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Ernst Joachim Weniger - One of the best experts on this subject based on the ideXlab platform.
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construction of new generalizations of wynn s epsilon and rho algorithm by solving finite difference equations in the transformation order
2019Co-Authors: Xiangke Chang, Yi He, Xingbiao Hu, Ernst Joachim WenigerAbstract:We construct new sequence transformations based on Wynn's epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn's epsilon and rho algorithm and of Osada's generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some Divergent Series.
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convergence analysis of the summation of the factorially Divergent euler Series by pade approximants and the delta transformation
2015Co-Authors: Riccardo Borghi, Ernst Joachim WenigerAbstract:Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging Series. However, our understanding of their theoretical properties is far from satisfactory. The Euler Series E ( z ) ~ ? n = 0 ∞ ( - 1 ) n n ! z n is a very important model for the ubiquitous factorially Divergent perturbation expansions in theoretical physics and for the Divergent asymptotic expansions for special functions. In this article, we analyze the summation of the Euler Series by Pade approximants and by the delta transformation, which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or Divergent Series. Our analysis is based on a very recent factorial Series representation of the truncation error of the Euler Series. We derive explicit expressions for the transformation errors of Pade approximants and of the delta transformation. A subsequent asymptotic analysis proves rigorously the convergence of both Pade and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Pade. This is in agreement with previous numerical results.
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nonlinear sequence transformations for the acceleration of convergence and the summation of Divergent Series
2003Co-Authors: Ernst Joachim WenigerAbstract:Slowly convergent Series and sequences as well as Divergent Series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of Divergent Series are discussed. Some of the sequence transformations of this report as for instance Wynn's $\epsilon$ algorithm or Levin's sequence transformation are well established in the literature on convergence acceleration, but the majority of them is new. Efficient algorithms for the evaluation of these transformations are derived. The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed. Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and Divergent Series, which are hopefully realistic models for a large part of the slowly convergent or Divergent Series that can occur in scientific problems and in applied mathematics.
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computation of the whittaker function of the second kind by summing its Divergent asymptotic Series with the help of nonlinear sequence transformations
1996Co-Authors: Ernst Joachim WenigerAbstract:The computation of the Whittaker function W κ,μ(z) via its Divergent asymptotic expansion is discussed. This Divergent Series can be summed with the help of Pade approximants, which convert its partial sums into rational functions [C. de Izarra, O. Vallee, J. Picart, and N. Tran Minh, Comput. Phys. 9, 318 (1995)]. However, this and related Divergent Series, as they for instance occur in special function theory or in quantum mechanical perturbation theory, can be summed much more efficiently by some alternative rational approximants that use explicit estimates for the truncation error [D. Levin, Int. J. Comput. Math. B 3, 371 (1973); E.J. Weniger, Comput. Phys. Rep. 10, 189 (1989)]. © 1996 American Institute of Physics.
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a convergent renormalized strong coupling perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator
1996Co-Authors: Ernst Joachim WenigerAbstract:Abstract The Rayleigh–Schrodinger perturbation Series for the energy eigenvalue of an anharmonic oscillator defined by the Hamiltonian Ĥ ( m ) ( β )= p 2 + x 2 + βx 2 m with m =2, 3, 4, … diverges quite strongly for every β ≠0 and has to summed to produce numerically useful results. However, a Divergent weak coupling expansion of that kind cannot be summed effectively if the coupling constant β is large. A renormalized strong coupling expansion for the ground state energy of the quartic, sextic, and octic anharmonic oscillator is constructed on the basis of a renormalization scheme introduced by F. Vinette and J. Cižek [ J. Math. Phys. 32 (1991), 3392]. This expansion, which is a power Series in a new effective coupling constant with a bounded domain, permits a convenient computation of the ground state energy in the troublesome strong coupling regime. It can be proven rigorously that the new expansion converges if the coupling constant is sufficiently large. Moreover, there is strong evidence that it converges for all physically relevant β ∈[0, ∞). The coefficients of the new expansion are defined by Divergent Series which can be summed efficiently with the help of a sequence transformation which uses explicit remainder estimates[E. J. Weniger, Comput. Phys. Rep. 10 (1989), 189].
Daniel Erman - One of the best experts on this subject based on the ideXlab platform.
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Divergent Series and serre s intersection formula for graded rings
2017Co-Authors: Daniel ErmanAbstract:Abstract On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a Divergent Series. But there are alternate geometric approaches for assigning (often fractional) intersection multiplicities in some singular settings. Our motivating question comes from Fulton, who asks whether an analytic continuation of the Divergent Series from Serre's formula can be related to these fractional multiplicities. We apply work of Avramov and Buchweitz to answer Fulton's question in the context of graded rings.
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Divergent Series and serre s intersection formula for graded rings
2015Co-Authors: Daniel ErmanAbstract:On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a Divergent Series. But there are alternate geometric approaches for assigning (often fractional) intersection multiplicities in some singular settings. Our motivating question comes from Fulton, who asks whether an analytic continuation of the Divergent Series from Serre's formula can be related to these fractional multiplicities. By applying work of Avramov and Buchweitz, we positively answer Fulton's question in the context of graded rings.
