Comparison of Convergence of General Stationary Iterative Methods for Singular Matrices

Marek, Ivo; Szyld, Daniel B.

doi:10.1137/s0895479800375989

Cited by 13 publications

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“…When the iteration matrices have spectral radius equal to one, as is usually the case for singular linear systems, the convergence rate is given by (3.2). Comparison theorems for these can be found in [30], [31]. Here we present a new comparison theorem, which we use in our context.…”

Section: Note Than When M Is Symmetric This Theorem Says That If 2m −mentioning

confidence: 99%

Schwarz Iterations for Symmetric Positive Semidefinite Problems

Nabben¹,

Szyld²

2007

SIAM J. Matrix Anal. & Appl.

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Abstract. Convergence properties of additive and multiplicative Schwarz iterations for solving linear systems of equations with a symmetric positive semidefinite matrix are analyzed. The analysis presented applies to matrices whose principal submatrices are nonsingular, i.e., positive definite. These matrices appear in discretizations of some elliptic partial differential equations, e.g., those with Neumann or periodic boundary conditions. [35], where analysis of convergence and properties for several variants of the methods are provided, both for symmetric positive definite and for nonsingular M -matrices. Recently, convergence properties were studied for singular systems arising in the solution of Markov chains, i.e., singular M -matrices with all principal submatrices being nonsingular [7], [32]. In particular, this theory applies to singular matrices with a one-dimensional nullspace, and to those representing irreducible Markov chains; see, e.g., [42]. We also mention the recent work on multiplicative Schwarz iterations for positive semidefinite operators [26], [28].In this paper, we extend the theory to the symmetric positive semidefinite case, with particular emphasis on the singular case (the analysis of the symmetric positive definite case is known; see, e.g., [1],[21, Ch. 11],[41],[44]). We study in particular the case when all principal submatrices are nonsingular, i.e., positive definite. This situation arises in practice, e.g., in the discretization of certain elliptic differential equations such as −Δu + u = f with Neumann or periodic boundary conditions; see, e.g., [5]. We show that in this case, the additive and multiplicative Schwarz iterations are convergent and we characterize the convergence factor γ for such methods (sections 4 and 5). We use the theory of matrix splittings (see section 3) to obtain these convergence properties. We remark that we do not use splittings to produce new

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Section: Note Than When M Is Symmetric This Theorem Says That If 2m −mentioning

confidence: 99%