On the Numerical Range of Operator Polynomials

An upper bound for the number of components of the numerical range of matrix polynomials is established. We also establish a necessary condition and a sufficient condition for the connectedness of the numerical range of a quadratic selfadjoint matrix polynomial. The boundary of the numerical range of linear matrix polynomials is also considered.     

The block numerical range of matrix polynomials

Block numerical ranges of matrix polynomials, especially the quadratic numerical range, are considered. The main results concern spectral inclusion, boundedness of the block numerical range, an estimate of the resolvent in terms of the quadratic numerical range, geometrical properties of the quadratic numerical range, and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions. As an application, we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials.     

On the connectedness of numerical range of matrix polynomials

An investigation on nonconnectedness of numerical range for monic matrix polynomials L(λ) is undertaking here. The distribution of eigenvalues of L(λ) to the components of numerical range and some other algebraic properties are also presented.     

Some results on numerical ranges and factorizations of matrix polynomials

Some new factorization theorems for monic matrix polynomials are obtained. These theorems are based on the numerical range having the number of connected components equal to the degree of the polynomial. For second degree polynomials, sufficient conditions are given for the numerical range to have two connected components.     

Birkhoff-James approximate orthogonality sets and numerical ranges

In this paper, the notion of Birkhoff-James approximate orthogonality sets is introduced for rectangular matrices and matrix polynomials. The proposed definition yields a natural generalization of standard numerical range and q-numerical range (and also of recent extensions), sharing with them several geometric properties.     

The stable computation of formal orthogonal polynomials

For many applications — such as the look-ahead variants of the Lanczos algorithm — a sequence of formal (block-)orthogonal polynomials is required. Usually, one generates such a sequence by taking suitable polynomial combinations of a pair of basis polynomials. These basis polynomials are determined by a look-ahead generalization of the classical three term recurrence, where the polynomial coefficients are obtained by solving a small system of linear equations. In finite precision arithmetic, the numerical orthogonality of the polynomials depends on a good choice of the size of the small systems; this size is usually controlled by a heuristic argument such as the condition number of the small matrix of coefficients. However, quite often it happens that orthogonality gets lost.We present a new variant of the Cabay-Meleshko algorithm for numerically computing pairs of basis polynomials, where the numerical orthogonality is explicitly monitored with the help of stability parameters. A corresponding error analysis is given. Our stability parameter is shown to reflect the condition number of the underlying Hankel matrix of moments. This enables us to prove the weak and strong stability of our method, provided that the corresponding Hankel matrix is well-conditioned.This work was partially supported by the HCM project ROLLS, under contract CHRX-CT93-0416.     

Structured eigenvalue condition numbers and linearizations for matrix polynomials

This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations. For all structures under consideration, we show that this cannot happen if the matrix polynomial is well scaled: there is always a structured linearization for which the structured eigenvalue condition number does not differ much. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.     

On a Compression of Normal Matrix Polynomials

In this article, we study a compression of normal matrices and matrix polynomials with respect to a given vector and its orthogonal complement. The numerical range of this compression satisfies special boundary properties, which are investigated in detail. The characteristic polynomial of the compression is also considered.     

Matrix polynomials with spectral radius equal to the numerical radius

In this paper we study a class of matrix polynomials with the property that spectral radius and numerical radius coincide. Special attention is paid to the spectrum on the boundary of the numerical range.     

On the Stability Radius of Matrix Polynomials

The stability radius of a matrix polynomial P ( λ) relative to an open region Ωof the complex plane and its relation to the numerical range of P ( λ) are investigated. Using an expression of the stability radius in terms of λon the boundary of Ωand ‖P ( λ) -1 ‖2 , a lower bound is obtained. This bound for the stability radius involves the distances of Ωto the connected components of the numerical range of P ( λ) and can be applied in conjunction with polygonal approximations of the numerical range. The special case of hyperbolic matrix polynomials is also considered.     

Fiedler companion linearizations for rectangular matrix polynomials

The development of new classes of linearizations of square matrix polynomials that generalize the classical first and second Frobenius companion forms has attracted much attention in the last decade. Research in this area has two main goals: finding linearizations that retain whatever structure the original polynomial might possess, and improving properties that are essential for accurate numerical computation, such as eigenvalue condition numbers and backward errors. However, all recent progress on linearizations has been restricted to square matrix polynomials. Since rectangular polynomials arise in many applications, it is natural to investigate if the new classes of linearizations can be extended to rectangular polynomials. In this paper, the family of Fiedler linearizations is extended from square to rectangular matrix polynomials, and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials. Fiedler linearizations are one of the most important classes of linearizations introduced in recent years, but their generalization to rectangular polynomials is nontrivial, and requires a completely different approach to the one used in the square case. To the best of our knowledge, this is the first class of new linearizations that has been generalized to rectangular polynomials.     

