Polynomial Matrix Research Articles

Skew-circulant matrix and critical points of polynomials

Skew-circulant matrix and critical points of polynomials

A novel numerical manner for non-linear coupled variable order reaction-diffusion equation

In this work, an efficient variable order Bernstein collocation technique, which is based on Bernstein polynomials, is applied to a non-linear coupled system of variable order reaction-diffusion equations with given initial and boundary conditions. The operational matrix of Bernstein polynomials is derived for variable order derivatives w.r.t. time and space. The Bernstein operational matrix and collocation technique are applied to the concerned non-linear physical model to achieve a system of non-linear algebraic equations, which are further solved by using Newton method. A few examples are presented to demonstrate the accuracy and stability of the scheme by comparing L2 and L? norm errors between the obtained numerical solutions and existing solutions. The important feature of this article is the graphical exhibitions of the effects of variable order derivatives on the solutions of the considered non-linear coupled reaction-diffusion equation for different particular cases.

Note on Exact Factorization Algorithm for Matrix Polynomials

Note on Exact Factorization Algorithm for Matrix Polynomials

Joint convergence of sample cross-covariance matrices

Suppose $X$ and $Y$ are $p\times n$ matrices each with mean $0$, variance $1$ and where all moments of any order are uniformly bounded as $p,n \to \infty$. Moreover, the entries $(X_{ij}, Y_{ij})$ are independent across $i,j$ with a common correlation $\rho$. Let $C=n^{-1}XY^*$ be the sample cross-covariance matrix. We show that if $n, p\to \infty, p/n\to y\neq 0$, then $C$ converges in the algebraic sense and the limit moments depend only on $\rho$. Independent copies of such matrices with same $p$ but different $n$, say $\{n_l\}$, different correlations $\{\rho_l\}$, and different non-zero $y$'s, say $\{y_l\}$ also converge jointly and are asymptotically free. When $y=0$, the matrix $\sqrt{np^{-1}}(C-\rho I_p)$ converges to an elliptic variable with parameter $\rho^2$. In particular, this elliptic variable is circular when $\rho=0$ and is semi-circular when $\rho=1$. If we take independent $C_l$, then the matrices $\{\sqrt{n_lp^{-1}}(C_l-\rho_l I_p)\}$ converge jointly and are also asymptotically free. As a consequence, the limiting spectral distribution of any symmetric matrix polynomial exists and has compact support.

Disturbance Observer-Based Robust Control Design for Uncertain Periodic Piecewise Time-Varying Systems With Disturbances

With the aid of disturbance observer strategy, this article aims to investigate the disturbance rejection and stabilization problems for periodic piecewise time-varying systems that are subject to time-varying delays, parameter uncertainties, nonlinear perturbations and exogenous disturbances. To be more specific, the periodic piecewise time-varying systems are built by segmenting the fundamental period of periodic systems into a limited number of subintervals. Further, the disturbances engendered from an exogenous system are estimated by deploying the disturbance observer and subsequently, on the premise of disturbance that is esimated, a robust controller protocol is constructed for the considered system. Moreover, by bridging the time-varying periodic piecewise Lyapunov-Krasovskii functional with a matrix polynomial lemma, a set of adequate criteria is framed, which confirms the asymptotic stability of the system that is being addressed. Subsequently, on the premise of established criteria, the design of periodic piecewise gain matrices of devised controller and configured observer are presented. Eventually, the importance and potential of the presented theoretical concepts are evidenced through offering a numerical illustration with the simulation results.

Utilization of the ExactMPF package for solving a discrete analogue of the nonlinear Schrödinger equation by the inverse scattering transform method

We are revisiting the problem of solving a discrete nonlinear Schrödinger equation by the inverse scattering transform method, by use of the recently developed ExactMPF package within MAPLE Software. ExactMPF allows for an exact Wiener–Hopf factorization of matrix polynomials regardless of the partial indices of the matrix. The package can be widely used in various problems, where Wiener–Hopf factorization as one of the effective mathematical tools is required, as its code has already been disclosed. The analysis presented in this paper contains not only numerical examples of its use, but is also supported by appropriate and accurate a priori estimations. The procedure itself guarantees that the ExactMPF package produces all computations arithmetically exactly , and a detailed numerical analysis of various aspects of the computational algorithm and approximation strategies is provided in the case of a finite initial impulse.

Stability Verification for Heterogeneous Complex Networks via Iterative SOS Programming

This letter is to investigate the stability verification for heterogeneous polynomial complex networks through iterative sum-of-squares programming approach. With polynomial Lyapunov functions, a global asymptotic stability criterion is established for the heterogeneous complex networks under the directed topology. Based on the proposed criterion, the stability verification problem is then reduced to a sum-of-squares optimization problem for solving polynomial matrix inequalities. To process non-convex terms of the polynomial matrix inequalities, we present an iterative sum-of-squares programming approach to turn them into the convex polynomial matrix inequalities, yielding polynomial Lyapunov functions effectively to realize the stability verification. Finally, a numerical example is given to show the fully automatic verification of global asymptotic stability for the heterogeneous polynomial complex networks, which demonstrates the theoretical and algorithmic advantages of the proposed method.

