Abstract
In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub’s conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.
Highlights
There is a large number of problems in Science and Engineering modeled by nonlinear equations f (z) = 0, where f : D ⊆ Rn → Rn is a vectorial real function defined in a convex set D, with n ≥ 1
Many known schemes designed for solving nonlinear equations or nonlinear systems can be obtained as particular cases of (4) by using different weight functions satisfying the conditions of Theorem 3
Ghorbanzadeh and Soleymani presented in [11] an iterative method solving nonlinear equations or nonlinear systems, which is a particular case of our scheme using the weight function
Summary
There is a large number of problems in Science and Engineering modeled by nonlinear equations f (z) = 0, where f : D ⊆ Rn → Rn is a vectorial real function defined in a convex set D, with n ≥ 1 These problems cannot be analytically solved, so we use iterative schemes. The weight function procedure is used to achieve this goal, which is able to increase the order of convergence of known methods (see [1]), achieving optimal schemes under the point of view of Kung-Traub’s conjecture [4]. The aim of this section is to check how, using complex dynamics tools, we can determine which elements of a given family have good stability properties and which have chaotic behavior
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