Method of optimal placement of approximation nodes

In this article we propose a method for optimizing the arrangement of approximation nodes and use Runge function as an example to implement this approach. The method is based on the idea of nonlinearity of space along the axes of Cartesian coordinate system.  To control the nonlinearity, we use a polynomial function with a parameter uniformly distributed over the segment $[0,1]$.  A comparative analysis of the following standard methods of selecting nodes for the approximation of Runge function was carried out: uniformly along the abscissa axis, uniformly along the ordinate axis, uniformly along the curve length, and by Chebyshev's nodes. To compare the Lagrange interpolation polynomials, we estimate the approximation errors of Runge's function. Graphs of the constructed Lagrange's polynomials for five and seven interpolation nodes selected in different ways are presented. To select the optimal arrangement of approximation nodes of the proposed method, we compile an objective function, whose minimization ensures optimal arrangement of nodes $x_i$ along the abscissa axis.  The arrangement of approximation nodes along the ordinate axis is determined by calculating the $y_i$ values basing on the original Runge's function. As a result, we found nodes that provide minimal deviations from the original approximated Runge's function.  The paper considers cases of five and seven approximation nodes.  To visualize the results obtained, we provide graphs of original Runge's function and of its approximation, indicating the optimal nodes found. This method is stable to increasing the number of nodes, whose arrangement is optimized each time and adapted to the original function.