Abstract
In order to find the zeros of nonlinear equations, in this paper, we propose a family of third-order and optimal fourth-order iterative methods. We have also obtained some particular cases of these methods. These methods are constructed through weight function concept. The multivariate case of these methods has also been discussed. The numerical results show that the proposed methods are more efficient than some existing third- and fourth-order methods.
Highlights
Newton’s iterative method is one of the eminent methods for finding roots of a nonlinear equation: f (x) = 0. (1)Recently, researchers have focused on improving the order of convergence by evaluating additional functions and first derivative of functions
In order to find the zeros of nonlinear equations, in this paper, we propose a family of third-order and optimal fourth-order iterative methods
In order to improve the order of convergence and efficiency index, many modified third-order methods have been obtained by using different approaches
Summary
Newton’s iterative method is one of the eminent methods for finding roots of a nonlinear equation:. Kung and Traub [4] presented a hypothesis on the optimality of the iterative methods by giving 2n−1 as the optimal order. It means that the Newton iteration by two function evaluations per iteration is optimal with 1.414 as the efficiency index. The concept of weight functions has been used to obtain different classes of third- and fourthorder methods; one can see [5,6,7] and the references therein. We employ some numerical examples and compare the performance of our proposed methods with some existing third- and fourth-order methods
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