We consider large scale ill-conditioned linear systems arising from discretization of ill-posed problems. Regularization is imposed through an (assumed known) upper bound constraint on the solution. An iterative scheme, requiring the computation of the smallest eigenvalue and corresponding eigenvector, is used to determine the proper level of regularization. In this paper we consider several co...

In a recent paper [Rojas, Santos, Sorensen: ACM ToMS 34 (2008), Article 11] an efficient method for solving the Large-Scale Trust-Region Subproblem was suggested which is based on recasting it in terms of a parameter dependent eigenvalue problem and adjusting the parameter iteratively. The essential work at each iteration is the solution of an eigenvalue problem for the smallest eigenvalue of t...

In the dense nonsymmetric eigenvalue problem, work has focused on the Hessenberg reduction and QR iteration, using efficient algorithms and fast, Level 3 BLAS. Comparatively, computation of eigenvectors performs poorly, limited to slow, Level 2 BLAS performance with little speedup on multi-core systems. It has thus become a dominant cost in the solution of the eigenvalue problem. To address thi...

Two inverse eigenvalue problems are discussed. First, given the eigenvalues and a weight vector an extended Hessenberg matrix is computed. This matrix represents the recurrences linked to a (rational) Arnoldi inverse problem. It is well-known that the matrix capturing the recurrence coefficients is of Hessenberg form in the standard Arnoldi case. Considering, however, rational functions and adm...

The paper is about a generalization of a classical eigenvalue-decomposition method originally developed for errors–in-variables linear system identification to handle an important class of nonlinear problems. A number of examples are presented to call the attention to the most critical part of the procedure turning the identification problem to a generalized eigenvalue-eigenvector calculation p...

We propose a verified computation method for eigenvalues in region and the corresponding eigenvectors of generalized Hermitian eigenvalue problems. The proposed uses complex moments to extract eigencomponents interest from random matrix Rayleigh$\unicode{x2013}$Ritz procedure project given problem into reduced problem. moment is by contour integral approximated using numerical quadrature. split...

In this paper an analogue of the Schwarz alternating method is considered to nd a minimal eigenvalue and its corresponding eigenvector of generalized symmetric eigenvalue problem. The technique suggested is based on decomposition of the original domain into overlapping subdomains and on consideration of local eigenvalue problems in subdomains. Both multiplicative and additive variants of the me...