Perron Complementation on Linear Systems Involving M-Matrices

We propose in this paper a generalized Perron complementation method for uncoupling a consistent linear system which involves an irreducible, either singular or nonsingular, M-matrix. We show that this uncoupling arises naturally from a regular splitting, which also leads to an efficient iterative scheme for solving the linear system.

On the strong P-regular splitting iterative methods for non-Hermitian linear systems

<abstract><p>The strong <italic>P</italic>-regular splitting is put forward and defined for iterative methods of non-Hermitian linear systems in the paper. The strong <italic>P</italic>-regular splitting combining SOR iterative methods and relaxed SOR iterative methods are established, and conditions guaranteeing the convergence are presented. Furthermore, two numerical experiments are done to illustrate the convergence and effectiveness of our iterative methods.</p></abstract>

Some New Results on Various Graph Energies of the Splitting Graph

The energy of a simple connected graph G is equal to the sum of the absolute value of eigenvalues of the graph G where the eigenvalue of a graph G is the eigenvalue of its adjacency matrix AG. Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graph S′G is a multiple of corresponding energy of a given graph G.

Properweak regular splitting and its application to convergence of alternating iterations

Theory of matrix splittings is a useful tool for finding the solution of a rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit the theory of weak regular splittings for rectangular matrices. Secondly, we propose an alternating iterative method for solving rectangular linear systems by using the Moore-Penrose inverse and discuss its convergence theory, by extending the work of Benzi and Szyld [Numererische Mathematik 76 (1997) 309-321; MR1452511]. Furthermore, a comparison result is obtained which ensures the faster convergence rate of the proposed alternating iterative scheme.

SEMI-IMPLICIT DIFFERENCE SCHEME FOR A TWO-DIMENSIONAL PARABOLIC EQUATION WITH AN INTEGRAL BOUNDARY CONDITION

In this paper, we consider a finite difference method for a class of twodimensional parabolic equations with integral boundary conditions. The semi-implicit difference scheme is considered. The stability of difference scheme is proved using the properties of the M-matrices, particularly, the regular splitting of an M-matrix. The numerical results of some examples are presented, that approve our theoretical investigations.

On the disintegration of (proto-)languages

Do languages split into dialects and subsequently into new languages at regular rates? Does such a regular splitting rate also apply to speech communities ancestral to the world's current language families? Do linguistic phylogenies exhibit intermediate levels (“genera”) which are somehow objectively identifiable? These questions are rarely raised, much less answered. In this article we present a simple method that provides insights into all the questions, drawing upon data from a world-wide sample of languages. It will be shown that splitting rates are approximately regular even if the languages studied are proto-languages spoken at very different points in prehistory and different places in the world. Ancestors of the world's linguistic families tend to have similar life-times. An intermediate transitional level corresponding to the point where genera appear can be objectively inferred from differences among descendant languages even without previously having established the structure of the phylogenetic tree.