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

Freund, Roland W.; Nachtigal, Noël M.

Numer. Math. 60, No. 3, 315-339 (1991).

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 \(\{v_ 1,v_ 2,\dots,v_ n\}\) and \(\{w_ 1,w_ 2,\dots,w_ n\}\), \(n=1,2,\dots,\) from: \(v_{j+1}=Av_ j-\alpha_ jv_ j-\beta_ jv_{j-1}\) and \(w_{j+1}=A^ Tv_ j-\alpha_ j w_ j-\gamma_ jw_{j-1}\) where the scalar coefficients are chosen to satisfy the biorthogonality condition \(w^ T_ kv_ l=d_ k\delta_{kl}\). The biconjugate gradient (BCG) method is a variant of the Lanczos’ method. Note that if \(w^ T_{n+1}v_{n+1}=0\), 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 \(v_ 1=r_ 0/\| r_ 0\|\), where \(r_ 0\) is the residual \(r_ 0=b-Ax_ 0\) 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.

Reviewer: A.Swift (Palmerston North)

MSC:

65F10	Iterative numerical methods for linear systems
65N22	Numerical solution of discretized equations for boundary value problems involving PDEs

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

Software:

ITSOL; Harwell-Boeing sparse matrix collection; SPARSKIT; CGS

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