimator.derivative (
z,
1
);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return (c - 1.) * (
cMinusHypergeometricEstimator.regularHypergeometric (z) -
hypergeometricEstimator.regularHypergeometric (z)
);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Hyper-geometric Gauss Contiguous Identity #5 Verifier
*
* @param a A
* @param b B
* @param c C
*
* @return The Hyper-geometric Gauss Contiguous Identity #5 Verifier
*/
public static final org.drip.function.definition.R1ToR1Property RelationIdentity5 (
final double a,
final double b,
final double c)
{
org.drip.function.definition.R2ToR1 logBetaEstimator =
org.drip.specialfunction.beta.LogGammaEstimator.Weierstrass (10000);
if (null == logBetaEstimator)
{
return null;
}
try
{
final org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator hypergeometricEstimator =
new org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator (
new org.drip.specialfunction.definition.HypergeometricParameters (
a,
b,
c
),
logBetaEstimator,
100000
);
final org.drip.specialfunction.definition.RegularHypergeometricEstimator
aMinusHypergeometricEstimator = new
org.drip.specialfunction.hypergeometric.GaussContiguousRelations
(hypergeometricEstimator).aMinus();
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.EQ,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return z * hypergeometricEstimator.derivative (
z,
1
);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return (
(c - a) * aMinusHypergeometricEstimator.regularHypergeometric (z) +
(a - c + b * z) * hypergeometricEstimator.regularHypergeometric (z)
) / (1. - z);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Hyper-geometric Gauss Contiguous Identity #6 Verifier
*
* @param a A
* @param b B
* @param c C
*
* @return The Hyper-geometric Gauss Contiguous Identity #6 Verifier
*/
public static final org.drip.function.definition.R1ToR1Property RelationIdentity6 (
final double a,
final double b,
final double c)
{
org.drip.function.definition.R2ToR1 logBetaEstimator =
org.drip.specialfunction.beta.LogGammaEstimator.Weierstrass (10000);
if (null == logBetaEstimator)
{
return null;
}
try
{
final org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator hypergeometricEstimator =
new org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator (
new org.drip.specialfunction.definition.HypergeometricParameters (
a,
b,
c
),
logBetaEstimator,
100000
);
final org.drip.specialfunction.definition.RegularHypergeometricEstimator
bMinusHypergeometricEstimator = new
org.drip.specialfunction.hypergeometric.GaussContiguousRelations
(hypergeometricEstimator).bMinus();
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.EQ,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return z * hypergeometricEstimator.derivative (
z,
1
);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return (
(c - b) * bMinusHypergeometricEstimator.regularHypergeometric (z) +
(b - c + a * z) * hypergeometricEstimator.regularHypergeometric (z)
) / (1. - z);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Hyper-geometric Gauss Contiguous Identity #7 Verifier
*
* @param a A
* @param b B
* @param c C
*
* @return The Hyper-geometric Gauss Contiguous Identity #7 Verifier
*/
public static final org.drip.function.definition.R1ToR1Property RelationIdentity7 (
final double a,
final double b,
final double c)
{
org.drip.function.definition.R2ToR1 logBetaEstimator =
org.drip.specialfunction.beta.LogGammaEstimator.Weierstrass (10000);
if (null == logBetaEstimator)
{
return null;
}
try
{
final org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator hypergeometricEstimator =
new org.drip.specialfunction.hypergeometric.EulerQuadratureEstimator (
new org.drip.specialfunction.definition.HypergeometricParameters (
a,
b,
c
),
logBetaEstimator,
100000
);
final org.drip.specialfunction.definition.RegularHypergeometricEstimator
cPlusHypergeometricEstimator = new
org.drip.specialfunction.hypergeometric.GaussContiguousRelations
(hypergeometricEstimator).cPlus();
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.EQ,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return hypergeometricEstimator.derivative (
z,
1
);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return (
(c - a) * (c - b) * cPlusHypergeometricEstimator.regularHypergeometric (z) +
c * (a + b - c) * hypergeometricEstimator.regularHypergeometric (z)
) / (c * (1. - z));
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
}