The BiCOR and CORS Iterative Algorithms for Solving Nonsymmetric Linear Systems

Abstract

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.

Symmetric Inverse-Based Multilevel Ilu Preconditioning for Solving Dense Complex Non-Hermitian Systems in Electromagnetics

“…We combined the inverse-based multilevel incomplete LU factorization with three Krylov methods, that are the restarted GMRES method [37], the recently developed CORS method [12] and the SQMR method [22]. We summarise the relative costs of the three solvers in Table 3.…”

Fast Preconditioned Krylov Methods for Boundary Integral Equations in Electromagnetic Scattering

“…We do not use preconditioning. In addition to restarted GMRES, we consider complex versions of iterative algorithms based on Lanczos biorthogonalization, such as BiCGSTAB (van der Vorst (1992)) and QMR (Freund & Nachtigal (1994)) and on the recently developed Lanczos biconjugate A-orthonormalization, such as BiCOR and CORS (Carpentieri et al (2011);Jing, Huang, Zhang, Li, Cheng, Ren, Duan, Sogabe & Carpentieri (2009)). We clearly observe the importance of the choice of the iterative method.…”

“…The BiCOR and CORS methods are introduced in Carpentieri et al (2011);Jing, Huang, Zhang, Li, Cheng, Ren, Duan, Sogabe & Carpentieri (2009 A significant amount of work has been devoted in the last years to design fast algorithms that can reduce the O(n 2 ) computational complexity for the M-V product operation required at each step of a Krylov method, such as the Fast Multipole Method (FMM) (Greengard & Rokhlin (1987); Rokhlin (1990)), the panel clustering method (Hackbush & Nowak (1989)), the H-matrix approach (Hackbush (1999)), wavelet techniques (Alpert et al (1993); Bond & Vavasis (1994)), the adaptive cross approximation method (Bebendorf (2000)), the impedance matrix localization method (Canning (1990)), the multilevel matrix decomposition algorithm (Michielssen & Boag (1996)) and others. In particular, the combination of iterative Krylov subspace solvers and FMM is a popular approach for solving integral equations.…”

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

“…lead to the BiCG method [12] and the BiCGCR2 method, respectively, while L n = A H K n (A H , r * 0 ) leads to the BiCR method [16,39] and the BiCOR method [17,18]. Moreover, we have the following condition by the definition of K n…”

“…Nevertheless, they tend to be too expensive to use for solving large-scale problems especially in terms of memory. Iterative solvers, namely the well-known class of Krylov subspace methods, can be an attractive alternative to also developed some efficient hybrid BiCOR variants, including CORS [17,18], GCORS [33], BiCORSTAB [17], BiCORSTAB2 [34] and GPBiCOR [35]. Many numerical experiments on practical applications have illustrated the robustness of the hybrid BiCR and hybrid BiCOR methods; refer, e.g., to [16-18, 32, 35, 36] for details.…”