QZ algorithm with two-sided generalized Rayleigh quotient shifts

We generalize the recently proposed two-sided Rayleigh quotient single-shift and the two-sided Grassmann-Rayleigh quotient double-shift used in the QR algorithm and apply the generalized versions to the QZ algorithm. With such shift strategies the QZ algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the QZ algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single-shift and Francis double-shift.