Bernard Candelpergher - One of the best experts on this subject based on the ideXlab platform.
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Ramanujan Summation of Divergent Series
2017Co-Authors: Bernard CandelpergherAbstract:In Chapter VI of his second Notebook Ramanujan introduce the Euler-MacLaurin formula to define the " constant " of a Series. When the Series is Divergent he uses this " constant " like a sum of the Series. We give a rigorous definition of Ramanujan summation and some properties and applications of it. These properties of the summation seems very unusual so in the last chapter we give a general algebraic view on summation of Series that unify Ramanujan summation with the classical summations procedures.
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ramanujan summation and the exponential generating function sum_ k 0 infty frac z k k zeta prime k
2010Co-Authors: Bernard Candelpergher, Gopalkrishna H Gadiyar, R PadmaAbstract:In the sixth chapter of his notebooks, Ramanujan introduced a method of summing Divergent Series which assigns to the Series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the Series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions ∑n≥1log n enz and \(\sum_{n\geq 1}H_{n}^{(j)}~e^{-nz}\) where \(H_{n}^{(j)}=\sum_{m=1}^{n}\frac{1}{m^{j}}\) . We find a surprising relation between the two sums when j=1 from which follows a formula that connects the derivatives of the Riemann zeta-function at the negative integers to the Ramanujan sum of the Divergent Euler sums ∑n≥1nkHn, k ≥ 0, where \(H_{n}=H_{n}^{(1)}\) . Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.
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ramanujan summation and the exponential generating function sum_ k 0 infty frac z k k zeta prime k
2009Co-Authors: Bernard Candelpergher, Gopalkrishna H Gadiyar, R PadmaAbstract:In the sixth chapter of his notebooks Ramanujan introduced a method of summing Divergent Series which assigns to the Series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the Series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from which follows a formula that connects the derivatives of the Riemann zeta - function at the negative integers to the Ramanujan summation of the Divergent Euler sums $\sum_{n\ge 1} n^kH_n, k \ge 0$, where $H_n= H_n^{(1)}$. Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.
Riccardo Borghi - One of the best experts on this subject based on the ideXlab platform.
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convergence analysis of the summation of the factorially Divergent euler Series by pade approximants and the delta transformation
2015Co-Authors: Riccardo Borghi, Ernst Joachim WenigerAbstract:Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging Series. However, our understanding of their theoretical properties is far from satisfactory. The Euler Series E ( z ) ~ ? n = 0 ∞ ( - 1 ) n n ! z n is a very important model for the ubiquitous factorially Divergent perturbation expansions in theoretical physics and for the Divergent asymptotic expansions for special functions. In this article, we analyze the summation of the Euler Series by Pade approximants and by the delta transformation, which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or Divergent Series. Our analysis is based on a very recent factorial Series representation of the truncation error of the Euler Series. We derive explicit expressions for the transformation errors of Pade approximants and of the delta transformation. A subsequent asymptotic analysis proves rigorously the convergence of both Pade and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Pade. This is in agreement with previous numerical results.
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decoding Divergent Series in nonparaxial optics
2011Co-Authors: Riccardo Borghi, Franco Gori, Giorgio Guattari, Massimo SantarsieroAbstract:A theoretical analysis aimed at investigating the Divergent character of perturbative Series involved in the study of free-space nonparaxial propagation of vectorial optical beams is proposed. Our analysis predicts a factorial divergence for such Series and provides a theoretical framework within which the results of recently published numerical experiments concerning nonparaxial propagation of vectorial Gaussian beams find a meaningful interpretation in terms of the decoding operated on such Series by the Weniger transformation.
Peter R Surjan - One of the best experts on this subject based on the ideXlab platform.
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analytic continuation approach to the resummation of Divergent Series in rayleigh schrodinger perturbation theory
2017Co-Authors: Zsuzsanna E Mihalka, Peter R SurjanAbstract:The method of analytic continuation is applied to estimate eigenvalues of linear operators from finite order results of perturbation theory even in cases when the latter is Divergent. Given a finite number of terms ${E}^{(k)},k=1,2,\ensuremath{\cdots}M$ resulting from a Rayleigh-Schr\"odinger perturbation calculation, scaling these numbers by ${\ensuremath{\mu}}^{k}$ ($\ensuremath{\mu}$ being the perturbation parameter) we form the sum $E(\ensuremath{\mu})={\ensuremath{\sum}}_{k}{\ensuremath{\mu}}^{k}{E}^{(k)}$ for small $\ensuremath{\mu}$ values for which the finite Series is convergent to a certain numerical accuracy. Extrapolating the function $E(\ensuremath{\mu})$ to $\ensuremath{\mu}=1$ yields an estimation of the exact solution of the problem. For Divergent Series, this procedure may serve as resummation tool provided the perturbation problem has a nonzero radius of convergence. As illustrations, we treat the anharmonic (quartic) oscillator and an example from the many-electron correlation problem.