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

A quasi-minimal residual variant of the BiCORSTAB method for nonsymmetric linear systems

The Biconjugate A-orthogonal residual stabilized method named as BiCORSTAB was proposed by Jing et al. (2009), where the numerical experiments therein demonstrate that the BiCORSTAB method converges more smoothly than the Bi-Conjugate Gradient stabilized (BiCGSTAB) method in some circumstances. In order to further stabilize the convergence behavior and hopefully to accelerate the convergence speed of the BiCORSTAB algorithm when it has erratic convergence curves, a quasi-minimal residual variant of the BiCORSTAB algorithm, named as QMRCORSTAB, will be developed and investigated for solving non-symmetric systems of linear equations borrowing the same further-smooth-effect idea for the QMRCGSTAB method. Numerical experiments on typical sets of both sparse and dense matrices will show that the proposed QMRCORSTAB method shares attractive smoother effect over its basic parent and also outperforms its counterpar