QR algorithm with two-sided Rayleigh quotient shifts

We introduce the two-sided Rayleigh quotient shift to the QR algorithm for non-Hermitian matrices to achieve a cubic local convergence rate. For the singly shifted case, the two-sided Rayleigh quotient iteration is incorporated into the QR iteration. A modified version of the method and its truncated version are developed to improve the efficiency. Based on the observation that the Francis double-shift QR iteration is related to a 2D Grassmann-Rayleigh quotient iteration, A doubly shifted QR algorithm with the two-sided 2D Grassmann-Rayleigh quotient double-shift is proposed. A modified version of the method and its truncated version are also developed. Numerical examples are presented to show the convergence behavior of the proposed algorithms. Numerical examples also show that the truncated versions of the modified methods outperform their counterparts including the standard Rayleigh quotient single-shift and the Francis double-shift.