Research Article A Generalized HSS Iteration Method for Continuous Sylvester Equations Xu Li, 1 Yu-Jiang Wu, 1 Ai-Li Yang, 1 and Jin-Yun Yuan 2 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China 2 Department of Mathematics, Federal University of Paran´ a, Centro Polit´ ecnico, CP 19.081, 81531-980 Curitiba, PR, Brazil Correspondence should be addressed to Xu Li; mathlixu@163.com and Yu-Jiang Wu; myjaw@lzu.edu.cn Received 20 August 2013; Accepted 13 December 2013; Published 12 January 2014 Academic Editor: Qing-Wen Wang Copyright © 2014 Xu Li et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish a generalized HSS (GHSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive defnite/semidefnite matrices. Te GHSS method is essentially a four-parameter iteration which not only covers the standard HSS iteration but also enables us to optimize the iterative process. An exact parameter region of convergence for the method is strictly proved and a minimum value for the upper bound of the iterative spectrum is derived. Moreover, to reduce the computational cost, we establish an inexact variant of the GHSS (IGHSS) iteration method whose convergence property is discussed. Numerical experiments illustrate the efciency and robustness of the GHSS iteration method and its inexact variant. 1. Introduction Consider the following continuous Sylvester equation:  +  = , (1) where ∈ C × , ∈ C × , and ∈ C × are given complex matrices. Assume that (i) , , and  are large and sparse matrices; (ii) at least one of  and  is non-Hermitian; (iii) both  and  are positive semi-defnite, and at least one of them is positive defnite. Since under assumptions (i)–(iii) there is no common eigen- value between  and −, we obtain from [1, 2] that the continuous Sylvester equation (1) has a unique solution. Obviously, the continuous Lyapunov equation is a special case of the continuous Sylvester equation (1) with = ∗ and  Hermitian, where  ∗ represents the conjugate transpose of the matrix . Tis continuous Sylvester equation arises in several areas of applications. For more details about the practical backgrounds of this class of problems, we refer to [2–15] and the references therein. Before giving its numerical scheme, we rewrite the continuous Sylvester equation (1) in the mathematically equivalent system of linear equations A = , (2) where A =⊗+  ⊗; the vectors  and  contain the con- catenated columns of the matrices  and , respectively, with ⊗ being the Kronecker product symbol and   representing the transpose of the matrix . However, it is quite expensive and ill-conditioned to use the iteration method to solve this variation of the continuous Sylvester equation (1). Tere is a large number of numerical methods for solving the continuous Sylvester equation (1). Te Bartels-Stewart and the Hessenberg-Schur methods [16, 17] are direct algo- rithms, which can only be applied to problems of reasonably small sizes. When the matrices  and  become large and sparse, iterative methods are usually employed for efciently and accurately solving the continuous Sylvester equation (1), for instance, the Smith’s method [18], the alternating direction implicit (ADI) method [19–22], and others [23–26]. Recently, Bai established the Hermitian and skew- Hermitian splitting (HSS) [4] iterative method for solving the continuous Sylvester equation (1), which is based on Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 578102, 9 pages http://dx.doi.org/10.1155/2014/578102