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

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

All Implemented Interfaces:

IterativeSolver

public class ConjugateGradientSquaredSolver
extends java.lang.Object
implements IterativeSolver

The Conjugate Gradient Squared method (CGS) is useful for solving non-symmetric n-by-n linear systems. This method is a variant of BiCG that applies the updating operations for the A-sequence and the A^T-sequences both to the same vectors. Ideally, this would double the convergence rate, but in practice convergence may be much more irregular than for BiCG, which may sometimes lead to unreliable results. A practical advantage is that the method does not need the multiplications with the transpose of the coefficient matrix. In some applications of CG methods, A is available only through some approximations but not explicitly. In such situations, the transpose of A, i.e., A^T is usually not available.

The rounding errors in this algorithm tend to be more damaging than in the standard BiCG algorithm.

Only left preconditioning is supported in this implementation.

See Also:

"Yousef Saad, “Conjugate Gradient Squared,” in Iterative Methods for Sparse Linear Systems, 2nd ed. 2000, ch. 7, sec. 7.4.1, p. 215-216."

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

ConjugateGradientSquaredSolver()

ConjugateGradientSquaredSolver(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

ConjugateGradientSquaredSolver

public ConjugateGradientSquaredSolver()

ConjugateGradientSquaredSolver

public ConjugateGradientSquaredSolver(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