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Geometric approach to the parallel sum of vectors and application to the vector \(\varepsilon \)-algorithm

Abstract

The paper aims at answering the following question, in the scalar as well in the vector case: What do the famous Aitken’s \(\Delta ^{2}\) and Wynn’s \(\varepsilon \)-algorithm exactly do with the terms of the input sequence? Inspecting the rules of these algorithms from a geometric point of view leads to change the question into another one: By what kind of geometric object can the parallel (or harmonic) sum be represented? Thus, the paper begins with geometric considerations on the parallel addition and the parallel subtraction of vectors, including equivalent definition, and properties derived from the new point of view. It is shown how the parallel sum of vectors is related to the Bézier parabola controlled by these vectors and to their interpolating equiangular spiral. In the second part, observing that Aitken’s \(\Delta ^{2}\) and Wynn’s \(\varepsilon \)-algorithm may be defined through hybrid sums, mixing standard and parallel sums, the consequences are drawn regarding the way one can define and analyze these algorithms. New explanatory rules are derived for the \(\varepsilon \)-algorithm, providing a better understanding of its basic step and of its cross-rule.

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Correspondence to Alain F. Berlinet.

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Berlinet, A.F. Geometric approach to the parallel sum of vectors and application to the vector \(\varepsilon \)-algorithm. Numer Algor 65, 783–807 (2014). https://doi.org/10.1007/s11075-013-9714-y

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Keywords

  • Parallel sum
  • Bézier parabola
  • Spiral similarity
  • Aitken’s \(\Delta ^{2}\)
  • \(\varepsilon \)-algorithm

Mathematical Subject Classifications (2010)

  • 65B99
  • 65B05