Lanczos-type variants of the COCR method for complex nonsymmetric linear systems Yan-Fei Jing a, * , Ting-Zhu Huang a , Yong Zhang a , Liang Li a , Guang-Hui Cheng a , Zhi-Gang Ren a , Yong Duan a , Tomohiro Sogabe b , Bruno Carpentieri c a School of Applied Mathematics/Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China b Department of Computational Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan c CRS4 Bioinformatics Laboratory, Edificio 3, Loc. Piscinamanna, 09010 Pula (CA), Italy article info Article history: Received 1 October 2008 Received in revised form 28 April 2009 Accepted 14 May 2009 Available online 20 May 2009 MSC: 65F10 Keywords: Physical problems CBiCG COCR Complex nonsymmetric matrices Lanczos-type variants abstract Motivated by the celebrated extending applications of the well-established complex Bicon- jugate Gradient (CBiCG) method to deal with large three-dimensional electromagnetic scattering problems by Pocock and Walker [M.D. Pocock, S.P. Walker, The complex Bi-con- jugate Gradient solver applied to large electromagnetic scattering problems, computational costs, and cost scalings, IEEE Trans. Antennas Propagat. 45 (1997) 140–146], three Lanczos- type variants of the recent Conjugate A-Orthogonal Conjugate Residual (COCR) method of Sogabe and Zhang [T. Sogabe, S.-L. Zhang, A COCR method for solving complex symmetric linear systems, J. Comput. Appl. Math. 199 (2007) 297–303] are explored for the solution of complex nonsymmetric linear systems. The first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the BiCR and CRS methods for complex systems presented in Sogabe’s Ph.D. Dissertation. And the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances. The presented algorithms are with the hope of obtain- ing smoother and, hopefully, faster convergence behavior in comparison with the CBiCG method as well as its two corresponding variants. This motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from The Univer- sity of Florida Sparse Matrix Collection by Davis. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction With respect to ‘‘the greatest influence on the development and practice of science and engineering in the 20th century” as written by Dongarra and Sullivan [3], Krylov subspace methods are considered as one of the ‘‘Top Ten Algorithms of the Century”. Preconditioned Krylov subsace methods are one of the most widespread and extensively accepted techniques for numerical solution of today’s large-scale linear systems of the form Ax ¼ b. Most of those linear systems arise from various fields of computational science and engineering. Such examples are electromagnetic applications in particular by discretiza- tions of, for instance, Helmholtz equations and Maxwell equations. Successive attempts and efforts have been made in the last half century for generalizations of the well-known Conjugate Gradient (CG) method by Hestenes and Stiefel [4] and the breakthrough Lanczos algorithm by Lanczos [5] for symmetric linear systems, leading to many advances in the development of Krylov subspace methods. The monograph by Brezinski [6] gives an elegant coverage of the intimate relations between 0021-9991/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2009.05.022 * Corresponding author. Fax: +86 28 83200131. E-mail addresses: yanfeijing@uestc.edu.cn (Y.-F. Jing), tingzhuhuang@126.com, tzhuang@uestc.edu.cn (T.-Z. Huang). Journal of Computational Physics 228 (2009) 6376–6394 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp