Document Type : Research Paper

Authors

• Vali Torkashvand ^{1, 2}
• Reza Ezzati ³

¹ ‎‎‎Young Researchers and Elite Club, Shahr-e-Qods Branch, Islamic Azad University Tehran Iran

² Farhangian University, Tehran, Iran

³ Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Abstract

The main goal of this work, obtaining a family of Steffensen-type iterative methods adaptive with memory for solving nonlinear equations, which uses three self-accelerating parameters. For this aim, we present a new scheme to construct the self-accelerating parameters and obtain a family of Steffensen-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 methods has increased from 4 to 8. Also, these methods possess very high computational efficiency. Another advantage of the new method is that they remove the severe condition $f'(x)$ in a neighborhood of the required root imposed on Newton's method. Numerical comparisons have made to show the performance of the proposed methods, as shown in the illustrative examples.‎

Keywords

• Nonlinear equations
• Newton's interpolatory polynomial
• Adaptive method with memory
• The order of convergence
• Self accelerating parameter‎‎