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| In [[mathematics]], '''series acceleration''' is one of a collection of [[sequence transformation]]s for improving the [[rate of convergence]] of a [[series (mathematics)|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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| == Definition ==
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| Given a sequence
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| :<math>S=\{ s_n \}_{n\in\N}</math>
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| having a limit
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| :<math>\lim_{n\to\infty} s_n = \ell,</math>
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| an accelerated series is a second sequence
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| :<math>S'=\{ s'_n \}_{n\in\N}</math>
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| which '''converges faster''' to <math>\ell</math> than the original sequence, in the sense that
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| :<math>\lim_{n\to\infty} \frac{s'_n-\ell}{s_n-\ell} = 0.</math>
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| If the original sequence is [[Divergent series|divergent]], the [[sequence transformation]] acts as an [[extrapolation method]] to the [[antilimit]] <math>\ell</math>.
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| The mappings from the original to the transformed series may be linear (as defined in the article [[sequence transformation]]s), or non-linear. In general, the non-linear sequence transformations tend to be more powerful.
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| == Overview ==
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| Two classical techniques for series acceleration are [[Euler's transformation of series]]<ref>{{AS ref|3, eqn 3.6.27|16}}</ref> and [[Kummer's transformation of series]].<ref>{{AS ref|3, eqn 3.6.26|16}}</ref> 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 [[Takebe Kenko|Katahiro 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 [[epsilon algorithm]] given by [[Peter Wynn (mathematician)|Peter Wynn]] in 1956, the [[Levin u-transform]], and the [[Wilf-Zeilberger-Ekhad method]] or [[WZ theory|WZ method]].
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| For alternating series, several powerful techniques, offering convergence rates from <math>5.828^{-n}</math> all the way to <math>17.93^{-n}</math> for a summation of <math>n</math> terms, are described by Cohen ''et al.''.<ref>[[Henri Cohen (number theorist)|Henri Cohen]], Fernando Rodriguez Villegas, and [[Don Zagier]],
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| "[http://www.math.utexas.edu/~villegas/publications/conv-accel.pdf Convergence Acceleration of Alternating Series]", ''Experimental Mathematics'', '''9''':1 (2000) page 3.</ref>
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| ==Euler's transform==
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| 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
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| :<math>\sum_{n=0}^\infty (-1)^n a_n = \sum_{n=0}^\infty (-1)^n
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| \frac {\Delta^n a_0} {2^{n+1}}</math>
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| where <math>\Delta</math> is the [[forward difference operator]]:
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| :<math>\Delta^n a_0 = \sum_{k=0}^n (-1)^k {n \choose k} a_{n-k}.</math>
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| 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 right hand side converges.
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| A particularly efficient numerical implementation of the Euler transform is the [[van Wijngaarden transformation]].<ref>William H. Press, ''et al.'', ''Numerical Recipes in C'', (1987) Cambridge University Press, ISBN 0-521-43108-5 (See section 5.1).</ref>
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| ==Conformal mappings==
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| A series
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| :S = <math>\sum_{n=0}^{\infty} a_n</math>
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| can be written as f(1), where the function f(z) is defined as
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| :<math>f(z) = \sum_{n=0}^{\infty} a_n z^{n}</math>
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| The function f(z) can have singularities in the complex plane (branch point singularities, poles or essential singularities), which limit the radius of convergence of the series. If the point z = 1 is close to or on the boundary of the disk of convergence, the series for S will converge very slowly. One can then improve the convergence of the series by means of a conformal mapping that moves the singularities such that the point that is mapped to z = 1, ends up deeper in the new disk of convergence.
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| The conformal transform <math>z = \Phi(w)</math> needs to be chosen such that <math>\Phi(0)=0</math>, and one usually chooses a function that has a finite derivative at w = 0. One can assume that <math>\Phi(1)=1</math> without loss of generality, as one can always rescale w to redefine <math>\Phi</math>. We then consider the function
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| :<math>g(w)= f\left(\Phi(w)\right)</math>
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| Since <math>\Phi(1)=1</math>, we have f(1) = g(1). We can obtain the series expansion of g(w) by putting <math>z=\Phi(w)</math> in the series expansion of f(z) because <math>\Phi(0)=0</math>; the first n terms of the series expansion for f(z) will yield the first n terms of the series expansion for g(w) if <math>\Phi'(0)\neq 0</math>. Putting w = 1 in that series expansion will thus yield a series such that if it converges, it will converge to the same value as the original series.
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| ==Non-linear sequence transformations==
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| Examples of such nonlinear sequence transformations are [[Padé approximant]]s, the [[Shanks transformation]], and [[Levin-type sequence transformation]]s.
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| 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 method]]s.
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| ===Aitken method===
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| ::''Main article: [[Aitken's delta-squared process]]'' | |
| A simple nonlinear sequence transformation is the Aitken extrapolation or delta-squared method,
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| :<math>\mathbb{A} : S \to S'=\mathbb{A}(S) = {(s'_n)}_{n\in\N}</math>
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| defined by
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| :<math>s'_n = s_{n+2} - \frac{(s_{n+2}-s_{n+1})^2}{s_{n+2}-2s_{n+1}+s_n}.</math>
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| 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]].
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| == See also ==
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| * [[Minimum polynomial extrapolation]]
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| * [[Van Wijngaarden transformation]]
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| ==References==
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| <references/>
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| * C. Brezinski and M. Redivo Zaglia, ''Extrapolation Methods. Theory and Practice'', North-Holland, 1991.
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| * G. A. Baker, Jr. and P. Graves-Morris, ''Padé Approximants'', Cambridge U.P., 1996.
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| * {{mathworld|urlname=ConvergenceImprovement|title=Convergence Improvement}}
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| * Herbert H. H. Homeier, ''Scalar Levin-Type Sequence Transformations'', Journal of Computational and Applied Mathematics, vol. 122, no. 1-2, p 81 (2000). {{cite doi|10.1016/S0377-0427(00)00359-9}}, {{arxiv|math/0005209}}.
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| [[Category:Numerical analysis]]
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| [[Category:Asymptotic analysis]]
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| [[Category:Summability methods]]
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| [[Category:Perturbation theory]]
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but you can do your buying at malls and Western designer outlets like celine Malaysia. p celine Malaysia is in a natural way situated in Kuala Lumpur, the country's best city centre. celine Malaysia presents a full selection of celine classics and new releases,
allowing Malaysian girls of all sorts of cultural backgrounds to accessorize their outfits. Right here is a guidebook to the most common clothing developments in Malaysia and the most effective celine bags to go with them. . Western dressing.
Obiously, Malaysia is not immune to the perading impact of Western preferred culture. The West has manufactured its mark in the Malaysian psyche in phrases of trend, and you will locate several of the younger girls dressing in European or American design garments like miniskirts and sleeeless tops as they emulate their faourite Western celebs.
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