krylov_GMRES: Generalized Minimal Residual method
In SolveLS: Iterative Methods for Solving Linear System of Equations

Description Usage Arguments Value References Examples

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.

1 2	lsolve.gmres(A, B, xinit = NA, reltol = 1e-05, maxiter = 1000, preconditioner = diag(ncol(A)), restart = (ncol(A) - 1), verbose = TRUE)

`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.

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.

\insertRef

saad_gmres:_1986SolveLS

## 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)