Error analysis of an algorithm for summing certain finite series

Abstract

An algorithm for summing the series , where the coefficients a n are assumed known, and the quantities p n , satisfy a linear three term recurrence relation, has been given by Clenshaw [1]. If we suppose that the p n satisfy the recurrence relation where α n and β n are, in general, functions of n , then P N may be found by constructing a sequence { b n } for n = N (−1)0, where the b satisfy the inhomogeneous recurrence relation with the conditions, The sum P N is then given by This result can be readily verified by multiplying each side of equation (1.2) by p n , summing from n = 0 to N , and making use of equations (1.1) and (1.3).