How to Find Matrix Modifications Keeping Essentially Unaltered a Selected Set of Eigenvalues

Linear Algebra Proceedings, CP5 in http://www.siam.org/meetings/la0

Eigenvalue patterned condition numbers: Toeplitz and Hankel cases

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

On the Nonnegative Solution of a Freud Three-Term Recurrence

This article deals with the sequence xi = {xi(n)}(n = 0, 1), ... defined by the three-term recurrence n = 4 xi(n)(xi(n - 1) + xi(n) + xi(n) (+ 1)), n = 1,2, ..., and by the initial conditions xi(0) = 0, xi(1) = Gamma(3/4)/Gamma(1/4). Owing both to connections between the xi(n)'s and orthonormal polynomials with respect to the weight function w:w(x) = exp(-x(4)) and to difficulties that arise when one attempts to compute its elements, the sequence xi has been studied by many authors. Properties xi have been shown and computational algorithms provided. In this paper we show further properties of xi. First we establish bounds for the departure of xi from the sequence to which it asymptotically converges. Then we prove that xi is an increasing sequence

Tridiagonal Toeplitz matrices: Properties and novel applications

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 E-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. Copyright (c) 2012 John Wiley & Sons, Ltd