%0 Journal Article
%T A Generalized HSS Iteration Method for Continuous Sylvester Equations
%A Xu Li
%A Yu-Jiang Wu
%A Ai-Li Yang
%A Jin-Yun Yuan
%J Journal of Applied Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/578102
%X 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 definite/semidefinite matrices. The 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 efficiency and robustness of the GHSS iteration method and its inexact variant. 1. Introduction Consider the following continuous Sylvester equation: where , , and 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-definite, and at least one of them is positive definite.Since under assumptions (i)¨C(iii) there is no common eigenvalue 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 . This 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¨C15] and the references therein. Before giving its numerical scheme, we rewrite the continuous Sylvester equation (1) in the mathematically equivalent system of linear equations where ; the vectors and contain the concatenated 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). There is a large number of numerical methods for solving the continuous Sylvester equation (1). The Bartels-Stewart and the Hessenberg-Schur methods [16, 17] are direct algorithms, 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 efficiently and accurately solving the continuous Sylvester equation (1), for instance, the Smith¡¯s method [18],
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