On the optimal arrangement of approximation nodes of the Runge’s function

O. V. Kotova

State Educational Institution of Higher Professional Education «Donbas National Academy of Civil Engineering and Architecture», Makeyevka

Abstract: Function approximation is an important tool in mathematical modeling. A special case of approximation is interpolation. To conduct the research, we will consider the Runge's function, which is defined on the segment $[-1,1]$: $ f(x)=\frac{1}{1+25x^2}.$ In this paper, a method for implementing the optimal arrangement of approximation nodes is proposed using the example of the Runge's function for a certain number of nodes. The essence of the method is that in order to select the optimal location of the approximation nodes, the target function $F$ is compiled, the minimization of which ensures the optimal location of the nodes $x_i$ along the abscissa axis. The location of the approximation nodes along the ordinate axis is determined by calculating the values $y_i$ based on the original function. Traditionally, to interpolate a function, already known coordinates of nodes are used to calculate polynomial coefficients. This approach provides limited opportunities to control the location of interpolation nodes, since in fact the location has to be determined at random. The paper proposes a method based on the nonlinearity of space idea along the axes of the Cartesian's coordinate system. To control this nonlinearity, a polynomial function with the parameter $t$ is used. As an interpolation polynomial we will use the Lagrange's polynomial $L_n(f,x)$, which at nodes $x_i$ corresponds to the values $y_i$. We construct the objective function $F$ as the sum of the squares of the difference between the Runge's function $f(x)$ and the Lagrange's polynomial $L_n(f,x)$: $F(x)=\sum\limits_{i=1}^{m+1} \big(f(\xi_i)-L_n(f,\xi_i)\big)^2,$ where the points $\xi_i$ are defined as follows: $\xi_i=x_1+\frac{i-1}{m}(x_n-x_1).$ The function $F(x)$ is a function of $n$ variables $x=(x_1,\ x_2,\ ..., \ x_n)$ and is constructed by analogy with the sum of squares of regression residuals in regression analysis. Minimization of the objective function $F(x)$ on the interval $[-1,1]$ allows us to determine the the coordinates of the approximation nodes values $x_i$ that ensure minimal deviations from the original Runge's function. The article presents the application of the optimal node arrangement method for seven, ten and fifteen nodes. The obtained interpolation polynomial is compared with the polynomial constructed on Chebyshev's nodes. The polynomial constructed on nodes by the proposed method approximates better than the polynomial constructed on Chebyshev's nodes. For this conclusion, the mean square errors of the interpolation curves are compared. The advantages of the proposed method for optimizing the arrangement of approximation nodes are: low values of the mean square error, the method is stable to an increase in the number of nodes, a significant decrease in the degree of approximating polynomials compared to other approximation methods without the need to use piecewise functions. The disadvantages of the proposed method include the use of numerical methods for minimizing the objective function, which, when implementing existing methods for finding the minimum values of a function of many variables, largely depend on the quality of the choice of the initial approximation.

Keywords: approximation, interpolation, Runge's function, approximation nodes, objective function, Chebyshev's nodes, mean square error.

UDC: 517.5 + 519.65

MSC: 41A10