On Stability of Linear Systems with Impulsive Action at the Matrix

2018 ◽  
Vol 230 (5) ◽  
pp. 673-676 ◽  
Author(s):  
N. I. Zhelonkina ◽  
A. N. Sesekin
Keyword(s):  
Linear Systems ◽  
Impulsive Action ◽  
The Matrix

DMRG Approach to Fast Linear Algebra in the TT-Format

2011 ◽  
Vol 11 (3) ◽  
pp. 382-393 ◽  
Author(s):  
Ivan Oseledets
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
Direct Method ◽  
Fast Algorithms ◽  
Approximate Solutions ◽  
Vector Product ◽  
The Matrix ◽  
Minimization Scheme

AbstractIn this paper, the concept of the DMRG minimization scheme is extended to several important operations in the TT-format, like the matrix-by-vector product and the conversion from the canonical format to the TT-format. Fast algorithms are implemented and a stabilization scheme based on randomization is proposed. The comparison with the direct method is performed on a sequence of matrices and vectors coming as approximate solutions of linear systems in the TT-format. A generated example is provided to show that randomization is really needed in some cases. The matrices and vectors used are available from the author or at http://spring.inm.ras.ru/osel

Solutions to the matrix equation X − AXB = CY+R and its application

2016 ◽  
Vol 40 (3) ◽  
pp. 995-1004 ◽  
Author(s):  
Caiqin Song ◽  
Guoliang Chen
Keyword(s):  
Linear Systems ◽  
Matrix Equation ◽  
Controller Design ◽  
Equivalent Form ◽  
Closed Form Solutions ◽  
Lyapunov Matrix Equation ◽  
Special Cases ◽  
The Matrix ◽  
Lyapunov Matrix ◽  
Low Dimensional

The solution of the nonhomogeneous Yakubovich matrix equation [Formula: see text] is important in stability analysis and controller design in linear systems. The nonhomogeneous Yakubovich matrix equation [Formula: see text], which contains the well-known Kalman–Yakubovich matrix equation and the general discrete Lyapunov matrix equation as special cases, is investigated in this paper. Closed-form solutions to the nonhomogeneous Yakubovich matrix equation are presented using the Smith normal form reduction. Its equivalent form is provided. Compared with the existing method, the method presented in this paper has no limit to the dimensions of an unknown matrix. The present method is suitable for any unknown matrix, not only low-dimensional unknown matrices, but also high-dimensional unknown matrices. As an application, parametric pole assignment for descriptor linear systems by PD feedback is considered.

scholarly journals Statistical applications for equivariant matrices

2001 ◽  
Vol 25 (1) ◽  
pp. 53-61
Author(s):  
S. H. Alkarni
Keyword(s):  
Linear Systems ◽  
Fourier Transforms ◽  
Toeplitz Matrices ◽  
Computational Cost ◽  
Special Kind ◽  
Circulant Matrices ◽  
Stationary Processes ◽  
Linear System Of Equations ◽  
The Matrix ◽  
A Finite Group

Solving linear system of equationsAx=benters into many scientific applications. In this paper, we consider a special kind of linear systems, the matrixAis an equivariant matrix with respect to a finite group of permutations. Examples of this kind are special Toeplitz matrices, circulant matrices, and others. The equivariance property ofAmay be used to reduce the cost of computation for solving linear systems. We will show that the quadratic form is invariant with respect to a permutation matrix. This helps to know the multiplicity of eigenvalues of a matrix and yields corresponding eigenvectors at a low computational cost. Applications for such systems from the area of statistics will be presented. These include Fourier transforms on a symmetric group as part of statistical analysis of rankings in an election, spectral analysis in stationary processes, prediction of stationary processes and Yule-Walker equations and parameter estimation for autoregressive processes.

scholarly journals A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-26 ◽  
Author(s):  
M. Ilić ◽  
I. W. Turner ◽  
V. Anh
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Poisson Equation ◽  
Matrix Function ◽  
Fractional Power ◽  
Analytic Solutions ◽  
Linear System Of Equations ◽  
Exact Analytic Solutions ◽  
The Matrix ◽  
Symmetric Positive Definite Matrices

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

Exponential Stability for Grey Linear Systems with Time-Varying Delay

2013 ◽  
Vol 760-762 ◽  
pp. 2258-2262
Author(s):  
Ji Chao Wang ◽  
Li Jun Song ◽  
Wei Liu ◽  
Jin Fang Han
Keyword(s):  
Exponential Stability ◽  
Linear Systems ◽  
Stability Problem ◽  
Measure Theory ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Matrix Measure ◽  
Time Varying Delay ◽  
The Matrix ◽  
Varying Delay

In this paper, the exponential stability problem of grey linear systems with time-varying delay is investigated. By using the matrix measure theory and differential inequality approach, some practical sufficient conditions for guaranteeing the exponential stability of the grey linear systems with time-varying delay are presented. The grey-matrix measure and norm are also introduced.

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