A transpose-free 1-norm quasi-minimal residual algorithm for non-Hermitian linear systems

The transpose-free quasi-minimal residual method (TFQMR) for solving general non-Hermitian linear systems makes use of the conjugate gradient squared algorithm (CGS) to generate the underlying Krylov subspaces. By incorporating the quasi-minimal residual approach, TFQMR produces a smooth and nearly monotone convergence behavior in the residual norm thus remedying the rather irregularly oscillating convergence behavior of CGS. While in TFQMR the quasi-minimal residual approach involves minimization in the Euclidean norm, this note proposes a different, yet similar approach by minimizing with respect to the 1-norm. It is shown that this new approach can be used to obtain a CGS-like iterative process based on short recurrences with a smooth and regular convergence behavior. The resulting transpose-free 1-norm quasi-minimal residual method (TFQMR_1) does not involve matrix-by-vector products with the transpose of the coefficient matrix and can be implemented very easily either from CGS or TFQMR. Numerical experiments are reported demonstrating that TFQMR_1 is particularly attractive for very large systems because the norm of the residual vector is explicitly available during the course of the computation