New implementation of QMR-type algorithms

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.