lsolve.qmr {Rlinsolve}

Quasi Minimal Residual Method

Description

Quasia-Minimal Resudial(QMR) method is another remedy of the BiCG which shows rather irregular convergence behavior. It adapts to solve the reduced tridiagonal system in a least squares sense and its convergence is known to be quite smoother than BiCG.

Usage

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

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

Freund RW, Nachtigal NM (1991). “QMR: a quasi-minimal residual method for non-Hermitian linear systems.” Numerische Mathematik, 60(1), 315–339. ISSN 0029-599X, 0945-3245.

Examples

## Not run: 
## Overdetermined System
set.seed(100)
A = matrix(rnorm(10*5),nrow=10)
x = rnorm(5)
b = A%*%x

out1 = lsolve.cg(A,b)
out2 = lsolve.bicg(A,b)
out3 = lsolve.qmr(A,b)
matout = cbind(matrix(x),out1$x, out2$x, out3$x);
colnames(matout) = c("true x","CG result", "BiCG result", "QMR result")
print(matout)

## End(Not run)