IntegrandGenerator.java
package org.drip.numerical.quadrature;
/*
* -*- 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>IntegrandGenerator</i> contains the Settings that enable the Generation of Integrand Quadrature and
* Weights for the Specified Orthogonal Polynomial Scheme. The References are:
*
* <br><br>
* <ul>
* <li>
* Abramowitz, M., and I. A. Stegun (2007): <i>Handbook of Mathematics Functions</i> <b>Dover Book
* on Mathematics</b>
* </li>
* <li>
* Gil, A., J. Segura, and N. M. Temme (2007): <i>Numerical Methods for Special Functions</i>
* <b>Society for Industrial and Applied Mathematics</b> Philadelphia
* </li>
* <li>
* Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (2007): <i>Numerical Recipes:
* The Art of Scientific Computing 3rd Edition</i> <b>Cambridge University Press</b> New York
* </li>
* <li>
* Stoer, J., and R. Bulirsch (2002): <i>Introduction to Numerical Analysis 3rd Edition</i>
* <b>Springer</b>
* </li>
* <li>
* Wikipedia (2019): Gaussian Quadrature https://en.wikipedia.org/wiki/Gaussian_quadrature
* </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/NumericalAnalysisLibrary.md">Numerical Analysis Library</a></li>
* <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/numerical/README.md">Numerical Quadrature, Differentiation, Eigenization, Linear Algebra, and Utilities</a></li>
* <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/numerical/quadrature/README.md">R<sup>1</sup> Gaussian Integration Quadrature Schemes</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class IntegrandGenerator
{
private double _lowerBound = java.lang.Double.NaN;
private double _upperBound = java.lang.Double.NaN;
private org.drip.function.definition.R1ToR1 _weightFunction = null;
private org.drip.numerical.quadrature.OrthogonalPolynomialSuite _orthogonalPolynomialSuite = null;
/**
* Construct the Gauss-Legendre Integrand Quadrature Generator
*
* @param orthogonalPolynomialSuite Orthogonal Polynomial Suite
*
* @return The Gauss-Legendre Integrand Quadrature Generator
*/
public static final IntegrandGenerator GaussLegendre (
final org.drip.numerical.quadrature.OrthogonalPolynomialSuite orthogonalPolynomialSuite)
{
try
{
return new IntegrandGenerator (
orthogonalPolynomialSuite,
org.drip.numerical.quadrature.WeightFunctionBuilder.Legendre(),
-1.,
1.
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Gauss-Jacobi Integrand Quadrature Generator
*
* @param orthogonalPolynomialSuite Orthogonal Polynomial Suite
* @param alpha Jacobi Alpha
* @param beta Jacobi Beta
*
* @return The Gauss-Jacobi Integrand Quadrature Generator
*/
public static final IntegrandGenerator GaussJacobi (
final org.drip.numerical.quadrature.OrthogonalPolynomialSuite orthogonalPolynomialSuite,
final double alpha,
final double beta)
{
try
{
return new IntegrandGenerator (
orthogonalPolynomialSuite,
org.drip.numerical.quadrature.WeightFunctionBuilder.Jacobi (
alpha,
beta
),
-1.,
1.
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Gauss-Chebyshev (Second-Kind) Integrand Quadrature Generator
*
* @param orthogonalPolynomialSuite Orthogonal Polynomial Suite
*
* @return The Gauss-Chebyshev (Second-Kind) Integrand Quadrature Generator
*/
public static final IntegrandGenerator GaussChebyshevSecondKind (
final org.drip.numerical.quadrature.OrthogonalPolynomialSuite orthogonalPolynomialSuite)
{
try
{
return new IntegrandGenerator (
orthogonalPolynomialSuite,
org.drip.numerical.quadrature.WeightFunctionBuilder.ChebyshevSecondKind(),
-1.,
1.
