Quadratic Matrix Polynomial Research Articles

Stability Analysis of Matrices and Single-Parameter Matrix Families on Special Regions

This paper proposes an efficient method for determining the D-stability of matrices using Kelley’s cutting-plane method. The study examines the D-stability of matrices in symmetric regions of the complex plane, defined by quadratic matrix inequalities (QMI) and polynomial functions. Using Kelley’s cutting-plane method, we present an iterative algorithm for determining the D-stability of a given matrix. Further, the robustness of single-parameter matrix families is analyzed, and a method is proposed for considering their D-stability. Through examples, we indicate the possible applicability of the proposed approach in addressing stability problems in linear systems.

Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models

AbstractStandard multiparameter eigenvalue problems (MEPs) are systems of linear ‐parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time‐invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.

Matrices whose eigenvalues are those of a quadratic matrix polynomial

It is shown how the eigenvalues of sparse matrices, composed of a small number of nonzero block rows and columns, can be obtained from quadratic matrix polynomials with coefficients of a size determined by the number of nonzero rows and columns. This can present advantages, such as revealing eigenvalue properties without the need to compute them, better upper bounds for eigenvalues than those derived directly from the matrix with a smaller computational effort, and lower bounds that cannot otherwise directly be derived from the given matrix. The focus will be on upper and lower eigenvalue bounds, which are useful when eigenvalues are computed iteratively or when an application requires them to be in a particular region of the complex plane.

A note on the computation of invariant pairs of quadratic matrix polynomials

An algorithm is proposed for the computation of invariant pairs of quadratic matrix polynomials associated with the eigenvalues inside a given circle in the complex plane. The invariant pair of interest is derived from an SVD of the spectral projector of a linearization of the quadratic matrix polynomial which is computed from two matrix integrals by a spectral dichotomy-type approach. The performance of the method is illustrated by numerical experiments.

Theoretical and computational properties of semi-infinite quasi-Toeplitz M-matrices

A quasi-Toeplitz matrix is a semi-infinite matrix of the form A=T(a)+E, where T(a) is a Toeplitz matrix with entries (T(a))i,j=aj−i, for aj−i∈C, i,j≥1 and E is a compact correction. Quasi-Toeplitz M-matrices are encountered in the study of quadratic matrix equations arising in the analysis of a 2-dimensional Quasi-Birth-Death (QBD) stochastic process. We investigate properties of such matrices and provide conditions under which a quasi-Toeplitz matrix is an M-matrix. We show that under a mild and easy-to-check condition, an invertible quasi-Toeplitz M-matrix has a unique square root that is an M-matrix possessing quasi-Toeplitz structure. The quasi-Toeplitz structure of the square root of M-matrices provides inspirations for proving spectral properties of the quadratic matrix polynomial L(λ)=λ2A1+λA0+A−1 having quasi-Toeplitz coefficients where A1,A−1≥0 and −A0 in addition is an M-matrix. Some issues concerning the computation of square root of quasi-Toeplitz M-matrices are discussed and numerical experiments are performed.

Locating eigenvalues of quadratic matrix polynomials

The location of the roots of a quadratic scalar polynomial may be identified from its coefficients. This paper shows that when the coefficients of the polynomial are square matrices, then appropriate generalizations of some of these statements hold for the eigenvalues of the resulting quadratic matrix polynomial. The locations of the eigenvalues are described with respect to the imaginary axis, the unit circle or the real line. The results lead to upper bounds on some important distances associated with quadratic matrix polynomials. The principal tool used is an eigenvalue localization technique using block Geršgorin sets applied to certain linearizations of these polynomials that come from well known vector spaces. New bounds on the eigenvalues of the matrix polynomial arising from these localizations are also presented.

Convergence analysis of vector extended locally optimal block preconditioned extended conjugate gradient method for computing extreme eigenvalues

AbstractThis article is concerned with the convergence analysis of an extension of the locally optimal preconditioned conjugate gradient method for the extreme eigenvalue of a Hermitian matrix polynomial that possesses some extended form of Rayleigh quotient. This work is a generalization of the analysis by Ovtchinnikov (SIAM J Numer Anal. 2008;46(5):2567–92.). As instances, the algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate. Also, numerical examples are given to illustrate the convergence behavior.

