Invariant Subspace Convergence for Arnoldi's Method

Mark Embree

Oxford University Computing Laboratory
Wolfson Building
Parks Road, Oxford OX1 3QD


Krylov subspace eigenvalue algorithms typically aim to efficiently calculate some subset of the spectrum with a special property, such as the rightmost eigenvalues or minimum modulus eigenvalues. What matrix properties determine the rate at which the Krylov subspace captures the invariant subspace associated with the desired eigenvalues? When a matrix is normal, convergence only depends upon the dimension of the invariant subspace, the distance between desired and undesired eigenvalues, and biases of the starting vector.

Non-normality leads to a more complicated situation. In this talk, we develop bounds on the angle between the Krylov subspace and the desired invariant subspace for general (non-derogatory) matrices. These bounds take the form of multiplicative constants based on the quality of the starting vector and the non-normality of the matrix, along with a polynomial approximation problem over the pseudospectra. In the context of restarted iterations, the bounds inform shift selection and choice of the basis dimension. They also describe convergence when the desired invariant subspace is associated with a defective eigenvalue, and when the desired eigenvalues are stable but the undesired eigenvalues are ill-conditioned. Examples illustrate the applicability of these bounds to a variety of situations.

This talk describes joint research with Christopher Beattie and John Rossi.