Summability methods sFrom Wikipedia, the free encyclopediaContents1 Series acceleration 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Eulers transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Conformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Non-linear sequence transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5.1 Aitken method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 SilvermanToeplitz theorem 42.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Summation by parts 53.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Newton series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Similarity with an integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 83.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8iChapter 1Series accelerationIn mathematics,series acceleration is one of a collection of sequence transformations for improving the rate ofconvergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they areused 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 seriesgives some of the classic, well-known hypergeometric series identities.1.1 DenitionGiven a sequenceS= {sn}nNhaving a limitlimnsn= ,an accelerated series is a second sequenceS= {sn}nNwhich converges faster to than the original sequence, in the sense thatlimnsn sn = 0.If the original sequence is divergent, the sequence transformation acts as an extrapolation method to the antilimit .The mappings from the original to the transformed series may be linear (as dened in the article sequence transfor-mations), or non-linear. In general, the non-linear sequence transformations tend to be more powerful.1.2 OverviewTwo classical techniques for series acceleration are Eulers transformation of series[1] and Kummers transformationof series.[2] Avariety 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 knownand used by Katahiro Takebe in 1722, the Aitken delta-squared process, introduced by Alexander Aitken in 1926 but12 CHAPTER 1. SERIES ACCELERATIONalso known and used by Takakazu Seki in the 18th century, the epsilon algorithm given by Peter Wynn in 1956, theLevin u-transform, and the Wilf-Zeilberger-Ekhad method or WZ method.For alternating series, several powerful techniques, oering convergence rates from 5.828nall the way to 17.93nfor a summation of n terms, are described by Cohen et al..[3]1.3 Eulers transformA basic example of a linear sequence transformation, oering improved convergence, is Eulers transform. It isintended to be applied to an alternating series; it is given byn=0(1)nan=n=0(1)nna02n+1where is the forward dierence operator:na0=nk=0(1)k(nk)ank.If the original series, on the left hand side, is only slowly converging, the forward dierences will tend to becomesmall quite rapidly; the additional power of two further improves the rate at which the right hand side converges.A particularly ecient numerical implementation of the Euler transform is the van Wijngaarden transformation.[4]1.4 Conformal mappingsA seriesn=0ancan be written as f(1), where the function f(z) is dened asf(z) =n=0anznThe function f(z) can have singularities in the complex plane (branch point singularities, poles or essential singular-ities), which limit the radius of convergence of the series. If the point z = 1 is close to or on the boundary of thedisk of convergence, the series for S will converge very slowly. One can then improve the convergence of the seriesby means of a conformal mapping that moves the singularities such that the point that is mapped to z = 1, ends updeeper in the new disk of convergence.The conformal transform z= (w) needs to be chosen such that (0) = 0 , and one usually chooses a function thathas a nite derivative at w = 0. One can assume that (1) = 1 without loss of generality, as one can always rescalew to redene . We then consider the functiong(w) = f ((w))Since (1) = 1 , we have f(1) = g(1). We can obtain the series expansion of g(w) by putting z= (w) in the seriesexpansion of f(z) because (0)=0 ; the rst n terms of the series expansion for f(z) will yield the rst n terms ofthe series expansion for g(w) if (0) = 0 . Putting w = 1 in that series expansion will thus yield a series such that ifit converges, it will converge to the same value as the original series.1.5. NON-LINEAR SEQUENCE TRANSFORMATIONS 31.5 Non-linear sequence transformationsExamples of such nonlinear sequence transformations are Pad approximants, the Shanks transformation, and Levin-type sequence transformations.Especially nonlinear sequence transformations often provide powerful numerical methods for the summation ofdivergent series or asymptotic series that arise for instance in perturbation theory, and may be used as highly ef-fective extrapolation methods.1.5.1 Aitken methodMain article: Aitkens delta-squared processA simple nonlinear sequence transformation is the Aitken extrapolation or delta-squared method,A : S S= A(S) = (sn)nNdened bysn= sn+2 (sn+2 sn+1)2sn+2 2sn+1 + sn.This transformation is commonly used to improve the rate of convergence of a slowly converging sequence; heuris-tically, it eliminates the largest part of the absolute error.1.6 See alsoMinimum polynomial extrapolationVan Wijngaarden transformation1.7 References[1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 3, eqn 3.6.27, Handbook of Mathematical Functions withFormulas, Graphs, and Mathematical Tables, New York: Dover, p. 16, ISBN 978-0486612720, MR 0167642.[2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 3, eqn 3.6.26, Handbook of Mathematical Functions withFormulas, Graphs, and Mathematical Tables, New York: Dover, p. 16, ISBN 978-0486612720, MR 0167642.[3] Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, "Convergence Acceleration of Alternating Series", Experi-mental 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 section5.1).C. Brezinski and M. Redivo Zaglia, Extrapolation Methods. Theory and Practice, North-Holland, 1991.G. A. Baker, Jr. and P. Graves-Morris, Pad Approximants, Cambridge U.P., 1996.Weisstein, Eric W., Convergence Improvement, MathWorld.Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations, Journal of Computational and AppliedMathematics, vol. 122, no. 1-2, p 81 (2000). Homeier, H. H. H. (2000). Scalar Levin-type sequence transfor-mations. Journal of Computational and Applied Mathematics 122: 81. doi:10.1016/S0377-0427(00)00359-9., arXiv:math/0005209.Chapter 2SilvermanToeplitz theoremIn mathematics, theSilvermanToeplitztheorem, rst proved by Otto Toeplitz, is a result in summability the-ory characterizing matrix summability methods that are regular.A regular matrix summability method is a matrixtransformation of a convergent sequence which preserves the limit.An innite matrix(ai,j)i,jN with complex-valued entries denes a regular summability method if and only if itsatises all of the following properties:limiai,j= 0 j Nlimij=0ai,j= 1supij=0|ai,j| < 2.1 ReferencesToeplitz, Otto (1911) "ber die lineare Mittelbildungen." Prace mat.-z., 22, 113118 (the original paper inGerman)Silverman, Louis Lazarus (1913) On the denition of the sum of a divergent series. University of MissouriStudies, Math. Series I, 1964Chapter 3Summation by partsAbel transformation redirects here. For another transformation, see Abel transform.In mathematics, summation by parts transforms the summation of products of sequences into other summations,often simplifying the computation or (especially) estimation of certain types of sums. The summation by parts formulais sometimes called Abels lemma or Abel transformation.3.1 StatementSuppose {fk} and {gk} are two sequences. Then,nk=mfk(gk+1 gk) = [fn+1gn+1 fmgm] nk=mgk+1(fk+1 fk).Using the forward dierence operator , it can be stated more succinctly asnk=mfkgk= [fn+1gn+1 fmgm] nk=mgk+1fk,Note that summation by parts is an analogue to the integration by parts formula,f dg= fg g df.Note also that although applications almost always deal with convergence of sequences, the statement is purely al-gebraic and will work in any eld. It will also work when one sequence is in a vector space, and the other is in therelevant eld of scalars.3.2 Newton seriesThe formula is sometimes given in one of these - slightly dierent - formsnk=0fkgk= f0nk=0gk +n1j=0(fj+1 fj)nk=j+1gk= fnnk=0gk n1j=0(fj+1 fj)jk=0gk,56 CHAPTER 3. SUMMATION BY PARTSwhich represent a special case ( M= 1 ) of the more general rulenk=0fkgk=M1i=0f(i)0G(i+1)i+nMj=0f(M)jG(M)j+M==M1i=0(1)if(i)niG(i+1)ni+ (1)MnMj=0f(M)jG(M)j;both result from iterated application of the initial formula. The auxiliary quantities are Newton series:f(M)j:=Mk=0(1)Mk(Mk)fj+kandG(M)j:=nk=j(k j + M 1M 1)gk,G(M)j:=jk=0(j k + M 1M 1)gk.A remarkable, particular ( M= n + 1 ) result is the noteworthy identitynk=0fkgk=ni=0f(i)0G(i+1)i=ni=0(1)if(i)niG(i+1)ni.Here, (nk) is the binomial coecient.3.3 MethodFor two given sequences (an) and (bn) , with n N , one wants to study the sum of the following series:SN=Nn=0 anbnIf we dene Bn=nk=0 bk, then for every n > 0, bn= Bn Bn1andSN= a0b0 +Nn=1an(Bn Bn1),SN= a0b0 a0B0 + aNBN+N1n=0Bn(an an+1).Finally SN= aNBN N1n=0Bn(an+1 an).This process, called an Abel transformation, can be used to prove several criteria of convergence for SN.3.4 Similarity with an integration by partsThe formula for an integration by parts is baf(x)g(x) dx = [f(x)g(x)]ba baf(x)g(x) dxBeside the boundary conditions, we notice that the rst integral contains two multiplied functions, one which isintegrated in the nal integral ( gbecomes g ) and one which is dierentiated ( fbecomes f).The process of the Abel transformation is similar, since one of the two initial sequences is summed ( bnbecomesBn) and the other one is dierenced ( anbecomes an+1 an).3.5. APPLICATIONS 73.5 ApplicationsIt is used to prove Kroneckers lemma, which in turn, is used to prove a version of the strong law of largenumbers under variance constraints.Summation by parts is frequently used to prove Abels theorem.If bn is a convergent series, andan a bounded monotone sequence, thenSN=Nn=0 anbn remains aconvergent series.The Cauchy criterion givesSM SN= aMBM aNBN+M1n=NBn(an+1 an)= (aM a)BM (aN a)BN+ a(BM BN) +M1n=NBn(an+1 an),where a is the limit of an . As bn is convergent, BN is bounded independently of N , say by B . As ana go tozero, so go the rst two terms. The third term goes to zero by the Cauchy criterion for bn . The remaining sum isbounded byM1n=N|Bn||an+1 an| BM1n=N|an+1 an| = B|aN aM|by the monotonicity of an , and also goes to zero as N .Using the same proof as above, one shows that1. if the partial sums BN form a bounded sequence independently of N ;2. if n=0|an+1 an| < (so that the sum M1n=N |an+1 an| goes to zero as N goes to innity) ; and3. if liman= 0then SN=Nn=0 anbn is a convergent series.In both cases, the sum of the series satises: |S| = |n=0 anbn| Bn=0|an+1 an|3.6 See alsoConvergent seriesDivergent seriesIntegration by partsCesro summationAbels theoremAbel sum formula3.7 ReferencesAbels lemma at PlanetMath.org.8 CHAPTER 3. SUMMATION BY PARTS3.8 Text and image sources, contributors, and licenses3.8.1 Text Series acceleration Source: https://en.wikipedia.org/wiki/Series_acceleration?oldid=544640922 Contributors: Michael Hardy, Loisel,Rich Farmbrough, Pt, Count Iblis, Linas, Planetneutral, Reyk, Lambiam, A. Pichler, Hair Commodore, Addbot, Yobot, Aliotra, R. 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