Vladimir Kostić

A new subclass of H-matrices

Class of H-matrices plays an important role in various scientific disciplines, in economics, for example. However, this class could be used in order to get various benefits in other linear algebra fields, like determinant estimation, Perron root estimation, eigenvalue localization, improvement of convergence area of relaxation methods, etc. For that reason, it seems important to find a subclass of H-matrices, as wide as possible, and expressed by explicit conditions, involving matrix elements only. One step forward in this direction, starting from Gudkov matrices, from one side, and S-SDD matrices, from the other side, will be presented in this paper.

A note on the convergence of the MSMAOR method for linear complementarity problems

SUMMARY Modulus-based splitting, as well as multisplitting iteration methods, for linear complementarity problems are developed by Zhong-Zhi Bai. In related papers (see Bai, Z.-Z., Zhang, L.-L.: Modulus-Based Synchronous Multisplitting Iteration Methods for Linear Complementarity Problems. Numerical Linear Algebra with Applications 20 (2013) 425–439, and the references cited therein), the problem of convergence for two-parameter relaxation methods (accelerated overrelaxation-type methods) is analyzed under the assumption that one parameter is greater than the other. Here, we will show how we can avoid this assumption and, consequently, improve the convergence area. Copyright

A simple generalization of Geršgorin's theorem

It is well known that the spectrum of a given matrix A belongs to the Geršgorin set Γ(A), as well as to the Geršgorin set applied to the transpose of A, Γ(AT). So, the spectrum belongs to their intersection. But, if we first intersect i-th Geršgorin disk Γi(A) with the corresponding disk

Eigenvalue Localization Refinements for Matrices Related to Positivity

\Gamma_i(A^T)

Further results on H-matrices and their Schur complements

, and then we make union of such intersections, which are, in fact, the smaller disks of each pair, what we get is not an eigenvalue localization area. The question is what should be added in order to catch all the eigenvalues, while, of course, staying within the set Γ(A) ∩ Γ(AT). The answer lies in the appropriate characterization of some subclasses of nonsingular H-matrices. In this paper we give two such characterizations, and then we use them to prove localization areas that answer this question.

On general principles of eigenvalue localizations via diagonal dominance

Eigenvalue localization results and methods for matrices with constant row or column sum are provided, together with the numerical examples that show the efficiency of the proposed methods. The extension of the results to other classes of matrices is additionally analyzed.

Max-norm bounds for the inverse of S-Nekrasov matrices

Abstract 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.

Geršgorin‐type localizations of generalized eigenvalues

This paper suggests a unifying framework for matrix spectra localizations that originate from different generalizations of strictly diagonally dominant matrices. Although a lot of results of this kind have been published over the years, in many papers same properties were proven for every specific localization area using basically the same techniques. For that reason, here, we introduce a concept of DD-type classes of matrices and show how to construct eigenvalue localization sets. For such sets we then prove some general principles and obtain as corollaries many singular results that occur in the literature. Moreover, obtained principles can be used to construct and use novel Geršgorin-like localization areas. To illustrate this, we first prove a new nonsingularity result and then use established principles to obtain the corresponding localization set and its several properties. In addition, some new results on eigenvalue separation lines and upper bounds for spectral radius are obtained, too.

New subclasses of block H-matrices with applications to parallel decomposition-type relaxation methods

Max norm bounds of the inverse of an H-matrix appear in a wide range of applications. Motivated by this fact, we will start with some preliminary estimations already obtained for S-SDD matrices, and present new, applicable and practical bounds for a wider class of S-Nekrasov matrices. Then, we will comment how the scaling technique proves that new bounds improve the known ones for Nekrasov matrices. Finally, we will illustrate new results by numerical examples.

On the choice of parameters in MAOR type splitting methods for the linear complementarity problem

We introduce several localization techniques for the generalized eigenvalues of a matrix pair, obtained via the famous Gersgorin theorem and its generalizations. Specifically, we address the techniques of computing and graphing of the obtained localization sets of a matrix pair. The work that follows involves much about nonnegative matrices, strictly diagonally dominant (SDD) matrices, H- and M-matrices. We show the utility of our results theoretically, as well as with numerical examples and graphs. Copyright