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


Author: E. Gekeler
Journal: Math. Comp. 26 (1972), 427-436
MSC: Primary 65B99
DOI: https://doi.org/10.1090/S0025-5718-1972-0314226-X
MathSciNet review: 0314226
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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.


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DOI: https://doi.org/10.1090/S0025-5718-1972-0314226-X
Keywords: $ \epsilon $-algorithm, acceleration of the convergence of sequences, quadratic convergent iterative method without differentiation, solution of equations
Article copyright: © Copyright 1972 American Mathematical Society