Numerical Methods for a Class of Quadratic Matrix Equations

Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of minimal nonnegative solution for this quadratic matrix equation,and then propose some numerical methods for solving it.Convergence analysis and numerical examples are given to verify the theories and the numerical methods of this paper.

ACCURATE COMPUTATION FOR IRREDUCIBLE NONSINGULAR M-MATRIX

Computing the eigenvalue of smallest modulus and its corresponding eigneveclor of an irreducible nonsingular M-matrix A is considered, It is shown that if the entries of A are known with high relative accuracy, its eigenvalue of smallest modulus and each component of the corresponding eigenvector will be determined to much higher accuracy than the standard perturbation theory suggests. An algorithm is presented to compute them with a small componentwise backward error, which is consistent with the perturbation results.

THE SPECTRAL PROPERTIES OF THE ITERATION MATRIX OF REGULAR SPLITTINGS FOR AN M-MATRIX

Let A=M-N be a regular splitting of an M-matrix. We study the spectral properties of the ineration matrix M-1N. Under a mild assumption on M-1 N. some necessary and sufficent conditions such that p(M-1N)=1 are obtained and the algebraic multiplicity and the index associated with eigenvalue 1 in M-1N are considered.

BANDED M-MATRIX SPLITTING PRECONDITIONER FOR RIESZ SPACE FRACTIONAL REACTION-DISPERSION EQUATION

Based on the Crank-Nicolson and the weighted and shifted Grunwald operators,we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the convergence of this implicit difference scheme.However,after estimating the condition number of the coefficient matrix of the discretized scheme,we find that this coefficient matrix is ill-conditioned when the spatial mesh-size is sufficiently small.To overcome this deficiency,we further develop an effective banded M-matrix splitting preconditioner for the coefficient matrix.Some properties of this preconditioner together with its preconditioning effect are discussed.Finally,Numerical examples are employed to test the robustness and the effectiveness of the proposed preconditioner.

The Asymptotic Stability for a Class of Nonlinear Large Sceale System

This paper an cited instances in illustration of the incorrectness of the criteria of asymptotic stability of a class of nonlinear large seale system that L_j·T·Grujie gave in paper [1] by the comparison theory and then corrected it,and has given the sufficient conditions of the asymptotic stability.

Optimality of Solution with Numerical Investigation for Coronavirus Epidemic Model

The novel coronavirus disease,coined as COVID-19,is a murderous and infectious disease initiated from Wuhan,China.This killer disease has taken a large number of lives around the world and its dynamics could not be controlled so far.In this article,the spatio-temporal compartmental epidemic model of the novel disease with advection and diffusion process is projected and analyzed.To counteract these types of diseases or restrict their spread,mankind depends upon mathematical modeling and medicine to reduce,alleviate,and anticipate the behavior of disease dynamics.The existence and uniqueness of the solution for the proposed system are investigated.Also,the solution to the considered system is made possible in a well-known functions space.For this purpose,a Banach space of function is chosen and the solutions are optimized in the closed and convex subset of the space.The essential explicit estimates for the solutions are investigated for the associated auxiliary data.The numerical solution and its analysis are the crux of this study.Moreover,the consistency,stability,and positivity are the indispensable and core properties of the compartmental models that a numerical design must possess.To this end,a nonstandard finite difference numerical scheme is developed to find the numerical solutions which preserve the structural properties of the continuous system.The M-matrix theory is applied to prove the positivity of the design.The results for the consistency and stability of the design are also presented in this study.The plausibility of the projected scheme is indicated by an appropriate example.Computer simulations are also exhibited to conclude the results.

Matrix Inequalities for the Fan Product and the Hadamard Product of Matrices

A new inequality on the minimum eigenvalue for the Fan product of nonsingular M-matrices is given. In addition, a new inequality on the spectral radius of the Hadamard product of nonnegative matrices is also obtained. These inequalities can improve considerably some previous results.

Preconditioned Iterative Method for Regular Splitting

Several preconditioners are proposed for improving the convergence rate of the iterative method derived from splitting. In this paper, the comparison theorem of preconditioned iterative method for regular splitting is proved. And the convergence and comparison theorem for any preconditioner are indicated. This comparison theorem indicates the possibility of finding new preconditioner and splitting. The purpose of this paper is to show that the preconditioned iterative method yields a new splitting satisfying the regular or weak regular splitting. And new combination preconditioners are proposed. In order to denote the validity of the comparison theorem, some numerical examples are shown.

