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Zbl 0754.65034
Freund, Roland W.; Nachtigal, Noël M.
QMR: A quasi-minimal residual method for non-Hermitian linear systems.
(English)
[J] Numer. Math. 60, No.3, 315-339 (1991). ISSN 0029-599X; ISSN 0945-3245/e

The basic Lanczos biorthogonal method [cf. {\it 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 $\{v\sb 1,v\sb 2,\dots,v\sb n\}$ and $\{w\sb 1,w\sb 2,\dots,w\sb n\}$, $n=1,2,\dots,$ from: $v\sb{j+1}=Av\sb j-\alpha\sb jv\sb j-\beta\sb jv\sb{j-1}$ and $w\sb{j+1}=A\sp Tv\sb j-\alpha\sb j w\sb j-\gamma\sb jw\sb{j-1}$ where the scalar coefficients are chosen to satisfy the biorthogonality condition $w\sp T\sb kv\sb l=d\sb k\delta\sb{kl}$. The biconjugate gradient (BCG) method is a variant of the Lanczos' method. Note that if $w\sp T\sb{n+1}v\sb{n+1}=0$, 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 {\it B. N. Parlett}, {\it D. R. Taylor}, {\it Z. A. Liu}, Math. Comput. 44, 105-124 (1985; Zbl 0564.65022)].\par 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 $v\sb 1=r\sb 0/\Vert r\sb 0\Vert$, where $r\sb 0$ is the residual $r\sb 0=b-Ax\sb 0$ of $x\sb 0$, an initial guess to the solution of the linear system. Implementation details are presented, together with further properties and an error bound.\par In conclusion, results of extensive numerical experiments with QMR and other iterative methods mentioned in the paper are presented.
[A.Swift (Palmerston North)]
MSC 2000:
*65F10 Iterative methods for linear systems
65N22 Solution of discretized equations (BVP of PDE)

Keywords: non-Hermitian linear systems; sparse linear systems; quasi-minimal residual method; Lanczos biorthogonal method; numerical instability; error bound; numerical experiments; iterative methods; biconjugate gradient method

Citations: Zbl 0564.65022

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