Robust Padé Approximation via SVD

Gonnet, Pedro; Güttel, Stefan; Trefethen, Lloyd N.

doi:10.1137/110853236

Cited by 108 publications

(127 citation statements)

References 30 publications

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“…Very recently, Gonnet et al suggested an efficient algorithm for the Padé approximations [30,31]. The algorithm detects and eliminates the spurious pole-zero pairs caused by the rounding errors by means of singular value decomposition for the Teoplitz matrix.…”

Section: Padé Approximationmentioning

confidence: 99%

“…In this subsection, we try to investigate the residues of the Padé approximation for some test functions. Up to now, the residue analysis has been mainly used for performance comparison between the different algorithms of the Padé approximation of the same order [30,31]. On the other hand, it seems that the study by using the information of the residue analysis is still rare in the Padé approximation [10,12].…”

Section: Numerical Accuracy and Spurious Polesmentioning

confidence: 99%

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A Numerical Test of Padé Approximation for Some Functions with Singularity

Yamada¹,

Ikeda

2014

International Journal of Computational Mathematics

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The aim of this study is to examine some numerical tests of Padé approximation for some typical functions with singularities such as simple pole, essential singularity, brunch cut, and natural boundary. As pointed out by Baker, it was shown that the simple pole and the essential singularity can be characterized by the poles of the Padé approximation. However, it was not fully clear how the Padé approximation works for the functions with the branch cut or the natural boundary. In the present paper, it is shown that the poles and zeros of the Padé approximated functions are alternately lined along the branch cut if the test function has branch cut, and poles are also distributed around the natural boundary for some lacunary power series and random power series which rigorously have a natural boundary on the unit circle. On the other hand, Froissart doublets due to numerical errors and/or external noise also appear around the unit circle in the Padé approximation. It is also shown that the residue calculus for the Padé approximated functions can be used to confirm the numerical accuracy of the Padé approximation and quasianalyticity of the random power series.

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Section: Padé Approximationmentioning

confidence: 99%

Section: Numerical Accuracy and Spurious Polesmentioning

confidence: 99%

A Numerical Test of Padé Approximation for Some Functions with Singularity

Yamada¹,

Ikeda

2014

International Journal of Computational Mathematics

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“…For example, the strength of the Borel-Padé-Laplace algorithm relies on is however also an obstacle in the speed of the method. Indeed, in some cases, it has to be combined to other techniques (such as singular value decomposition method [1,18]) to avoid the apparition of a singular matrix. As an alternative, factorial series may be used [15,27,28].…”

Section: Resultsmentioning

confidence: 99%

“…In a first test, a source term is added such that the solution is periodic in time. More precisely, u = cos(πt)ū (18) whereū has a Poiseuille profile. The approximate solution given by the series resummation approach is presented in Figure 7 where N = 8.…”

Section: Application To Fluid Mechanics Problemsmentioning

confidence: 99%

Borel-Laplace summation method used as time integration scheme

et al. 2014

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Abstract.A time integration method for the resolution of ordinary and partial differential equations is proposed. The method consists in computing a formal solution as a (eventually divergent) time series. Next, the Borel resummation method is applied to deduce an (sectorial) analytical solution. The speed and spectral properties of the scheme are analyzed through some examples. Applications to fluid mechanics are presented.Résumé. On propose une méthode numérique d'intégration temporelle d'équations différentielles ou aux dérivées partielles. Cette méthode consiste d'abordà calculer une solution sous forme de série formelle, dont le rayon de convergence peutêtre nul. Ensuite, la méthode de resommation de BorelLaplace est utilisée pour déduire une solution analytique (dans un secteur) de l'équation. La rapidité et les propriétés géométriques du schéma sont analyséesà travers quelques exemples. Des applications en mécanique des fluides sont présentées.

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“…Conversely, we will see by the following linear algebra that if m + n = N , then (2.4) has a solution such that p − f q N = 0, called a linearised solution. Furthermore, (2.4) can be used to define more general solutions in the case that n + m < N , called linearised least-squares solutions for the rational interpolation problem [8]. The notion of a least-squares problem is a general one, in which the objective is to minimise a sum of squares.…”

Section: The Proposed Methodmentioning

confidence: 99%

Computing Complex Singularities of Differential Equations with Chebfun

2013

SIURO

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Abstract. Given a solution to an ordinary differential equation (ODE) on a time interval, the solution for complex-valued time may be of interest, in particular whether the solution is singular at some complex time value. How can the solution be approximated in the complex plane using only the data on the interval? A polynomial approximation of the solution always fails to capture singularities; to extrapolate solutions with singularities, approximation with rational functions is more appropriate. In this paper, a robust form of rational interpolation and least-squares approximation, due to Pachón, Gonnet et al., is discussed and tested. It is found that the method avoids the issue of spurious poles found by many standard rational approximations, but that it is not suitable when a high degree of accuracy is required.

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Robust Padé Approximation via SVD

Cited by 108 publications

References 30 publications

A Numerical Test of Padé Approximation for Some Functions with Singularity

A Numerical Test of Padé Approximation for Some Functions with Singularity

Borel-Laplace summation method used as time integration scheme

Computing Complex Singularities of Differential Equations with Chebfun

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