The basic element method

Piecewise polynomial approximation of the sixth order with automatic knots detection

“…In the basic element method [8], polynomials are based on a three point grid in which variables and are functionally related to the independent variable x by a special rule of the cross ratio of four points of the straight line: = The functions of such a ratio depend on algebraic distances and on the order of points in the quadruple. For the quadruple with the fixed first point x, only three different functions relative to arguments τ, α, and β [7] exist: (1) where and…”

“…Theorem. Coefficients of the polynomial of the fifth degree presenting by its decomposition by degrees on the interval (8) are expressed by the following formulas: (9) where and are calculated by formulas (4)-(5), Proof. Formulas (9) follow from the equality of the right hand parts of Eqs.…”

“…Formulas (9) follow from the equality of the right hand parts of Eqs. (7) and (8). They are obtained taking into account (1), (3)-(5) after are written in the coordinate form and group ing the multipliers with the respective degrees Corollary 1.…”

“…In this work the PPA method of the sixth order based on polynomials of the fifth order presented in the form of basic elements (BEM) [7,8] is proposed. The use of BEM polynomials in PPA problems provides a benefit both in the accuracy of estimates and in the length of local supports.…”

“…The use of BEM polynomials in PPA problems provides a benefit both in the accuracy of estimates and in the length of local supports. In [8] the advantages of the use of BEM in solving problems of approximation of functions and smoothing data in terms of increasing the constancy of computations and reducing the computational complexity.…”