On the solution of systems of equations by the epsilon algorithm of Wynn

Abstract

The ϵ \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 x i + 1 = F ( x i ) {{\text {x}}_{i + 1}} = F({{\text {x}}_i}) of an analytic function F : R n ⊃ D → R n F:{{\text {R}}^n} \supset D \to {{\text {R}}^n} provides a quadratically convergent iterative method; furthermore, no differentiation of F F is needed. Some examples illustrate the numerical performance of this method and show that convergence can be obtained even when F F is not contractive near the fixed point. A modification of the method is discussed and illustrated.