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4288 results in Numerical analysis and computational science

Page 164 of 215

  • 19 - The Levin ℒ- and Sidi S-Transformations

  • from II - Sequence Transformations
  • Avram Sidi, Technion - Israel Institute of Technology, Haifa
  • Book:
    Practical Extrapolation Methods
    Published online:
    25 February 2010
    Print publication:
    05 June 2003, pp 363-374

  • Summary

    Introduction

    In this and the next few chapters, we discuss some nonlinear sequence transformations that have proved to be effective on some or all types of logarithmic, linear, and factorial sequences {Am} for which {ΔAm} ∈ b(1). We show how these transformations are derived, and we provide a thorough analysis of their convergence and stability with respect to columns in their corresponding tables, as we did for the iterated Δ2-process, the iterated Lubkin transformation, and the Shanks transformation. (Analysis of the diagonal sequences turns out to be very difficult, and the number of meaningful results concerning this has remained very small.)

    We recall that the sequences mentioned here are in either b(1)/LOG or b(1)/LIN or b(1)/FAC described in Definition 15.3.2. In the remainder of this work, we use the notation of this definition with no changes, as we did in previous chapters.

    Before proceeding further, let us define

    Consequently, we also have

    The Levin L-Transformation

    Derivation of the L-Transformation

    We mentioned in Section 6.3 that the Levin—Sidi d(1)-transformation reduces to the Levin u-transformation when the Rl in Definition 6.2.2 are chosen to be Rl = l + 1. We now treat the Levin transformations in more detail.

  • 15 - The Euler Transformation, Aitken Δ2-Process, and Lubkin W-Transformation

  • from II - Sequence Transformations
  • Avram Sidi, Technion - Israel Institute of Technology, Haifa
  • Book:
    Practical Extrapolation Methods
    Published online:
    25 February 2010
    Print publication:
    05 June 2003, pp 279-296

  • Summary

    Introduction

    In this chapter, we begin the treatment of sequence transformations. As mentioned in the Introduction, a sequence transformation operates on a given sequence {An} and produces another sequence {Ân} that hopefully converges more quickly than the former. We also mentioned there that a sequence transformation is useful only when Ân is constructed from a finite number of the Ak.

    Our purpose in this chapter is to review briefly a few transformations that have been in existence longer than others and that have been applied successfully in various situations. These are the Euler transformation, which is linear, the Aitken Δ2-process and Lubkin W-transformation, which are nonlinear, and a few of the more recent generalizations of the latter two. As stated in the Introduction, linear transformations are usually less effective than nonlinear ones, and they have been considered extensively in other places. For these reasons, we do not treat them in this book. The Euler transformation is an exception to this in that it is one of the most effective of the linear methods and also one of the oldest acceleration methods. What we present here is a general version of the Euler transformation known as the Euler—Knopp transformation. A good source for this transformation on which we have relied is Hardy [123].

Page 164 of 215