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

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

All Implemented Interfaces:

IterativeSolver

public class ConjugateGradientNormalErrorSolver
extends java.lang.Object
implements IterativeSolver

The Conjugate Gradient method (CG) can be applied on the normal equations. The Congugate Gradient Normal Error method (CGNE) is used to solve under-determined systems of linear equations, Ax = b. This algorithm applies CG to solve

(AA^T)y = b

for y, and then compute the solution

x = A^Ty

The equivalent symmetric system is of the form:

 [ I  A ] [ r ] = [ b ]
 [ AT O ] [ x ]   [ 0 ]
 

with r = b - Ax, arises from the standard necessary conditions satisfied by the solution of the constrained optimization problem,

minimize || r - b ||² subject to A^Tr = 0

When the coefficient matrix A is non-symmetric and nonsingular, the normal equations matrix AA^T will be symmetric and positive definite, and hence CG can be applied. The convergence may be slow, since the spectrum of AA^T will be less favorable than the spectrum of A.

Only left preconditioning is supported in this implementation.

See Also:

"Yousef Saad, “Preconditioned CG for the Normal Equations,” in Iterative Methods for Sparse Linear Systems, 2nd ed. 2000, ch. 9, sec. 9.5, p. 259-260."

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

ConjugateGradientNormalErrorSolver()

ConjugateGradientNormalErrorSolver(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

ConjugateGradientNormalErrorSolver

public ConjugateGradientNormalErrorSolver()

ConjugateGradientNormalErrorSolver

public ConjugateGradientNormalErrorSolver(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