Quadratic realizability of palindromic matrix polynomials: the real case

Let be a list consisting of structural data for a matrix polynomial; here is a sublist consisting of powers of irreducible (monic) scalar polynomials over the field , and is a sublist of nonnegative integers. For an arbitrary such , we give easy-to-check necessary and sufficient conditions for to be the list of elementary divisors and minimal indices of some real T-palindromic quadratic matrix polynomial. For a list satisfying these conditions, we show how to explicitly build a real T-palindromic quadratic matrix polynomial having as its structural data; that is, we provide a T-palindromic quadratic realization of over . A significant feature of our construction differentiates it from related work in the literature; the realizations constructed here are direct sums of blocks with low bandwidth, that transparently display the spectral and singular structural data in the original list .

Eigenvalue embedding problem for quadratic regular matrix polynomials with symmetry structures

In this paper, we consider structure-preserving eigenvalue embedding problem (SEEP) for quadratic regular matrix polynomials with symmetry structures. First, we determine perturbations of a quadratic matrix polynomial, unstructured or structured, such that the perturbed polynomials reproduce a desired invariant pair while maintaining the invariance of another invariant pair of the unperturbed polynomial. If the latter is unknown, it is referred to as no spillover perturbation. Then we use these results for solving the SEEP for structured quadratic matrix polynomials that include: symmetric, Hermitian, ⋆-even and ⋆-odd quadratic matrix polynomials. Finally, we show that the obtained analytical expressions of perturbations can realize existing results for structured polynomials that arise in real-world applications, as special cases.

Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials

Abstract An inexact variant of inverse subspace iteration is used to find a small invariant pair of a large quadratic matrix polynomial. It is shown that linear convergence is preserved provided the inner iteration is performed with increasing accuracy. A preconditioned block GMRES solver is employed as inner iteration. The preconditioner uses the strategy of “tuning” which prevents the inner iteration from increasing and therefore results in a substantial saving in costs. The accuracy of the computed invariant pair can be improved by the addition of a post-processing step involving very few iterations of Newton’s method. The effectiveness of the proposed approach is demonstrated by numerical experiments.

Spectral theory for self-adjoint quadratic eigenvalue problems - a review

Many physical problems require the spectral analysis of quadratic matrix polynomials $M\lambda^2+D\lambda +K$, $\lambda \in \mathbb{C}$, with $n \times n$ Hermitian matrix coefficients, $M,\;D,\;K$. In this largely expository paper, we present and discuss canonical forms for these polynomials under the action of both congruence and similarity transformations of a linearization and also $\lambda$-dependent unitary similarity transformations of the polynomial itself. Canonical structures for these processes are clarified, with no restrictions on eigenvalue multiplicities. Thus, we bring together two lines of attack: (a) analytic via direct reduction of the $n \times n$ system itself by $\lambda$-dependent unitary similarity and (b) algebraic via reduction of $2n \times 2n$ symmetric linearizations of the system by either congruence (Section 4) or similarity (Sections 5 and 6) transformations which are independent of the parameter $\lambda$. Some new results are brought to light in the process. Complete descriptions of associated canonical structures (over $\mathbb{R}$ and over $\mathbb{C}$) are provided -- including the two cases of real symmetric coefficients and complex Hermitian coefficients. These canonical structures include the so-called sign characteristic. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics are studied here and connections between them are clarified. In particular, we consider which of the linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory (Sections 7 and 9).

Observer-based output reachable set synthesis for periodic piecewise time-varying systems

In this paper, the output reachable set synthesis problem for periodic piecewise time-varying systems has been addressed for the first time. Generalized periodic time-varying conditions of the closed-loop system to guarantee the system output bounded by a collection of finite ellipsoids are developed. The decoupling of time-varying observer and controller gains is achieved by a technical lemma . Moreover, by using time-varying matrix functions in multiple linear-interpolative forms and the relaxed quadratic matrix polynomial definiteness results, sufficient condition is developed to determine the periodic time-varying observer and controller with reduced conservatism. A tractable iterative algorithm is proposed to obtain the periodic time-varying observer and controller gains. A numerical example is presented to illustrate the validity of proposed methods.

