Information on the thesis

Constructed and justified new methods for the numerical solution of nonlinear functional equations.

The first section carried out a brief overview of the work related to research methods for solving nonlinear functional equations with a differentiable and non-differentiable operator, namely the method of simple iteration, Newton’s method and its modifications, iterative difference methods. A review of the main choices of step factor, which simplifies the choice of the initial approximation.

The second section presents the three-step method based on Newton’s method, accelerated Newton’s method, the method of simple iteration. The basic idea of ​​these methods is to find two intermediate approximation, and the basic approach to the problem is, at least, on the line which connects the intermediate approach. A built three-step method based on Gauss-Newton method for solving nonlinear over determined systems. The convergence of the proposed methods and set their speed of convergence. It is shown that the rate of convergence of the three-step method is not less than the convergence of the corresponding basic mathods.

The third section presents the three-step methods with memory based on Newton’s method and the Gauss-Newton for the solution of nonlinear functional equations and nonlinear over determined systems respectively. These methods are used to re-found the information in the subsequent iteration. Also built three-step method based on the method with the rate of convergence. The convergence of the proposed methods and set their speed of convergence. A comparison of these basic methods. It has been shown that these methods in terms of the number of calculations, give the best results for both routine tasks and for tasks that are degenerate at the solution of the Jacobi matrix.

In the fourth section, proposed modifications difference of all the three-step methods, namely methods based on the chord, Steffensen, linear interpolation. Theoretical studies of these methods have proved their convergence and studied the speed of convergence.

Numerical experiments were conducted and three-step comparison of the difference of their modifications with the fundamental methods of methods. The advantage of the three-step method in comparison with the basis, in the sense of reducing the amount of computation.

Keywords: nonlinear functional equations, the problem of least squares, Newton’s method, the chord method, Steffensen method, linear interpolation, Gauss-Newton method, a system of nonlinear equations.