Keywords: nonsymmetric linear systems, sparse matrices, Krylov subspace methods, quasi-minimal residual methods,.

The quasi-minimal residual methods, these are QMR (Freund and Nachtigal [3]), TFQMR (Freund [4]) and QMRCGSTAB (Chan et al [1]), are biorthogonalization methods for solving nonsymmetric linear systems of equations which improve the irregular behaviour of BiCG, CGS and BiCGSTAB algorithms [6], respectively. They are based on the quasi-minimization of the residual using the standard Givens rotations that lead to methods with short term recurrences.

In this paper, the quasi-minimization problem is solved using a similar procedure to that developed in [5] for the minimization problem arising in GMRES method. It consists of a direct solver which provides new versions of QMR-type methods, the so called 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 not only reduce the number of iterations but also reach convergence in some cases where the standard algorithms do not work well.

On the other hand, we study the effect of preconditioning, for example with Jacobi, ILU, SSOR or sparse approximate inverse [7], and reordering [2] on the performance of these algorithms is studied.

Finally, some numerical experiments are solved in order to compare the results obtained by standard and modified algorithms.

1

T.F. Chan, E. Gallopoulos, V. Simonsini, T. Szeto and C.H. Tong, "A Quasi-minimal residual variant of the BI-CGSTAB algorithm for nonsymmetric systems", SIAM J. Sci. Comput., 15, 2, 338-347, 1994. doi:10.1137/0915023

2

E. Flórez, D. García, L. González and G. Montero, "The effect of ordering on Sparse Approximate Inverse Preconditioners for Nonsymmetric", Advances in Engineering Software, 33, 611-619, 2002. doi:10.1016/S0965-9978(02)00070-4

3

R.W. Freund and N.M. Nachtigal, "QMR: a quasi-minimal residual method for non-Hermitian linear systems", Numerische Math., 60, 315-339, 1991. doi:10.1007/BF01385726

4

R. W. Freund, " A Transpose-Free Quasi-Minimal Residual algorithm for non-Hermitian Linear Systems", SIAM J. Sci. Comput., 14, 470-482. 1993. doi:10.1137/0914029

5

M. Galán, G. Montero and G. Winter, "A direct solver for the least square problems arising from GMRES(k)", Com. Num. Meth. Eng., 10, 743-749, 1994. doi:10.1002/cnm.1640100909

6

G. Montero and A. Suárez, "Left-Right preconditioning versions of BCG-like methods". Neural, Parallel & Scientific Computations, 3, 4, 487-501, 1995.

7

G. Montero, L. González, E. Flórez, M.D. García and A. Suárez, "Approximate Inverse Computation Using Frobenius Inner product", Num. Lin. Al. Appl., 9, 239-247, 2002. doi:10.1002/nla.269

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