A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems Randolph E. Bank Tony F. Chan Article

Received: 05 November 1993 Revised: 27 January 1994 DOI :
10.1007/BF02141258

Cite this article as: Bank, R.E. & Chan, T.F. Numer Algor (1994) 7: 1. doi:10.1007/BF02141258
Abstract The Bi-Conjugate Gradient (BCG) algorithm is the simplest and most natural generalization of the classical conjugate gradient method for solving nonsymmetric linear systems. It is well-known that the method suffers from two kinds of breakdowns. The first is due to the breakdown of the underlying Lanczos process and the second is due to the fact that some iterates are not well-defined by the Galerkin condition on the associated Krylov subspaces. In this paper, we derive a simple modification of the BCG algorithm, the Composite Step BCG (CSBCG) algorithm, which is able to compute all the well-defined BCG iterates stably, assuming that the underlying Lanczos process does not break down. The main idea is to skip over a step for which the BCG iterate is not defined.

Keywords Biconjugate gradients nonsymmetric linear systems

AMS(MOS) subject classification 65N20 65F10 Communicated by C. Brezinski

The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.

The work of this author was supported by the Office of Naval Research under contracts N00014-90J-1695 and N00014-92J-1890, the Department of Energy under contract DE-FG03-87ER25307, the National Science Foundation under contracts ASC 90-03002 and 92-01266, and the Army Research Office under contract DAAL03-91-G-0150. Part of this work was completed during a visit to the Computer Science Dept., The Chinese University of Hong Kong.

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CrossRef © J.C. Baltzer AG, Science Publishers 1994

Authors and Affiliations Randolph E. Bank Tony F. Chan 1. Department of Mathematics University of California at San Diego La Jolla USA 2. Department of Mathematics University of California at Los Angeles Los Angeles USA