On the Solution of Systems of Equations by the Epsilon Algorithm of Wynn

Abstract

The $\epsilon$-algorithm has been proposed by Wynn on a number of occasions as a convergence acceleration device for vector sequences; however, little is known concerning its effect upon systems of equations. In this paper, we prove that the algorithm applied to the Picard sequence ${{\text {x}}_{i + 1}} = F({{\text {x}}_i})$ of an analytic function $F:{{\text {R}}^n} \supset D \to {{\text {R}}^n}$ provides a quadratically convergent iterative method; furthermore, no differentiation of $F$ is needed. Some examples illustrate the numerical performance of this method and show that convergence can be obtained even when $F$ is not contractive near the fixed point. A modification of the method is discussed and illustrated.