Two new three and four parametric with memory methods for solving nonlinear equations
الموضوعات : مجله بین المللی ریاضیات صنعتی
T. Lotfi
1
(Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran.)
P. Assari
2
(Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran.)
الکلمات المفتاحية: Nonlinear equation, With memory method, R-order of convergence, Self accelerating parameter, Efficiency index,
ملخص المقالة :
In this study, based on the optimal free derivative without memory methods proposed by Cordero et al. [A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation, Mathematical and Computer Modeling. 57 (2013) 1950-1956], we develop two new iterative with memory methods for solving a nonlinear equation. The first has two steps with three self-accelerating parameters, and the second has three steps with four self-accelerating parameters. These parameters are calculated using information from the current and previous iteration so that the presented methods may be regarded as the with memory methods. The self-accelerating parameters are computed applying Newton's interpolatory polynomials. Moreover, they use three and four functional evaluations per iteration and corresponding R-orders of convergence are increased from 4 ad 8 to 7.53 and 15.51, respectively. It means that, without any new function calculations, we can improve convergence order by $93\%$ and $96\%$. We provide rigorous theories along with some numerical test problems to confirm theoretical results and high computational efficiency.
