On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

Abstract

We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each iteration. Convergence including the order of convergence, the radius of convergence, and error bounds is presented. Theoretical results are verified through numerical experimentation. Stability of the proposed class is analyzed and presented by means of using new dynamics tool, namely, the convergence plane. Performance is exhibited by implementing the methods on nonlinear systems of equations, including those resulting from the discretization of the boundary value problem. In addition, numerical comparisons are made with the existing techniques of the same order. Results show the better performance of the proposed techniques than the existing ones.

A Fast Derivative-Free Iteration Scheme for Nonlinear Systems and Integral Equations

“…It is here assumed that A (α) is not singular and e (n) = x (n) − α is called the error at the n-th iterate and [6,21]:…”

Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations

“…There is another important class of multistep methods based on Jarratt methods or Jarratt-type methods [20][21][22]. Such methods have been extensively studied in the literature; see [23][24][25][26][27][28] and references therein. In particular, Alzahrani et al [23] have recently proposed a class of sixth order methods for approximating solution of H(x) = 0 using a Jarratt-like composite scheme.…”