NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2007; 14:217–235 Published online 29 January 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nla.528 Modified Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems Liang Li 1 , Ting-Zhu Huang 1, ∗, † and Xing-Ping Liu 2 1 School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, People’s Republic of China 2 State Key Lab of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of China SUMMARY To further study the Hermitian and non-Hermitian splitting methods for a non-Hermitian and positive- definite matrix, we introduce a so-called lopsided Hermitian and skew-Hermitian splitting and then establish a class of lopsided Hermitian/skew-Hermitian (LHSS) methods to solve the non-Hermitian and positive-definite systems of linear equations. These methods include a two-step LHSS iteration and its inexact version, the inexact Hermitian/skew-Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter . Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright 2007 John Wiley & Sons, Ltd. Received 25 April 2006; Revised 11 November 2006; Accepted 27 November 2006 KEY WORDS: non-Hermitian positive-definite matrix; skew-Hermitian matrix; splitting; iteration 1. INTRODUCTION Many problems in scientific computing require to solve the system of linear equations Ax = b (1) ∗ Correspondence to: Ting-Zhu Huang, School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, People’s Republic of China. † E-mail: tzhuang@uestc.edu.cn Contract/grant sponsor: NCET of China; contract/grant number: NCET-04-0893 Contract/grant sponsor: Applied Basic Research Foundations of Sichuan Province; contract/grant number: 05JY029- 068-2 Copyright 2007 John Wiley & Sons, Ltd.