Ilse C. F. Ipsen

A Note on Preconditioning Nonsymmetric Matrices

The preconditioners for indefinite matrices of KKT form in [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput., 21 (2000), pp. 1969--1972] are extended to general nonsymmetric matrices.

Computing an Eigenvector with Inverse Iteration

The purpose of this paper is two-fold: to analyze the behavior of inverse iteration for computing a single eigenvector of a complex square matrix and to review Jim Wilkinsons contributions to the development of the method. In the process we derive several new results regarding the convergence of inverse iteration in exact arithmetic. nIn the case of normal matrices we show that residual norms decrease strictly monotonically. For eighty percent of the starting vectors a single iteration is enough. nIn the case of non-normal matrices, we show that the iterates converge asymptotically to an invariant subspace. However, the residual norms may not converge. The growth in residual norms from one iteration to the next can exceed the departure of the matrix from normality. We present an example where the residual growth is exponential in the departure of the matrix from normality. We also explain the often significant regress of the residuals after the first iteration: it occurs when the non-normal part of the matrix is large compared to the eigenvalues of smallest magnitude. In this case computing an eigenvector with inverse iteration is exponentially ill conditioned (in exact arithmetic). nWe conclude that the behavior of the residuals in inverse iteration is governed by the departure of the matrix from normality rather than by the conditioning of a Jordan basis or the defectiveness of eigenvalues.

The idea behind Krylov methods

1. INTRODUCTION. We explain why Krylov methods make sense, and why it is natural to represent a solution to a linear system as a member of a Krylov space. In particular we show that the solution to a nonsingular linear system Ax = b lies in a Krylov space whose dimension is the degree of the minimal polynomial of A. Therefore, if the minimal polynomial of A has low degree then the space in which a Krylov method searches for the solution can be small. In this case a Krylov method has the opportunity to converge fast. When the matrix is singular, however, Krylov methods can fail. Even if the linear system does have a solution, it may not lie in a Krylov space. In this case we describe a class of right-hand sides for which a solution lies in a Krylov space. As it happens, there is only a single solution that lies in a Krylov space, and it can be obtained from the Drazin inverse. Our discussion demonstrates that eigenvalues play a central role when it comes to ensuring existence and uniqueness of Krylov solutions; they are not merely an artifact of convergence analyses.

Relative perturbation techniques for singular value problems

A technique is presented for deriving bounds on the relative change in the singular values of a real matrix (or the eigenvalues of a real symmetric matrix) due to a perturbation, as well as bounds on the angles between the unperturbed and perturbed singular vectors (or eigenvectors). The class of perturbations considered consists of all

On Rank-Revealing Factorisations

delta B

GMREs and the minimal polynomial

for which

Relative perturbation results for matrix eigenvalues and singular values

B + delta B = D_L BD_R

PageRank Computation, with Special Attention to Dangling Nodes

for some nonsingular matrices

Systolic Networks for Orthogonal Decompositions

D_L

Uniform Stability of Markov Chains

and