Improving the convergence order of Steffensen’s method from two to four and its dynamic

In this paper, the degree of convergence of Newton’s method has been increased from two to four using two function evaluations. For this purpose,the weakness of Newton’s method, derivative calculation has been eliminated with a proper approximation of the previous data. Then, by entering two selfaccelerating parameters, the family new with-memory methods with Steffensen-Like memory with convergence orders of ۲.۴۱, ۲.۶۱, ۲.۷۳, ۳.۵۶, ۳.۹۰, ۳.۹۷, and ۴ are made. This goal has been achieved by approximating the self-accelerator parameters by using the secant method and Newton interpolation polynomials.Finally, we have examined the dynamic behavior of the proposed methods for solving polynomial equations.

Keywords:

With-memory method , Accelerator parameter , Basin of attraction , Efficiency index , Newton’s interpolatory polynomial