DigammaEqualityLemma.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>DigammaEqualityLemma</i> contains the Verifiable Equality Lemmas of the Digamma 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 DigammaEqualityLemma
{
/**
* Generate the Digamma (0, 1) Reflection Formula Verifier
*
* @return The Digamma (0, 1) Reflection Formula Verifier
*/
public static final org.drip.function.definition.R1ToR1Property ReflectionFormula()
{
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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
{
if (!org.drip.numerical.common.NumberUtil.IsValid (z))
{
throw new java.lang.Exception
("DigammaEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
}
return abramowitzStegun2007.evaluate (1. - z) - abramowitzStegun2007.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
("DigammaEqualityLemma::ReflectionFormula::evaluate => Invalid Inputs");
}
return java.lang.Math.PI / java.lang.Math.tan (java.lang.Math.PI * z);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Gaussian Finite Summation Identity Verifier #1
*
* @return The Gaussian Finite Summation Identity Verifier #1
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity1()
{
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity1::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r <= m; ++r)
{
summation = summation + abramowitzStegun2007.evaluate (((double) r) / ((double) m));
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity1::evaluate => Invalid Inputs");
}
return -m * (org.drip.specialfunction.gamma.Definitions.EULER_MASCHERONI +
java.lang.Math.log (m));
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Gaussian Finite Summation Identity Verifier #2
*
* @param k k
*
* @return The Gaussian Finite Summation Identity Verifier #2
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity2 (
final int k)
{
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m || m <= k)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity2::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r < m; ++r)
{
double z = ((double) r) / ((double) m);
summation = summation + abramowitzStegun2007.evaluate (z) *
java.lang.Math.cos (2. * java.lang.Math.PI * k * z);
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m || m <= k)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity2::evaluate => Invalid Inputs");
}
return m * java.lang.Math.log (2. * java.lang.Math.sin (java.lang.Math.PI * k / m)) +
org.drip.specialfunction.gamma.Definitions.EULER_MASCHERONI;
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Gaussian Finite Summation Identity Verifier #3
*
* @param k k
*
* @return The Gaussian Finite Summation Identity Verifier #3
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity3 (
final int k)
{
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m || m <= k)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity3::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r < m; ++r)
{
double z = ((double) r) / ((double) m);
summation = summation + abramowitzStegun2007.evaluate (z) *
java.lang.Math.sin (2. * java.lang.Math.PI * k * z);
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m || m <= k)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity3::evaluate => Invalid Inputs");
}
return 0.5 * java.lang.Math.PI * (2. * k - m);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Blagouchine Finite Summation Identity Verifier #4
*
* @param k k
*
* @return The Blagouchine Finite Summation Identity Verifier #4
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity4 (
final int k)
{
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m || m <= k)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity4::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 0; r < m; ++r)
{
double z = (2. * r + 1.) / (2. * m);
summation = summation + abramowitzStegun2007.evaluate (z) *
java.lang.Math.cos (2. * java.lang.Math.PI * k * z);
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m || m <= k)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity4::evaluate => Invalid Inputs");
}
return m * java.lang.Math.log (java.lang.Math.tan (0.5 * java.lang.Math.PI * k / m));
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Blagouchine Finite Summation Identity Verifier #5
*
* @param k k
*
* @return The Blagouchine Finite Summation Identity Verifier #5
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity5 (
final int k)
{
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m || m <= k)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity5::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 0; r < m; ++r)
{
double z = (2. * r + 1.) / (2. * m);
summation = summation + abramowitzStegun2007.evaluate (z) *
java.lang.Math.sin (2. * java.lang.Math.PI * k * z);
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m || m <= k)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity5::evaluate => Invalid Inputs");
}
return -0.5 * java.lang.Math.PI * m;
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Blagouchine Finite Summation Identity Verifier #6
*
* @return The Blagouchine Finite Summation Identity Verifier #6
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity6()
{
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity6::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r < m; ++r)
{
double z = ((double) r) / ((double) m);
summation = summation + abramowitzStegun2007.evaluate (z) /
java.lang.Math.tan (java.lang.Math.PI * z);
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity6::evaluate => Invalid Inputs");
}
return -java.lang.Math.PI * (m - 1.) * (m - 2.) / 6.;
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Blagouchine Finite Summation Identity Verifier #7
*
* @return The Blagouchine Finite Summation Identity Verifier #7
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity7()
{
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity7::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r < m; ++r)
{
double z = ((double) r) / ((double) m);
summation = summation + z * abramowitzStegun2007.evaluate (z);
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity7::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r < m; ++r)
{
double z = ((double) r) / ((double) m);
summation = summation + z / java.lang.Math.tan (java.lang.Math.PI * z);
}
return -0.5 * org.drip.specialfunction.gamma.Definitions.EULER_MASCHERONI * (m - 1.)
