High-Order Iterative Methods
Ioannis Konstantinos Argyros · 2021
In this chapter, we develop a sixth order Steffensen-type method with one parameter to solve systems of equations. Our study&s;s novelty lies in the fact that two types of local convergence are established under weak conditions including computable error bounds and uniqueness of the results. The performance of our methods is discussed and compared to other schemes using similar information. Finally, very large systems of equations (100 × 100 and 200 × 200) are solved to test the theoretical results and compare them favorably to earlier works. Introduction: Numerous problems from the Biology, Chemistry, Economics, Engineering, Mathematics, Physics are converted to a mathematical expression of the following form F ( u ) = 0. (7.1) Here, F: Ω ⊂ B → B, is differentiable, B is a Banach space and Ω is nonempty and open. Closed form solutions are rarely found, so iterative methods are used converging to the solution u ∗. In particular, we propose the following new scheme yp = up –[ up + F ( up ), up ; F] −1 F ( up ) zp = up −λ[ up + F ( up ), up ; F] −1 (F ( up )+ F ( yp ))−(1−λ)[ up , yp ; F] −1 F ( up ) up +1 = zp –[ zp + F ( zp ), zp ; F] −1 F ( zp ), (7.2), where u 0 ∈ Ω is an initial point and λ ∈ R is a free parameter. In addition to this, [·,·; F ] : Ω × Ω → l(B, B) is a divided difference of order one. We shall present two convergence analyses. Later we present the advantages over other methods using similar information.