The Generalization Known Methods to Approximate Various Sets of Discreet Data

In the process of mathematical modeling a necessity often arises to smoothly approximate various dependencies which are defined discretely or graphically. Especially if the value of such dependencies are obtained as a result of complex experiments or cumbersome calculations. The inverse transform of continuous simulated objects in discrete digital format that is used for storage and computer processing also requires a certain ordering. It is assumed beforehand that the smooth approximation of discrete set of points on the plane is performed by linear analytical model. Interpolation conditions lead to a system of linear equations with a square matrix. When interpolating polynomials by breaking and rearranging the terms of a power series one can get such basic functions as Lagrange polynomials or Bernstein polynomials. Other methods of interpolation are Newton polynomials, Aitken iterative process, etc. However, these methods realize only some particular cases of all possible approximations of discrete data by arbitrary basis functions and are mainly focused on manual calculations. In computer calculations, it is desirable to find a general algorithm for solutions in order to avoid programming many particular cases. The problem of generalization of existing methods for approximation of discrete data sets (generalized algorithm) and bringing these discrete data to a common form (a discrete unified structure) is considered.

discrete structure, the linear analytical model interpolation conditions, approximation, collocation method, Galerkin method