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A classical approach for solving quadratic eigenvalue problem (QEP) is via linearization. However, it can suffer from numerical instability when the norms of coefficient matrices vary widely. Two strategies of Betcke’s balancing for heavily damped QEP are considered. One strategy is to balance matrix polynomial before linearizing (called balancing–linearizing for short). The other strategy is first to linearize the matrix polynomial, then balancing (called linearizing–balancing for short). We analyze the backward error of approximate eigenpairs computed by these two strategies, and find that the backward error of approximate eigenpairs by linearizing–balancing is smaller than that by balancing–linearizing under relatively mild conditions. Numerical experiments are presented to demonstrate the advantages of linearizing-balancing.

In this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results.

In this paper, for a certain class of Kukles polynomial systems of arbitrary degree n with an invariant ellipse, we show that for certain values of the parameters, the system has an upper bound of limit cycles, where one of the limit cycle is given by an invariant ellipse as an algebraic limit cycle. Writing the system as a perturbation of a Hamiltonian system, we show that the first Poincare–Melnikov integral of the system is a polynomial whose coefficients are the Lyapunov quantities. The maximum number of simple zeros of this polynomial, gives the maximum number of the global limit cycles and the multiplicity of the origin as a root of the polynomial, minus one, gives the maximum weakness that may have the weak focus at the origin.

We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, therefore the whole series, converge to zero much more rapidly compared to the Taylor expansion of the same order. Furthermore, using numerical analysis using sixth-order polynomial expansion, we demonstrate that the Legendre polynomial approximation yields an error at least an order of magnitude smaller than the analogous Taylor series approximation. This strongly suggests that Legendre expansions, instead of Taylor expansions, should be used when global accuracy is important., 6 pages, 1 figure

We consider in this work planar polynomial differential systems having a polynomial first integral. We prove that these systems can be obtained from a linear system through a polynomial change of variables., Comment: 9 pages, 0 figures

Recently, Psarrakos obtained a lower bound, which depends on a real parameter ν, for the distance between the pseudospectrum of a matrix polynomial and a given point that lies out of it. In this note, we study the best choice of the parameter. As a consequence, the best lower bound among those given by Psarrakos is obtained.

In this note, we obtain a lower bound for the distance between the pseudospectrum of a matrix polynomial and a given point that lies out of it, generalizing a known result on pseudospectra of matrices.

For when the Sylvester matrix equation has a unique solution, this work provides a closed form solution, which is expressed as a polynomial of known matrices. In the case of non-uniqueness, the solution set of the Sylvester matrix equation is a subset of that of a deduced equation, which is a system of linear algebraic equations.

This work deals with the asymptotics of normalized Laguerre matrix polynomials of a complex matrix parameter for n / x = o ( 1 ) and x / n = O ( 1 ) as n → ∞ .

The coefficients of the characteristic polynomial of an n × n matrix are derived in terms of the eigenvalues and in terms of the elements of the matrix. The connection between the two expressions allows the sum of the products of all sets of k eigenvalues to be calculated using cofactors of the matrix.

In this paper, a new condition for the controllability of higher-order linear dynamical systems is obtained. The suggested test contains rank conditions of suitably defined matrices and is based on the notion of compound matrices and the Binet-Cauchy formula.

A certain multiple integral evaluated by Chikuse [1, p. 402, equation (11)] is evaluated in this paper by two different methods. The value of this integral is expressed in terms of a single matrix argument zonal polynomial unlike Chikuse [1], who expresses this value in terms of a zonal polynomial of two matrix arguments.

In our previous paper [1], we observed that generalized Vandermonde determinants of the form V n ; ν ( x 1 ,…, x s ) = | x i μ k |, 1 ≤ i , k ≤ n , where the x i are distinct points belonging to an interval [ a, b ] of the real line, the index n stands for the order, the sequence μ consists of ordered integers 0 ≤ μ 1 2 n , can be factored as a product of the classical Vandermonde determinant and a Schur function . On the other hand, we showed that when x = x n , the resulting polynomial in x is a Schur function which can be factored as a two-factors polynomial: the first is the constant ∏ i =1 n −1 x i μ 1 times the monic polynomial ∏ i =1 n −1 ( x − x i , while the second is a polynomial P M ( x ) of degree M = m n −1 − n + 1. In this paper, we first present a typical application in which these factorizations arise and then we discuss a condition under which the polynomial P M ( x ) is monic .

Using the concept of Jordan chain of L(λ) = Alλl + Al−1λl−1 + … + A0, where the Ai are n × n matrices with complex entries, we obtain a necessary and sufficient condition for the existence of S ϵ C n×n such that L(S) = 0. A way of finding those matrices, usually called solvents of L(λ), is given as well.

This paper is concerned with the representation of system behaviors by equations involving polynomial shift operators. In particular, the question of the elimination of latent (i.e., auxiliary) variables from an ARMA representation is considered for the case of multidimensional systems.

The purpose of this note is to re-establish, with a new and simple proof, a theorem which in fact represents the main results of [1].

In this note we show that Laguerre's method for finding the root of a polynomial when applied to the characteristic polynomial of a matrix is the same as the method of traces for determining an eigenpair of that matrix.

This paper deals with the asymptotic expression of parametric matrix integrals related to Laguerre's matrix polynomials using path integration of matrix valued functions.

In this paper we introduce the concept of rectangular co-solution of a polynomial matrix equation which permit us to obtain a description of the general solution of systems of higher order differential equations with constant coefficients in a way analogous to the scalar case.

In this paper we introduce the concept of co-solution of a polynomial matrix equation which permits us to obtain necessary and sufficient conditions so that a set of solutions be a complete set.

A factorization x n I n = ( xI n - A 1 )···( xI n - A n ) where each A i is an n × n matrix with minimal polynomial x n is presented and a uniqueness result is proved for n = 2,3.

In this work, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the limit cycle problem for a general Lienard polynomial system with an arbitrary (but finite) number of singular points.

We present a general method for the exact computation of the number of zeros of a complex polynomial inside the unit disk, assuming that the polynomial does not vanish on the unit circle. We prove the existence of a polynomial sequence. This sequence involves a reduced number of arithmetic operations and the growth of intermediate coefficients remains controlled. We study the singular case where the constant term of a polynomial of this sequence vanishes.

In this note, we study the relation between the existence of algebraic invariants and integrability for planar polynomial systems. It is proved, under certain genericity conditions, that if the sum of the degrees of the algebraic invariants exceeds the degree of the polynomial system by one, then the system is integrable.