Синтез нейросетвого регулятора для линеаризованной модели объекта – два перевернутых маятника на тележке

This article demonstrates the procedure for the synthesis of a neural network controller for a linearized model of the plant «two inverted pendulums on a cart». A feature of this plant can be considered its nonsquare matrix transfer function and the number of input effects is less than the number of output effects. The task of the control is to stabilize the angles of the inverted pendulums in a stable position and transfer the position of the cart to a set value. A distinctive feature of the demonstrated procedure for the synthesis of a neuroregulator is the deterministic choice of architecture and initialization of the weighting coefficients of the neural network. Data on the choice of architecture and on the values of initialized weighting coefficients for the neural network controller are obtained based on information about the transfer function of the controller obtained by the modal method using the polynomial matrix decomposition of the system. Recommendations for structural transformations of a neural network regulator containing recurrent connections are given. They are necessary for further training of a neural network controller with a deterministic approach to initialization of weight coefficients used in the demonstrated synthesis procedure. As a result of the complexity of the structure and further training of the neural network controller obtained, it is possible to increase the efficiency of the automatic control system compared with the system using the controller obtained by the modal method.

О выборе архитектуры нейрорегулятора

In practice, the choice of the type of neural network is carried out empirically based on an experience of an investigator and many training attempts. At the same time, the excessive complexity of the neural network leads to an increase in its training time, and in some cases, to the impossibility of learning at all. Thus, the justification of the choice of an artificial neural network structure and/or its preliminary calculation based on other models is an urgent task. An equally important task is the choice of an initial weighting coefficients of an neural network, the choice of which determines the speed of convergence of search algorithms. This paper demonstrates several approaches to solving the problem of choosing an architecture and initializing a weighting coefficients of a neural network. One of them is carried out on the basis of a previously calculated function using Petri nets. This approach is demonstrated for solving various tasks, which include the implementation of functions using previously defined neural network models of the simplest logical operations "and", "or", etc. An approach is given that allows optimizing an architecture of a neural network that solves the problem of approximating functions of one and several variables. The principle of determining an architecture and initial weight coefficients is also used in the tasks of training neural networks with reinforcement. A separate section is devoted to the formation of a methodology for determining an architecture and initialization of a weighting coefficients of a neural network of the controller based on information about the controller obtained by a modal method using a polynomial matrix decomposition of a system. The problem of synthesis of a neural network controller for an object model containing nonlinearities and nonparametric uncertainties in the control channel is solved.

Differential equations for certain hybrid special matrix polynomials

The main aim of this article is to find the matrix recurrence relation and shift operators for the Gould-Hopper-Laguerre-Appell matrix polynomials. The matrix differential, matrix integro-differential and matrix partial differential equations are derived for these polynomials via factorization method. Certain examples are constructed in order to illustrate the applications of the results.

On the moments of the moments of the characteristic polynomials of Haar distributed symplectic and orthogonal matrices

We establish formulae for the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices in terms of certain lattice point count problems. This allows us to establish asymptotic formulae when the matrix-size tends to infinity in terms of the volumes of certain regions involving continuous Gelfand–Tsetlin patterns with constraints. The results we find differ from those in the unitary case considered previously.

On the theory of operator interpolation in spaces of rectangular matrixes

The problem of constructing and studying interpolation operator polynomials of an arbitrary fixed degree, defined in spaces of rectangular matrices, which would be generalisations of the corresponding interpolation formulas in the case of square matrices, is considered. Linear interpolation formulas of various structures are constructed for rectangular matrices. Matrix polynomials, with respect to which the resulting interpolation formulas are invariant, are indicated. As a generalisation of linear formulas, formulas for quadratic interpolation and interpolation by polynomials of arbitrary fixed degree in the space of rectangular matrices are constructed. Particular cases of the obtained formulas are considered: when square matrices are chosen as nodes or when the values of the interpolated function are square matrices, as well as the case when both of these conditions are satisfied. For the last variant, the possibilities of different and identical matrix orders for nodes and function values are explored. The obtained results are based on the application of some well-known provisions of the theory of matrices and the theory of interpolation of scalar functions. The presentation of the material is illustrated by a number of examples.