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Gauss-Chebyshev (First-Kind) Integrand Quadrature Generator
*
* @param orthogonalPolynomialSuite Orthogonal Polynomial Suite
*
* @return The Gauss-Chebyshev (First-Kind) Integrand Quadrature Generator
*/
public static final IntegrandGenerator GaussChebyshevFirstKind (
final org.drip.numerical.quadrature.OrthogonalPolynomialSuite orthogonalPolynomialSuite)
{
try
{
return new IntegrandGenerator (
orthogonalPolynomialSuite,
org.drip.numerical.quadrature.WeightFunctionBuilder.ChebyshevFirstKind(),
-1.,
1.
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Gauss-Laguerre Integrand Quadrature Generator
*
* @param orthogonalPolynomialSuite Orthogonal Polynomial Suite
*
* @return The Gauss-Laguerre Integrand Quadrature Generator
*/
public static final IntegrandGenerator GaussLaguerre (
final org.drip.numerical.quadrature.OrthogonalPolynomialSuite orthogonalPolynomialSuite)
{
try
{
return new IntegrandGenerator (
orthogonalPolynomialSuite,
org.drip.numerical.quadrature.WeightFunctionBuilder.Laguerre(),
-1.,
1.
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Generalized Gauss-Laguerre Integrand Quadrature Generator
*
* @param orthogonalPolynomialSuite Orthogonal Polynomial Suite
* @param alpha Generalized Laguerre Alpha
*
* @return The Generalized Gauss-Laguerre Integrand Quadrature Generator
*/
public static final IntegrandGenerator GeneralizedGaussLaguerre (
final org.drip.numerical.quadrature.OrthogonalPolynomialSuite orthogonalPolynomialSuite,
final double alpha)
{
try
{
return new IntegrandGenerator (
orthogonalPolynomialSuite,
org.drip.numerical.quadrature.WeightFunctionBuilder.GeneralizedLaguerre (alpha),
-1.,
1.
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Gauss-Hermite Integrand Quadrature Generator
*
* @param orthogonalPolynomialSuite Orthogonal Polynomial Suite
*
* @return The Gauss-Hermite Integrand Quadrature Generator
*/
public static final IntegrandGenerator GaussHermite (
final org.drip.numerical.quadrature.OrthogonalPolynomialSuite orthogonalPolynomialSuite)
{
try
{
return new IntegrandGenerator (
orthogonalPolynomialSuite,
org.drip.numerical.quadrature.WeightFunctionBuilder.Hermite(),
-1.,
1.
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* IntegrandGenerator Constructor
*
* @param orthogonalPolynomialSuite Orthogonal Polynomial Suite
* @param weightFunction Weight Function
* @param lowerBound Lower Bound
* @param upperBound Upper Bound
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public IntegrandGenerator (
final org.drip.numerical.quadrature.OrthogonalPolynomialSuite orthogonalPolynomialSuite,
final org.drip.function.definition.R1ToR1 weightFunction,
final double lowerBound,
final double upperBound)
throws java.lang.Exception
{
if (null == (_orthogonalPolynomialSuite = orthogonalPolynomialSuite) ||
null == (_weightFunction = weightFunction) ||
!org.drip.numerical.common.NumberUtil.IsValid (_lowerBound = lowerBound) ||
!org.drip.numerical.common.NumberUtil.IsValid (_upperBound = upperBound) ||
_lowerBound >= _upperBound)
{
throw new java.lang.Exception ("IntegrandGenerator Constructor => Invalid Inputs");
}
}
/**
* Retrieve the Orthogonal Polynomial Suite
*
* @return The Orthogonal Polynomial Suite
*/
public org.drip.numerical.quadrature.OrthogonalPolynomialSuite orthogonalPolynomialSuite()
{
return _orthogonalPolynomialSuite;
}
/**
* Retrieve the Weight Function
*
* @return The Weight Function
*/
public org.drip.function.definition.R1ToR1 weightFunction()
