com.imsl.test.example.math

Class ConjugateGradientEx2

java.lang.Object

com.imsl.test.example.math.ConjugateGradientEx2

All Implemented Interfaces:

ConjugateGradient.Function, ConjugateGradient.Preconditioner

public class ConjugateGradientEx2
extends Object
implements ConjugateGradient.Preconditioner

Solves a sparse linear system using the conjugate gradient method with preconditioning.

In this example, two different preconditioners are used to find the solution of a sparse positive definite linear system. The system occurs in a finite difference solution of Laplace's equation on a regular \(c \times c\) grid, where we use \(c = 50\). The matrix is $$ A = E(c^2, c) $$. For the first solution, Jacobi preconditioning with preconditioner \(M={\rm{diag}}(A)\) is used. The required iteration number and maximum absolute error are printed. Next, a more complicated preconditioning matrix, consisting of the symmetric tridiagonal part of A, is used. Again, the iteration number and the maximum absolute error are printed. The iteration number is substantially reduced.

See Also:

Code, Output

Constructor Summary

Constructors

Constructor and Description
ConjugateGradientEx2(int n, int c)

Method Summary

Modifier and Type	Method and Description
void	amultp(double[] p, double[] z) A user-supplied function which computes z=Ap.
static void	main(String[] args)
void	preconditioner(double[] r, double[] z) Used to compute \(z = M^{-1}r \) where M is the preconditioning matrix and r and z are arrays of length n, the order of matrix M.

Methods inherited from class java.lang.Object

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

Constructor Detail

ConjugateGradientEx2

public ConjugateGradientEx2(int n,
                            int c)
                     throws SparseCholesky.NotSPDException

Throws:

SparseCholesky.NotSPDException

Method Detail

amultp

public void amultp(double[] p,
                   double[] z)

Description copied from interface: ConjugateGradient.Function

A user-supplied function which computes z=Ap.

Specified by:

amultp in interface ConjugateGradient.Function

Parameters:

p - an input double vector of length dimension of A.

z - an output double vector containing the matrix-vector product Ap.

preconditioner

public void preconditioner(double[] r,
                           double[] z)

Description copied from interface: ConjugateGradient.Preconditioner

Used to compute \(z = M^{-1}r \) where M is the preconditioning matrix and r and z are arrays of length n, the order of matrix M.

Specified by:

preconditioner in interface ConjugateGradient.Preconditioner

Parameters:

r - an input double array of length n generated during the implementation of the solve method.

z - an output double array of length n.

main

public static void main(String[] args)
                 throws Exception

Throws:

Exception