Two New Three and Four Parametric With Memory Methods For Solving Nonlinear ‎Equations‎

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.