Abstract. In this paper, we investigate a method for restarting the Lanczos method for approximating the matrix-vector product f(A)b, where A ∈ Rn×n is a symmetric matrix. For analytic f we derive a novel restart function that identifies the error in the Lanczos approximation. The restart procedure is then generated by a restart formula using a sequence of these restart functions. We present an...

In this paper, we present a new approach for the solution to a series of slightly perturbed symmetric eigenvalue problems (A + BSi BT) t = X E, 0 5 i 5 m, where A = AT f 72 nxn, B E 7Znx*, and Si = q E 7ZPx*, p < n. The matrix B is assumed to have full column rank, The main idea of our approach lies in a specific choice of starting vectors used in the block Lanczos algorithm so that the effect ...

An implicitly restarted symplectic Lanczos method for the symplectic eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. The inherent numerical diiculties of the symplectic Lanczos method are addressed by inexpensive implicit restarts. The method is used to compute some eigenvalues and eigenvectors of large and sparse symplectic matrices.

A numerical algorithm is presented to solve the constrained weighted energy problem from potential theory. As one of the possible applications of this algorithm, we study the convergence properties of the rational Lanczos iteration method for the symmetric eigenvalue problem. The constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated...

A deflated and restarted Lanczos algorithm to solve hermitian linear systems, and at the same time compute eigenvalues and eigenvectors for application to multiple right-hand sides, is described. For the first right-hand side, eigenvectors with small eigenvalues are computed while simultaneously solving the linear system. Two versions of this algorithm are given. The first is called Lan-DR and ...

Recently, the authors have proposed a new Krylov subspace iteration, the quasi-minimal residual algorithm (QMR), for solving non-Hermitian linear systems. In the original implementation of the QMR method, the Lanczos process with look-ahead is used to generate basis vectors for the underlying Krylov subspaces. In the Lanczos algorithm, these basis vectors are computed by means of three-term rec...

In this paper we study the relationship between spectral decomposition, orthogonal rational functions and the rational Lanczos algorithm, based on a simple identity for rational Krylov sequences.

Estimating upper bounds of the spectrum of large Hermitian matrices has long been a problem with both theoretical and practical significance. Algorithms that can compute tight upper bounds with minimum computational cost will have applications in a variety of areas. We present a practical algorithm that exploits k-step Lanczos iteration with a safeguard step. The k is generally very small, say ...