A fixed point technique to refine a simple approximate eigenvalue and a corresponding eigenvector
Lalita N. Deshpande, Balmohan V. Limaye · Numerical Functional Analysis and Optimization · 1989
Let T be a bounded operator on a Banach space X. Let λ0 be a nonzero simple eigenvalue of a ‘nearby’ operator T0 and let ⊘0 be a corresponding eigenvector. Several modified versions of a fixed point scheme are given for iteratively refining the initial approximations λ0 and ⊘0 of an eigenvalue λ of T and a corresponding eigenvector ⊘ Convergence of these schemes is proved by considering error bounds for the iterates. These bounds hold if a compact operator T is approximated in the norm or in a Collectively compact manner by a sequence (T0) of bounded operators, and λ0 and ⊘0 are eigenelements of Tn0 for a fixed n0 of ‘moderate’ size. Numerical examples are no included to illustrate the performation of various iteration schemes.