Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.

/pdf/computational-methods-for-linear-matrix-equations-29c8rmvfvt.pdf

Computational Methods for Linear Matrix Equations

In this study, the authors consider distributed computation of the Stein equations with set constraints, where each agent or node knows a few rows or columns of coefficient matrices. By formulating an equivalent distributed optimisation problem, they propose a projection-based algorithm to seek least-squares solutions to the constrained Stein equation over a multi-agent system network. Then, they rigorously prove the convergence of the proposed algorithm to a least-squares solution for any initial condition, and moreover, provide a simplified distributed algorithm with an exponential convergence rate for the case without constraints.

https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-cta.2019.0140

Distributed optimisation design for solving the Stein equation with constraints

We are concerned with stability of numerical methods for delay differential systems of neutral type. In particular, delay-dependent stability of numerical methods is investigated. By means of the H-matrix norm, a necessary and sufficient condition for the asymptotic stability of analytic solution of linear neutral differential systems is derived. Then, based on the argument principle, sufficient conditions for delay-dependent stability of Runge–Kutta and linear multi-step methods are presented, respectively. Furthermore, two algorithms are provided for checking delay-dependent stability of analytical and numerical solutions, respectively. Numerical examples are given to illustrate the main results.

/pdf/delay-dependent-stability-of-numerical-methods-for-delay-4641025gtu.pdf

Delay-dependent stability of numerical methods for delay differential systems of neutral type

This paper is concerned with the numerical solutions of the general matrix equation $\sum ^{p}_{i=1}{\sum ^{s_{i}}_{j=1} }\,\,{A_{ij}X_{i}}{B_{ij}} = C$ , and the general discrete-time periodic matrix equations $\sum ^{p}_{i=1}\sum ^{s_{i}}_{j=1} (A_{i,j,k}X_{i,k}B_{i,j,k}+ C_{i,j, k}X_{i,k+1}D_{i,j,k}) = M_{k}$ , for $k = 1, 2, \ldots ,t$ , which include the well-known Lyapunov, Stein, and Sylvester matrix equations that appear in a wide range of applications in engineering and mechanical problems. Recently the generalized product-type BiCOR method, denoted as GPBiCOR, has been originally proposed to solve the nonsymmetric linear systems $Ax = b$ , and its significant convergence performance has been confirmed in many numerical results. By applying the Kronecker product and the vectorization operator, we develop a matrix form of the GPBiCOR method to approximate the solutions of the above-mentioned general matrix equation and general discrete-time periodic matrix equations. We present the theoretical background of the extended GPBiCOR method and its main computational aspects. Furthermore, several numerical examples of matrix equations arising in different applications are considered to exhibit the accuracy and the efficiency of the proposed method as compared with other popular methods in the literature.

The GPBiCOR Method for Solving the General Matrix Equation and the General Discrete-Time Periodic Matrix Equations

We are concerned with bounds of the matrix eigenvalues and its exponential Combining the Lyapunov equation with the weighted logarithmic matrix norm technique, four sequences are presented to locate eigenvalues of a matrix Based on the relations between the real parts of the eigenvalues and the weighted logarithmic matrix norms, we derive both lower and upper bounds of the matrix exponential, which complement and improve the existing results in the literature Some numerical examples are also given

Bounds of the matrix eigenvalues and its exponential by Lyapunov equation

This paper is concerned with bounds of eigenvalues of a complex matrix. Both lower and upper bounds of modulus of eigenvalues are given by the Stein equation. Furthermore, two sequences are presented which converge to the minimal and the maximal modulus of eigenvalues, respectively. We have to point out that the two sequences are not recommendable for practical use for finding the minimal and the maximal modulus of eigenvalues.

http://www.kybernetika.cz/content/2010/4/655/paper.pdf

Qiao Zhu

Papers

Bounds of modulus of eigenvalues based on stein equation