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Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational...

The structured partial quadratic inverse eigenvalue problem (SPQIEP) is to construct the structured quadratic matrix polynomial using the partial eigendata. The structures arising in physical appli...

This paper investigates the problem of non-fragile H ∞ control for periodic piecewise time-varying systems. Based on a Lyapunov function with continuous time-varying Lyapunov matrix polynomial, and...

This paper proposes a design method of model predictive control (MPC) for multi-input multi-output (MIMO) plants with time-delay by using an interactor matrix and a sequential procedure to solve th...

In this paper, the stability of matrix polynomials is investigated. First, upper and lower bounds are derived for the eigenvalues of a matrix polynomial. The bounds are based on the spectra...

We show how l -ifications, namely, lower order matrix polynomials with the same eigenvalues as a given complex square matrix polynomial, can be used in combination with other recent results, namely...

We proposed a polynomial approximation-based approach to solve a specific type of chance-constrained optimization problem that can be equivalently transformed into a convex programme. This type of ...

This paper is concerned with the polynomial filtering problem for a class of nonlinear systems with quantisations and missing measurements. The nonlinear functions are approximated with polynomials of a chosen degree and the approximation errors are described as low-order polynomial terms with norm-bounded coefficients. The transmitted outputs are quantised by a logarithmic quantiser and are also subject to randomly missing measurements governed by a Bernoulli distributed sequence taking values on 0 or 1. Dedicated efforts are made to derive an upper bound of the filtering error covariance in the simultaneous presence of the polynomial approximation errors, the quantisations as well as the missing measurements at each time instant. Such an upper bound is then minimised through designing a suitable filter gain by solving a set of matrix equations. The filter design algorithm is recursive and therefore applicable for online computation. An illustrative example is exploited to show the effectiveness ...

We present a method for finding partial fraction decompositions of rational functions with linear or quadratic factors in the denominators by means of repeated polynomial divisions. This method does not involve differentiation or solving linear equations for obtaining the unknown partial fraction coefficients, which is very suitable for either hand or machine calculations. Some illustrative examples are provided.

In this paper, the global asymptotic stabiliser design of rational systems is studied in detail. To develop the idea, the state equations of the system are transformed to a new coordinate via polynomial transformation and the state feedback control law. This in turn is followed by the satisfaction of the linear growth condition (i.e. Lipschitz at zero). Based on a linear matrix inequality solution, the system in the new coordinate is globally asymptotically stabilised and then, leading to the global asymptotic stabilisation of the primary system. The polynomial transformation coefficients are derived by solving the bilinear matrix inequality problem. To confirm the capability of this method, three examples are highlighted.

The current contribution presents exact third-order polynomial expressions of matrix functions that arise in the computation of the Kolmogoroff–Nagumo mean of a set of optical transference matrices, that belong to the affine symplectic group .

In this paper, we study the growth of the codimensions of a finite dimensional -superalgebra over a field of characteristic zero. We prove that has polynomial growth if and only if any finite dimensional -superalgebra satisfying the same -graded identities of has an explicit decomposition into suitable -superalgebras. We also give such a characterization by studying the decomposition of the -cocharacter of . In this case, the main tool is the representation theory of the product of symmetric groups.

A state feedback is proposed to analyse the stability of a matrix polynomial in closed loop. First, it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable. Following the ideas of the Routh–Hurwitz stability procedure for scalar polynomials, certain continued-fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the Euclid’s division algorithm which leads to an extension of the well-known Routh–Hurwitz stability criteria but this time in terms of matrix coefficients. After that, stability of the closed-loop matrix polynomial is guaranteed based on a Corollary of a Lyapunov Theorem. The sufficient stability conditions are: (i) The matrices of one column of the presented array must be symmetric and positive definite and (ii) the matrices of the cascade realisation must satisfy a commutative condition. These stability conditions are also necessary for matrix polynomial of second order...

The paper presents characterizations for an evolution family with the polynomial growth to be polynomially stable using the Perron method. To do this we associate the evolution family to an integral equation, which can be obtained by integrating an inhomogeneous differential equation in Banach spaces. We also give similar results of the polynomial instability by extending techniques from the polynomial stability.

In this paper, we consider global observability decompositions of autonomous polynomial systems by using commutative algebra and algebraic geometry. In contrast to the local observability decomposi...

An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This algorithm is a modification of some well known and widely used algorithms. A novel feature is an approximate calculation of divided differences for the Newton interpolating polynomial in a special way. This modification does not require to reorder the Schur triangular form and to solve Sylvester equations., Comment: 11 pages

We consider the performance analysis of the natural frequency-based radar target detection. By making the Lagrange polynomial approximation of the standard normal distribution, the probability of detection for an augmented input vector can be recursively calculated. We present the bound of the error due to the Lagrange polynomial approximation, and it is illustrated that the actual error is within the derived error bound. We also present how to determine the optimal first-order Lagrange polynomial.

