org.apache.commons.math.linear
Class ConjugateGradient

java.lang.Object
  org.apache.commons.math.linear.IterativeLinearSolver
      org.apache.commons.math.linear.PreconditionedIterativeLinearSolver
          org.apache.commons.math.linear.ConjugateGradient

public class ConjugateGradient
extends PreconditionedIterativeLinearSolver

This is an implementation of the conjugate gradient method for RealLinearOperator. It follows closely the template by Barrett et al. (1994) (figure 2.5). The linear system at hand is A · x = b, and the residual is r = b - A · x.

Default stopping criterion

A default stopping criterion is implemented. The iterations stop when || r || ≤ δ || b ||, where b is the right-hand side vector, r the current estimate of the residual, and δ a user-specified tolerance. It should be noted that r is the so-called updated residual, which might differ from the true residual due to rounding-off errors (see e.g. Strakos and Tichy, 2002).

Iteration count

In the present context, an iteration should be understood as one evaluation of the matrix-vector product A · x. The initialization phase therefore counts as one iteration.

Exception context

Besides standard DimensionMismatchException, this class might throw NonPositiveDefiniteOperatorException if the linear operator or the preconditioner are not positive definite. In this case, the ExceptionContext provides some more information

• key "operator" points to the offending linear operator, say L,
• key "vector" points to the offending vector, say x, such that x^T · L · x < 0.

References

Barret et al. (1994)

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. M. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM

Strakos and Tichy (2002)

Z. Strakos and P. Tichy, On error estimation in the conjugate gradient method and why it works in finite precision computations, Electronic Transactions on Numerical Analysis 13: 56-80, 2002

Since:

3.0

Version:

$Id: ConjugateGradient.java 1179488 2011-10-06 02:14:20Z celestin $

Nested Class Summary

static class

ConjugateGradient.ConjugateGradientEvent The type of all events fired by this implementation of the Conjugate Gradient method.

Field Summary

static java.lang.String	OPERATOR Key for the exception context.
static java.lang.String	VECTOR Key for the exception context.

Constructor Summary

ConjugateGradient(int maxIterations, double delta, boolean check)
Creates a new instance of this class, with default stopping criterion.

ConjugateGradient(IterationManager manager, double delta, boolean check)
Creates a new instance of this class, with default stopping criterion and custom iteration manager.

Method Summary

boolean	getCheck() Returns true if positive-definiteness should be checked for both matrix and preconditioner.
RealVector	solveInPlace(RealLinearOperator a, InvertibleRealLinearOperator m, RealVector b, RealVector x0) Returns an estimate of the solution to the linear system A · x = b.

Methods inherited from class org.apache.commons.math.linear.PreconditionedIterativeLinearSolver

checkParameters, solve, solve, solve, solve, solveInPlace

Methods inherited from class org.apache.commons.math.linear.IterativeLinearSolver

checkParameters, getIterationManager

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

OPERATOR

public static final java.lang.String OPERATOR

Key for the exception context.

See Also:

Constant Field Values

VECTOR

public static final java.lang.String VECTOR

Key for the exception context.

See Also:

Constant Field Values

Constructor Detail

ConjugateGradient

public ConjugateGradient(int maxIterations,
                         double delta,
                         boolean check)

Creates a new instance of this class, with default stopping criterion.

Parameters:

maxIterations - Maximum number of iterations.

delta - δ parameter for the default stopping criterion.

check - true if positive definiteness of both matrix and preconditioner should be checked.

ConjugateGradient

public ConjugateGradient(IterationManager manager,
                         double delta,
                         boolean check)

Creates a new instance of this class, with default stopping criterion and custom iteration manager.

Parameters:

manager - Custom iteration manager.

delta - δ parameter for the default stopping criterion.

check - true if positive definiteness of both matrix and preconditioner should be checked.

Method Detail

getCheck

public final boolean getCheck()

Returns true if positive-definiteness should be checked for both matrix and preconditioner.

Returns:

true if the tests are to be performed.

solveInPlace

public RealVector solveInPlace(RealLinearOperator a,
                               InvertibleRealLinearOperator m,
                               RealVector b,
                               RealVector x0)
                        throws NullArgumentException,
                               NonSquareOperatorException,
                               DimensionMismatchException,
                               MaxCountExceededException

Returns an estimate of the solution to the linear system A · x = b. The solution is computed in-place (initial guess is modified).

Specified by:

solveInPlace in class PreconditionedIterativeLinearSolver

Parameters:

a - Linear operator A of the system.

m - Preconditioner (can be null).

b - Right-hand side vector.

x0 - Initial guess of the solution.

Returns:

A reference to x0 (shallow copy) updated with the solution.

Throws:

NullArgumentException - if one of the parameters is null.

NonSquareOperatorException - if a or m is not square.

DimensionMismatchException - if m, b or x0 have dimensions inconsistent with a.

MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction.