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

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

All Implemented Interfaces:

IterativeSolver

public class ConjugateGradientSolver
extends java.lang.Object
implements IterativeSolver

The Conjugate Gradient method (CG) is useful for solving symmetric n-by-n linear systems. The method derives its name from the fact that it generates a sequence of conjugate (or orthogonal) vectors. These vectors are the residuals of the iterates. They are also the gradients of a quadratic functional, the minimization of which is equivalent to solving the linear system. CG is an extremely effective method when the coefficient matrix is symmetric positive definite, since storage for only a limited number of vectors is required.

If A is symmetric, positive-definite and square, the algorithm solves

Ax = b

Note that if the coefficient matrix A passed into the algorithm is not symmetric positive-definite, the algorithm behaves unexpectedly.

This algorithm can still be used to solve systems where A is not symmetric, not positive-definite, and even not square. For over-determined systems, use CGNR. For under-determined systems, use CGNE.

Only left preconditioning is supported in this implementation. It requires in addition that the preconditioner is symmetric and positive definite.

See Also:

"Yousef Saad, “The Conjugate Gradient Algorithm,” in Iterative Methods for Sparse Linear Systems, 2nd ed. 2000, ch. 6, sec. 6.7, p. 174-181."

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

ConjugateGradientSolver()

ConjugateGradientSolver(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

ConjugateGradientSolver

public ConjugateGradientSolver()

ConjugateGradientSolver

public ConjugateGradientSolver(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