Interpolatory four-parametric adaptive method with memory for solving nonlinear equations

Abstract

The adaptive technique enables us to achieve the highest efficiency index theoretically and practically. The idea of introducing an adaptive self-accelerator (via all the old information for Steffensen-type methods) is new and efficient to obtain the highest efficiency index. In this work, we have used four self-accelerating parameters and have increased the order of convergence from $8$ to $16$, i. e. any new function evaluations improve the convergence order up to $100\%$. The numerical results are compared without and with memory methods. It confirms that the proposed methods have more efficiency index.

Keywords

• nonlinear equations
• adaptive method with memory
• $r$-order convergence
• self accelerating parameter