Fast algorithms, based on the unsymmetric look-ahead Lanczos and the Arnoldi process, are developed for the estimation of the functional (f)= uf(A)v for xed u; v and A, where A∈Rn×n is a large7 scale unsymmetric matrix. Numerical results are presented which validate the comparable accuracy of both approaches. Although the Arnoldi process reaches convergence more quickly in some cases, it 9 has ...

We review numerical methods for computing eigenvalues of matrices. We start by considering the computation of the dominant eigenpair of a general dense matrix using the power method, and then generalize to orthogonal iterations and the QR iteration with shifts. We also consider divide-and-conquer algorithms for tridiagonal matrices. The second part of this survey involves the computation of eig...

Abstract. We investigate several generalizations of the harmonic and refined Rayleigh–Ritz method. These may be practical when one is interested in eigenvalues close to one of two targets (for instance, when the eigenproblem has Hamiltonian structure such that eigenvalues come in pairs or quadruples), or in rightmost eigenvalues close to (for instance) the imaginary axis. Our goal is to develop...

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields. The singularities are described by eigenpairs of a corresponding operator pencil on a subdomain of the sphere. The solution approach is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-...

The need to evaluate expressions of the form f(A) or f(A)b, where f is a nonlinear function, A is a large sparse n × n matrix, and b is an n-vector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process. Applications of ...

In this addendum to an earlier paper by the author, it is shown how to compute a Krylov decomposition corresponding to an arbitrary Rayleigh-Quotient. This decomposition can be used to restart an Arnoldi process, with a selection of the Ritz vectors corresponding to the Rayleigh quotient. ABSTRACT In this addendum to an earlier paper by the author, it is shown how to compute a Krylov decomposit...

A linear algebraic theory called the 'multiple Arnoldi method' is presented and realizes large-scale (order-N) electronic structure calculations with generalized eigenvalue equations. A set of linear equations, in the form of (zS - H)x = b, are solved simultaneously with multiple Krylov subspaces. The method is implemented in a simulation package ELSES (www.elses.jp) with tight-binding-form Ham...

By introducing the second order Krylov subspace, a method for the reduction of second order systems is proposed leading to a reduced system of the same structure. This generalization of Krylov subspace involves two matrices and some starting vectors and the reduced order model is found by applying a projection directly to the second order model without any conversion to state space. A numerical...