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:

IterativeSolver

public class MinimalResidualSolver
extends java.lang.Object
implements IterativeSolver

The Minimal Residual method (MINRES) is useful for solving symmetric n-by-n linear systems (possibly indefinite or singular).

When the matrix A is Hermitian, the Arnoldi algorithm used in GMRES can be simplified to a 3-term recurrence known as Lanczos algorithm. Thus, the approximate solution can be computed without saving all the orthonormal basis vectors generated.

If 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.IterativeSolver

IterativeSolver.ConvergenceFailure, IterativeSolver.Problem

Constructor Summary

MinimalResidualSolver()

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

Constructor Detail

MinimalResidualSolver

public MinimalResidualSolver()

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