Abstract
The paper contains a comparison of the accuracy of the restoration of elementary functions by the values in the nodes for algorithms of low-degree piecewise-polynomial interpolation. The test results 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 different 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 unexpectedly appeared the most accurate in one of the examples. A new problem arises: to find a local interpolation algorithm, accurately restoring any rational functions of the second order.
Highlights
The problem of recovering a piecewise analytic on the segment [a, b] real function f (x) by the vector ⃗y = (y0, . . . , yn) from its n + 1 values yi = f (xi) at the points x0 < · · · < xn, . . . , n of the real line has a multi-millennial history [1] and various generalizations and applications
We focus on the interpolation algorithms available with the source code, capable to recover simple dependencies by a few specified values
We beleive that identifying the weak points of algorithms from this point of view can lead to the formulation of new problems, the solution of which could improve the quality of interpolation in other applications
Summary
We focus on the interpolation algorithms available with the source code, capable to recover simple dependencies by a few specified values. This context naturally arises in the analysis of scarce experimental data. We beleive that identifying the weak points of algorithms from this point of view can lead to the formulation of new problems, the solution of which could improve the quality of interpolation in other applications. This investigation focuses on two practically significant interpolation quality criteria:. To identify the effect of the interpolant’s smoothness, we consider a simple local quadratic continuous (not necessarily smooth) interpolating spline formula and evaluate its interpolation accuracy on the examples of simple elementary functions, analyzing specific situations arising during comparison
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