Asymptotic behavior of the $s$-step method of steepest descent for eigenvalue problems in Hilbert space

On the example of the Rayleigh functional a new approach is developed to the study of the asymptotic behavior of the $s$-step method, based on the proof of the existence of limit iteration parameters of the method in even (odd) iterations. This approach may be used to analyze the asymptotic behavior of the $s$-step method in the optimization of arbitrary sufficiently smooth functionals defined on a Hilbert space.