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Program Systems: Theory and Applications, 2018, Volume 9, Issue 4, Pages 93–116 (Mi ps316)  

Mathematical Foundations of Programming

Numerical evaluation of the interpolation accuracy of simple elementary functions

S. V. Znamenskij

Ailamazyan Program Systems Institute of Russian Academy of Sciences

Abstract: Comparison of the accuracy of the restoration of elementary functions by the values in the nodes was made for algorithms of low-degree piecewise-polynomial interpolation. The test results clearly demonstrate in graphical form the advantages and disadvantages of the widely used cubic interpolation splines.
The comparison revealed that, contrary to popular belief, the smoothness of the interpolant is not directly related to the accuracy of the approximation. In the 20 disparate examples considered, the piecewise quadratic interpolation is rarely and only slightly inferior in the form of the used classical cubic splines, often by orders of magnitude better than many of them.
In several examples the high interpolation error of simple functions on a fixed grid appears to be almost independent of the degree of the algorithm and the smoothness of the interpolant. The piecewise-linear interpolation unexpectally appeared the most accurate in one of examples.
A new problem is presented: to find a local interpolation algorithm, exactly restoring any rational functions of the second order.

Key words and phrases: local interpolation, spline interpolation, convexity preserving, recovery precision.

DOI: https://doi.org/10.25209/2079-3316-2018-9-4-93-116

Full text: PDF file (2811 kB)
References: PDF file   HTML file

UDC: 519.652.3
MSC: 65D05; 65D07, 68W99
Received: 05.10.2018
24.10.2018
Accepted: 14.11.2018
Language:

Citation: S. V. Znamenskij, “Numerical evaluation of the interpolation accuracy of simple elementary functions”, Program Systems: Theory and Applications, 9:4 (2018), 93–116

Citation in format AMSBIB
\Bibitem{Zna18}
\by S.~V.~Znamenskij
\paper Numerical evaluation of the interpolation accuracy of simple elementary functions
\jour Program Systems: Theory and Applications
\yr 2018
\vol 9
\issue 4
\pages 93--116
\mathnet{http://mi.mathnet.ru/ps316}
\crossref{https://doi.org/10.25209/2079-3316-2018-9-4-93-116}


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