On the convergence of the Newton-Raphson method and some of its generalizations

Abstract

The article aims to improve the solving algorithms and methods of increasing the speed of convergence of the solution of nonlinear algebraic systems or even ill-conditioned, as well as some generalizations of the proposed method, in the sense of broadening the conditions of its application. These algebraic systems come, for example, from discretization with the method of finite elements of some boundary value problems that also contain nondifferentiable terms given by some boundary conditions and which are smoothed using the regularization methods. To solve these algebraic systems, an incremental-iterative algorithm is chosen, which involved a great computational effort, but proved to be useful. These proposed algorithms can simulate the evolution in time of some processes, such as quasistatic or dynamic cases, and the article proves that the use of the Newton- Raphson method and generalizations lead to an increase in the speed of convergence with a decrease in the calculation effort and to broadening the conditions of applicability.