Is a general -matrix?

Linear Algebra and its Applications, 1990

Ahac and Olesky have shown that by using the most diagonaldominat column as a pivot element for M-matrices was numerically stable. The purpose of this paper is to show that this method of selecting a pivoting is stable when finding the incomplete factorization of an H-matrix.

Computing, 2007

Hierarchical matrices (H-matrices) approximate matrices in a data-sparse way, and the approximate arithmetic for H-matrices is almost optimal. In this paper we present an algebraic approach for constructing H-matrices which combines multilevel clustering methods with H-matrix arithmetic to compute the H-inverse, H-LU, and the H-Cholesky factors of a matrix. Then the H-inverse, H-LU or H-Cholesky factors can be used as preconditioners in iterative methods to solve systems of linear equations. The numerical results show that this method is efficient and greatly speeds up convergence compared to other approaches, such as JOR or AMG, for solving some large, sparse linear systems, and is comparable to other H-matrix constructions based on Nested Dissection.

Linear Algebra and its Applications, 2008

Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M-matrix. H-matrices such that their comparison matrices are non-singular are well studied in the literature. In this paper, we study characterizations of H-matrices with singular or nonsingular comparison matrix. In particular, we analyze the case when A is irreducible and then give insights into the reducible case. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, from these characterizations, a partition of the general H-matrix set in three classes is obtained.

Demonstratio Mathematica, 1997

Numerical Algorithms, 2016

The Linear Complementarity Problem (LCP), with an H + −matrix coefficient, is solved by using the new "(Projected) Matrix Analogue of the AOR (MAAOR)" iterative method; this new method constitutes an extension of the "Generalized AOR (GAOR)" iterative method. In this work two sets of convergence intervals of the parameters involved are determined by the theories of "Perron-Frobenius" and of "Regular Splittings". It is shown that the intervals in question are better than any similar convergence intervals found so far by similar iterative methods. A deeper analysis reveals that the "best" values of the parameters involved are those of the (projected) scalar Gauss-Seidel iterative method. A theoretical comparison of the "best" (projected) Gauss-Seidel and the "best" modulus-based splitting Gauss-Seidel method is in favor of the former method. A number of numerical examples support most of our theoretical findings.

Applied Mathematics and Computation, 2008

It is well-known [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. J. 29 (104) (1979) 246-251, [1]] that the Schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant. Also, if a matrix is an H-matrix, then its Schur complement is an H-matrix, too [J. Liu, Y. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Algebra Appl. 389 (2004) 365-380, [8]]. Recent research showed that the same type of statement holds for some special subclasses of H-matrices, see, for example, Liu et al. [J. Liu, Y. Huang, F. Zhang, The Schur complements of generalized doubly diagonally dominant matrices, Linear Algebra Appl. 378 (2004) 231-244]. The aim of this paper is to show that the proof of these results can be significantly simplified by using ''scaling'' approach as in Zhang et al. [F. Zhang et al., The Schur Complement and its Applications, Springer, New York, 2005] and to give another invariance result of this type.

Numerical Linear Algebra with Applications, 2009

It is well known that the Schur complement of some H‐matrices is an H‐matrix. In this paper, the Schur complement of any general H‐matrix is studied. In particular, it is proved that the Schur complement, if it exists, is an H‐matrix and the class to which the Schur complement belongs is studied. In addition, results are given for singular irreducible H‐matrices and for the Schur complement of nonsingular irreducible H‐matrices. Copyright © 2009 John Wiley & Sons, Ltd.

Proc. Appl. Math. Mech., 2003

A short overview of H 2-matrices H 2-matrices are a refined version of hierarchical matrices, which in turn are based on the panel-clustering approach for approximating certain integral operators. Their main advantage, as compared with their predecessors, is that the associated algorithms are of significantly lower complexity while providing the same order of accuracy. Under certain conditions, it is even possible to reach optimal complexity.

2003

In this paper, several new simple criteria for nonsingular H-matrices are obtained by making use of elements of matrices only. Advantages are illustrated by numerical examples. And a necessary condition for nonsingular H-matrices is also presented.

This paper describes many of the standard numerical methods used in Linear Algebra. Topics include Gaussian Elimination, LU and QR Factorizations, The Singular Value Decomposition, Eigenvalues and Eigenvectors via the QR Method with Shifts or the Divide-and-Conquer Method, and the Conjugate Gradient and Lanczos Iterative Methods.