Distributions of zeros and poles of $N$-point  Padé approximants  to complex-symmetric functions defined at complex points

R. Jedynak; J.   Gilewicz

doi:10.37863/umzh.v73i8.333

R. Jedynak K. Pulaski Univ. Technology and Humanities, Radom, Poland
J. Gilewicz K. Pulaski Univ. Technology and Humanities, Radom, Poland

DOI: https://doi.org/10.37863/umzh.v73i8.333

Keywords: .

Abstract

UDC 517.5

The knowledge of the location of zeros and poles Padé and $N$-point Padé approximations to a given function $f$ provides much valuable information about the function being studied.
In general PAs reproduce the exact zeros and poles of considered function, but, unfortunately, some spurious zeros and poles appear randomly.
Then, it is clear that the control of the position of poles and zeros becomes essential for applications of Padé approximation method.
The numerical examples included in the paper show how necessary for the convergence of PA is the knowledge of the position of their zeros and poles.
We relate our research of localization of poles and zeros of PA and NPA in the case of Stieltjes functions because we are interested in the efficiency of numerical application of these approximations.
These functions belong to the class of complex-symmetric functions.
The PA and NPA to the Stieltjes functions in different regions of the complex plane is also analyzed.
It is expected that the appropriate selection of the complex point for the definition of approximant can improve it with respect to the traditional choice of $\zeta = 0.$
All considered cases are graphically illustrated.
Some unique numerical results presented in the paper, which are sufficiently regular should motivate the reader to reflect on them.

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