Approximation of analytic functions: a method of enhanced convergence

Bruno, Oscar P.; Reitich, Fernando

doi:10.1090/s0025-5718-1994-1240654-9

Cited by 22 publications

(11 citation statements)

References 18 publications

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“…The optimal value of the parameters, however, should be chosen so as to obtain the fastest convergence for the series of (21). If the complex numbers 6j denote singularities of the functions B r (6), then A and a should minimize, for any given 5, the expression max (22) see [7]. A pair of optimal parameters A and a can be obtained, without any knowledge of the set of singularities of B r , simply by seeking parameters that yield the fastest numerical convergence.…”

Section: Numerical Results and Comparison With Experimental Datamentioning

confidence: 99%

“…Indeed, we have shown [7] that the relative arrangement of the singularities of an analytic function is closely related to the numerical conditioning of the corresponding Pade approximation. A conformal change of variables on a function B(6) can lead to a dramatic improvement in the conditioning of the corresponding Pade problem.…”

Section: 25mentioning

confidence: 97%

“…Since the main numerical weakness of Pade approximation is its ill conditioning, it is reasonable to expect that its use in conjunction with enhanced convergence would lead to improvement in the calculation of the diffraction efficiencies. A great deal of improvement has been obtained, as a matter of fact, by using this idea in some simple approximation problems [7]. However, there is a requisite that needs to be met: one needs to use accurate values of the series of the composite functions.…”

Section: 25mentioning

confidence: 99%

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Numerical solution of diffraction problems: a method of variation of boundaries

1993

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We have recently introduced a method of variation of boundaries for the solution of diffraction problems. This method, which is based on a theorem of analyticity of the electromagnetic field with respect to variations of the interfaces, has been successfully applied in problems of diffraction of light by perfectly conducting gratings. In this paper we continue our investigation of diffraction problems. Using our previous results on analytic dependence with respect to the grating groove-depth, we present a new numerical algorithm which applies to dielectric gratings. We also incorporate Pade approximation in our numerics. This addition enlarges the domain of applicability of our methods, and it results in computer codes which can predict more accurately the response of diffraction gratings in the resonance region. In many cases, results are obtained which are several orders of magnitude more accurate than those given by other methods available at present, such as the integral or differential formalisms.We present a variety of numerical applications, including examples for several types of grating profiles and for wavelengths of light ranging from microwaves to ultraviolet, and we compare our results with experimental data. We also use Pade approximants to gain insight on the analytic structure and the spectrum of singularities of the fields as functions of the groove-depth. Finally, we discuss some connections between Pade approximation and another summation mechanism, enhanced convergence, which we introduced earlier. It is argued that, provided certain numerical difficulties can be overcome, the performance of our algorithms could be further improved by a combination of these summation methods.

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Section: Numerical Results and Comparison With Experimental Datamentioning

confidence: 99%

Section: 25mentioning

confidence: 97%

Section: 25mentioning

confidence: 99%

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Numerical solution of diffraction problems: a method of variation of boundaries

1993

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“…In spite of this, one typically obtains a rational function that is far more accurate than the coefficients of the denominator polynomial. Partial explanations and further discussions of this are given in [5], [19]. Especially since we are using only matrices of moderate size, we have not found the ill-conditioning to be a concern.…”