{
return _weightFunction;
}
/**
* Retrieve the Lower Integration Bound
*
* @return The Lower Integration Bound
*/
public double lowerBound()
{
return _lowerBound;
}
/**
* Retrieve the Upper Integration Bound
*
* @return The Upper Integration Bound
*/
public double upperBound()
{
return _upperBound;
}
/**
* Generate the Integral of the Weight Function Over the Bounds
*
* @return The Integral of the Weight Function Over the Bounds
*
* @throws java.lang.Exception Thrown if it cannot be computed
*/
public double weightFunctionIntegral()
throws java.lang.Exception
{
return _weightFunction.integrate (
_lowerBound,
_upperBound
);
}
/**
* Generate the Weight at the specified Node for the specified Orthogonal Polynomial
*
* @param x X Node
* @param degree Orthogonal Polynomial Degree
*
* @return The Weight at the specified Node for the specified Orthogonal Polynomial
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double nodeWeight (
final double x,
final int degree)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (x))
{
throw new java.lang.Exception ("IntegrandGenerator::nodeWeight => Invalid Inputs");
}
if (0 > degree)
{
return 0.;
}
final org.drip.numerical.quadrature.OrthogonalPolynomial orthogonalPolynomialN =
_orthogonalPolynomialSuite.orthogonalPolynomial (degree);
final org.drip.numerical.quadrature.OrthogonalPolynomial orthogonalPolynomialNMinusOne =
_orthogonalPolynomialSuite.orthogonalPolynomial (degree - 1);
if (null == orthogonalPolynomialN || null == orthogonalPolynomialNMinusOne)
{
throw new java.lang.Exception ("IntegrandGenerator::nodeWeight => Invalid Inputs");
}
double weightIntegrand = new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
double pNMinusOne = orthogonalPolynomialNMinusOne.evaluate (z);
return _weightFunction.evaluate (z) * pNMinusOne * pNMinusOne;
}
}.integrate (
_lowerBound,
_upperBound
);
return orthogonalPolynomialN.degreeCoefficient() * weightIntegrand / (
orthogonalPolynomialNMinusOne.degreeCoefficient() *
orthogonalPolynomialNMinusOne.evaluate (x) *
orthogonalPolynomialN.derivative (
x,
1
)
);
}
/**
* Compute the Loaded Inner Product between the Polynomial identified by their Degrees
*
* @param degree1 Polynomial Degree #1
* @param degree2 Polynomial Degree #2
*
* @return The Loaded Inner Product
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double loadedInnerProduct (
final int degree1,
final int degree2)
throws java.lang.Exception
{
if (0 > degree1 || 0 > degree2)
{
return 0.;
}
final org.drip.numerical.quadrature.OrthogonalPolynomial orthogonalPolynomial1 =
_orthogonalPolynomialSuite.orthogonalPolynomial (degree1);
final org.drip.numerical.quadrature.OrthogonalPolynomial orthogonalPolynomial2 =
_orthogonalPolynomialSuite.orthogonalPolynomial (degree2);
if (null == orthogonalPolynomial1 || null == orthogonalPolynomial2)
{
throw new java.lang.Exception ("IntegrandGenerator::loadedInnerProduct => Invalid Inputs");
}
return new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return z * _weightFunction.evaluate (z) * orthogonalPolynomial1.evaluate (z) *
orthogonalPolynomial2.evaluate (z);
}
}.integrate (
_lowerBound,
_upperBound
);
}
/**
* Compute the Unloaded Inner Product between the Polynomial identified by their Degrees
*
* @param degree1 Polynomial Degree #1
* @param degree2 Polynomial Degree #2
*
* @return The Unloaded Inner Product
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double unloadedInnerProduct (
final int degree1,
