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44 results in Practical Extrapolation Methods

Page 2 of 3

  • 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].

  • 13 - Acceleration of Convergence of Fourier and Generalized Fourier Series by the d-Transformation: The Complex Series Approach with APS

  • from I - The Richardson Extrapolation Process and Its Generalizations
  • Avram Sidi, Technion - Israel Institute of Technology, Haifa
  • Book:
    Practical Extrapolation Methods
    Published online:
    25 February 2010
    Print publication:
    05 June 2003, pp 253-262

  • Summary

    Introduction

    In this chapter, we extend the treatment we gave to power series in the preceding chapter to Fourier series and their generalizations, whether convergent or divergent. In particular, we are concerned with Fourier cosine and sine series, orthogonal polynomial expansions, series that arise from Sturm—Liouville problems, such as Fourier—Bessel series, and other general special function series.

    Several convergence acceleration methods have been used on such series, with limited success. An immediate problem many of these methods face is that they do not produce any acceleration when applied to Fourier and generalized Fourier series. The transformations of Euler and of Shanks discussed in the following chapters and the d-transformation are exceptions. See the review paper by Smith and Ford [318] and the paper by Levin and Sidi [165]. With those methods that do produce acceleration, another problem one faces in working with such series is the lack of stability and acceleration near points of singularity of the functions that serve as limits or antilimits of these series. Recall that the same problem occurs in dealing with power series.

    In this chapter, we show how the d-transformation can be used effectively to accelerate the convergence of these series. The approach we are about to propose has two main ingredients that can be applied also with some of the other sequence transformations.

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