GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
CGS, a fast Lanczos-type solver for nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific and Statistical Computing
How fast are nonsymmetric matrix iterations
SIAM Journal on Matrix Analysis and Applications
An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices
SIAM Journal on Scientific Computing
A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems
SIAM Journal on Scientific Computing
Residual smoothing techniques for iterative methods
SIAM Journal on Scientific Computing
An implementation of the QMR method based on coupled two-term recurrences
SIAM Journal on Scientific Computing
A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems
SIAM Journal on Scientific Computing
Iterative solution methods
A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems
SIAM Journal on Scientific Computing
Orderings for Incomplete Factorization Preconditioning of Nonsymmetric Problems
SIAM Journal on Scientific Computing
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
The effect of orderings on sparse approximate inverse preconditioners for non-symmetric problems
Advances in Engineering Software - Engineering computational technology
Reducing the bandwidth of sparse symmetric matrices
ACM '69 Proceedings of the 1969 24th national conference
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
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Quasi-minimal residual algorithms, these are QMR, TFQMR and QMRCGSTAB, are biorthogonalisation methods for solving nonsymmetric linear systems of equations which improve the irregular behaviour of BiCG, CGS and BiCGSTAB algorithms, respectively. They are based on the quasi-minimisation of the residual using the standard Givens rotations that lead to iterations with short term recurrences. In this paper, these quasi-minimisation problems are solved using a different direct solver which provides new versions of QMR-type methods, the modified QMR methods (MQMR). MQMR algorithms have different convergence behaviour in finite arithmetic although are equivalent to the standard ones in exact arithmetic. The new implementations may reduce the number of iterations in some cases. In addition, we study the effect of reordering and preconditioning with Jacobi, ILU, SSOR or sparse approximate inverse preconditioners on the performance of these algorithms. Some numerical experiments are solved in order to compare the results obtained by standard and modified algorithms.