NEWTON - KRYLOV - NEDZHIBOV METHOD SOLUTIONS TO A SYSTEM OF NONLINEAR EQUATIONS WITH QUADRATIC CONVERGENCE RATE

Abstract

Many problems in practical and engineering applications lead to finding solutions to a system of nonlinear equations with large numbers of variables and equations. The problem of finding the exact solution to a system of nonlinear equations is not always feasible, especially for large-scale systems. Therefore, finding approximate solutions to this class of equations is great importance. Since the appearance of the Newton's method, there have been many methods proposed by scientists to address this issue with the support of computer software. Research and development of improved methods for this algorithm are always of interest and focus for scientists. In this paper, we investigate the problem of finding approximate solutions to a system of nonlinear equations using the Newton-Krylov-Nedzhibov method. The convergence of the iterative method can only be confirmed through practical experiments. Using the properties of Krylov space and the properties of the mapping that satisfies the Lipschitz condition, we prove the convergence of the method. Moreover, we present the solution to a nonlinear equation system using Mathlab software.