A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems

The biconjugate gradient method (BCG) for solving general non-Hermitian linear systems $Ax = b$ and its transpose-free variant, the conjugate gradients squared algorithm (CGS), both typically exhibit a rather irregular convergence behavior with wild oscillations in the residual norm. Recently, Freund and Nachtigal proposed a BCG-like approach, the quasi-minimal residual method (QMR), that remedies this problem for BCG and produces smooth convergence curves. However, like BCG, QMR requires matrix-vector multiplications with both the coefficient matrix A and its transpose AT. In this note, it is demonstrated that the quasi-minimal residual approach can also be used to obtain a smoothly convergent CGS-like algorithm that does not involve matrix-vector multiplications with AT. It is shown that the resulting transpose-free QMR method (TFQMR) can be implemented very easily by changing only a few lines in the standard CGS algorithm. Finally, numerical experiments are reported.