## Journal of Applied Mathematics

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

#### Abstract

The Biconjugate $A$-Orthogonal Residual (BiCOR) method carried out in finite precision arithmetic by means of the biconjugate $A$-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 $A$-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.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 408167, 16 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807858

Digital Object Identifier
doi:10.1155/2013/408167

Mathematical Reviews number (MathSciNet)
MR3032247

Zentralblatt MATH identifier
1268.65045

#### Citation

Jing, Yan-Fei; Huang, Ting-Zhu; Carpentieri, Bruno; Duan, Yong. Exploiting the Composite Step Strategy to the Biconjugate $A$ -Orthogonal Residual Method for Non-Hermitian Linear Systems. J. Appl. Math. 2013 (2013), Article ID 408167, 16 pages. doi:10.1155/2013/408167. https://projecteuclid.org/euclid.jam/1394807858

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