GammaInequalityLemma.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>GammaInequalityLemma</i> contains the Verifiable Inequality Lemmas of the Gamma Function. The
* References are:
*
* <br><br>
* <ul>
* <li>
* Blagouchine, I. V. (2014): Re-discovery of Malmsten's Integrals, their Evaluation by Contour
* Integration Methods, and some Related Results <i>Ramanujan Journal</i> <b>35 (1)</b> 21-110
* </li>
* <li>
* Borwein, J. M., and R. M. Corless (2017): Gamma Function and the Factorial in the Monthly
* https://arxiv.org/abs/1703.05349 <b>arXiv</b>
* </li>
* <li>
* Davis, P. J. (1959): Leonhard Euler's Integral: A Historical Profile of the Gamma Function
* <i>American Mathematical Monthly</i> <b>66 (10)</b> 849-869
* </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): Gamma Function https://en.wikipedia.org/wiki/Gamma_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 GammaInequalityLemma
{
/**
* Construct the Asymptotic Upper Approximate
*
* @param alpha Alpha
*
* @return The Asymptotic Upper Approximate
*/
public static final org.drip.function.definition.R1ToR1Property AsymptoticUpperApproximate (
final double alpha)
{
if (!org.drip.numerical.common.NumberUtil.IsValid (alpha))
{
return null;
}
try
{
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.GTE,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double s)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (s))
{
throw new java.lang.Exception
("GammaInequalityLemma::AsymptoticUpperApproximate::evaluate => Invalid Inputs");
}
return org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400).evaluate
(s + alpha);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double s)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (s))
{
throw new java.lang.Exception
("GammaInequalityLemma::AsymptoticUpperApproximate::evaluate => Invalid Inputs");
}
return alpha * java.lang.Math.log (s) +
org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400).evaluate (s);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Exponentially Convex Inequality Verifier
*
* @param z1 z1
* @param z2 z2
*
* @return The Exponentially Convex Inequality Verifier
*/
public static final org.drip.function.definition.R1ToR1Property ExponentiallyConvex (
final double z1,
final double z2)
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z1) ||
!org.drip.numerical.common.NumberUtil.IsValid (z2))
{
return null;
}
try
{
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.LTE,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double t)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (t) || 0. > t || 1. < t)
{
throw new java.lang.Exception
("GammaInequalityLemma::ExponentiallyConvex::evaluate => Invalid Inputs");
}
return org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400).evaluate
(t * z1 + (1. - t) * z2);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double t)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (t) || 0. > t || 1. < t)
{
throw new java.lang.Exception
("GammaInequalityLemma::ExponentiallyConvex::evaluate => Invalid Inputs");
}
org.drip.specialfunction.loggamma.InfiniteSumEstimator weierStrass =
org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);
return t * weierStrass.evaluate (z1) + (1. - t) * weierStrass.evaluate (z2);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Spaced Point Convex Inequality Verifier
*
* @param y y
*
* @return The Spaced Point Convex Inequality Verifier
*/
public static final org.drip.function.definition.R1ToR1Property SpacedPointConvex (
final double y)
{
if (!org.drip.numerical.common.NumberUtil.IsValid (y))
{
return null;
}
final org.drip.specialfunction.loggamma.InfiniteSumEstimator weierStrass =
org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);
try
{
final double logGammaY = weierStrass.evaluate (y);
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.GT,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double x)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (x) || x >= y)
{
throw new java.lang.Exception
("GammaInequalityLemma::SpacedPointConvex::evaluate => Invalid Inputs");
}
return (logGammaY - weierStrass.evaluate (x)) / (y - x);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double x)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (x))
{
throw new java.lang.Exception
("GammaInequalityLemma::SpacedPointConvex::evaluate => Invalid Inputs");
}
org.drip.specialfunction.gamma.EulerIntegralSecondKind eulerIntegralSecondKind =
