Block diagonal dominance of matrices revisited: Bounds for the norms of inverses and eigenvalue inclusion sets

Abstract

We generalize the bounds on the inverses of diagonally dominant matrices obtained in [16] from scalar to block tridiagonal matrices. Our derivations are based on a generalization of the classical condition of block diagonal dominance of matrices given by Feingold and Varga in [11]. Based on this generalization, which was recently presented in [3], we also derive a variant of the Gershgorin Circle Theorem for general block matrices which can provide tighter spectral inclusion regions than those obtained by Feingold and Varga.Key words. block matrices, block diagonal dominance, block tridiagonal matrices, decay bounds for the inverse, eigenvalue inclusion regions, Gershgorin Circle Theorem AMS subject classifications. 15A45, 65F151. Introduction. Matrices that are characterized by off-diagonal decay, or more generally "localization" of their entries, appear in applications throughout the mathematical and computational sciences. The presence of such localization can lead to computational savings, since it allows to (closely) approximate a given matrix by using its significant entries only, and discarding the negligible ones according to a pre-established criterion. In this context it is then of great practical interest to know a priori how many and which of these entries can be discarded as insignificant. Many authors have therefore studied decay rates for different matrix classes and functions of matrices; see, e.g., [2,4,5,6,7,10,14,18]. For an excellent survey of the current state-of-the-art we refer to [1].An important example in this context is given by the (nonsymmetric) diagonally dominant matrices, and in particular the diagonally dominant tridiagonal matrices, which were studied, e.g., in [15,16]. As shown in these works, the entries of the inverse decay with an exponential rate along a row or column, depending on whether the given matrix is row or column diagonally dominant; see [1, Section 3.2] for a more general treatment of decay bounds for the inverse and further references. Our main goal in this paper is to generalize results of [16] from scalar to block tridiagonal matrices. In order to do so, we use a generalization of the classical definition of block diagonal dominance of Feingold and Varga [11] to derive bounds and decay rates for the block norms of the inverse of block tridiagonal matrices. We also show how to improve these bounds iteratively (Section 2). Moreover, we obtain a new variant of the Gershgorin Circle Theorem for general block matrices (Section 3). Throughout this paper we assume that · is a given submultiplicative matrix norm. *

Analysis of the multiplicative Schwarz method for matrices with a special block structure

“…see Definition A.1 in Appendix A. Analogously to the row block diagonally dominant case described above, a proof of Theorem A.2 along the lines of the proof of [3,Theorem 2.6] shows that if A H and A h satisfy the conditions (4.14). Then…”

“…Motivated by an analysis for a one-dimensional convectiondiffusion model problem in [4], we have studied the convergence of the multiplicative Schwarz method for matrices with a special block structure. After deriving a general expression for the convergence factor of the method, we have focussed on block tridiagonal matrices, and applied recent results on block diagonal dominance from [3] in order to obtain quantitative error bounds that are valid from the first iteration. In our analysis we did not use any of the usual assumptions on the matrices in this context, such as symmetry, or the M -or H-matrix properties.…”

“…, m. Such a matrix A can be obtained, for instance, by discretizing a boundary value problem like (5.1), but with nonconstant coefficients. An analysis of the multiplicative Schwarz method for such a matrix A following the approach in this paper is still possible, since the results from [3] on block diagonal dominance are formulated for general block tridiagonal matrices; see also Appendix A. A generalization of Theorem 4.4 to A with blocks (6.1) would require that the conditions (4.3) hold in every block row, and then, analogously to (4.5)-(4.6), every block row in A H or A h would give a parameter η H,i or η h,i , respectively.…”

“…Both A H and A h satisfy all assumptions of [3,Theorem 2.6]. A minor modification of the first equation in the proof of that theorem (namely multiplying both sides of this equation by C h or B H before taking norms) shows that the blocks of the inverses, i.e., A −1 H = [Z (H) ij ] and A −1 h = [Z (h) ij ], satisfy…”

“…We point out that the model problems studied in [4] are of the form (1.1)-(1.2) with N = 1. While (usual) diagonal dominance of tridiagonal matrices is one of the main tools in [4], the derivation of error bounds here relies on recent results on block diagonal dominance of block tridiagonal matrices from [3].…”