com.numericalmethod.suanshu.matrix.doubles.matrixtype.sparse.solver.iterative.nonstationary
Class ConjugateGradientNormalResidualSolver

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.matrixtype.sparse.solver.iterative.nonstationary.ConjugateGradientNormalResidualSolver

All Implemented Interfaces:

IterativeSolver

public class ConjugateGradientNormalResidualSolver
extends java.lang.Object
implements IterativeSolver

The Conjugate Gradient method (CG) can be applied on the normal equations. The Conjugate Gradient Normal Residual method (CGNR) is used to solve over-determined systems of linear equations, Ax = b. This algorithm uses CG to solve

(A^TA)x = A^Tb

The equivalent symmetric system is of the form:

 [ O  A ] [ Ax ] = [  b  ]
 [ AT O ] [  x ]   [ ATb ]
 

This solver can also be applied to square but non-symmetric systems, Ax = b.

When the coefficient matrix A is non-symmetric and nonsingular, the normal equations matrix A^TA will be symmetric and positive definite, and hence CG can be applied. The convergence may be slow, since the spectrum of A^TA will be less favorable than the spectrum of A.

Only left preconditioning is supported in this implementation.

See Also:

"Yousef Saad, “Preconditioned CG for the Normal Equations,” in Iterative Methods for Sparse Linear Systems, 2nd ed. 2000, ch. 9, sec. 9.5, p. 259-260."

Nested Class Summary

Nested classes/interfaces inherited from interface com.numericalmethod.suanshu.matrix.doubles.matrixtype.sparse.solver.iterative.IterativeSolver

IterativeSolver.ConvergenceFailure, IterativeSolver.Problem

Field Summary

static int

DEFAULT_RESIDUAL_REFRESH_RATE The algorithm recomputes the residual as b - Axi once per this number of iterations

Constructor Summary

ConjugateGradientNormalResidualSolver()

ConjugateGradientNormalResidualSolver(int residualRefreshRate)
The solver recomputes the residual as b - Ax_i once per this number of iterations

Method Summary

Vector	solve(IterativeSolver.Problem problem) Solve iteratively Ax = b until the solution is close enough, i.e., the norm of residual (b - Ax) is less than or equal to the specified iteration.
Vector	solve(IterativeSolver.Problem problem, IterationMonitor monitor) Solve iteratively Ax = b until the solution is close enough, i.e., the norm of residual (b - Ax) is less than or equal to the specified iteration.

Methods inherited from class java.lang.Object

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

Field Detail

DEFAULT_RESIDUAL_REFRESH_RATE

public static final int DEFAULT_RESIDUAL_REFRESH_RATE

The algorithm recomputes the residual as b - Ax_i once per this number of iterations

See Also:

Constant Field Values

Constructor Detail

ConjugateGradientNormalResidualSolver

public ConjugateGradientNormalResidualSolver()

ConjugateGradientNormalResidualSolver

public ConjugateGradientNormalResidualSolver(int residualRefreshRate)

The solver recomputes the residual as b - Ax_i once per this number of iterations

Parameters:

residualRefreshRate - the number of iterations before the next refresh

Method Detail

solve

public Vector solve(IterativeSolver.Problem problem)
             throws IterativeSolver.ConvergenceFailure

Description copied from interface: IterativeSolver

Solve iteratively

Ax = b

until the solution is close enough, i.e., the norm of residual (b - Ax) is less than or equal to the specified iteration.

Specified by:

solve in interface IterativeSolver

Parameters:

problem - the problem of solving Ax = b

Returns:

the computed solution for the problem

Throws:

IterativeSolver.ConvergenceFailure - if the algorithm fails to converge

solve

public Vector solve(IterativeSolver.Problem problem,
                    IterationMonitor monitor)
             throws IterativeSolver.ConvergenceFailure

Description copied from interface: IterativeSolver

Solve iteratively

Ax = b

until the solution is close enough, i.e., the norm of residual (b - Ax) is less than or equal to the specified iteration.

In each iteration, the newly computed iterate is added to the IterationMonitor for statistics or diagnostic purpose.

Specified by:

solve in interface IterativeSolver

Parameters:

problem - the problem of solving Ax = b

monitor - an IterationMonitor instance

Returns:

the computed solution for the problem

Throws:

IterativeSolver.ConvergenceFailure - if the algorithm fails to converge