An extension of Olver’s method for the numerical solution of linear recurrence relations

Abstract

An algorithm is developed for computing the solution of a class of linear recurrence relations of order greater than two when unstable error propagation prevents the required solution being found by direct forward recurrence. By abandoning an appropriate number of initial conditions the original problem may be replaced by an inexact but well-conditioned boundary value problem, and in certain circumstances the solution of this new problem is a good approximation to the required solution of the original problem. The required solution of this reposed problem is generated using an algorithm based on Gaussian elimination, and a technique developed by Olver is extended to estimate automatically the truncation error of the proposed algorithm.

Some thoughts on solution in series

“…The determination of intermediate solutions of difference equations has received surprisingly little attention in the literature, although the basic approach is given by Oliver [2], Cash [3] and the references cited there. For the example cited above we set a 0 = 1 and assume that a r is negligible for r>R. We then write down equation (6) for r = -1,0,1,..., 7?…”

“…The fact that a r is negligible to 8 decimal places for r ^ 12 does not guarantee that the final column in table 2 is accurate to 8 decimal places, although it happens to be so in this example. In order to be sure of the accuracy we have to carry out a detailed error analysis, described in references [2] and [3], which involves additional computation.…”