On condition numbers of polynomial eigenvalue problems

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented.     

Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification

The spectral properties of Hermitian matrix polynomials with real eigenvalues have been extensively studied, through classes such as the definite or definitizable pencils, definite, hyperbolic, or quasihyperbolic matrix polynomials, and overdamped or gyroscopically stabilized quadratics. We give a unified treatment of these and related classes that uses the eigenvalue type (or sign characteristic) as a common thread. Equivalent conditions are given for each class in a consistent format. We show that these classes form a hierarchy, all of which are contained in the new class of quasidefinite matrix polynomials. As well as collecting and unifying existing results, we make several new contributions. We propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line. By analyzing their effect on eigenvalue type, we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient, which is important for treating definite and quasidefinite polynomials. We also give a sufficient and necessary condition for a quasihyperbolic matrix polynomial to be strictly isospectral to a real diagonal quasihyperbolic matrix polynomial of the same degree, and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial, thereby identifying an important new class of diagonalizable matrix polynomials.     

Theoretical analysis of numerical integration in Galerkin meshless methods

In this paper, we study effects of numerical integration on Galerkin meshless methods for solving elliptic partial differential equations with Neumann boundary conditions. The shape functions used in the meshless methods reproduce linear polynomials. The numerical integration rules are required to satisfy the so-called zero row sum condition of stiffness matrix, which is also used by Babuška et al. (Int. J. Numer. Methods Eng. 76:1434–1470, 2008). But the analysis presented there relies on a certain property of the approximation space, which is difficult to verify. The analysis in this paper does not require this property. Moreover, the Lagrange multiplier technique was used to handle the pure Neumann condition. We have also identified specific numerical schemes, diagonal elements correction and background mesh integration, that satisfy the zero row sum condition. The numerical experiments are carried out to verify the theoretical results and test the accuracy of the algorithms.     

On backward errors of structured polynomial eigenproblems solved by structure preserving linearizations

We derive explicit computable expressions of structured backward errors of approximate eigenelements of structured matrix polynomials including symmetric, skew-symmetric, Hermitian, skew-Hermitian, even and odd polynomials. We determine minimal structured perturbations for which approximate eigenelements are exact eigenelements of the perturbed polynomials. We also analyze structured pseudospectra of a structured matrix polynomial and establish a partial equality between unstructured and structured pseudospectra. Finally, we analyze the effect of structure preserving linearizations of structured matrix polynomials on the structured backward errors of approximate eigenelements and show that structure preserving linearizations which minimize structured condition numbers of eigenvalues also minimize the structured backward errors of approximate eigenelements.     

Matrix polynomials and the index problem for elliptic systems

The main new results of this paper concern the formulation of algebraic conditions for the Fredholm property of elliptic systems of P.D.E.'s with boundary values, which are equivalent to the Lopatinskii condition. The Lopatinskii condition is reformulated in a new algebraic form (based on matrix polynomials) which is then used to study the existence of homotopies of elliptic boundary value problems. The paper also contains an exposition of the relevant parts of the theory of matrix polynomials and the theory of elliptic systems of P.D.E.'s.

     

Do Orthogonal Polynomials Dream of Symmetric Curves?

The complex or non-Hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion. The problem of the limit zero distribution for the non-Hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation. Thus, the second motivation of this paper is to discuss some “mysterious” configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while others are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.     

Generalized global conjugate gradient squared algorithm

In this paper, a generalized global conjugate gradient squared method for solving nonsymmetric linear systems with multiple right-hand sides is presented. The method can be derived by using products of two nearby global BiCG polynomials and formal orthogonal polynomials, of which global CGS and global BiCGSTAB are just particular cases. We also show to apply the method for solving the Sylvester matrix equation. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.     

A New Approach for Solving a Class of Delay Fractional Partial Differential Equations

In this article, a new numerical approach has been proposed for solving a class of delay time-fractional partial differential equations. The approximate solutions of these equations are considered as linear combinations of Müntz–Legendre polynomials with unknown coefficients. Operational matrix of fractional differentiation is provided to accelerate computations of the proposed method. Using Padé approximation and two-sided Laplace transformations, the mentioned delay fractional partial differential equations will be transformed to a sequence of fractional partial differential equations without delay. The localization process is based on the space-time collocation in some appropriate points to reduce the fractional partial differential equations into the associated system of algebraic equations which can be solved by some robust iterative solvers. Some numerical examples are also given to confirm the accuracy of the presented numerical scheme. Our results approved decisive preference of the Müntz–Legendre polynomials with respect to the Legendre polynomials.     

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N⁸) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N^d+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009