A note on the approximation of two-sided Rayleigh quotient to eigenvalues

We consider the two-sided Rayleigh quotient introduced by Ostrowski as convergence acceleration tool for eigenvalue solvers. An interesting property of the two-sided Rayleigh quotient is established: its approximation to eigenvalues depends on the eigenvalue condition number. This fact is valid not only for matrix but also for matrix polynomial, multiparameter system, and periodic matrix pairs.