STRUCTURES OF CIRCULANT INVERSE M-MATRICES

In this paper, we present a useful result on the structures of circulant inverse Mmatrices. It is shown that if the n × n nonnegative circulant matrix A = Circ[c0, c1,… , c（n- 1）] is not a positive matrix and not equal to c0I, then A is an inverse M-matrix if and only if there exists a positive integer k, which is a proper factor of n, such that cjk 〉 0 for j=0,1…, [n-k/k], the other ci are zero and Circ[co, ck,… , c（n-k）] is an inverse M-matrix. The result is then extended to the so-called generalized circulant inverse M-matrices.     

An equivalence between two algorithms for a class of quadratic programming problems with mmatrices *

In this paper the behaviour of the gradient projection method for the Quadratic Programming problems with M-Matrices and lower bounds on variables is analysed. It is shown that the Chandbasekaran method for solving such problems is simply a realization of the Newton projection method if for the latter one the starting point has components equal to lower bounds on variables.     

Conditions for positive definiteness of M-matrices

The present paper concentrates on conditions that are necessary and sufficient for M-matrices to be positive definite. The obtained results can be used in the analysis of productivity of the Leontief input-output model.     

A note on matrics with given diagonal entries

Mirsky and Erdos and Mine have given criteria for the existence of a non-negative square matrix with prescribed diagonal entries and row and column sums; it is shown that this result follows from the supply-demand theorem for flows in networks.     

Rounding Errors in Solving Block Hessenberg Systems

A rounding error analysis is presented for a divide-and-conquer algorithm to solve linear systems with block Hessenberg matrices. Conditions are derived under which the algorithm computes a stable solution. The algorithm is shown to be stable for block diagonally dominant matrices and for M-matrices.

     

M—矩阵的等价表征

本文引进了按环路弱不约非零元素对角占优的概念，讨论了Ｍ－矩阵的等价条件，给出了Ｍ－矩阵的两个等价表征，改进与推广了（１），（２），（５），（９）的相应结果。     

A positive finite-difference model in the computational simulation of complex biological film models

In this work, we design a linear, two-step, finite-difference method to approximate the solutions of a biological system that describes the interaction between a microbial colony and a surrounding substrate. The model is a system of four partial differential equations with nonlinear diffusion and reaction, and the colony is formed by an active portion, an inert component and the contribution of extracellular polymeric substances. In this work, we extend the computational approach proposed by Eberl and Demaret [A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, Electr. J. Differ. Equ. 15 (2007) pp. 77–95], in order to design a numerical technique to approximate the solutions of a more complicated model proposed in the literature. As we will see in this work, this approach guarantees that positive and bounded initial solutions will evolve uniquely into positive and bounded, new approximations. We provide numerical simulations to evince the preservation of the positive character of solutions.     

A comparison result for multisplittings and waveform relaxation methods

We show that certain multisplitting iterative methods based on overlapping blocks yield faster convergence than corresponding nonoverlapping block iterations, provided the coefficient matrix is an M-matrix. This result can be used to compare variants of the waveform relaxation algorithm for solving initial value problems. The methods under consideration use the same discretization technique, but are based on multisplittings with different overlaps. Numerical experiments on the Intel iPSC/860 hypercube are included.     

Approximate factorizations with modified S/P consistently ordered M-factors

Preconditioned iterative methods are widely used to solve linear systems such as those arising from the finite element formulation of boundary value problems and approximate factorizations are widely used as preconditioners. The ordering of the unknowns is therefore an important issue because it has a strong influence on the convergence behaviour of the iteration method while it is also a decisive aspect for their parallel implementation. Consistent orderings are attractive for parallel implementations and it has been shown that some subclasses of these orderings also enhance the convergence behaviour of the associated iteration methods. This has in particular been shown for the so-called S/P consistent orderings. A wider definition of this class of orderings has recently been proposed and we investigate here how approximate factorizations should be implemented when using such more general orderings (still called S/P consistent) in order to keep their expected high convergence properties. A simple practical conclusion is suggested, supported by both theoretical and numerical arguments.     

论几类广义严格对角占优矩阵

1引言 设A=（a_η）∈Cm~(3n),若存在正对角阵D.使得AD为严格对角占优矩阵，则A称为广义严格对角占优矩阵，记作A∈SGDDM.