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

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

All Implemented Interfaces:

IterativeSolver

public class QuasiMinimalResidualSolver
extends java.lang.Object
implements IterativeSolver

The Quasi-Minimal Residual method (QMR) is useful for solving non-symmetric n-by-n linear systems. It applies a least-squares solve and update to the BiCG residuals, thereby smoothing out the irregular convergence behavior of BiCG. Also, QMR largely avoids the breakdown that can occur in BiCG. On the other hand, it does not effect a true minimization of either the error or the residual, and while it converges smoothly, it does not essentially improve on the BiCG in terms of the number of iteration steps.

This current implementation does not have a look ahead mechanism.

Split preconditioning (M = M₁M₂) is implemented.

See Also:

"Yousef Saad, “Quasi-Minimal Residual Algorithm,” in Iterative Methods for Sparse Linear Systems, 2nd ed. 2000, ch. 7, sec. 7.3.2, p. 211-212."

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

QuasiMinimalResidualSolver()

QuasiMinimalResidualSolver(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

QuasiMinimalResidualSolver

public QuasiMinimalResidualSolver()

QuasiMinimalResidualSolver

public QuasiMinimalResidualSolver(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