Abstract
The aim of this paper is to present a new semi-local convergence analysis for Newton’s method in a Banach space setting. The novelty of this paper is that by using more precise Lipschitz constants than in earlier studies and our new idea of restricted convergence domains, we extend the applicability of Newton’s method as follows: The convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. These advantages are obtained using the same information as before, since new Lipschitz constant are tighter and special cases of the ones used before. Numerical examples and applications are used to test favorable the theoretical results to earlier ones.
Highlights
In this study we are concerned with the problem of approximating a locally unique solution z∗ of equation G ( x ) = 0, (1)where G is a Fréchet-differentiable operator defined on a nonempty, open convex subset D of a Banach space E1 with values in a Banach space E2 .Many problems in Computational disciplines such us Applied Mathematics, Optimization, Mathematical Biology, Chemistry, Economics, Medicine, Physics, Engineering and other disciplines can be solved by means of finding the solutions of equations in a form like Equation (1) using MathematicalModelling [1,2,3,4,5,6,7]
Where G is a Fréchet-differentiable operator defined on a nonempty, open convex subset D of a Banach space E1 with values in a Banach space E2
We shall increase the convergence region by finding a more precise domain where the iterates {zn } lie leading to smaller Lipschitz constants which in turn lead to a tighter convergence analysis for Newton’s method than before
Summary
In this study we are concerned with the problem of approximating a locally unique solution z∗. A very important problem in the study of iterative procedures is the convergence region. We shall increase the convergence region by finding a more precise domain where the iterates {zn } lie leading to smaller Lipschitz constants which in turn lead to a tighter convergence analysis for Newton’s method than before. This technique can apply to improve the convergence domain of other iterative methods in an analogous way. The rest of the paper is structured as follows: In Section 2 we present the semi-local convergence analysis of Newton’s method Equation (2).
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