Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scalar Levin-Type Sequence Transformations

Herbert H. H. Homeier

Abstract

Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Transformations that depend not only on the sequence elements or partial sums $s_n$ but also on an auxiliary sequence of so-called remainder estimates $ω_n$ are of Levin-type if they are linear in the $s_n$, and nonlinear in the $ω_n$. Known Levin-type sequence transformations are reviewed and put into a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.

Scalar Levin-Type Sequence Transformations

Abstract

Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Transformations that depend not only on the sequence elements or partial sums but also on an auxiliary sequence of so-called remainder estimates are of Levin-type if they are linear in the , and nonlinear in the . Known Levin-type sequence transformations are reviewed and put into a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.

Paper Structure

This paper contains 96 sections, 30 theorems, 329 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Definitions and Notations
  3. General Definitions
  4. Sets
  5. Natural Numbers:
  6. Integer Numbers:
  7. Real Numbers and Vectors:
  8. Complex Numbers:
  9. Vectors with nonvanishing components:
  10. Polynomials:
  11. Sequences:
  12. Sequences with nonvanishing terms:
  13. Special Functions and Symbols
  14. Gamma function
  15. Factorial:
  16. ...and 81 more sections

Key Result

Theorem 1

Let $\phi{}_{n,m}^{({k})}$ for $m=0,\dots,k-1$ be the $k$ linearly independent solutions of the linear $(k+1)$--term recurrence (eqPrec). The kernel of $\mathcal{T}[\Lambda{}_{}^{({k})}](\{\!\!\{s_n\}\!\!\},\{\!\!\{\omega_n\}\!\!\})$ is given by all sequences $\{\!\!\{\sigma_n\}\!\!\}$ with (anti)li

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Corollary 1
  • ...and 20 more