In this paper, we present a numerical method based on the polynomial scaling functions to solve the second-order one-space-dimensional hyperbolic telegraph equation. The method consists of expanding the approximate solution as the elements of polynomial scaling functions. The operational matrix of derivative for polynomial scaling functions is developed. Using the operational matrix of derivative, the problem reduces to a set of algebraic linear equations. An estimation of error bound for this method is investigated. Two numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces considerable accurate results among the existing scaling functions.

This paper presents a solution of the suboptimal regulator problem for a class of discrete-time nonlinear polynomial systems. The solution is obtained by reducing the control problem to the corresponding one. A general solution has been obtained for a polynomial of an arbitrary order; then, finite-dimensional regulator equations are derived explicitly for a second-order polynomial. Numerical simulations have been carried out to show effectiveness of the proposed method.

In this article, we present a new method to estimate the asymptotic stability regions for a class of nonlinear systems via composite homogeneous polynomial Lyapunov functions, where these nonlinear systems are approximated as a convex hull of some linear systems. Since level set of the composite homogeneous polynomial Lyapunov functions is a union set of several homogeneous polynomial functions, the composite homogeneous polynomial Lyapunov functions are nonconservative compared with quadratic or homogeneous polynomial Lyapunov functions. Numerical examples are used to illustrate the effectiveness of our method.

In this paper, we first prove that if the following differential equationadmits a meromorphic function with finitely many poles, where and is a differential polynomial in with degree and rational functions as its coefficients, is a non-zero rational function and is a non-constant polynomial, then has the form and where is a rational function and is a polynomial with With this in hand, we prove if is a transcendental entire function, is a polynomial of degree , then assumes every complex number infinitely many times, except a possible value . On the other hand, if assumes the complex value finitely many times, then and .

A method for the computation of inverses of a given matrix is derived, based on the full-rank decomposition of an appropriate matrix . As a corollary, a new method considering the advantages of full-rank decomposition is developed. It is then specialized to the set of polynomial matrices. Therefore, an algorithm for efficient symbolic computation of inverses of a polynomial matrix is proposed. An additional diagonal matrix yields to avoiding the computation of entries containing square roots of polynomials, therefore increasing the algorithm’s performances. Some implementation details and comparative processing times to other similar methods are provided, illustrating the algorithm’s efficiency.

This paper describes a method for extracting a lower degree polynomial embedded in a pair of polynomials. The method is then successfully applied to reconstruct an original quartic equation, which was transformed into two quartic equations through a quadratic Tschirnhaus transformation. We solve some numerical examples which illustrate the usefulness of the method.

We establish an explicit formula for the integral of the product of several Appell polynomials.

Submitted by Fátima Piedade (fpiedade@sa.isel.pt) on 2016-04-15T16:11:26Z No. of bitstreams: 1 Spectral and weak polynomial completeness for the product of nonsingular matrices.pdf: 373748 bytes, checksum: d82b3db796975209236850499e9d9190 (MD5) Made available in DSpace on 2016-04-15T16:11:26Z (GMT). No. of bitstreams: 1 Spectral and weak polynomial completeness for the product of nonsingular matrices.pdf: 373748 bytes, checksum: d82b3db796975209236850499e9d9190 (MD5) Previous issue date: 2015-10-03

For any pattern of the specified entries, an explicit finite list of polynomial inequalities in the specified entries is given that characterizes the completability of a partial positive matrix to a TP matrix. Each polynomial happens to be a monomial, and an algorithm, whose input is the pattern, is given for finding the polynomials. The method uses some new partial orders on matrices and the logarithmic method to reduce the question to determining the generators of a certain finitely generated, pointed cone.

In this paper, we discuss α-type polynomial sets and also generalized α -type polynomial sets of type zero in two variables. Some properties of certain polynomials have also been shown, in support of α-type zero in two variables.

Our goal is to deal with a family of quasi-symmetric semiclassical orthogonal polynomial sequences of class two through the study of the differential functional equation fulfilled by its corresponding regular form. Up to a linear transformation, we determine all polynomial sequences of this family. The recurrence coefficients and integral representations are established.

A matrix over a field is minimal if for every matrix that commutes with , the centralizer of is a subset of the centralizer of . In this paper, we study the minimal matrices over the field with two elements. We characterize the minimal matrices with their minimal polynomial of the form , where is an irreducible polynomial and . We also characterize all minimal matrices with spectrum in .

