# SuanShu, a Java numerical and statistical library

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

• java.lang.Object
• All Implemented Interfaces:
IterativeLinearSystemSolver

public class ConjugateGradientSolver
extends Object
implements IterativeLinearSystemSolver
The Conjugate Gradient method (CG) is useful for solving a symmetric n-by-n linear system. 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 function, 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 as storage for only a limited number of vectors is required. For a coefficient matrix that is not symmetric, not positive-definite, and even not square, there are solvers using the CG method. For example, CGNR solves an over-determined system; CGNE solves an under-determined system.

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

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

Only left preconditioning is supported in this implementation. The preconditioner must be symmetric and positive definite.

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

IterativeLinearSystemSolver.Solution
• ### Field Summary

Fields
Modifier and Type Field and Description
static int DEFAULT_RESIDUAL_REFRESH_RATE
The algorithm recomputes the residual as b - Axi once per this number of iterations
• ### Constructor Summary

Constructors
Constructor and Description
ConjugateGradientSolver(int maxIteration, Tolerance tolerance)
Construct a Conjugate Gradient (CG) solver.
ConjugateGradientSolver(PreconditionerFactory leftPreconditionerFactory, int residualRefreshRate, int maxIteration, Tolerance tolerance)
Construct a Conjugate Gradient (CG) 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
• ### Field Detail

• #### DEFAULT_RESIDUAL_REFRESH_RATE

public static final int DEFAULT_RESIDUAL_REFRESH_RATE
The algorithm recomputes the residual as b - Axi once per this number of iterations
Constant Field Values
• ### Constructor Detail

public ConjugateGradientSolver(PreconditionerFactory leftPreconditionerFactory,
int residualRefreshRate,
int maxIteration,
Tolerance tolerance)
Construct a Conjugate Gradient (CG) solver.
Parameters:
leftPreconditionerFactory - constructs a new left preconditioner
residualRefreshRate - the number of iterations before the next refresh
maxIteration - the maximum number of iterations
tolerance - the convergence threshold

public ConjugateGradientSolver(int maxIteration,
Tolerance tolerance)
Construct a Conjugate Gradient (CG) 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