New results on sliding mode control for polynomial fuzzy singular systems with time‐varying delay

AbstractThis article focuses on the problem of sliding mode control (SMC) for polynomial fuzzy singular systems with the matched uncertainty, time‐varying delay and different control input matrices. An integral sliding surface with different control input matrices and polynomial matrices is established. The existence of the equivalent controller can be determined according to the given sum‐of‐squares conditions. By choosing an appropriate augmented Lyapunov–Krasovskii functional and utilizing a generalized free‐matrix‐based integral inequality, a less conservative admissibility condition of sliding mode dynamics is proposed in the form of sum‐of‐squares. It is noteworthy that due to the choice of augmented Lyapunov–Krasovskii functional, the traditional methods cannot transform the nonlinear terms in the admissibility condition of sliding mode dynamics into linear terms. Here, several inequalities are introduced to overcome this difficulty. Furthermore, the SMC law is proposed to ensure the reachability of the sliding surface. The simulation results show that the proposed method is less conservative and effective.

A polynomial eigenvalue test for checking DLMI constraints

In order to obtain an optimization problem with a finite number of variables and constraints, Differential Linear Matrix Inequalities (DLMIs) are usually recast into the form of a system of LMIs. To this end, a typical approach consists in parameterizing the solution as a linear combination of known functions, with weights to be optimized. The constraints are then imposed on a finite set of discrete time points. This technical communiqué is concerned with the problem of checking whether a matrix polynomial solution obtained in this manner satisfies the original DLMI constraints. To this end, a novel test based on the concept of polynomial eigenvalues is presented. The proposed test is illustrated through a numerical example based on the H∞ synthesis of a sampled-data controller.

Random matrix theory and moments of moments of L-functions

In this paper, we give an analytic proof of the asymptotic behavior of the moments of moments of the characteristic polynomials of random symplectic and orthogonal matrices. We therefore obtain alternate, integral expressions for the leading order coefficients previously found by Assiotis, Bailey and Keating. We also discuss the conjectures of Bailey and Keating for the corresponding moments of moments of [Formula: see text]-functions with symplectic and orthogonal symmetry. Specifically, we show that these conjectures follow from the shifted moments conjecture of Conrey, Farmer, Keating, Rubinstein and Snaith.

Glasner property for unipotently generated group actions on tori

A theorem of Glasner from 1979 shows that if \(A \subset \mathbb{T}= \mathbb{R}/\mathbb{Z}\) is infinite, then for each ϵ > 0 there exists an integer n such that nA is ϵ-dense and Berend—Peres later showed that in fact one can take n to be of the form f(m) for any non-constant f(x) ∈ ℤ[x]. Alon and Peres provided a general framework for this problem that has been used by Kelly—Lê and Dong to show that the same property holds for various linear actions on \({\mathbb{T}^d}\). We complement the result of Kelly—Lê on the ϵ-dense images of integer polynomial matrices in some subtorus of \({\mathbb{T}^d}\) by classifying those integer polynomial matrices that have the Glasner property in the full torus \({\mathbb{T}^d}\). We also extend a recent result of Dong by showing that if Γ ≤ SLd(ℤ) is generated by finitely many unipotents and acts irreducibly on ℝd, then the action \(\Gamma \curvearrowright {\mathbb{T}^d}\) has a uniform Glasner property.

A general matrix series inversion pair and associated polynomials

Abstract In the present work, a pair of general inverse matrix series relations is established, and thereby a general class of matrix polynomials is introduced. This class generalizes the extended Jacobi polynomials and their particular cases such as the polynomials of Brafman, Jacobi, Chebyshev, and Legendre. It is further shown that this pair also gives rise to the matrix forms of the Wilson polynomials and the Racah polynomials. For these polynomials, the generating matrix function relations as well as the matrix summation formulas are deduced from their respective inverse pairs. Certain inverse pairs belonging to the Gould classes and the Legendre-Chebyshev classes due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968] are also extended to matrix forms.

Certain results on hybrid relatives of the Sheffer polynomials

The multi-variable special matrix polynomials have been identified significantly both in mathematical and applied frameworks. Due to its usefulness and various applications, a variety of its extensions and generalizations have been investigated and presented. The purpose of the paper is intended to study and emerge with a new generalization of Hermite matrix based Sheffer polynomials by involving integral transforms and some known operational rules. Their properties and quasi-monomial nature are also established. Further, these sequences are expressed in determinant forms by utilizing the relationship between the Sheffer sequences and Riordan arrays. An analogous study of these results is also carried out for certain members belonging to generalized Hermite matrix based Sheffer polynomials.

Stieltjes Property of Quasi-Stable Matrix Polynomials

In this paper, basing on the theory of matricial Hamburger moment problems, we establish the intrinsic connections between the quasi-stability of a monic or comonic matrix polynomial and the Stieltjes property of a rational matrix-valued function built from the even–odd split of the original matrix polynomial. As applications of these connections, we obtain some new criteria for quasi-stable matrix polynomials and Hurwitz stable matrix polynomials, respectively.

Backward error analysis of specified eigenpairs for sparse matrix polynomials

AbstractThis article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include ‐symmetric, ‐skew‐symmetric, Hermitian, skew Hermitian, ‐even, ‐odd, ‐even, ‐odd, ‐palindromic, ‐anti‐palindromic, ‐palindromic, and ‐anti‐palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real‐life applications.