Generalizing the Telescoping Procedures for Power Series to Reduce the Degree of Polynomials

Abstract

By using the Telescoping Procedures for Power Series (TPPS), the degree of any given polynomial can be decreased by one degree on the interval [-1; 1], with only a slight accuracy loss, the reason for this is that, the (TPPS) depends on the Chebyshev polynomials, which have a minimum maximum-absolute value that is distributed uniformly over [-1; 1]. In this paper, we generalized the (TPPS) with the aim to reduce the degree of an arbitrary polynomial for any degree and in any interval [a; b]. The generalization was in the form of a theorem and the theorem was proven using the Induction. The new technique called Generalized Telescoping Procedures for Power Series (GTPPS). We also conducted a study of the error bound of the theorem in the intervals [-1; 1], and [a; b]. The authors believe that this paper paves the way for a comprehensive set of researches and applications in the fields of approximation theory, optimization theory, computer simulation and other computation techniques to issues in numerous scientific disciplines.