Section: Accuracy and Time Comparisonsmentioning

confidence: 99%

A numerical methodology for the Painlevé equations

Fornberg

Weideman

2011

Journal of Computational Physics

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Abstract. The six Painlevé transcendents P I -P V I have both applications and analytic properties that make them stand out from most other classes of special functions. Although they have been the subject of extensive theoretical investigations for about a century, they still have a reputation for being numerically challenging. In particular, their extensive pole fields in the complex plane have often been perceived as 'numerical mine fields'. In the present work, we note that the Painlevé property in fact provides the opportunity for very fast and accurate numerical solutions throughout such fields. When combining a Taylor/Padé-based ODE initial value solver for the pole fields with a boundary value solver for smooth regions, numerical solutions become available across the full complex plane. We focus here on the numerical methodology, and illustrate it for the P I equation. In later studies, we will concentrate on mathematical aspects of both the P I and the higher Painlevé transcendents.Key words. Painlevé transcendents, P I equation, Taylor series method, Padé approximation, Chebyshev collocation method.1. Introduction. The six Painlevé transcendents were discovered about 100 years ago [4], [13], [23], [24], and have since received extensive attention. They satisfy nonlinear second order ODEs, cannot be reduced to other types special functions, and obey the Painlevé property of all its solutions being free from movable branch points (however, the solutions may have movable poles or movable isolated essential singularities) [18], [21]. They arise in many branches of physics, including classical and quantum spin models, scattering theory, self-similar solutions to nonlinear dispersive wave PDEs, string theory and random matrix theory. For a summary of their applications, with extensive references, we refer to Sections 32.13 -32.16 in the NIST Handbook of Mathematical Functions [21].The attention that the Painlevé functions have received over the last century has primarily been limited to their analytic properties and applications. The numerical computation of these functions, by contrast, has received comparatively little attention. With the exceptions of [8], [20], [22] (discussed later), numerical studies have so far mostly focused on pole-free intervals along the real line [17], [21], [25] or on a polefree region in the complex plane [11]. A possible explanation for this state of affairs is that the large pole fields in the complex plane associated with the Painlevé ODEs may be thought of as a challenge to standard numerical methods for the integration of ODEs.We argue in the present work that these pole fields provide, in fact, especially advantageous integration opportunities. A 'pole friendly' algorithm based on Padé approximation is devised that is capable of integrating accurately and stably through pole fields for distances in the 10 5 − 10 6 range. This allows numerical studies not only of 'near-fields', but also of transition processes towards 'far fields', for which an extensive body of asymptotic...

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“…On one hand, it allows us to perform a direct comparison (of stability, domain of applicability, and so forth) with the more classical methods. On the other hand, it opens the possibility for incorporating analytic continuation mechanisms to accelerate and/or enhance the convergence of the series; see, e.g., (2,3). The study of this latter possibility, and of its potential impact on a rather general setting of preconditioned spectral approximations (to accelerate the convergence of the underlying Neumann series), entails a further investigation of the analytic structure of the solution as a function of the perturbation parameter outside the disk of convergence of the series (see (1,2)), and will therefore be left for future work.…”

Section: Introductionmentioning

confidence: 99%

Stability of High-Order Perturbative Methods for the Computation of Dirichlet–Neumann Operators

Nicholls¹,

Reitich²

2001

Journal of Computational Physics

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117

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In this paper we present results on the stability of perturbation methods for the evaluation of Dirichlet-Neumann operators (DNO) defined on domains that are viewed as complex deformations of a basic, simple geometry. In such cases, geometric perturbation methods, based on variations of the spatial domains of definition, have long been recognized to constitute efficient and accurate means for the approximation of DNO and, in fact, several numerical implementations have been previously proposed. Inspired by our recent analytical work, here we demonstrate that the convergence of these algorithms is, quite generally, limited by numerical instability. Indeed, we show that these standard perturbative methods for the calculation of DNO suffer from significant ill-conditioning which is manifest even for very smooth boundaries, and whose severity increases with boundary roughness. Moreover, and again motivated by our previous work, we introduce an alternative perturbative approach that we show to be numerically stable. This approach can be interpreted as a reformulation of classical perturbative algorithms (in suitable independent variables), and thus it allows for both direct comparison and the possibility of analytic continuation of the perturbation series. It can also be related to classical (preconditioned) spectral approaches and, as such, it retains, in finite arithmetic, the spectral convergence properties of classical perturbative methods, albeit at a higher computational cost (as it does not take advantage of possible dimensional reductions). Still, as we show, an alternative approach such as the one we propose may be mandated in cases where substantial information is contained in high-order harmonics and /or perturbation coefficients of the solution.

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Approximation of analytic functions: a method of enhanced convergence

Cited by 22 publications

References 18 publications

Numerical solution of diffraction problems: a method of variation of boundaries

Numerical solution of diffraction problems: a method of variation of boundaries

A numerical methodology for the Painlevé equations

Stability of High-Order Perturbative Methods for the Computation of Dirichlet–Neumann Operators

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