A family of Newton-type iterative methods using some special self-accelerating parameters

Abstract

In this paper, a family of Newton-type iterative methods with memory is obtained for solving nonlinear equations, which uses some special self-accelerating parameters. To this end, we first present two optimal fourth-order iterative methods with memory for solving nonlinear equations. Then we give a novel way to construct the self-accelerating parameter and obtain a family of Newton-type iterative methods with memory. The self-accelerating parameters have the properties of simple structure and easy calculation, which do not increase the computational cost of the iterative methods. The convergence order of the new iterative method is increased from 4 to $2+\sqrt{7}\approx \text{4}\text{.64575}$. 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 new methods. Experiment results show that, compared with the existing methods, the new iterative methods with memory have the advantage of costing less computing time.

KEYWORDS:

• Iterative method with memory
• self-accelerating parameter
• root-finding
• Newton method
• convergence order

2010 AMS SUBJECT CLASSIFICATIONS:

• 65H05
• 65B99

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The project was supported by the National Natural Science Foundation of China [Nos. 11547005 and 61572082], Doctor Startup Foundation of Liaoning Province of China [No. 201501196] and the Educational Commission Foundation of Liaoning Province of China [No. L2015012].