19 - The Levin ℒ- and Sidi S-Transformations

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 {A_m} for which {ΔA_m} ∈ 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 R_l in Definition 6.2.2 are chosen to be R_l = l + 1. We now treat the Levin transformations in more detail.