On the numerical evaluation of linear recurrences

We present some remarks on the numerical evaluation of recurrence relations. Rounding error bounds are presented of the numerical scheme and some numerical examples are given, in particular, we analyse conversion recurrences from different families of orthogonal polynomials, the limit case of Jacobi–Sobolev polynomials, random recurrences and perturbed Gegenbauer polynomials. In all these examples the theoretical bounds give sharp relative rounding error estimations. The parallel evaluation of recurrences are also considered and numerical tests on a Cray T3D are presented.