new org.drip.specialfunction.gamma.EulerIntegralSecondKind (null);
return eulerIntegralSecondKind.derivative (
x,
1
) - weierStrass.evaluate (x);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Logarithmically Convex Inequality Verifier
*
* @return The Logarithmically Convex Inequality Verifier
*/
public static final org.drip.function.definition.R1ToR1Property LogarithmicConvex()
{
final org.drip.specialfunction.loggamma.InfiniteSumEstimator weierStrass =
org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);
try
{
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.GT,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("GammaInequalityLemma::LogarithmicConvex::evaluate => Invalid Inputs");
}
return java.lang.Math.log (
new org.drip.specialfunction.gamma.EulerIntegralSecondKind (null).derivative (
z,
2
)
) + weierStrass.evaluate (z);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("GammaInequalityLemma::LogarithmicConvex::evaluate => Invalid Inputs");
}
return java.lang.Math.log (
new org.drip.specialfunction.gamma.EulerIntegralSecondKind (null).derivative (
z,
1
)
);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Gautschi Left Inequality Verifier
*
* @param s s
*
* @return The Gautschi Left Inequality Verifier
*/
public static final org.drip.function.definition.R1ToR1Property GautschiLeft (
final double s)
{
if (!org.drip.numerical.common.NumberUtil.IsValid (s) || 0. >= s || 1. <= s)
{
return null;
}
final org.drip.specialfunction.loggamma.InfiniteSumEstimator weierStrass =
org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);
try
{
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.LT,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("GammaInequalityLemma::GautschiLeft::evaluate => Invalid Inputs");
}
return (1. - s) * java.lang.Math.log (z);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("GammaInequalityLemma::GautschiLeft::evaluate => Invalid Inputs");
}
return weierStrass.evaluate (z + 1) - weierStrass.evaluate (z + s);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Gautschi Right Inequality Verifier
*
* @param s s
*
* @return The Gautschi Right Inequality Verifier
*/
public static final org.drip.function.definition.R1ToR1Property GautschiRight (
final double s)
{
if (!org.drip.numerical.common.NumberUtil.IsValid (s) || 0. >= s || 1. <= s)
{
return null;
}
final org.drip.specialfunction.loggamma.InfiniteSumEstimator weierStrass =
org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);
try
{
return new org.drip.function.definition.R1ToR1Property (
org.drip.function.definition.R1ToR1Property.LT,
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("GammaInequalityLemma::GautschiRight::evaluate => Invalid Inputs");
}
return weierStrass.evaluate (z + 1) - weierStrass.evaluate (z + s);
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("GammaInequalityLemma::GautschiRight::evaluate => Invalid Inputs");
}
return (1. - s) * java.lang.Math.log (z + 1.);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Jensen Multi-Point Interpolant Convexity Verification
*
* @param multiPoint2D Multi-Point 2D
*
* @return Jensen Multi-Point Interpolant Convexity Verification
*/
public static final org.drip.function.definition.R1PropertyVerification JensenMultiPointInterpolant (
final org.drip.numerical.common.Array2D multiPoint2D)
{
if (null == multiPoint2D)
{
return null;
}
final org.drip.specialfunction.loggamma.InfiniteSumEstimator weierStrass =
org.drip.specialfunction.loggamma.InfiniteSumEstimator.Weierstrass (1638400);
double[] xArray = multiPoint2D.x();
double[] aArray = multiPoint2D.y();
double interpolantDenominator = 0.;
double interpolantNumerator = 0.;
int count = aArray.length;
double rValue = 0.;
for (int index = 0; index < count; ++index)
{
interpolantNumerator = interpolantNumerator + aArray[index] * xArray[index];
interpolantDenominator = interpolantDenominator + aArray[index];
}
double interpolantDenominatorInverse = 1. / interpolantDenominator;
try
{
double lValue = weierStrass.evaluate (interpolantNumerator* interpolantDenominatorInverse);
for (int index = 0; index < count; ++index)
{
rValue = rValue + aArray[index] * weierStrass.evaluate (xArray[index]);
}
return new org.drip.function.definition.R1PropertyVerification (
lValue,
rValue = rValue * interpolantDenominatorInverse,
lValue <= rValue
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
}