On the sensitivity of Lanczos recursions to the spectrum
This paper is concerned with the inverse eigenvalue problem of finding a symmetric tridiagonal matrix \(T\) having a prespecified set \(\Lambda\) of eigenvalues and a prespecified first row \(e^T S\) of the orthogonal eigenvector matrix \(S\). In particular, the individual sensitivities of the diagonal and (positive) off-diagonal entries of \(T\) with respect to perturbations in \(\Lambda\) and \(S\) are investigated. For this purpose, the matrices \(T\) and \(S\) along with their perturbations are related to each other via a projected Sylvester equation. Linearizing this equation yields a first order perturbation expansion, which is employed to derive relative error bounds for the off-diagonal entries and absolute error bounds for the diagonal entries of \(T\). As an important application of the obtained results, stability estimates for the coefficients produced by the Lanczos process applied to a symmetric matrix are given. The relevance of these estimates is discussed in detail for discrete Sturm-Liouville problems.
scientific article; zbMATH DE number 2136674
scientific article