The Effect of Aggressive Early Deflation on the Convergence of the QR Algorithm

Aggressive early deflation has proven to significantly enhance the convergence of the QR algorithm for computing the eigenvalues of a nonsymmetric matrix. One purpose of this paper is to point out that this deflation strategy is equivalent to extracting converged Ritz vectors from certain Krylov subspaces. As a special case, the single-shift QR algorithm enhanced with aggressive early deflation corresponds to a Krylov subspace method whose starting vector undergoes a Rayleigh-quotient iteration. It is shown how these observations can be used to derive improved convergence bounds for the QR algorithm.