Abstract
In this paper we propose a parallel two-stage iteration algorithm for solving large-scale continuous Sylvester equations. By splitting the coefficient matrices, the original linear system is transformed into a symmetric linear system which is then solved by using the SYMMLQ algorithm. In order to improve the relative parallel efficiency, an adjusting strategy is explored during the iteration calculation of the SYMMLQ algorithm to decrease the degree of the reduce-operator from two to one communications at each step. Moreover, the convergence of the iteration scheme is discussed, and finally numerical results are reported showing that the proposed method is an efficient and robust algorithm for this class of continuous Sylvester equations on a parallel machine.
Highlights
IntroductionWith large sparse matrices A ∈ Rn×n , B ∈ Rn×n , X ∈ Rn×n , F ∈ Rn×n and with A, B positive definite is a common task in numerical linear algebra
The solution of the continuous Sylvester equation AX + XB = F (1)with large sparse matrices A ∈ Rn×n, B ∈ Rn×n, X ∈ Rn×n, F ∈ Rn×n and with A, B positive definite is a common task in numerical linear algebra
In this paper we have proposed a parallel algorithm of two-stage iteration for solving large-scale continuous Sylvester equations with the combination of the HSS iteration method and the SYMMLQ algorithm
Summary
With large sparse matrices A ∈ Rn×n , B ∈ Rn×n , X ∈ Rn×n , F ∈ Rn×n and with A, B positive definite is a common task in numerical linear algebra. A and B into triangular or Hessenberg form [21] by an orthogonal similarity transformation and to solve the resulting system of linear equations directly by a back-substitution process This method is not applicable in large-scale problems due to the prohibitive computational issue. Bai et al [24] proposed Hermitian and skew-Hermitian splitting (HSS) iteration methods for solving systems of linear equations with non-Hermitian positive definite form. This has been studied widely and generalized in [25,26,27,28]. A Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices was discussed in [29].
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