Global Convergence of Hessenberg Shifted QR I: Exact Arithmetic

Rapid convergence of the shifted QR algorithm on symmetric matrices was shown more than 50 years ago. Since then, despite significant interest and its practical relevance, an understanding of the dynamics and convergence properties of the shifted QR algorithm on nonsymmetric matrices has remained elusive. We introduce a new family of shifting strategies for the Hessenberg shifted QR algorithm. We prove that when the input is a diagonalizable Hessenberg matrix H of bounded eigenvector condition number \(\kappa _V(H)\)—defined as the minimum condition number of V over all diagonalizations \(VDV^{-1}\) of H—then the shifted QR algorithm with a certain strategy from our family is guaranteed to converge rapidly to a Hessenberg matrix with a zero subdiagonal entry, in exact arithmetic. Our convergence result is nonasymptotic, showing that the geometric mean of certain subdiagonal entries of H decays by a fixed constant in every QR iteration. The arithmetic cost of implementing each iteration of our strategy scales roughly logarithmically in the eigenvector condition number \(\kappa _V(H)\), which is a measure of the nonnormality of H. The key ideas in the design and analysis of our strategy are: (1) we are able to precisely characterize when a certain shifting strategy based on Ritz values stagnates. We use this information to design certain “exceptional shifts” which are guaranteed to escape stagnation whenever it occurs. (2) We use higher degree shifts (of degree roughly \(\log \kappa _V(H)\)) to dampen transient effects due to nonnormality, allowing us to treat nonnormal matrices in a manner similar to normal matrices.