Obstacles to a Satisfying Convergence Theory for Restarted Arnoldi with Exact Shifts

The restarted Arnoldi method iteratively applies polynomial filters to enhance the orientation of the starting vector toward the desired invariant subspace.  At each step the roots of these filters are the Ritz values that least resemble the desired eigenvalues.  For Hermitian eigenvalue problems, Sorensen proved that this procedure must lead to convergence (under mild and natural conditions).  However, one can construct non-Hermitian examples where a filter polynomial root falls precisely on a desired eigenvalue, making convergence impossible in theory.  To handle such examples, we require a deeper understanding of how  Ritz eigenvalue estimates are distributed in the numerical range.   Given a set of points {theta_1, ..., theta_k} in the numerical range of the n-by-n matrix A, does there exist some n-by-k matrix Q with orthonormal columns for which the spectrum of Q'*A*Q is {theta_1, ..., theta_k}?

This seminar is part of the Recent Progress and Open Directions in Matrix Computations series.