Abstract
In this paper we present a two-step solver for nonlinear equations with a nondifferentiable operator. This method is based on two methods of order of convergence 1 + 2 . We study the local and a semilocal convergence using weaker conditions in order to extend the applicability of the solver. Finally, we present the numerical example that confirms the theoretical results.
Highlights
A plethora of real-life applications from various areas, including Computational Science and Engineering, are converted via mathematical modeling to equations valued on abstract spaces such as n-dimensional Euclidean, Hilbert, Banach, and other spaces [1,2]
That is why we use our ideas of the center-Lipschitz condition, in combination with the notion of the restricted convergence region, to present local as well as semilocal improvements leading to the extension of the applicability of iterative methods
Extending the choice of initial points without imposing additional, more restrictive, conditions than before is extremely important in computational sciences. This difficult task has been achieved by defining a convergence region where the iterates lie, that is more restricted than before, ensuring the Lipschitz constants are at least as small as in previous works
Summary
A plethora of real-life applications from various areas, including Computational Science and Engineering, are converted via mathematical modeling to equations valued on abstract spaces such as n-dimensional Euclidean, Hilbert, Banach, and other spaces [1,2]. Researchers face the great challenge of finding a solution x∗ in the closed form of the equation This task is generally very difficult to achieve. One considers methods that mix Newton and secant steps to increase the order of convergence. This is our first objective in this paper. That is why we use our ideas of the center-Lipschitz condition, in combination with the notion of the restricted convergence region, to present local as well as semilocal improvements leading to the extension of the applicability of iterative methods. Two-step methods have some advantages over one-step methods They usually require fewer number of iterations for finding an approximate solution. Numerical results for method (2) were presented in [10,12]
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