ABSTRACT
Nonlinear equations are ubiquitous in many areas of applied mathematics and play vital roles in a number of applications such as science and engineering. Nonlinear equations are usually difficult to solve analytically, therefore a numerical method is needed. This chapter defines a self-accelerating parameter, which is calculated during the iterative process using Newton’s interpolating polynomial. Accelerating technique relies on information from the current and the previous iterative step, defining in this way two-point methods with memory. The chapter devotes to modifications of the two steps of Guo and Qian methods and is concerned with developing new with-memory methods. It aims at accelerating convergence without imposing further functional evaluations per cycle and attempts to prove that the R-order of convergence of a new derivative-free methodsin Equation.
