Abstract
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.
Highlights
Consider the following continuous Sylvester equation: AX + XB = C, (1)where A ∈ Cm×m, B ∈ Cn×n, and C ∈ Cm×n are given complex matrices
We perform numerical tests to exhibit the superiority of generalized Hermitian and skewHermitian splitting (HSS) (GHSS) and inexact variant of the GHSS (IGHSS) to HSS and inexact HSS (IHSS) when they are used as solvers for solving the continuous Sylvester equation (1), in terms of iteration numbers
All sub-problems involved in each step of the HSS and GHSS iteration methods are solved exactly by the method in [16]
Summary
Where A ∈ Cm×m, B ∈ Cn×n, and C ∈ Cm×n are given complex matrices. Assume that (i) A, B, and C are large and sparse matrices;. Since under assumptions (i)–(iii) there is no common eigenvalue between A and −B, we obtain from [1, 2] that the continuous Sylvester equation (1) has a unique solution. The continuous Lyapunov equation is a special case of the continuous Sylvester equation (1) with B = A∗ and C Hermitian, where A∗ represents the conjugate transpose of the matrix A. 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 [2,3,4,5,6,7,8,9,10,11,12,13,14,15] and the references therein
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