| Item type: | Article | ||||
|---|---|---|---|---|---|
| Journal or Publication Title: | Journal of Computational and Applied Mathematics | ||||
| Publisher: | ELSEVIER SCIENCE BV | ||||
| Place of Publication: | AMSTERDAM | ||||
| Volume: | 122 | ||||
| Number of Issue or Book Chapter: | 1-2 | ||||
| Page Range: | pp. 81-147 | ||||
| Date: | 2000 | ||||
| Institutions: | Chemistry and Pharmacy > Institut für Physikalische und Theoretische Chemie | ||||
| Identification Number: |
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| Keywords: | RICHARDSON EXTRAPOLATION PROCESS; NONLINEAR CONVERGENCE ACCELERATORS; GROUND-STATE ENERGY; INFINITE-SERIES REPRESENTATIONS; OCTIC ANHARMONIC-OSCILLATOR; EFFICIENT EVALUATION; PERTURBATION-SERIES; LINEAR CONVERGENCE; OVERLAP INTEGRALS; FINITE CLUSTER; convergence acceleration; extrapolation; summation of divergent series; stability analysis; hierarchical consistency; iterative sequence transformation; Levin-type transformations; algorithm; linear convergence; logarithmic convergence; Fourier series; power series; rational approximation | ||||
| Dewey Decimal Classification: | 500 Science > 510 Mathematics 500 Science > 540 Chemistry & allied sciences | ||||
| Status: | Published | ||||
| Refereed: | Yes, this version has been refereed | ||||
| Created at the University of Regensburg: | Yes | ||||
| Item ID: | 74083 |
Abstract
Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. The basic idea is to construct from a given sequence {{s(n)}} a new sequence {{s(n)'}} = T({{s(n)}}) where each s(n)' depends on a finite number of elements s(n1), ..., s(nm). Often, the s(n) are the partial sums of an infinite ...
Abstract
Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. The basic idea is to construct from a given sequence {{s(n)}} a new sequence {{s(n)'}} = T({{s(n)}}) where each s(n)' depends on a finite number of elements s(n1), ..., s(nm). Often, the s(n) are the partial sums of an infinite series. The aim is to find transformations such that {{s(n)'}} converges faster than (or sums) {{s(n)}}. Transformations T{{s(n)}}, {{w(n)}}) that depend not only on the sequence elements or partial sums s(n) but also on an auxiliary sequence of the so-called remainder estimates w(n) are of Levin-type if they are linear in the s(n), and nonlinear in the w(n). Such remainder estimates provide an easy-to-use possibility to use asymptotic information on the problem sequence for the construction of highly efficient sequence transformations. As shown first by Levin, it is possible to obtain such asymptotic information easily for large classes of sequences in such a way that the w(n) are simple functions of a few sequence elements s(n). Then, nonlinear sequence transformations are obtained. Special cases of such Levin-type transformations belong to the most powerful currently known extrapolation methods for scalar sequences and series. Here, we review known Levin-type sequence transformations and put them in 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. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 65B05; 65B10; 65B15; 40A05; 40A25; 42C15.
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