- 0.5 * m * java.lang.Math.log (m)
- 0.5 * java.lang.Math.PI * summation;
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Blagouchine Finite Summation Identity Verifier #8
*
* @param l l
*
* @return The Blagouchine Finite Summation Identity Verifier #8
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity8 (
final double l)
{
if (!org.drip.numerical.common.NumberUtil.IsValid (l))
{
return null;
}
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity8::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r < m; ++r)
{
double z = ((double) r) / ((double) m);
summation = summation + abramowitzStegun2007.evaluate (z) *
java.lang.Math.cos ((2. * l + 1.) * java.lang.Math.PI * z);
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity8::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r < m; ++r)
{
double z = ((double) r) / ((double) m);
double theta = 2. * java.lang.Math.PI * z;
summation = summation + r * java.lang.Math.sin (theta) / (
java.lang.Math.cos (theta) -
java.lang.Math.cos ((2. * l + 1.) * java.lang.Math.PI / m)
);
}
return -1. * java.lang.Math.PI / m * summation;
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Blagouchine Finite Summation Identity Verifier #9
*
* @param l l
*
* @return The Blagouchine Finite Summation Identity Verifier #9
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity9 (
final double l)
{
if (!org.drip.numerical.common.NumberUtil.IsValid (l))
{
return null;
}
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity9::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r < m; ++r)
{
double z = ((double) r) / ((double) m);
summation = summation + abramowitzStegun2007.evaluate (z) *
java.lang.Math.sin ((2. * l + 1.) * java.lang.Math.PI * z);
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity9::evaluate => Invalid Inputs");
}
double summation = 0.;
double twoLPlus1PiByM = (2. * l + 1.) * java.lang.Math.PI / m;
double cos2LPlus1PiByM = java.lang.Math.cos (twoLPlus1PiByM);
for (int r = 1; r < m; ++r)
{
double z = ((double) r) / ((double) m);
double theta = java.lang.Math.PI * z;
summation = summation + java.lang.Math.log (java.lang.Math.sin (theta)) /
(java.lang.Math.cos (2. * theta) - cos2LPlus1PiByM);
}
return java.lang.Math.sin (twoLPlus1PiByM) * summation -
(
org.drip.specialfunction.gamma.Definitions.EULER_MASCHERONI +
java.lang.Math.log (2. * m)
) / java.lang.Math.tan (0.5 * twoLPlus1PiByM);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Blagouchine Finite Summation Identity Verifier #10
*
* @return The Blagouchine Finite Summation Identity Verifier #10
*/
public static final org.drip.function.definition.R1ToR1Property SummationIdentity10()
{
final org.drip.specialfunction.digamma.CumulativeSeriesEstimator abramowitzStegun2007 =
org.drip.specialfunction.digamma.CumulativeSeriesEstimator.AbramowitzStegun2007 (1638400);
try
{
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 m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity10::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int r = 1; r < m; ++r)
{
double digamma = abramowitzStegun2007.evaluate ((double) r) / ((double) m);
summation = summation + digamma * digamma;
}
return summation;
}
},
new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double m)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (m) || 0. >= m)
{
throw new java.lang.Exception
("DigammaEqualityLemma::SummationIdentity10::evaluate => Invalid Inputs");
}
double summation = 0.;
for (int l = 1; l < m; ++l)
{
double logSinPiLByM = java.lang.Math.log (
java.lang.Math.sin (
java.lang.Math.PI * l / m
)
);
summation = summation + logSinPiLByM * logSinPiLByM;
}
double log2 = java.lang.Math.log (2.);
return
(m - 1.) * org.drip.specialfunction.gamma.Definitions.EULER_MASCHERONI *
org.drip.specialfunction.gamma.Definitions.EULER_MASCHERONI +
m * (
2. * org.drip.specialfunction.gamma.Definitions.EULER_MASCHERONI +
java.lang.Math.log (4. * m)
) * java.lang.Math.log (m) -
log2 * log2 * m * (m - 1.) +
java.lang.Math.PI * java.lang.Math.PI / 12. * (2. - 3. * m + m * m) +
m * summation;
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
}