SolveLS (version 1.0)

lsolve.gmres: Generalized Minimal Residual method

Description

GMRES is a generic iterative solver for a nonsymmetric system of linear equations. As its name suggests, it approximates the solution using Krylov vectors with minimal residuals.

Usage

lsolve.gmres(A, B, xinit = NA, reltol = 1e-05, maxiter = 1000,
  preconditioner = diag(ncol(A)), restart = (ncol(A) - 1), verbose = TRUE)

Arguments

A

an \((m\times n)\) dense or sparse matrix. See also sparseMatrix.

B

a vector of length \(m\) or an \((m\times k)\) matrix (dense or sparse) for solving \(k\) systems simultaneously.

xinit

a length-\(n\) vector for initial starting point. NA to start from a random initial point near 0.

reltol

tolerance level for stopping iterations.

maxiter

maximum number of iterations allowed.

preconditioner

an \((n\times n)\) preconditioning matrix; default is an identity matrix.

restart

the number of iterations before restart.

verbose

a logical; TRUE to show progress of computation.

Value

a named list containing

x

solution; a vector of length \(n\) or a matrix of size \((n\times k)\).

iter

the number of iterations required.

errors

a vector of errors for stopping criterion.

References

saad_gmres:_1986SolveLS

Examples

# NOT RUN {
## Overdetermined System
A = matrix(rnorm(10*5),nrow=10)
x = rnorm(5)
b = A%*%x

out1 = lsolve.cg(A,b)
out3_1 = lsolve.gmres(A,b,restart=2)
out3_2 = lsolve.gmres(A,b,restart=3)
out3_3 = lsolve.gmres(A,b,restart=4)
matout = cbind(matrix(x),out1$x, out3_1$x, out3_2$x, out3_3$x);
colnames(matout) = c("true x","CG", "GMRES(2)", "GMRES(3)", "GMRES(4)")
print(matout)


# }