NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2010; 17:139–154 Published online 3 August 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nla.663 On the convergence of general stationary iterative methods for range-Hermitian singular linear systems Naimin Zhang 1 and Yi-Min Wei 2, 3, ∗, † 1 School of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, People’s Republic of China 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China 3 Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, People’s Republic of China SUMMARY General stationary iterative methods with a singular matrix M for solving range-Hermitian singular linear systems are presented, some convergence conditions and the representation of the solution are also given. It can be verified that the general Ortega–Plemmons theorem and Keller theorem for the singular matrix M still hold. Furthermore, the singular matrix M can act as a good preconditioner for solving range-Hermitian linear systems. Numerical results have demonstrated the effectiveness of the general stationary iterations and the singular preconditioner M. Copyright 2009 John Wiley & Sons, Ltd. Received 8 September 2008; Revised 17 June 2009; Accepted 18 June 2009 KEY WORDS: range-Hermitian; singular linear systems; matrix splittings; semi-convergence; general stationary iterative methods; singular preconditioners 1. INTRODUCTION Consider the iterative solution of a singular linear system of equations Ax = b (1) ∗ Correspondence to: Yi-Min Wei, School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China. † E-mail: ymwei@fudan.edu.cn, yimin.wei@gmail.com Contract/grant sponsor: Zhejiang Provincial Natural Science Foundation of China; contract/grant number: Y606009 Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 10871051 Contract/grant sponsor: Shanghai Education Committee; contract/grant number: 08SG01 Contract/grant sponsor: Shanghai Science and Technology Committee; contract/grant number: 09DZ2272900 Copyright 2009 John Wiley & Sons, Ltd.