Efficient Derivative-Free Class of Seventh Order Method for Non-differentiable Equations

Abstract

Many applications from a wide variety of disciplines in the natural sciences and also in engineering are reduced to solving of an equation or a system of equations in a correspondingly chosen abstract area. For most of these problems, the solutions are found iterative, because their analytic versions are difficult to find or impossible. This article encompasses efficient, derivatives-free, high-convergence iterative methods. Convergence of two types: Local and Semi-local areas will be investigated under the conditions of the ϕ, ψ-continuity utilizing operators on the method. The new method can also be applied to other methods, using inverses of the linear operator or the matrix.

Extended Newton-Traub Method of Order Six

“…Evaluating the local and semi-local characteristics of iterative techniques offers valuable insights into convergence traits, error limits, and the area of uniqueness for solutions [10][11][12][13][14]. Numerous research endeavors have concentrated on exploring the local and semi-local convergence of effective iterative approaches, yielding noteworthy outcomes like convergence radii, error approximations, and the broadened applicability of these methods [15][16][17][18][19].…”