Abstract
The study of the dynamics and the analysis of local convergence of an iterative method, when approximating a locally unique solution of a nonlinear equation, is presented in this article. We obtain convergence using a center-Lipschitz condition where the ball radii are greater than previous studies. We investigate the dynamics of the method. To validate the theoretical results obtained, a real-world application related to chemistry is provided.
Highlights
A well known problem is that of approximating a locally unique solution x ∗ of equation F ( x ) = 0, (1)where F is a differentiable function defined on a nonempty convex subset D of S with values in Ω, where Ω can be R or C
Once we propose to find the solution iteratively, it is mandatory to study the convergence of the method. This convergence is usually seen in two different ways, which gives rise to two different categories, the semilocal convergence analysis and the local convergence analysis
The first of these, the semilocal convergence analysis, is based on information around an initial point, which will provide us with criteria that will ensure the convergence of an iteration procedure
Summary
A well known problem is that of approximating a locally unique solution x ∗ of equation. We must realize that there are a lot of iterative methods to approximate solutions of nonlinear equations defined in R or C [32,35,36,37,38] These studies show that if the initial point x0 is close enough to the solution x ∗ , the sequence { xn } converges to x ∗. The local results do not provide us with information about the radius of the convergence ball for the corresponding method. We can use the same technique with other different methods
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