Two New Classes of Nonlinear Transformations for Accelerating the Convergence of Infinite Integrals and Series

Levin, David; Sidi, Avram

doi:10.1016/0096-3003(81)90028-x

Cited by 132 publications

(133 citation statements)

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“…To remedy this problem, it was proposed in [5] and [4] to apply GREP not to the whole sequence but to a subsequence {A κn } of {A n } with some integer κ ≥ 1. This choice of the subsequence has been called arithmetic progression sampling (APS for short) in [19,Chapter 10].…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

“…It is discussed rigorously in [19,Chapter 12,Section 12.7] for m = 1, where a numerical example is also provided. In conjunction with the d (m) transformation for arbitrary m ≥ 1, APS is applied to power series, Fourier series, and generalized Fourier series in [5], [14], [19 …”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

“…Before we continue, we would like to note that, because m ≥ 1 is an arbitrary integer, F (m) includes most functions that arise in applications, hence is a very comprehensive set. Some special cases of GREP (m) are the D transformation for computing infinite-range integrals and the d transformation for summing infinite series, both of which are due to Levin and Sidi [5], and the D, D, W , and mW transformations of the author [11], [12], [13], [17], for infinite-range (mostly oscillatory) integrals. The summation of power series and generalized Fourier series by the d transformation is the subject of Sidi and Levin [20] and Sidi [14].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic analysis of a generalized Richardson extrapolation process on linear sequences

Sidi

2009

Math. Comp.

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Abstract. In this work, we give a detailed convergence and stability analysis for the author's generalized Richardson extrapolation process GREP (m) as this is being applied to linearly convergent or divergent infinite sequences {A n }, whereThe quantity we would like to compute is A, whether it is the limit or antilimit of {A n }. Such sequences arise, for example, as partial sums of power series and of Fourier series of functions that have algebraic and/or logarithmic branch singularities. Specifically, we define the GREP (m) approximation A (m,j) n to A, with n = (n 1 , . . . , n m ) and α > 0, via the linear systemsn k , whereβ ki are additional unknowns. We study the convergence and stability properties of A (m,j)is a true convergence acceleration method for the sequences considered. In addition, we show that GREP (m) is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to GREP (m) with m > 1.

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Section: Proofs Of Main Resultsmentioning

confidence: 99%

Section: Proofs Of Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Asymptotic analysis of a generalized Richardson extrapolation process on linear sequences

Sidi

2009

Math. Comp.

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“…Traditional quadrature rules and summation techniques have failed to provide accurate approximations to such integrals. Numerous methods and techniques were developed for improving convergence of these challenging integrals [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and extremely efficient methods were introduced such as numerical steepest descent, Filon-type and Levin-type methods. Unfortunately, their application to complicated integrals is extremely challenging.…”

Section: Introductionmentioning

confidence: 99%

A Generalized Technique in Numerical Integration

Safouhi

2018

EPJ Web Conf.

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Abstract. Integration by parts is one of the most popular techniques in the analysis of integrals and is one of the simplest methods to generate asymptotic expansions of integral representations. The product of the technique is usually a divergent series formed from evaluating boundary terms; however, sometimes the remaining integral is also evaluated. Due to the successive differentiation and anti-differentiation required to form the series or the remaining integral, the technique is difficult to apply to problems more complicated than the simplest. In this contribution, we explore a generalized and formalized integration by parts to create equivalent representations to some challenging integrals. As a demonstrative archetype, we examine Bessel integrals, Fresnel integrals and Airy functions.

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“…Following Levin and Sidi [4], Sidi [10], [12], we give the definition below: Definition 1.1. We shall say that a function a(x), defined for x > a > 0, belongs to the set A(y\ if it is infinitely differentiable for all x > a, and if, as x -> oo, it has a Poincaré-type asymptotic expansion of the form 00 (i.i) «W~xT2«¡A'.…”

Section: Introduction Recently Levin and Sidimentioning

confidence: 99%

The numerical evaluation of very oscillatory infinite integrals by extrapolation

Sidi¹

1982

Math. Comp.

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Abstract. Recently the author has given two modifications of a nonlinear extrapolation method due to Levin and Sidi, which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity. In this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity. The new method is applied to a number of complicated integrals, among them the solution to a problem in viscoelasticity. Some convergence results for this method are presented.

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Two New Classes of Nonlinear Transformations for Accelerating the Convergence of Infinite Integrals and Series

Cited by 132 publications

References 9 publications

Asymptotic analysis of a generalized Richardson extrapolation process on linear sequences

Asymptotic analysis of a generalized Richardson extrapolation process on linear sequences

A Generalized Technique in Numerical Integration

The numerical evaluation of very oscillatory infinite integrals by extrapolation

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