Package org.drip.numerical.quadrature

Class IntegrandGenerator

java.lang.Object

org.drip.numerical.quadrature.IntegrandGenerator

public class IntegrandGenerator
extends java.lang.Object

IntegrandGenerator contains the Settings that enable the Generation of Integrand Quadrature and Weights for the Specified Orthogonal Polynomial Scheme. The References are:

• Abramowitz, M., and I. A. Stegun (2007): Handbook of Mathematics Functions Dover Book on Mathematics
• Gil, A., J. Segura, and N. M. Temme (2007): Numerical Methods for Special Functions Society for Industrial and Applied Mathematics Philadelphia
• Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (2007): Numerical Recipes: The Art of Scientific Computing 3rd Edition Cambridge University Press New York
• Stoer, J., and R. Bulirsch (2002): Introduction to Numerical Analysis 3rd Edition Springer
• Wikipedia (2019): Gaussian Quadrature https://en.wikipedia.org/wiki/Gaussian_quadrature

• Module = Computational Core Module
• Library = Numerical Analysis Library
• Project = Numerical Quadrature, Differentiation, Eigenization, Linear Algebra, and Utilities
• Package = R¹ Gaussian Integration Quadrature Schemes

Author:

Lakshmi Krishnamurthy

Constructor Summary

Constructors

Constructor	Description
IntegrandGenerator(OrthogonalPolynomialSuite orthogonalPolynomialSuite, R1ToR1 weightFunction, double lowerBound, double upperBound)	IntegrandGenerator Constructor

Method Summary

Modifier and Type	Method	Description
static IntegrandGenerator	GaussChebyshevFirstKind(OrthogonalPolynomialSuite orthogonalPolynomialSuite)	Construct the Gauss-Chebyshev (First-Kind) Integrand Quadrature Generator
static IntegrandGenerator	GaussChebyshevSecondKind(OrthogonalPolynomialSuite orthogonalPolynomialSuite)	Construct the Gauss-Chebyshev (Second-Kind) Integrand Quadrature Generator
static IntegrandGenerator	GaussHermite(OrthogonalPolynomialSuite orthogonalPolynomialSuite)	Construct the Gauss-Hermite Integrand Quadrature Generator
static IntegrandGenerator	GaussJacobi(OrthogonalPolynomialSuite orthogonalPolynomialSuite, double alpha, double beta)	Construct the Gauss-Jacobi Integrand Quadrature Generator
static IntegrandGenerator	GaussLaguerre(OrthogonalPolynomialSuite orthogonalPolynomialSuite)	Construct the Gauss-Laguerre Integrand Quadrature Generator
static IntegrandGenerator	GaussLegendre(OrthogonalPolynomialSuite orthogonalPolynomialSuite)	Construct the Gauss-Legendre Integrand Quadrature Generator
static IntegrandGenerator	GeneralizedGaussLaguerre(OrthogonalPolynomialSuite orthogonalPolynomialSuite, double alpha)	Construct the Generalized Gauss-Laguerre Integrand Quadrature Generator
GolubWelsch	generateRecurrenceMatrix()	Generate the Cross Polynomial Recurrence Matrix to be used in the Golub-Welsch Algorithm
Array2D	gilSeguraTemme2007()	Generate the Quadrature Nodes and Scaled Weights Using the Gil, Segura, and Temme (2007) Scheme
double	golubWelschA(int degree)	Generate the Golub-Welsch Matrix A Entry
double	golubWelschB(int degree)	Generate the Golub-Welsch Matrix B Entry
double	loadedInnerProduct(int degree1, int degree2)	Compute the Loaded Inner Product between the Polynomial identified by their Degrees
double	lowerBound()	Retrieve the Lower Integration Bound
double	nodeWeight(double x, int degree)	Generate the Weight at the specified Node for the specified Orthogonal Polynomial
OrthogonalPolynomialSuite	orthogonalPolynomialSuite()	Retrieve the Orthogonal Polynomial Suite
double	unloadedInnerProduct(int degree1, int degree2)	Compute the Unloaded Inner Product between the Polynomial identified by their Degrees
double	upperBound()	Retrieve the Upper Integration Bound
R1ToR1	weightFunction()	Retrieve the Weight Function
double	weightFunctionIntegral()	Generate the Integral of the Weight Function Over the Bounds

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

IntegrandGenerator

public IntegrandGenerator(OrthogonalPolynomialSuite orthogonalPolynomialSuite,
R1ToR1 weightFunction,
double lowerBound,
double upperBound)
                   throws java.lang.Exception

IntegrandGenerator Constructor

Parameters:

orthogonalPolynomialSuite - Orthogonal Polynomial Suite

weightFunction - Weight Function

lowerBound - Lower Bound

upperBound - Upper Bound

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

Method Details

GaussLegendre

public static final IntegrandGenerator GaussLegendre(OrthogonalPolynomialSuite orthogonalPolynomialSuite)

Construct the Gauss-Legendre Integrand Quadrature Generator

Parameters:

orthogonalPolynomialSuite - Orthogonal Polynomial Suite

Returns:

The Gauss-Legendre Integrand Quadrature Generator

GaussJacobi

public static final IntegrandGenerator GaussJacobi(OrthogonalPolynomialSuite orthogonalPolynomialSuite,
double alpha,
double beta)

