Series acceleration

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In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities.

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[edit] Definition

Given a sequence

S=\{ s_n \}_{n\in\N}

having a limit

\lim_{n\to\infty} s_n = \ell,

an accelerated series is a second sequence

S'=\{ s'_n \}_{n\in\N}

which converges faster than the original sequence, in the sense that

\lim_{n\to\infty} \frac{s'_n-\ell}{s_n-\ell} = 0.

If the original sequence is divergent, the sequence transformation acts as extrapolation method to the antilimit \ell.

The mappings from the original to the transformed series may be linear (as defined in the article sequence transformations), or non-linear. In general, the non-linear sequence transformations tend to be more powerful.

[edit] Overview

Two classical techniques for series acceleration are Euler's transformation of series[1] and Kummer's transformation of series[2]. A variety of much more rapidly convergent and special-case tools have been developed in the 20th century, including Richardson extrapolation, introduced by Lewis Fry Richardson in the early 20th century but also known and used by Hirotaka Takebe in 1722, the Aitken delta-squared process, introduced by Alexander Aitken in 1926 but also known and used by Takakazu Seki in the 18th century, the e-algorithm given by Peter Wynn in 1956, the Levin u-transform, and the Wilf-Zeilberger-Ekhad method or WZ method.

For alternating series, several powerful techniques, offering convergence rates from 5.828 − n all the way to 17.93 − n for a summation of n terms, are described by Cohen et al. [3].

[edit] Euler's transform

A basic example of a linear sequence transformation, offering improved convergence, is Euler's transform. It is intended to be applied to an alternating series; it is given by

\sum_{n=0}^\infty (-1)^n a_n = \sum_{n=0}^\infty (-1)^n \frac {\Delta^n a_0} {2^{n+1}}

where Δ is the forward difference operator:

\Delta^n a_0 = \sum_{k=0}^n (-1)^k {n \choose k} a_{n-k}.

If the original series, on the left hand side, is only slowly converging, the forward differences will tend to become small quite rapidly; the additional power of two further improves the rate at which the left hand side converges.

A particularly efficient numerical implementation of the Euler transform is the van Wijngaarden transformation[4]

[edit] Non-linear sequence transformations

Examples of such nonlinear sequence transformations are Padé approximants and Levin-type sequence transformations.

Especially nonlinear sequence transformations often provide powerful numerical methods for the summation of divergent series or asymptotic series that arise for instance in perturbation theory, and may be used as highly effective extrapolation methods.

[edit] Aitken method

Main article: Aitken's delta-squared process

A simple nonlinear sequence transformation is the Aitken extrapolation or delta-squared method,

\mathbb{A} : S \to S'=\mathbb{A}(S) = {(s'_n)}_{n\in\N}

defined by

s'_n = s_{n+2} - \frac{(s_{n+2}-s_{n+1})^2}{s_{n+2}-2s_{n+1}+s_n}.

This transformation is commonly used to improve the rate of convergence of a slowly converging sequence; heuristically, it eliminates the largest part of the absolute error.

[edit] See also

[edit] References

  1. ^ Abramowitz, Milton & Stegun, Irene A., eds. (1965), “Chapter 3, eqn 3.6.27”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .
  2. ^ Abramowitz, Milton & Stegun, Irene A., eds. (1965), “Chapter 3, eqn 3.6.26”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .
  3. ^ Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, "Convergence Acceleration of Alternating Series", Experimental Mathematics, 9:1 (2000) page 3.
  4. ^ William H. Press, et al, Numerical Recipes in C, (1987) Cambridge University Press, ISBN 0-521-43108-5 (See section 5.1)