krylov_CHEBY: Chebyshev Method
In SolveLS: Iterative Methods for Solving Linear System of Equations

Description Usage Arguments Value References Examples

Chebyshev method - also known as Chebyshev iteration - avoids computation of inner product, enabling distributed-memory computation to be more efficient at the cost of requiring a priori knowledge on the range of spectrum for matrix A.

1 2	lsolve.cheby(A, B, xinit = NA, reltol = 1e-05, maxiter = 10000, preconditioner = diag(ncol(A)), adjsym = TRUE, 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.
`adjsym`	a logical; `TRUE` to symmetrize the system by transforming the system into normal equation, `FALSE` otherwise.
`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

gutknecht_chebyshev_2002SolveLS

## Overdetermined System
A = matrix(rnorm(10*5),nrow=10)
x = rnorm(5)
b = A%*%x

out1 = lsolve.sor(A,b,w=0.5)
out2 = lsolve.cheby(A,b)
matout = cbind(x, out1$x, out2$x);
colnames(matout) = c("original x","SOR result", "Chebyshev result")
print(matout)