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Journal Articles Numerical Algorithms Year : 2023

On the kernel of vector ε-algorithm and related topics

Abstract

The vector ε-algorithm of Wynn is a powerful method for accelerating the convergence of vector sequences. Its kernel is the set of sequences which are transformed into constant sequences whose terms are their limits or antilimits. In 1971, a sufficient condition characterizing sequences in this kernel was given by McLeod. In this paper, we prove that such a condition is not necessary. Moreover, using Clifford algebra, we give a formula for the vector ε 2-transformation, which is formally the same as in the scalar case, up to operations in a Clifford algebra. Hence, Aitken's ∆ 2 process is extended in this way to vectors. Then, we derive the explicit algebraic and geometric expressions of sequences of the kernel of the ε 2transformation. We also formulate a conjecture concerning the explicit algebraic expression of kernel of the vector ε-algorithm.
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Dates and versions

hal-04355268 , version 1 (20-12-2023)

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Claude Brezinski, Michela Redivo-Zaglia, Ahmed Salam. On the kernel of vector ε-algorithm and related topics. Numerical Algorithms, 2023, Numerical Methods and Scientific Computing CIRM, Luminy, France 8-12 November 2021, 92 (1), pp.207-221. ⟨10.1007/s11075-022-01358-z⟩. ⟨hal-04355268⟩
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