This paper proposes some sufficient conditions based on the computation of polynomial and piecewise polynomial storage functions for -gain analysis of discrete-time piecewise affine or piecewise polynomial systems. The computation of such storage functions is performed by means of convex optimisation techniques via the sum-of-squares decomposition of multivariate polynomials.

We show that the property of being a (weakly) admissible mesh for multivariate polynomials is preserved by small perturbations on real and complex Markov compacts. Applications are given to smooth transformations of polynomial meshes and to polynomial interpolation.

We consider the occurrence of unimodular eigenvalues for palindromic eigenvalue problems associated with the matrix polynomial where A i *= A n − i with M * ≡ M T, M H or . From the properties of palindromic eigenvalues and their characteristic polynomials, we show that eigenvalues are not generically excluded from the unit circle, thus occurring quite often, except for the complex transpose case when P n is complex and M * ≡ M T. This behaviour is observed in numerical simulations and has important implications on several applications such as the vibration of fast trains, surface acoustic wave filters, stability of time-delay systems and crack modelling.

Let k[x 1,…, x n ] be the polynomial algebra over a field k. We describe polynomial graded subalgebras of k[x 1,…, x n ], containing , where p 1, ⋅, p n are prime numbers.

Low-rank updated matrices are of crucial importance in many applications. Recently the relationship between the characteristic polynomial and the spectrum of a given matrix A and those of its specially structured rank-k updated matrix has become a hot topic. Many researchers consider the eigenproblem of a matrix of the form under the assumption that the columns of U k or V k are right or left eigenvectors corresponding to some non-defective eigenvalues of A. However, in many low-rank updated eigenproblems, this assumption does not hold. In this article, we investigate the low-rank updated eigenproblem without such a constraint; that is, our low-rank updates U k , V k ∈ ℂ n×k can be any complex matrices such that is a rank-k matrix. We first consider the relationship between the characteristic polynomial of a diagonalizable matrix and that of its rank-k update. We then focus on two special cases of k = 1 and k = 2. Moreover, the spectral relationship between a diagonalizable matrix and its rank-1 and rank...

Let Λ be the four subspace algebra. We show that for any Λ-module M there exists an algorithm (up to the problem of finding roots of the so-called characteristic polynomial of M) with relatively low polynomial complexity of determining multiplicities of all direct summands of M. Moreover, we give a fully algorithmic criterion for deciding if two Λ-modules M and N are isomorphic.

In this article, we study the minimal polynomials of parametric matrices. Using the concept of (comprehensive) Grobner systems for parametric ideals, we introduce the notion of a minimal polynomial system for a parametric matrix, i.e. we decompose the space of parameters into a finite set of cells and for each cell we give the corresponding minimal polynomial of the matrix. We also present an algorithm for computing a minimal polynomial system for a given parametric matrix.

In the theory of the separation of roots of algebraic equations, the well-known Routh–Hurwitz–Fujiwara theorem enables us to separate the complex roots of a polynomial with complex coefficients in terms of the inertia of a related Hermitian matrix. Unfortunately, it fails if the polynomial has a nontrivial factor which is symmetric with respect to the imaginary axis. In this article, we present a method to overcome the fault and formulate the inertia of a scalar polynomial with complex coefficients in terms of the inertia of several Hermitian matrices based on a factorization of a monic symmetric polynomial into products of monic symmetric polynomials with only simple roots in the complex plane and on computing the inertia of each factor by means of a subtle perturbation.

We derive two ovals of Cassini, each containing all the zeros of a polynomial. The computational cost to obtain these ovals is similar to that of the Brauer set for the companion matrix of a polynomial, although they are frequently smaller. Their derivation is based on the Gershgorin set for an appropriate polynomial of the companion matrix.

An orthogonal polynomial sequence with respect to a regular form (linear functional) u is said to be semi-classical if there exist a monic polynomial Φ and a polynomial Ψ, with deg Ψ≥1, such that (Φ u)′+Ψ u=0. Recently, all semi-classical monic orthogonal polynomial sequences of class one satisfying a three-term recurrence relation with β n =(−1) n β0, n≥0, β0∈ℂ∖{0} have been determined (see [B. Bouras and A. Alaya, A large family of semi-classical polynomials of class one, Integral Transforms Spec. Funct. 18 (2007), pp. 913–931]). In this paper, the sequences of the above family such that their corresponding Stieltjes function S(u)(z)=−∑ n≥0⟨ u, x n ⟩/z n+1 satisfies a quadratic relation of the form BS 2(u)+CS(u)+D=0, where B, C, D are polynomials, are described.

Let k be a field of characteristic zero. We say that a polynomial endomorphism F: k n → k n has a gap after component m if the homogeneous components F (j) are equal to 0 for j = m + 1,…, n(m + 1) +1. We prove the equivalence of the Jacobian conjecture and the following condition: if polynomial automorphism has a gap after component m, then the sum forms an automorphism.

Let F q be a finite field of characteristic two, S be a nonsingular non-alternate symmetric matrix over F q and Ps n (F q , S) be the associated pseudo-symplectic group. Let Ps n (F q , S) act linearly on the polynomial ring F q [x 1,…, x n ]. In this note, we find an explicit set of generators of the ring of invariants of Ps n (F q , S) for n = 2, 4 and 2ν +1. In particular, the results assert that the ring of invariants of Ps 4(F q , S) is not a polynomial algebra but is an example of hypersurface and the ring of invariants of Ps 2ν+1(F q , S) is a complete intersection.

We establish properties of and propose a conjecture concerning ∑ m (S(x+m))2, where S is a piecewise polynomial cardinal spline in .

We show that using the constrained Rayleigh quotient method to find the eigenvalues of matrix polynomials in different polynomial bases is equivalent to applying the Newton method to certain functions. We find those functions explicitly for a variety of polynomial bases including monomial, orthogonal, Newton, Lagrange and Bernstein bases. In order to do so, we provide explicit symbolic formulas for the right and left eigenvectors of the generalized companion matrix pencils for matrix polynomials expressed in those bases. Using the properties of the Newton basis, we also find two different formulas for the companion matrix pencil corresponding to the Hermite interpolation. We give pairs of explicit LU factors associated with these pencils. Additionally, we explicitly find the right and left eigenvectors for each of these pencils.

In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices.

Using Hermite's formulation of polynomial stability conditions, static output feedback (SOF) controller design can be formulated as a polynomial matrix inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming problem that can be solved (locally) with PENNON, an implementation of a penalty and augmented Lagrangian method. Typically, Hermite SOF PMI problems are badly scaled and experiments reveal that this has a negative impact on the overall performance of the solver. In this note we recall the algebraic interpretation of Hermite's quadratic form as a particular Bezoutian and we use results on polynomial interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an alternative to the conventional power basis. Numerical experiments on benchmark problem instances show the improvement brought by the approach, in terms of problem scaling, number of iterations and convergence behaviour of PENNON.

A general quadratic decomposition of polynomial sequences is carried out. This generalizes the works of L. Carlitz, Boll. Un. Mat. Ital. 16 (1961), pp. 386–390; L.M. Chihara and T.S. Chihara, J. Math. Anal. Appl. 126 (1987), pp. 275–291; T.S. Chihara, Boll. Un. Mat. Ital. 19 (1964), pp. 451–459 and that of P. Maroni, Rev. Math. Pura ed Appl. (6) (1990), pp. 19–53. We deal with the transformation , where instead of that where p and q are zero. We give the general quadratic decomposition of a monic polynomial sequence. The characterization of the cases where is orthogonal are given.

This article is concerned with robust stability of uncertain discrete-time linear systems. The matrix defining the linear system (system matrix) is assumed to depend affinely on a set of time-invariant unknown parameters lying on a known polytope. Robust stability is investigated by checking whether a certain integer power κ of the uncertain system matrix has spectral norm less than one. This peculiar stability test is shown to be equivalent to the positivity analysis of a homogeneous symmetric matrix polynomial with precisely known coefficients and degree indexed by κ. A unique feature is that no extra variables need to be added to the problems being solved. Numerical experiments reveal that the value of κ needed to test robust stability is mostly independent of the system dimension but grows sharply as the eigenvalues of the uncertain system approach the unit circle. By identifying the proposed stability test with a particular choice of a parameter-dependent Lyapunov function, extra variables can be int...

Let v be a valuation of a field K having value group ℤ. It is known that a polynomial x n + a n−1 x n−1 + … +a 0 satisfying with v(a 0) coprime to n, is irreducible over K. Such a polynomial is referred to as an Eisenstein–Dumas polynomial with respect to v. In this article, we give necessary and sufficient conditions so that some translate g(x + a) of a given polynomial g(x) belonging to K[x] is an Eisenstein–Dumas polynomial with respect to v. In fact, an analogous problem is dealt with for a wider class of polynomials, viz. Generalized Schonemann polynomials with coefficients over valued fields of arbitrary rank.

A new analytical method for solving an initial value problem (IVP) for the system of crystal optics with polynomial data and a polynomial inhomogeneous term is suggested. The found solution of the IVP is a polynomial. Theoretical and computational analysis of polynomial solutions and their comparison with non-polynomial solutions corresponding to smooth data are given. The applicability of polynomial solutions to physical processes is discussed. An implementation of this method has been made by symbolic computations in Maple 10.