An algorithmic decomposition system for second order recurrence relations

Abstract

A new technique of accurate stable computation of sequences satisfying non-homogeneous three term recurrence relations is presented. A decomposition system consisting of one non-linear and two linear first order recurrence relations is obtained. Forward or backward recursion directions of the linear relations provide additional flexibility in computation. This leads to an integrated system of three algorithms which can accurately compute the desired solution in each region of the solution set to the original second order relations. Applications are well known and numerical examples include Bessel functions of the second kind, Anger-Weber functions, and Lommel functions.