## 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.
Reviewer: J.Mandel (Denver)

### MSC:

 65F10 Iterative numerical methods for linear systems

Zbl 0754.65034
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