Construct the Gauss-Jacobi Integrand Quadrature Generator

Parameters:

orthogonalPolynomialSuite - Orthogonal Polynomial Suite

alpha - Jacobi Alpha

beta - Jacobi Beta

Returns:

The Gauss-Jacobi Integrand Quadrature Generator

GaussChebyshevSecondKind

public static final IntegrandGenerator GaussChebyshevSecondKind(OrthogonalPolynomialSuite orthogonalPolynomialSuite)

Construct the Gauss-Chebyshev (Second-Kind) Integrand Quadrature Generator

Parameters:

orthogonalPolynomialSuite - Orthogonal Polynomial Suite

Returns:

The Gauss-Chebyshev (Second-Kind) Integrand Quadrature Generator

GaussChebyshevFirstKind

public static final IntegrandGenerator GaussChebyshevFirstKind(OrthogonalPolynomialSuite orthogonalPolynomialSuite)

Construct the Gauss-Chebyshev (First-Kind) Integrand Quadrature Generator

Parameters:

orthogonalPolynomialSuite - Orthogonal Polynomial Suite

Returns:

The Gauss-Chebyshev (First-Kind) Integrand Quadrature Generator

GaussLaguerre

public static final IntegrandGenerator GaussLaguerre(OrthogonalPolynomialSuite orthogonalPolynomialSuite)

Construct the Gauss-Laguerre Integrand Quadrature Generator

Parameters:

orthogonalPolynomialSuite - Orthogonal Polynomial Suite

Returns:

The Gauss-Laguerre Integrand Quadrature Generator

GeneralizedGaussLaguerre

public static final IntegrandGenerator GeneralizedGaussLaguerre(OrthogonalPolynomialSuite orthogonalPolynomialSuite,
double alpha)

Construct the Generalized Gauss-Laguerre Integrand Quadrature Generator

Parameters:

orthogonalPolynomialSuite - Orthogonal Polynomial Suite

alpha - Generalized Laguerre Alpha

Returns:

The Generalized Gauss-Laguerre Integrand Quadrature Generator

GaussHermite

public static final IntegrandGenerator GaussHermite(OrthogonalPolynomialSuite orthogonalPolynomialSuite)

Construct the Gauss-Hermite Integrand Quadrature Generator

Parameters:

orthogonalPolynomialSuite - Orthogonal Polynomial Suite

Returns:

The Gauss-Hermite Integrand Quadrature Generator

orthogonalPolynomialSuite

public OrthogonalPolynomialSuite orthogonalPolynomialSuite()

Retrieve the Orthogonal Polynomial Suite

Returns:

The Orthogonal Polynomial Suite

weightFunction

public R1ToR1 weightFunction()

Retrieve the Weight Function

Returns:

The Weight Function

lowerBound

public double lowerBound()

Retrieve the Lower Integration Bound

Returns:

The Lower Integration Bound

upperBound

public double upperBound()

Retrieve the Upper Integration Bound

Returns:

The Upper Integration Bound

weightFunctionIntegral

public double weightFunctionIntegral()
                              throws java.lang.Exception

Generate the Integral of the Weight Function Over the Bounds

Returns:

The Integral of the Weight Function Over the Bounds

Throws:

java.lang.Exception - Thrown if it cannot be computed

nodeWeight

public double nodeWeight(double x,
int degree)
                  throws java.lang.Exception

Generate the Weight at the specified Node for the specified Orthogonal Polynomial

Parameters:

x - X Node

degree - Orthogonal Polynomial Degree

Returns:

The Weight at the specified Node for the specified Orthogonal Polynomial

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

loadedInnerProduct

public double loadedInnerProduct(int degree1,
int degree2)
                          throws java.lang.Exception

Compute the Loaded Inner Product between the Polynomial identified by their Degrees

Parameters:

degree1 - Polynomial Degree #1

degree2 - Polynomial Degree #2

Returns:

The Loaded Inner Product

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

unloadedInnerProduct

public double unloadedInnerProduct(int degree1,
int degree2)
                            throws java.lang.Exception

Compute the Unloaded Inner Product between the Polynomial identified by their Degrees

Parameters:

degree1 - Polynomial Degree #1

degree2 - Polynomial Degree #2

Returns:

The Unloaded Inner Product

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

golubWelschA

public double golubWelschA(int degree)
                    throws java.lang.Exception

Generate the Golub-Welsch Matrix A Entry

Parameters:

degree - The Orthogonal Polynomial Degree

Returns:

The Golub-Welsch Matrix A Entry

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

golubWelschB

public double golubWelschB(int degree)
                    throws java.lang.Exception

Generate the Golub-Welsch Matrix B Entry

Parameters:

degree - The Orthogonal Polynomial Degree

Returns:

The Golub-Welsch Matrix B Entry

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

generateRecurrenceMatrix

public GolubWelsch generateRecurrenceMatrix()

Generate the Cross Polynomial Recurrence Matrix to be used in the Golub-Welsch Algorithm

Returns:

The Cross Polynomial Recurrence Matrix to be used in the Golub-Welsch Algorithm

gilSeguraTemme2007

public Array2D gilSeguraTemme2007()

Generate the Quadrature Nodes and Scaled Weights Using the Gil, Segura, and Temme (2007) Scheme

Returns:

The Quadrature Nodes and Scaled Weights