Survey of numerical stability issues in convergence acceleration

Authors:
Avram Sidi
Affiliations:
Computer Science Department, Technion - Israel Institute of Technology, Haifa 32000, Israel
Venue:
Applied Numerical Mathematics
Year:
2010

Citing 12
Cited 1

On a generalization of the Richardson extrapolation process

Numerische Mathematik
On rates of acceleration of extrapolation methods for oscillatory infinite integrals

BIT
Extension and completion of Wynn's theory on convergence of columns of the epsilon table

Journal of Approximation Theory
On condition numbrs of some quasi-linear transformations

Journal of Computational and Applied Mathematics
Further convergence and stability results for the generalized Richardson extrapolation process GREP(1) with an application to the D(1)-transformation for infinite intergals

Journal of Computational and Applied Mathematics - Numerical evaluation of integrals
Extrapolation methods and derivatives of limits of sequences

Mathematics of Computation
HURRY: An Acceleration Algorithm for Scalar Sequences and Series

ACM Transactions on Mathematical Software (TOMS)
On condition numbers of the shanks transformation

Journal of Computational and Applied Mathematics
The Richardson Extrapolation Process with a Harmonic Sequence of Collocation Points

SIAM Journal on Numerical Analysis
A convergence and stability study of the iterated Lubkin transformation and the θ-algorithm

Mathematics of Computation
New convergence results on the generalized Richardson extrapolation process GREP (1) for logarithmic sequences

Mathematics of Computation
Practical Extrapolation Methods: Theory and Applications

Practical Extrapolation Methods: Theory and Applications

A comparative study of numerical steepest descent, extrapolation, and sequence transformation methods in computing semi-infinite integrals

Numerical Algorithms

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Abstract

An important issue that arises in application of convergence acceleration (extrapolation) methods is that of stability in the presence of floating-point arithmetic. This issue turns out to be critical because numerical instability is inherent, even built in, when convergence acceleration methods are applied to certain types of sequences that occur commonly in practice. If methods are applied without taking this issue into account, the attainable accuracy is limited, and eventually destroyed completely, as more terms are added in the process. Therefore, it is important to understand the origin of the problem and to propose practical ways to solve it effectively. In this work, we present a general discussion of the issue of stability within the context of a generalization of the Richardson extrapolation process, and review some of the recent developments that have taken place in the theoretical study of many of the known acceleration methods. We discuss approaches that have been proposed to cope with built-in instabilities when applying various methods, and illustrate the effectiveness of these strategies with some numerical examples.