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
A general method for approximating polynomial solutions of second-order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized Bessel polynomials, and an algorithm is derived for simultaneously finding their zeros. Then a comparison with several alternative algorithms is carried out. It shows that the computational problem of approximating the zeros of the generalized Bessel polynomials is not an easy matter at all and that the only algorithm able to give an accurate solution seems to be the one presented in this paper. Subject Classification (1991): 65F15, 65F30, 65H20
Mathematics
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