Convergence acceleration on a general class of power series

Gustafson, Sven-Åke

doi:10.1007/bf02252194

Cited by 32 publications

(35 citation statements)

References 11 publications

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“…We shall only show how to obtain (29) since, using Lemma 1, we can establish (30) in a similar way. Let 5 and 5(z) denote the matrices in (14) and (18) we conclude, from Lemma 3,that KiR) = KÍDzA¡-mDz) ~ KiiDb_aTaSDz)'H''+,iDb_aTaSDz)). G…”

Section: Remark 1)mentioning

confidence: 90%

“…As in known in the case of the Euler transform [23,14,15], conformai transformations can extend the region of convergence of power series and can enhance substantially the convergence rates inside the circles of convergence. In § §3 and 4 we show that they can also produce a rather dramatic improvement in the conditioning of Padé approximation.…”

Section: Introductionmentioning

confidence: 93%

“…In other words, errors (¿c") in the coefficients of (13) as follows from equations (14). We then define the condition number of the denominator value problem as the I2 condition number k"(z) of the matrix in (18).…”

Section: Enhanced Seriesmentioning

confidence: 99%

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Approximation of analytic functions: a method of enhanced convergence

Bruno¹,

Reitich²

1994

Math. Comp.

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Abstract. We deal with a method of enhanced convergence for the approximation of analytic functions. This method introduces conformai transformations in the approximation problems, in order to help extract the values of a given analytic function from its Taylor expansion around a point. An instance of this method, based on the Euler transform, has long been known; recently we introduced more general versions of it in connection with certain problems in wave scattering. In §2 we present a general discussion of this approach.As is known in the case of the Euler transform, conformai transformations can enlarge the region of convergence of power series and can enhance substantially the convergence rates inside the circles of convergence. We show that conformai maps can also produce a rather dramatic improvement in the conditioning of Padé approximation. This improvement, which we discuss theoretically for Stieltjes-type functions, is most notorious in cases of very poorly conditioned Padé problems. In many instances, an application of enhanced convergence in conjunction with Padé approximation leads to results which are many orders of magnitude more accurate than those obtained by either classical Padé approximants or the summation of a truncated enhanced series.

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Section: Remark 1)mentioning

confidence: 90%

Section: Introductionmentioning

confidence: 93%

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Approximation of analytic functions: a method of enhanced convergence

Bruno¹,

Reitich²

1994

Math. Comp.

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“…It is natural to consider the transformation Ez in (1.1) for z+-I as a generalized Euler transformation applied to power series ( [2,3,5,19]). Recursion formulas for the computation are derived by Wynn [22].…”

Section: Introductionmentioning

confidence: 99%

Numerical application of Euler's series transformation and its generalizations

Niethammer

1980

Numer. Math.

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Summary.A nonlinear generalization /~z of Euler's series transformation is compared with the (linear) Euler-Knopp transformation E~ and a twoparametric method E~,~. It is shown how to apply E, or E,,p to compute the value f(zo) of a function f from the power series at 0 iff is holomorphic in a half plane or in the 'cut' plane. Both E, and E~,~ are superior to /~=. A compact recursive algorithm is given for computing E~ and E,,p.

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“…We investigate the stability of the convergence acceleration methods in Sections 2 and 3 of [9] and prove Theorem 2. Let bn be the polynomial of degree n -1 obtained by developing (1 + zs)~x in a Taylor expansion around s = t and retaining the first n terms.…”

mentioning

confidence: 99%

On Stable Calculation of Linear Functionals

Gustafson¹

1979

Mathematics of Computation

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Abstract.In this paper we discuss the recurrent task of evaluating a linear functional defined by (generally infinitely many) linear constraints. We develop a theory for the stability of this problem and suggest a regularization procedure, based on orthogonal expansions. Simple and efficient computational schemes for evaluating the functional numerically are given. As a particular instance of the problem (1) and (2) is uniquely determined by the sequence cr = L{ar), r = 1, 2, . . . .Proof. Let bn be the polynomial of degree less than n which approximates b best in the maximum norm. bn is uniquely determined and II b -bn II -► 0 when n -► °°. Hence L{b) = limn_>00Z-(ôn), and the conclusion follows. Q.E.D.However, by practical calculations cr are known only with a finite accuracy and only finitely,many of the conditions (2) may be taken into account. Hence, a certain error is introduced in the calculated value of L{b) which is determined by approximating b (directly or indirectly) with linear combinations of a,, a2, . . . , an. The purpose of this paper is to extend and generalize the results in [6] and [7] as well as to de-

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Convergence acceleration on a general class of power series

Cited by 32 publications

References 11 publications

Approximation of analytic functions: a method of enhanced convergence

Approximation of analytic functions: a method of enhanced convergence

Numerical application of Euler's series transformation and its generalizations

On Stable Calculation of Linear Functionals

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