# SuanShu, a Java numerical and statistical library

com.numericalmethod.suanshu.algebra.linear.matrix.doubles.matrixtype.sparse.solver.iterative.nonstationary

## Class MinimalResidualSolver

• java.lang.Object
• com.numericalmethod.suanshu.algebra.linear.matrix.doubles.matrixtype.sparse.solver.iterative.nonstationary.MinimalResidualSolver
• All Implemented Interfaces:
IterativeLinearSystemSolver

public class MinimalResidualSolver
extends 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.

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

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

IterativeLinearSystemSolver.Solution
• ### Constructor Summary

Constructors
Constructor and Description
MinimalResidualSolver(int maxIteration, Tolerance tolerance)
Construct a MINRES solver.
MinimalResidualSolver(PreconditionerFactory leftPreconditionerFactory, int maxIteration, Tolerance tolerance)
Construct a MINRES solver.
• ### Method Summary

All Methods
Modifier and Type Method and Description
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