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

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.

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 |