BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

In the present paper, we introduce a new extension of the conjugate residual (CR) for solving non-Hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient (BiCG), biconjugate residual (BiCR) and biconjugate A-orthogonal residual (BiCOR) methods. The proposed Krylov subspace method, referred to as the BiCGCR2 method, is based on short-term vector recurrences and is mathematically equivalent to both BiCR and BiCOR. We demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than BiCG, BiCR and BiCOR. Hence, it may be exploited for the development of new variants of non-optimal Krylov subspace methods

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

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

The BICOR and CORS iterative algorithms for solving nonsymmetric linear systems

We present two iterative algorithms for solving real nonsymmetric and complex non-Hermitian linear systems of equations and that were developed from variants of the nonsymmetric Lanczos method. In this paper, we give the theoretical background of the two iterative methods and discuss their main computational aspects. Using a large number of numerical experiments, we analyze their convergence properties, and we also compare them with other popular nonsymmetric iterative solvers in use today