Padé approximants and noise: A case of geometric series

Gilewicz, Jacek; Pindor, M.

doi:10.1016/s0377-0427(97)00185-4

Cited by 46 publications

(38 citation statements)

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“…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%

“…The problem of constructing the -transform ( ) of a finite time-series is a standard problem in mathematics [10][11][12][13][14]. For example, it is shown that for a sum of oscillating damped signals, the -transform associated with the timeseries can be characterized by a sum of the poles of the Padé approximated function.…”

Section: Froissart Doubletsmentioning

confidence: 99%

“…The position of each pole is simply linked to the damping factor and the frequency of each of the oscillators. Also, it is important to note that all these poles lie strictly outside the unit circle because it corresponds to the damping [10][11][12][13]. In addition, we will consider quasianalyticity property of the random power series by the residue analysis of the Padé approximation.…”

Section: Froissart Doubletsmentioning

confidence: 99%

“…In particular, there have been several important examples in physics, to which the Padé approximation was applied, such as summation of the divergent Rayleigh-Schrodinger perturbation series in scattering theory [4], critical phenomena in statistical physics [5][6][7][8][9], denoising from noisy data of time-series [10][11][12][13][14], and detection of singularity of phase space trajectories of Hamiltonian dynamical systems [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

“…In addition, we can see an interesting example concerning the Padé approximation in noisy data analysis [10][11][12][13][14]. The power series with finite random coefficient "almost always" has a natural boundary on the unit circle in the complex plane [20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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: Numerical Accuracy and Spurious Polesmentioning

confidence: 99%