Weniger Transformation Research Articles

Numerical computation of diffraction catastrophes with codimension eight

A computational strategy, aimed at evaluating diffraction catastrophes belonging to the X(9) family is presented. The approach proposed is based on the use of power series expansions, suitably derived for giving meaningful representation of the whole (0)X(9) subfamily, jointly with a powerful sequence transformation algorithm, the so-called Weniger transformation. The convergence features of the above series expansions are investigated, and several numerical experiments are carried out to assess the effectiveness of the retrieving action of the Weniger transformation, as well as the ease of implementation of the whole approach.

Optimizing diffraction catastrophe evaluation

A theoretical analysis is proposed, aimed at investigating the character of those power series expansions recently considered for the evaluation of several types of diffraction catastrophes. A hyperlinear convergence is found to be the signature for such expansions, so that the results of the numerical experiments recently carried out find a meaningful interpretation in terms of the accelerating action operated by the Weniger transformation. As an important by-product of our analysis, simple criteria, aimed at numerically optimizing the diffraction catastrophe evaluations, are provided through analytical expressions.

Vacuum electron acceleration driven by a tightly focused radially polarized Gaussian beam

Electron acceleration in vacuum driven by a tightly focused radially polarized Gaussian beam has been studied in detail. Weniger transformation method is used to eliminate the divergence of the radially polarized electromagnetic field derived from the Lax series approach. And, electron dynamics in an intense radially polarized Gaussian beam is analyzed by using the Weniger transformation field. The roles of the initial phase of the electromagnetic field and the injection angle, position and energy of electron in energy gain of electron have been studied in detail.

Evaluation of cuspoid and umbilic diffraction catastrophes of codimension four

The evaluation of the two diffraction catastrophes of codimension four, namely, the butterfly and the parabolic umbilic, is here proposed by means of a simple computational approach developed in the past to characterize the whole hierarchy of the structurally stable diffraction patterns produced by optical diffraction in three-dimensional space. In particular, after expanding the phase integral representations of butterfly and parabolic umbilic in terms of (slowly) convergent power series, the retrieving action of the Weniger transformation on them is investigated through several numerical experiments. We believe that the methodology and the results presented here could also be of help for the dissemination of catastrophe optics to the widest scientific audience.

Decoding divergent series in nonparaxial optics

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.

Study of a generalized Levin–Weniger convergence accelerator of numerical series: Particular cases and applications to multi-electron integrals in atomic physics

In this work the general Levin transformation for a new remainder estimate is studied and compared to the u-transformation for some particular test cases. Also, the Weniger transformation and a generalization of both are studied specifically for the three-electron problem in atomic physics. In addition, we study the problem of the instability of these convergence accelerators and discuss attempts to reduce them.

On the numerical evaluation of umbilic diffraction catastrophes

A simple computational approach is proposed for the evaluation of umbilic diffraction catastrophes which, together with cuspoids, describe the whole hierarchy of the structurally stable diffraction patterns that can be produced by optical diffraction. In this paper, after expanding the double integral representations of hyperbolic and elliptic umbilics as convergent power series, the action of the Weniger transformation on them is studied. Exact expressions for the "on-axis" umbilic field have also been found, which extend previously published results to complex values of the control parameter. Numerical experiments aimed at giving evidence of the effectiveness and implementative ease of the approach are eventually presented.

Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. Higher-order transformations

The use of hyperasymptotics (H) and the Weniger transformation (WT) has been proposed, in a joint fashion, for decoding the divergent asymptotic series generated by the steepest descent on a wide class of saddle-point integrals evaluated across Stokes sets [R. Borghi, Phys. Rev. E 78, 026703 (2008)]. In the present sequel, the full development of the hyperasymptotic-Weniger transformation (H-WT) up to the second order in H is derived. Numerical experiments, carried out on several classes of saddle-point integrals, including the swallowtail diffraction catastrophe, show the effectiveness of the second-level H-WT, in particular when the integrals are evaluated beyond the asymptotic realm.