Solution of the symmetric band partial inverse eigenvalue problem for the damped mass spring system

ABSTRACT 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 applications include symmetry, band (tridiagonal, diagonal, pentagonal) etc. The construction of the structured matrix polynomial is the most difficult aspect of this problem and the research on structured inverse eigenvalue problem is rare. In this paper, the symmetric band partial quadratic inverse eigenvalue problem (SBPQIEP) for the damped mass spring system is considered. This problem concerns in finding the symmetric band matrices , and C with bandwidth p from m ( ) prescribed eigenpairs so that the corresponding quadratic matrix polynomial has the given eigenpairs as its eigenvalues and eigenvectors. In general, SBPQIEP is very hard to solve due to the additional band structure constraint. We propose a novel method, based on the matrix-vectorization and Kronecker product of matrices for solving this problem. Furthermore, explicit expressions for general solutions are presented. Numerical experiments on a spring mass problem are presented to illustrate the applicability and the practical usefulness of the proposed method.

Prime decomposition of quadratic matrix polynomials

<abstract> We study the prime decomposition of a quadratic monic matrix polynomial. From the prime decomposition of a quadratic matrix polynomial, we obtain a formula of the general solution to the corresponding second-order differential equation. For a quadratic matrix polynomial with pairwise commuting coefficients, we get a sufficient condition for the existence of a prime decomposition. </abstract>

Note on Finding an Optimal Deflation for Quadratic Matrix Polynomials

Note on Finding an Optimal Deflation for Quadratic Matrix Polynomials

Computing the distance to continuous-time instability of quadratic matrix polynomials

A bisection method is used to compute lower and upper bounds on the distance from a quadratic matrix polynomial to the set of quadratic matrix polynomials having an eigenvalue on the imaginary axis. Each bisection step requires to check whether an even quadratic matrix polynomial has a purely imaginary eigenvalue. First, an upper bound is obtained using Frobenius-type linearizations. It takes into account rounding errors but does not use the even structure. Then, lower and upper bounds are obtained by reducing the quadratic matrix polynomial to a linear palindromic pencil. The bounds obtained this way also take into account rounding errors. Numerical illustrations are presented.

Sur la distance à l'instabilité de polynômes matriciels quadratiques

A bisection method is developed for computing the distance to instability of quadratic matrix polynomials. The computation takes rounding errors into account.

Quadratic realizability of palindromic matrix polynomials

Let L=(L1,L2) be a list consisting of a sublist L1 of powers of irreducible (monic) scalar polynomials over an algebraically closed field F, and a sublist L2 of nonnegative integers. For an arbitrary such list L, we give easily verifiable necessary and sufficient conditions for L to be the list of elementary divisors and minimal indices of some T-palindromic quadratic matrix polynomial with entries in the field F. For L satisfying these conditions, we show how to explicitly construct a T-palindromic quadratic matrix polynomial having L as its structural data; that is, we provide a T-palindromic quadratic realization ofL. Our construction of T-palindromic realizations is accomplished by taking a direct sum of low bandwidth T-palindromic blocks, closely resembling the Kronecker canonical form of matrix pencils. An immediate consequence of our in-depth study of the structure of T-palindromic quadratic polynomials is that all even grade T-palindromic matrix polynomials have a T-palindromic strong quadratification. Finally, using a particular Möbius transformation, we show how all of our results can be easily extended to quadratic matrix polynomials with T-even structure.

Decoupling of second-order linear systems by isospectral transformation

We consider the class of real second-order linear dynamical systems that admit real diagonal forms with the same eigenvalues and partial multiplicities. The nonzero leading coefficient is allowed to be singular, and the associated quadratic matrix polynomial is assumed to be regular. We present a method and algorithm for converting any such n-dimensional system into a set of n mutually independent second-, first-, and zeroth-order equations. The solutions of these two systems are related by a real, time-dependent, and nonlinear n-dimensional transformation. Explicit formulas for computing the $$2n \times 2n$$ real and time-invariant equivalence transformation that enables this conversion are provided. This paper constitutes a complete solution to the problem of diagonalizing a second-order linear system while preserving its associated Jordan canonical form.

On the Geršgorin-type localizations for nonlinear eigenvalue problems

Since nonlinear eigenvalue problems appear in many applications, the research on their proper treatment has drawn a lot of attention lately. Therefore, there is a need to develop computationally inexpensive ways to localize eigenvalues of nonlinear matrix-valued functions in the complex plane, especially eigenvalues of quadratic matrix polynomials. Recently, few variants of the Geršgorin localization set for more general eigenvalue problems, matrix pencils and nonlinear ones, were developed and investigated. Here, we introduce a more general approach to Geršgorin-type sets for nonlinear case using diagonal dominance, prove some properties of such sets and show how they perform on several problems in engineering.