Abstract
This paper stresses the theoretical nature of constructing the optimal derivative-free iterations. We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence. We also establish the connection of derivative-free and derivative presence three-point iterations. The use of the sufficient convergence conditions allows us to design wide class of optimal derivative-free iterations. The proposed family of iterations includes not only existing methods but also new methods with a higher order of convergence.
Highlights
At present, there are a lot of iterative methods for solving nonlinear equations and systems of equations
We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence
We establish the connection of derivative-free and derivative presence three-point iterations
Summary
There are a lot of iterative methods for solving nonlinear equations and systems of equations (see [1] [2] [3] and reference therein). The derivative-free methods are necessary when the derivative of the function f is unavailable or expensive to obtain. The derivative-free two and three-point methods with better convergence properties were developed (see [4]-[19] and references therein). It should be pointed out that most of these methods were proposed mainly for the concrete choice of parameters (see Table 1). A systematic theory or an approach for constructing derivative-free methods is still needed. It is of interest and necessity to develop a global theory.
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