Numerical stabilization of recurrence relations with vanishing solutions

Abstract

Homogeneous recurrence relations exhibit a highly numerical unstable behaviour in step-by-step evaluation of succesive terms. It is pointed out that this is a result of the presence of vanishing solutions, which are always added to initial values for the recursion scheme, due to finite machine accuracy. Stabilization of the recursion is shown to be identical with resolving these vanishing contributions with sufficient accuracy. To this end, explicit analytical expressions for these solutions, as products of continued fractions, are given. Application of these vanishing solutions enables us to construct the self-consistent, numerical stable general solution of the recursion relation.