Abstract
In this paper, we obtain two iterative methods with memory by using inverse interpolation. Firstly, using three function evaluations, we present a two-step iterative method with memory, which has the convergence order 4.5616. Secondly, a three-step iterative method of order 10.1311 is obtained, which requires four function evaluations per iteration. Herzberger’s matrix method is used to prove the convergence order of new methods. Finally, numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods.
Highlights
Solving nonlinear equations is one of the most important problems in scientific computation.Since 1960’s, many multipoint iterative methods have been proposed for solving nonlinear equations of the form f ( x ) = 0
Two new iterative methods with memory are proposed for solving nonlinear equations, which are constructed by using inverse interpolation method
In order to improve the computational efficiency of iterative method, we construct a three-step iterative method by using inverse interpolation rational polynomial
Summary
Solving nonlinear equations is one of the most important problems in scientific computation. Inverse interpolation method and self-accelerating parameter method are two effective ways to construct iterative method with memory. Have proposed some derivative free iterative methods with one self-accelerating parameter for solving nonlinear equations. We [4,5,6] have obtained some Newton type iterative methods with memory using one simple self-accelerating parameter, which is constructed by the iterative sequences. The self-accelerating parameter will be very complex if it is constructed by high order interpolation polynomial. In order to save computing time, we should construct the self-accelerating parameter with simple structure. Two new iterative methods with memory are proposed for solving nonlinear equations, which are constructed by using inverse interpolation method. The basins of attraction of existing methods and our methods are presented and compared to illustrate their performance
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