Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations

Abstract

In this study, we extended the applicability of a derivative-free algorithm to encompass the solution of operators that may be either differentiable or non-differentiable. Conditions weaker than the ones in earlier studies are employed for the convergence analysis. The earlier results considered assumptions up to the existence of the ninth order derivative of the main operator, even though there are no derivatives in the algorithm, and the Taylor series on the finite Euclidian space restricts the applicability of the algorithm. Moreover, the previous results could not be used for non-differentiable equations, although the algorithm could converge. The new local result used only conditions on the divided difference in the algorithm to show the convergence. Moreover, the more challenging semi-local convergence that had not previously been studied was considered using majorizing sequences. The paper included results on the upper bounds of the error estimates and domains where there was only one solution for the equation. The methodology of this paper is applicable to other algorithms using inverses and in the setting of a Banach space. Numerical examples further validate our approach.

Keywords

• three step eighth order algorithm
• convergence
• divided differences
• differentiable- non-differentiable equation