Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems

We propose a verified computation method for partial eigenvalues of a Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a contour integral-type eigensolver, can reduce a given eigenproblem into a generalized eigenproblem of block Hankel matrices whose entries consist of complex moments. In this study, we evaluate all errors in computing the complex moments. We derive a truncation error bound of the quadrature. Then, we take numerical errors of the quadrature into account and rigorously enclose the entries of the block Hankel matrices. Each quadrature point gives rise to a linear system, and its structure enables us to develop an efficient technique to verify the approximate solution. Numerical experiments show that the proposed method outperforms a standard method and infer that the proposed method is potentially efficient in parallel.Comment: 15 pages, 4 figures, 1 tabl

The structured distance to singularity of a symmetric tridiagonal Toeplitz matrix

This paper is concerned with the distance of a symmetric tridiagonal Toeplitz matrix $T$ to the variety of similarly structured singular matrices, and with determining the closest matrix to $T$ in this variety. Explicit formulas are presented, that exploit the analysis of the sensitivity of the spectrum of $T$ with respect to structure-preserving perturbations of its entries.Comment: 16 pages, 5 Figure

Complementary results for the spectral analysis of matrices in Galerkin methods with GB-splines

We collect some new results relative to the study of the spectral analysis of matrices in Galerkin methods based on generalized B-splines with high smoothness

An improved Newton iteration for the generalized inverse of a matrix, with applications

The purpose here is to clarify and illustrate the potential for the use of variants of Newton's method of solving problems of practical interest on highly personal computers. The authors show how to accelerate the method substantially and how to modify it successfully to cope with ill-conditioned matrices. The authors conclude that Newton's method can be of value for some interesting computations, especially in parallel and other computing environments in which matrix products are especially easy to work with