The rise and fall of the vector epsilon algorithm

Coope, I. D.; Graves‐Morris, P. R.

doi:10.1007/bf02108462

Cited by 13 publications

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“…Finally, let us mention that a confluent form of the vector ε-algorithm exists and that a confluent form of the topological ε-algorithm was given in [27]; see also [79] where a functional ε-algorithm is discussed.…”

Section: The Confluent Casementioning

confidence: 99%

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The genesis and early developments of Aitken’s process, Shanks’ transformation, the ε–algorithm, and related fixed point methods

Brezinski

Redivo–Zaglia

2018

Numer Algor

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In this paper, we trace back the genesis of Aitken's ∆ 2 process, and Shanks' sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions, and Padé approximants. We go back to the roots of these methods, and analyze the original contributions. New and detailed explanations on the building and properties of Shanks' transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis.

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Section: The Confluent Casementioning

confidence: 99%

“…Their kernel is that of Shanks' transformation but, now, the condition is again necessary and sufficient as in the scalar case [295, Thm III.1]; see [64] for a survey. The vector valued rational approximants of Graves-Morris are related to the vector ε-algorithm [117,118,121,122]; see also [79].…”

Section: Successorsmentioning

confidence: 99%