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QMR: A quasi-minimal residual method for non-Hermitian linear systems. (English) Zbl 0754.65034

The basic Lanczos biorthogonal method [cf. C. Lanczos, J. Res. Natl. Bur. Stand. 49, 33-53 (1952; MR 14.501)] for the solution of the linear system $Ax=b$, $A$ non-Hermitian, generates sequences $\left\{{v}_{1},{v}_{2},\cdots ,{v}_{n}\right\}$ and $\left\{{w}_{1},{w}_{2},\cdots ,{w}_{n}\right\}$, $n=1,2,\cdots ,$ from: ${v}_{j+1}=A{v}_{j}-{\alpha }_{j}{v}_{j}-{\beta }_{j}{v}_{j-1}$ and ${w}_{j+1}={A}^{T}{v}_{j}-{\alpha }_{j}{w}_{j}-{\gamma }_{j}{w}_{j-1}$ where the scalar coefficients are chosen to satisfy the biorthogonality condition ${w}_{k}^{T}{v}_{l}={d}_{k}{\delta }_{kl}$. The biconjugate gradient (BCG) method is a variant of the Lanczos’ method. Note that if ${w}_{n+1}^{T}{v}_{n+1}=0$, the above process must be terminated to prevent division by zero at the next step. So-called look- ahead variants of BCG attempt to overcome this difficulty [cf B. N. Parlett, D. R. Taylor, Z. A. Liu, Math. Comput. 44, 105-124 (1985; Zbl 0564.65022)].

This paper presents the quasi-minimal residual (QMR) approach, a generalization of BCG which overcomes the tendency to numerical instability. It incorporates the $n$th iteration of the look-ahead BCG, starting with ${v}_{1}={r}_{0}/\parallel {r}_{0}\parallel$, where ${r}_{0}$ is the residual ${r}_{0}=b-A{x}_{0}$ of ${x}_{0}$, an initial guess to the solution of the linear system. Implementation details are presented, together with further properties and an error bound.

In conclusion, results of extensive numerical experiments with QMR and other iterative methods mentioned in the paper are presented.

##### MSC:
 65F10 Iterative methods for linear systems 65N22 Solution of discretized equations (BVP of PDE)
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