The Second Order Spectrum and Optimal Convergence

A new optimal method of fourth-order convergence for solving nonlinear equations

In this paper, we present a fourth order method for computing simple roots of nonlinear equations by using suitable Taylor and weight function approximation. The method is based on Weerakoon-Fernando method [S. Weerakoon, G.I. Fernando, A variant of Newton's method with third-order convergence, Appl. Math. Lett. 17 (2000) 87-93]. The method is optimal, as it needs three evaluations per iterate,...

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 nonline...

An ${cal O}(h^{8})$ optimal B-spline collocation for solving higher order boundary value problems

As we know the approximation solution of seventh order two points boundary value problems based on B-spline of degree eight has only ${cal O}(h^{2})$ accuracy and this approximation is non-optimal. In this work, we obtain an optimal spline collocation method for solving the general nonlinear seventh order two points boundary value problems. The ${cal O}(h^{8})$ convergence analysis, mainly base...

The second order spectrum and optimal convergence

A NOTE ON "A SIXTH ORDER METHOD FOR SOLVING NONLINEAR EQUATIONS"

In this study, we modify an iterative non-optimal without memory method, in such a way that is becomes optimal. Therefore, we obtain convergence order eight with the some functional evaluations. To justify our proposed method, some numerical examples are given.