Journal of Applied Mathematics
Volume 2013 (2013), Article ID 408167, 16 pages
Research Article

Exploiting the Composite Step Strategy to the Biconjugate -Orthogonal Residual Method for Non-Hermitian Linear Systems

1School of Mathematical Sciences, Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
2Institute of Mathematics and Computing Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, The Netherlands

Received 15 October 2012; Accepted 19 December 2012

Academic Editor: Zhongxiao Jia

Copyright © 2013 Yan-Fei Jing et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Biconjugate -Orthogonal Residual (BiCOR) method carried out in finite precision arithmetic by means of the biconjugate -orthonormalization procedure may possibly tend to suffer from two sources of numerical instability, known as two kinds of breakdowns, similarly to those of the Biconjugate Gradient (BCG) method. This paper naturally exploits the composite step strategy employed in the development of the composite step BCG (CSBCG) method into the BiCOR method to cure one of the breakdowns called as pivot breakdown. Analogously to the CSBCG method, the resulting interesting variant, with only a minor modification to the usual implementation of the BiCOR method, is able to avoid near pivot breakdowns and compute all the well-defined BiCOR iterates stably on the assumption that the underlying biconjugate -orthonormalization procedure does not break down. Another benefit acquired is that it seems to be a viable algorithm providing some further practically desired smoothing of the convergence history of the norm of the residuals, which is justified by numerical experiments. In addition, the exhibited method inherits the promising advantages of the empirically observed stability and fast convergence rate of the BiCOR method over the BCG method so that it outperforms the CSBCG method to some extent.