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

Preconditioning complex symmetric linear systems

A new polynomial preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to Conjugate Orthogonal Conjugate Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. Moreover, to reduce the computational cost, an inexact variant based on incomplete Cholesky decomposition or orthogonal polynomials is proposed. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the method completes the description of the preconditioner.Comment: 26 pages, 4 figures, 4 table

BiCR variants of the hybrid BiCG methods for solving linear systems with nonsymmetric matrices

We propose Bi-Conjugate Residual (BiCR) variants of the hybrid Bi-Conjugate Gradient (BiCG) methods (referred to as the hybrid BiCR variants) for solving linear systems with nonsymmetric coefficient matrices. The recurrence formulas used to update an approximation and a residual vector are the same as those used in the corresponding hybrid BiCG method, but the recurrence coefficients are different; they are determined so as to compute the coefficients of the residual polynomial of BiCR. From our experience it appears that the hybrid BiCR variants often converge faster than their BiCG counterpart. Numerical experiments show that our proposed hybrid BiCR variants are more effective and less affected by rounding errors. The factor in the loss of convergence speed is analyzed to clarify the difference of the convergence between our proposed hybrid BiCR variants and the hybrid BiCG method