final int degree2)
throws java.lang.Exception
{
if (0 > degree1 || 0 > degree2)
{
return 0.;
}
final org.drip.numerical.quadrature.OrthogonalPolynomial orthogonalPolynomial1 =
_orthogonalPolynomialSuite.orthogonalPolynomial (degree1);
final org.drip.numerical.quadrature.OrthogonalPolynomial orthogonalPolynomial2 =
_orthogonalPolynomialSuite.orthogonalPolynomial (degree2);
if (null == orthogonalPolynomial1 || null == orthogonalPolynomial2)
{
throw new java.lang.Exception ("IntegrandGenerator::unloadedInnerProduct => Invalid Inputs");
}
return new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double z)
throws java.lang.Exception
{
return _weightFunction.evaluate (z) * orthogonalPolynomial1.evaluate (z) *
orthogonalPolynomial2.evaluate (z);
}
}.integrate (
_lowerBound,
_upperBound
);
}
/**
* Generate the Golub-Welsch Matrix A Entry
*
* @param degree The Orthogonal Polynomial Degree
*
* @return The Golub-Welsch Matrix A Entry
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double golubWelschA (
final int degree)
throws java.lang.Exception
{
return loadedInnerProduct (
degree,
degree
) / unloadedInnerProduct (
degree,
degree
);
}
/**
* Generate the Golub-Welsch Matrix B Entry
*
* @param degree The Orthogonal Polynomial Degree
*
* @return The Golub-Welsch Matrix B Entry
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double golubWelschB (
final int degree)
throws java.lang.Exception
{
return unloadedInnerProduct (
degree,
degree
) / unloadedInnerProduct (
degree - 1,
degree - 1
);
}
/**
* Generate the Cross Polynomial Recurrence Matrix to be used in the Golub-Welsch Algorithm
*
* @return The Cross Polynomial Recurrence Matrix to be used in the Golub-Welsch Algorithm
*/
public org.drip.numerical.quadrature.GolubWelsch generateRecurrenceMatrix()
{
int size = _orthogonalPolynomialSuite.size();
double[][] golubWelschMatrix = new double[size][size];
for (int row = 0; row < size; ++row)
{
for (int column = 0; column < size; ++column)
{
golubWelschMatrix[row][column] = column == row + 1 ? 1. : 0.;
}
}
try
{
for (int row = 0; row < size; ++row)
{
golubWelschMatrix[row][row] = loadedInnerProduct (
row,
row
) / unloadedInnerProduct (
row,
row
);
if (0 < row)
{
golubWelschMatrix[row][row - 1] = unloadedInnerProduct (
row,
row
) / unloadedInnerProduct (
row - 1,
row - 1
);
}
}
return new org.drip.numerical.quadrature.GolubWelsch (golubWelschMatrix);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate the Quadrature Nodes and Scaled Weights Using the Gil, Segura, and Temme (2007) Scheme
*
* @return The Quadrature Nodes and Scaled Weights
*/
public org.drip.numerical.common.Array2D gilSeguraTemme2007()
{
org.drip.numerical.quadrature.GolubWelsch golubWelsch = generateRecurrenceMatrix();
if (null == golubWelsch)
{
return null;
}
org.drip.numerical.common.Array2D nodesAndUnscaledWeights = golubWelsch.nodesAndUnscaledWeights();
if (null == nodesAndUnscaledWeights)
{
return null;
}
double[] unscaledWeightArray = nodesAndUnscaledWeights.y();
double[] nodeArray = nodesAndUnscaledWeights.x();
int size = nodeArray.length;
double[] scaledWeightArray = new double[size];
try
{
double weightFunctionIntegral = weightFunctionIntegral();
for (int nodeIndex = 0; nodeIndex < size; ++nodeIndex)
{
scaledWeightArray[nodeIndex] = unscaledWeightArray[nodeIndex] * weightFunctionIntegral;
}
}
catch (java.lang.Exception e)
{
e.printStackTrace();
return null;
}
return org.drip.numerical.common.Array2D.FromArray (
nodeArray,
scaledWeightArray
);
}
}