Integrable lattices and convergence acceleration algorithms

Abstract

We show that a well-known convergence acceleration scheme, the ϵ-algorithm, when viewed as a two-variable difference equation, is nothing but the discrete Korteweg-de Vries lattice equation. The complete integrability of the latter confers to the ϵ-algorithm interesting properties among which the singularity confinement is outstanding. In fact, this property is used in order to derive the generalizations of the ϵ-accelerator leading to the most general form of the ϱ-algorithm. A new acceleration algorithm based on the modified Korteweg-de Vries lattice equation is also derived.