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

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

All Implemented Interfaces:

IterativeLinearSystemSolver

public class MinimalResidualSolver
extends java.lang.Object
implements IterativeLinearSystemSolver

The Minimal Residual method (MINRES) is useful for solving a symmetric n-by-n linear system (possibly indefinite or singular). When the coefficient matrix A is Hermitian, the Arnoldi algorithm used in GMRES can be simplified to a 3-term recurrence known as Lanczos algorithm. Thus, an approximate solution can be computed without saving all the orthonormal basis vectors generated. When A is singular, MINRES returns a least-squares solution with small |Ar| (where r = b - Ax).

Only left preconditioning is supported in this implementation.

See Also:

"Anne Greenbaum, "Algorithm 4P," Iterative methods for solving linear systems, ch.8, pp.122."

Nested Class Summary

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

IterativeLinearSystemSolver.Solution

Constructor Summary

MinimalResidualSolver(int maxIteration, Tolerance tolerance)
Construct a MINRES solver.

MinimalResidualSolver(PreconditionerFactory leftPreconditionerFactory, int maxIteration, Tolerance tolerance)
Construct a MINRES solver.

Method Summary

IterativeLinearSystemSolver.Solution	solve(LSProblem problem)
IterativeLinearSystemSolver.Solution	solve(LSProblem problem, IterationMonitor<Vector> monitor) Solves iteratively Ax = b until the solution converges, i.e., the norm of residual (b - Ax) is less than or equal to the threshold.

Methods inherited from class java.lang.Object

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

Constructor Detail

MinimalResidualSolver

public MinimalResidualSolver(PreconditionerFactory leftPreconditionerFactory,
                             int maxIteration,
                             Tolerance tolerance)

Construct a MINRES solver.

Parameters:

leftPreconditionerFactory - constructs a new left preconditioner

maxIteration - the maximum number of iterations

tolerance - the convergence threshold

MinimalResidualSolver

public MinimalResidualSolver(int maxIteration,
                             Tolerance tolerance)

Construct a MINRES solver.

Parameters:

maxIteration - the maximum number of iterations

tolerance - the convergence threshold

Method Detail

solve

public IterativeLinearSystemSolver.Solution solve(LSProblem problem)
                                           throws ConvergenceFailure

Throws:

ConvergenceFailure

solve

public IterativeLinearSystemSolver.Solution solve(LSProblem problem,
                                                  IterationMonitor<Vector> monitor)
                                           throws ConvergenceFailure

Description copied from interface: IterativeLinearSystemSolver

Solves iteratively

Ax = b

until the solution converges, i.e., the norm of residual (b - Ax) is less than or equal to the threshold.

Specified by:

solve in interface IterativeLinearSystemSolver

Parameters:

problem - a system of linear equations

monitor - an iteration monitor

Returns:

an (approximate) solution to the linear problem

Throws:

ConvergenceFailure - if the algorithm fails to converge