# 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 BiconjugateGradientSolver
extends Object
implements IterativeLinearSystemSolver
The Biconjugate Gradient method (BiCG) is useful for solving non-symmetric n-by-n linear systems. It generates two CG-like sequences of vectors, one based on a system with the original coefficient matrix A, and one on At. They are made mutually orthogonal, or "bi-orthogonal". This method is useful when the matrix is non-symmetric and nonsingular. However, convergence may be irregular, and there is a possibility that the method will break down. BiCG , like CG, uses limited storage. It requires a multiplication with the coefficient matrix and with its transpose at each iteration.

Only left preconditioning is supported in this implementation.

"Yousef Saad, "The Biconjugate Gradient Algorithm," in Iterative Methods for Sparse Linear Systems, 2nd ed. 2000, ch. 7, sec. 7.3.1, p. 210-211."

• ### 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
BiconjugateGradientSolver(int maxIteration, Tolerance tolerance)
Construct a Biconjugate Gradient (BiCG) solver.
BiconjugateGradientSolver(PreconditionerFactory leftPreconditionerFactory, int residualRefreshRate, int maxIteration, Tolerance tolerance)
Construct a Biconjugate Gradient (BiCG) 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 BiconjugateGradientSolver(PreconditionerFactory leftPreconditionerFactory,
int residualRefreshRate,
int maxIteration,
Tolerance tolerance)
Construct a Biconjugate Gradient (BiCG) 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 BiconjugateGradientSolver(int maxIteration,
Tolerance tolerance)
Construct a Biconjugate Gradient (BiCG) 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