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

Numerical enclosure for each eigenvalue in generalized eigenvalue problem

AbstractAn algorithm for enclosing all eigenvalues in generalized eigenvalue problem Ax=λBx is proposed. This algorithm is applicable even if A∈Cn×n is not Hermitian and/or B∈Cn×n is not Hermitian positive definite, and supplies n error bounds while the algorithm previously developed by the author supplies a single error bound. It is proved that the error bounds obtained by the proposed algorithm are equal or smaller than that by the previous algorithm. Computational cost for the proposed algorithm is similar to that for the previous algorithm. Numerical results show the property of the proposed algorithm

Guaranteed inclusions for the complex generalized eigenproblem

Es wird eine Methode zur Berechnung garantierter Schranken für die Lösung des komplexen allgemeinen Eigenproblems beschrieben. Die Methode erweitert einen ähnlichen Ansatz für allgemeine nichtlineare Gleichungssysteme in der Art, daß für den vorliegenden speziellen Fall weitgehende Folgerungen aus schwächeren Voraussetzungen gezogen wrden können.A method is described which produces guaranteed bounds for a solution of the generalized complex eigenproblem. The method extends a similar approach for general systems of nonlinear equations to the special case of complex pencils, where under weaker assumptions stronger assertions can be proved