Let F(z) be a vector-valued function F C CN, which is analytic at z 0 and meromorphic in a neighborhood of z 0, and let its Maclaurin series be given. In a recent work [J. Approx. Theory, 76 (1994), pp. 89-111] by the author, vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series, were developed, and some of their convergence properties were analyzed in ...

For a bounded linear operator A on a Hilbert space H we study local spectral sets and their relation to the spectrum of the local operator. By the local operator we mean A restricted to a Krylov subspace spanfb; Ab; A 2 b; :::g for a generic b 2 H. Moreover we investigate the relation between (A), the spectrum of A, and the spectrum of the local operator. Also spectral properties of the spectru...

In recent years, Krylov subspace methods have become popular tools for computing reduced order models of high order linear time invariant systems. The reduction can be done by applying a projection from high order to lower order space using bases of some subspaces called input and output Krylov subspaces. One aim of this paper is describing the invariancies of reduced order models using these m...

Consider the linear system Ax = b, where A ∈ CN×N is a singular matrix. In the present work we propose a general framework within which Krylov subspace methods for Drazininverse solution of this system can be derived in a convenient way. The Krylov subspace methods known to us to date treat only the cases in which A is hermitian and its index ind(A) is unity necessarily. In the present work A i...

Two approaches for approximating the solution of large-scale Lyapunov equations are considered: the alternating direction implicit (ADI) iteration and projective methods by Krylov subspaces. A link between them is presented by showing that the ADI iteration can always be identified by a Petrov-Galerkin projection with rational block Krylov subspaces. Then a unique Krylov-projected dynamical sys...

In the sequel, we will construct the reduced order models Gr(s) through Krylov projection methods. Toward this end, we construct matrices V ∈ R and Z ∈ R that span certain Krylov subspaces with the property that Z V = Ir. The reduced order model Gr(s) will then be obtained as Ar = Z T AV, Br = Z T B, and Cr = CV. (2) The corresponding oblique projection is given by V Z . Many researchers have w...

GMRES is a popular iterative method for the solution of large linear systems of equations with a square nonsymmetric matrix. The method generates a Krylov subspace in which an approximate solution is determined. We present modifications of the GMRES and the closely related RRGMRES methods that allow augmentation of the Krylov subspaces generated by these methods by a user-supplied subspace. We ...

and {ωk}k=1 ∈ C is a sequence of n distinct shifts. For example, shifted linear systems arise in model order reduction as well as in the geophysical exploration of both acoustic and elastic waves. In our application, we focus on wave propagation through elastic media in a frequency-domain formulation. This formulation has specific advantages when modeling visco-elastic effects. In order to impr...

LOW-RANK SOLUTION METHODS FOR LARGE-SCALE LINEAR MATRIX EQUATIONS Stephen D. Shank DOCTOR OF PHILOSOPHY Temple University, May, 2014 Professor Daniel B. Szyld, Chair We consider low-rank solution methods for certain classes of large-scale linear matrix equations. Our aim is to adapt existing low-rank solution methods based on standard, extended and rational Krylov subspaces to solve equations w...

Abstract. We consider the task of updating a matrix function f(A) when the matrix A ∈ Cn×n is subject to a low-rank modification. In other words, we aim at approximating f(A+D)− f(A) for a matrix D of rank k n. The approach proposed in this paper attains efficiency by projecting onto tensorized Krylov subspaces produced by matrix-vector multiplications with A and A∗. We prove the approximations...