GLOBAL INTERPOLATION BY A POINT POLYNOMIC GEOMETRIC COMPOSITION OF THREE POINTS, AMONG THESE TWO-TIME

Abstract

The article shows the sequence of performing parameterization, along the coordinate axis, of the original discretely presented line (DPL) and provides in general form an interpolation point polynomial. A possible embodiment of the interpolation of a discretely presented curve (DPC) is considered, which is formed by three points that have turned into a triple point and the parameter values for this option are provided. It is indicated that with the appearance of multiple points on the DPL in the constituent elements of a parametric point polynomial, uncertainties arise. It is proved that all these uncertainties are revealed, the boundaries of which, at nodal points, is zero or one. It is indicated that with the appearance of multiple points on the DPL in the constituent elements of the parametric Lagrange polynomial, uncertainties arise. It is proved that all these uncertainties are revealed, the boundaries of which, at nodal points, are zero or one. It is shown that the uncertainties that arise with the appearance of multiple points on the LPL are not an obstacle to global interpolation using a parametric point polynomial. That is, for any composition of three points, the construction and recording structure of a parametric point polynomial remains unchanged. At the same time, there are no restrictions on creating a composition of three points. It is proved that all these uncertainties are revealed, the boundaries of which, at nodal points, are zero or one. It is shown that the uncertainties that arise with the appearance of multiple points on the LPL are not an obstacle to global interpolation using a parametric polynomial in the Lagrange form. This fact is proved in this article. A composite numerical matrix is provided according to which interpolation occurs. The elements of this compositional matrix is the value of the characteristic functions of the interpolant at the nodal points. It is shown that the elements of the composite interpolation matrix do not change in the presence of any geometric composition of three points. Only the status of these elements can be changed. In one case, their values are accurate, and in the other, they can be the boundary to which the value of the characteristic function of the interpolation point parametric polynomial follows.