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].