Global Convergence of Hessenberg Shifted QR I: Dynamics

Rapid convergence of the shifted QR algorithm on symmetric matrices was shown more than fifty years ago. Since then, despite significant interest and its practical relevance, an understanding of the dynamics of the shifted QR algorithm on nonsymmetric matrices has remained elusive. We give a family of shifting strategies for the Hessenberg shifted QR algorithm with provably rapid global convergence on nonsymmetric matrices of bounded nonnormality, quantified in terms of the condition number of the eigenvector matrix -- the convergence is linear with a constant rate, and for a well-chosen strategy from this family, the computational cost of each QR step scales roughly logarithmically in the eigenvector condition number. The key ideas in the analysis are: (1) to use certain higher degree shifts to dampen transient effects due to nonnormality (2) to characterize and detect slowly converging trajectories with respect to such shifts in terms of certain symmetry considerations, and to break this symmetry explicitly in the shifting strategy. We perform our analysis in exact arithmetic. In the companion papers [Global Convergence of Hessenberg Shifted QR II: Numerical Stability and Deflation], [Global Convergence of Hessenberg Shifted QR III: Approximate Ritz Values via Shifted Inverse Iteration], we show that our shifting strategies can be implemented efficiently in finite arithmetic.