The eigenvalues and eigenvectors of tridiagonal Toeplitz matrices are known in closed form. This property is in the first part of the paper used to investigate the sensitivity of the spectrum. Explicit expressions for the structured distance to the closest normal matrix, the departure from normality, and the "-pseudospectrum are derived. The second part of the paper discusses applications of the theory to inverse eigenvalue problems, the construction of Chebyshev polynomial-based Krylov subspace bases, and Tikhonov regularization.TRIDIAGONAL TOEPLITZ MATRICES 303 detail in [13][14][15]. Our interest in tridiagonal Toeplitz matrices stems from the possibility of deriving explicit formulas for quantities of interest and from the many applications of these matrices.This paper is organized as follows. The eigenvalue sensitivity is investigated in Sections 2-6. Numerical illustrations also are provided. The latter part of this paper describes a few applications that are believed to be new. We consider an inverse eigenvalue problem in Section 7, where we also introduce a minimization problem, whose solution is a trapezoidal tridiagonal Toeplitz matrix. The latter matrices can be applied as regularization matrices in Tikhonov regularization. This application is described in Section 8. Section 9 is concerned with the construction of nonorthogonal Krylov subspace bases based on the recursion formulas for suitably chosen translated and scaled Chebyshev polynomials. The use of such bases in Krylov subspace methods for the solution of large linear systems of equations or for the computation of a few eigenvalues of a large matrix is attractive in parallel computing environments that do not allow efficient execution of the Arnoldi process for generating an orthonormal basis; see [16][17][18] for discussions. We describe how tridiagonal Toeplitz matrices can be applied to determine a suitable interval on which the translated and scaled Chebyshev polynomials are required to be orthogonal. Concluding remarks can be found in Section 10.

We continue the study started in [Noschese and Pasquini, Eigenvalue condition numbers: zero-structured versus traditional. J. Comput. Appl. Math. 185 (2006) 174-189] concerning the sensitivity of simple eigenvalues of a matrix A to perturbations in A that belong to a chosen subspace of matrices. In [Noschese and Pasquini, Eigenvalue condition numbers: zero-structured versus traditional. J. Comput. Appl. Math. 185 (2006) 174-189] the zero-structured perturbations have been considered. Here we focus on patterned perturbations, and the cases of the Toeplitz and of the Hankel matrices are investigated in detail. Useful expressions of the absolute patterned condition number of the eigenvalue lambda and of the analogue of the matrix yx(H), which leads to the traditional condition number of)., are given. MATLAB codes are defined to compare traditional, zero-structured and patterned condition numbers. A report on significant numerical tests is included. (C) 2006 Elsevier B.V. All rights reserved

We discuss questions of eigenvalue conditioning. We study in some depth relationships between the classical theory of conditioning and the theory of the zero-structured conditioning, and we derive from the existing theory formulae for the mathematical objects involved. Then an algorithm to compare the zero-structured individual condition numbers of a set of simple eigenvalues with the traditional ones is presented. Numerical tests are reported to highlight how the algorithm provides interesting information about eigenvalue sensitivity when the perturbations in the matrix have an arbitrarily assigned zero-structure. Patterned matrices (Toeplitz and Hankel) will be investigated in a forthcoming paper (Eigenvalue patterned condition numbers: Toeplitz and Hankel cases, Tech. Rep. 3, Mathematics Department, University of Rome ' La Sapienza', 2005.). (c) 2005 Elsevier B.V. All rights reserved

Spectral properties of normal (2k + 1)- banded Toeplitz matrices of order n, with k <= left perpendicularn/ 2right perpendicular, are described. Formulas for the distance of (2k + 1)-banded Toeplitz matrices to the algebraic variety of similarly structured normal matrices are presented

Abstract. This paper is concerned with the solution of large-scale linear discrete ill-posed problems with error-contaminated data. Tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. This matrix should be chosen to promote the recovery of known important features of the desired solution, such as smoothness and monotonicity. We describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem. The constructed regularization matrix is the closest matrix in the Frobenius norm with a prescribed null space to a given matrix. Numerical examples illustrate the performance of the regularization matrices so obtained.

One of the properties of interest in the analysis of networks is global communicability, i.e., how easy or difficult it is, generally, to reach nodes from other nodes by following edges. Different global communicability measures provide quantitative assessments of this property, emphasizing different aspects of the problem. This paper investigates the sensitivity of global measures of communicability to local changes. In particular, for directed, weighted networks, we study how different global measures of communicability change when the weight of a single edge is changed; or, in the unweighted case, when an edge is added or removed. The measures we study include the total network communicability, based on the matrix exponential of the adjacency matrix, and the Perron network communicability, defined in terms of the Perron root of the adjacency matrix and the associated left and right eigenvectors. Finding what local changes lead to the largest changes in global communicability has many potential applications, including assessing the resilience of a system to failure or attack, guidance for incremental system improvements, and studying the sensitivity of global communicability measures to errors in the network connection data.

The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [GO11] for the unstructured case, but their extension to structured pseudospectra and analysis presents several difficulties. Natural generalizations of the algorithms, allowing to draw significant sections of the structured pseudospectra in proximity of extremal points are also discussed. Since no algorithms are available in the literature to draw such structured pseudospectra, the approach we present seems promising to extend existing software tools (Eigtool [Wri02], Seigtool [KKK10]) to structured pseudospectra representation for Toeplitz matrices. We discuss local convergence properties of the algorithms and show some applications to a few illustrative examples.2010 Mathematics Subject Classification. 65F15, 65L07.

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small-matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra