4 research outputs found
The structured distance to singularity of a symmetric tridiagonal Toeplitz matrix
This paper is concerned with the distance of a symmetric tridiagonal Toeplitz
matrix to the variety of similarly structured singular matrices, and with
determining the closest matrix to in this variety. Explicit formulas are
presented, that exploit the analysis of the sensitivity of the spectrum of
with respect to structure-preserving perturbations of its entries.Comment: 16 pages, 5 Figure
The Generalized Schur Algorithm and Some Applications
The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the Q R and L U factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur algorithm. We show that it helps to prove some theoretical properties of the R factor of the Q R factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based algorithm for computing the null-space of polynomial matrices