Method of summation of some slowly convergent series

A new method of summation of slowly convergent series is proposed. It may be successfully applied to the summation of generalized and basic hypergeometric series, as well as some classical orthogonal polynomial series expansions. In some special cases, our algorithm is equivalent to Wynn’s epsilon algorithm, Weniger transformation [E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10 (1989) 189–371] or the technique recently introduced by Čížek et al. [J. Čížek, J. Zamastil, L. Skála, New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field, Journal of Mathematical Physics 44 (3) (2003) 962–968]. In the case of trigonometric series, our method is very similar to the Homeier’s H transformation, while in the case of orthogonal series — to the K transformation. Two iterated methods related to the proposed method are considered. Some theoretical results and several illustrative numerical examples are given.

Simulation of Gaussian laser beams and electron dynamics by Weniger transformation method

For an electron accelerated by a tightly focused Gaussian laser beam, its dynamics are usually simulated through the field obtained by Lax approach [Phys. Rev. A 11, 1365 (1975)]. However, as Lax series field (LSF) is not always convergent, the obtained results are usually inaccurate and even illogical. Here we report that the divergence of LSF can be eliminated by using Weniger transformation, and the electron dynamics simulated by this new field are logical and accurate.

Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals

When the Weniger transformation is systematically used in resumming the asymptotic series arising from the application of the steepest descent method can display dramatic numerical instabilities that prevent it from improving the accuracy achievable via superasymptotics. In the present paper an explanation of such pathologies through the concept of resurgence, introduced by Berry and Howls several years ago within the context of hyperasymptotics, is proposed. In particular, the way the topology of the whole set of the saddles influences the resummation capabilities of the Weniger transformation is here investigated for the integrals defining the lowest-order cuspoid diffraction catastrophes. Eventually, a powerful and easily implementable resummation scheme, based on a joint use of the Weniger transformation and hyperasymptotics, is proposed for taking care of the above pathologies. Such a joint action seems to encompass the main virtues of both approaches, and the preliminary numerical results obtained from its application show that relative errors several orders of magnitude smaller than those achievable via superasymptotics can be achieved with modest implementation efforts.

On the numerical evaluation of cuspoid diffraction catastrophes

A simple computational approach is proposed for the evaluation of cuspoid diffraction catastrophes, which are fundamental tools for the description of wave fields in proximity to stable caustics. In particular, on representing the Pearcey and the swallowtail functions in terms of convergent series expansions, the action of the Weniger transformation on them is numerically studied for both real and complex values of the cuspoid arguments. Numerical experiments, aimed at giving evidence of the effectiveness and implementative ease of the proposed method, together with some quantitative comparisons to previously published results, are also presented.

Summing Pauli asymptotic series to solve the wedge problem

The use of the asymptotic treatment for the wedge diffraction problem established long ago by Pauli [Phys. Rev.54, 924 (1938)] is here revisited and proposed in the character of a powerful computational tool for accurately retrieving the total electromagnetic field even in the near zone. After proving its factorial divergent character, the Pauli series is summed through the Weniger transformation, a nonlinear resummation scheme particularly efficient in the case of factorial divergence. Numerical results are carried out to show the accuracy and effectiveness of the proposed approach.

Evaluation of diffraction catastrophes by using Weniger transformation

Weniger transformation is a powerful nonlinear sequence transformation that, when applied to the sequence of the partial sums of a divergent or a slowly convergent series, can convert it to a fast-converging sequence. Weniger transformation is not yet well known in optics. Diffraction catastrophes are fundamental tools for evaluating an optical field in proximity to caustics and singularities. The action of the Weniger transformation on the power series representation of diffraction catastrophes is numerically studied for two particular cases, corresponding to the Airy and the Pearcey functions. The obtained results clearly show that Weniger transformation could become a computational tool of great importance for summing several types of series expansions in optics.

New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field

The perturbation series for the ground state energy of the hydrogen atom in the external magnetic field is summed via the sequence transformations. The formula for the large-order behavior of the partial sums of the series is derived. From this formula a new general sequence transformation is suggested. This transformation contains free parameters that can be further optimized. It is shown that if the renormalization approach is used, the optimal choice of these parameters leads to the previously suggested Weniger transformation.