Using decomposition of the nonlinear operator for solving non‐differentiable problems

Abstract

Starting from the decomposition method for operators, we consider Newton‐like iterative processes for approximating solutions of nonlinear operators in Banach spaces. These iterative processes maintain the quadratic convergence of Newton's method. Since the operator decomposition method has its highest degree of application in non‐differentiable situations, we construct Newton‐type methods using symmetric divided differences, which allow us to improve the accessibility of the methods. Experimentally, by studying the basins of attraction of these methods, we observe an improvement in the accessibility of the derivative‐free iterative processes that are normally used in these non‐differentiable situations, such as the classic Steffensen's method. In addition, we study both the local and semilocal convergence of the considered Newton‐type methods.

Enhancing the Convergence Order from p to p + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices

“…This observation reinforces the practical viability of our technique and its potential to significantly accelerate numerical computations, supporting its relevance in academic problem solving. Now, we focus on the detailed study of a nonlinear elliptic initial value and contour problem, which was previously treated in [20]. To enrich and improve the analysis, we introduce an additional term in the equation, namely |u|, which makes the problem nondifferentiable.…”