A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. (English) Zbl 0781.65022

Freund, Roland W.

SIAM J. Sci. Comput. 14, No. 2, 470-482 (1993).

Author’s summary: 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, R. W. Freund and N. M. Nachtigal [Numer. Math. 60, No. 3, 315-339 (1991; Zbl 0754.65034)] 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 \(A^ T\).
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 multiplication with \(A^ T\). It is shown that the resulting transpose-free QMR method can be implemented very easily by changing only a few lines in the standard CGS algorithm. Finally, numerical experiments are reported on.

Reviewer: J.Mandel (Denver)

MSC:

65F10

Iterative numerical methods for linear systems

Keywords:

biconjugate gradient method; non-Hermitian linear systems; transpose- free; conjugate gradients squared algorithm; quasi-minimal residual method; smooth convergence curves; numerical experiments

Citations:

Zbl 0754.65034