GLOBAL STABILITY IN HOPFIELD NEURAL NETWORKS WITH DISTRIBUTED TIME DELAYS

In this paper, without assuming the boundedness, monotonicity and differentiability of the activation functions, the conditions ensuring existence, uniqueness, and global asymptotical stability of the equilibrium point of Hopfield neural network models with distributed time delays are studied. Using M-matrix theory and constructing proper Liapunov functionals, the sufficient conditions for global asymptotic stability are obtained.

REMARKS ON BRUALDI'S THEOREM FOR SPECTRUM INCLUSION REGION OF AN IRREDUCIBLE MATRIX

This paper obtains a necessary and sufficient condition for an irreducible complex matrix whose comparison matrix is a singular M-matrix to be singular. This is used to establish a necessary and sufficient condition for a boundary point of Brualdi’s inclusion region of the eigenvalues of an irreducible complex matrix to be an eigenvalue.

Necessary and Sufficient Condition for Generalized Diagonal Dominance Matrices

In the paper,a necessary and sufficeent condition for generalized diagonal domiance matrices is given.Further, the relations among all generalized positive definite matrices are shown, also,some flaws and mistakes in the references are corrected.

On a Comparison Property of M-matrices

In this paper, we present a spectral property of a comparison matrix, which improves and generalize the comparison theorem of nonsingular M-matrices.

Extended consensus analysis of multi-agent systems with switching topologies

Theoretical analysis of consensus for networked multi-agent systems with switching topologies was conducted.Supposing that information-exchange topologies of networked system are dynamic,a modified linear protocol is proffered which is more practical than existing ones.The definition of trajectory consensus is given and a new consensus protocol is exhibited such that multi-agent system achieves trajectory consensus.In addition,a formation control strategy is designed.A common Lyapunov function is proposed to analyze the consensus convergence of networked multi-agent systems with switching topologies.Simulations are provided to demonstrate the effectiveness of the theoretical results.

Non-Singularity Conditions for Two Z-Matrix Types

A real square matrix whose non-diagonal elements are non-positive is called a Z-matrix. This paper shows a necessary and sufficient condition for non-singularity of two types of Z-matrices. The first is for the Z-matrix whose row sums are all non-negative. The non-singularity condition for this matrix is that at least one positive row sum exists in any principal submatrix of the matrix. The second is for the Z-matrix which satisfies where . Let be the ith row and the jth column element of , and be the jth element of . Let be a subset of which is not empty, and be the complement of if is a proper subset. The non-singularity condition for this matrix is such that or such that for? . Robert Beauwens and Michael Neumann previously presented conditions similar to these conditions. In this paper, we present a different proof and show that these conditions can be also derived from theirs.

INFLUENCE OF NOISE AND DELAY ON REACTION-DIFFUSION RECURRENT NEURAL NETWORKS

In this paper, the influence of the noise and delay upon the stability property of reaction-diffusion recurrent neural networks （RNNs） with the time-varying delay is discussed. The new and easily verifiable conditions to guarantee the mean value exponential stability of an equilibrium solution are derived. The rate of exponential convergence can be estimated by means of a simple computation based on these criteria.

ON A PROPERTY OF SOLUTIONS OF M-MATRIXEQUATIONS

For M-matrix equations, we provide a necessary and sufficient condition for that the solution of the equations has the property p, which improves and generalizes the corresponding results of [1] and [2].

Some New Inequalities Involving the Hadamard Product of an M-matrix and Its Inverse

For the Hadamard product of an M-matrix and its inverse, some new lower bounds on the minimum eigenvalue are given. These bounds can improve considerably some previous results.

SOME INVERSE M-MATRIX PROBLEMS

An upper bound and a lower bound for a0 are given such that aI+B∈M-1 for a>a0 and aI+BM-1 for a≤a0, where B is a nonnegative matrix and satisfies that for any positive constant β,βI+B is a power invariant zero pattern matrix.

EXISTENCE OF STABLE EQUILIBRIUM FOR A CLASS OF MULTI-SPECIES ECOSYSTEM

In this paper, we discuss the existence of the stable equilibrium in multispecies Scheoner ecosystem. By means of Banach fixed point theorem, we obtain a sufficent condition for the system to have a unique positive equilibrium. We also discuss the global stability of the equilibrium.

General Decay Pathwise Stability of Neutral Stochastic Differential Equations with Unbounded Delay

Without the linear growth condition, by the use of Lyapunov function, this paper estab- lishes the existence^and-uniqueness theorem of global solutions to a class of neutral stochastic differen- tim equations with unbounded delay, and examines the pathwise stability of this solution with general decay rate. As an application of our results, this paper also considers in detail a two-dimensional unbounded delay neutral stochastic differential equation with polynomial coefficients.