Method of iterpolatory type functions in Banach spaces

Abstract

We study the problem on how to reuse previously calculated values of
approximated function in the scheme of iteratively refining sequence of
approximations provided by the interpolatory type formulas. The proposed
method is based upon a collocation of the current–step’s interpolant to
the given function on the set of points where the function values were
already evaluated during the calcualtion of previous-step’s approximations.
Theoretical considerations are supported by several numerical examples.