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 general matrices has remained elusive. We give a family of shifting strategies for the Hessenberg shifted QR algorithm with provably rapid global convergence on 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 nearly logarithmically in the eigenvector condition number. We perform our analysis in exact arithmetic. In the companion paper [Global Convergence of Shifted QR II: Numerical Stability and Deflation], we show that our shifting strategies can be implemented efficiently in finite arithmetic.