In order to find an approximate solution of an eigenvalue problem Tφ=λφ for a bounded operator T on a Banach space, one often considers a sequence (T_{n}) of bounded operators which converge to T in some sense and then one solves T_{n}φ_{n}=λ_{n}φ_{n}. When the convergence of (T_{n}) to T is slow, one needs to accelerate the convergence of (λ_{n}) to λ and of φ_{n} to φ. This can be accomplished by considering a higher order spectral analysis involving an approximate solution of a polynomial eigenvalue problem for T. Error estimates in the case of a nonzero simple eigenvalue of T will be given, and a scheme of implementing this procedure by a suitable matrix formulation will be discussed.