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A hierarchically consistent, iterative sequence transformation

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Abstract

Recently, the author proposed a new nonlinear sequence transformation, the iterative\(J\) transformation, which was shown to provide excellent results in several applications (Homeier [15]). In the present contribution, this sequence transformation is derived by a hierarchically consistent iteration of some basic transformation. Hierarchical consistency is proposed as an approach to control the well-known problem that the basic transformation can be generalized in many ways. Properties of the\(J\) transformation are studied. It is of similar generality as the well-knownE algorithm (Brezinski [3], Håvie [18]). It is shown that the\(J\) transformation can be implemented quite easily. In addition to the defining representation, there are alternative algorithms for its computation based on generalized differences. The kernel of the\(J\) transformation is derived. The expression for the kernel is relatively compact and does not depend on any lower-order transforms. It is shown that several important other sequence transformations can be computed in an economical way using the\(J\) transformation.

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Communicated by C. Brezinski

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Homeier, H.H.H. A hierarchically consistent, iterative sequence transformation. Numer Algor 8, 47–81 (1994). https://doi.org/10.1007/BF02145696

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Keywords

  • Convergence acceleration
  • extrapolation
  • summation
  • hierarchical consistency
  • model sequences
  • iterative transformations
  • Levin-type transformations
  • E algorithm
  • generalized difference schemes
  • operator method

AMS subject classification

  • AMS(MOS)
  • 65B05
  • 65B10