Abstract
Developed here are sixteenth-order simple-root-finding optimal methods with generic weight functions. Their numerical and dynamical aspects are investigated with the establishment of a main theorem describing the desired optimal convergence. Special cases with polynomial and rational weight functions have been extensively studied for applications to real-world problems. A number of computational experiments clearly support the underlying theory on the local convergence of the proposed methods. In addition, to investigate the relevant global convergence, we focus on the dynamics of the developed methods, as well as other known methods through the visual description of attraction basins. Finally, we summarized the results, discussion, conclusion, and future work.
Highlights
The governing equations of real-world natural phenomena are often described by nonlinear equations whose exact solutions are infeasible due to their inherent complexities
For the sake of comparison with the new optimal family of methods to be proposed in this paper, we introduce existing three optimal sixteenth-order Equations [12,13,14] respectively given by Equations (1), (2), and (4) below
The main theorem is established by describing the error equation and the asymptotic error constant with relationships among generic weight functions Q f (s), K f (s, u), and J f (s, u, v): f ( j) (α)
Summary
The governing equations of real-world natural phenomena are often described by nonlinear equations whose exact solutions are infeasible due to their inherent complexities. The classical second-order Newton’s method is best known as the numerical root-finder for the governing equations. Many authors [1,2,3,4,5,6,7,8,9,10,11] have developed higher-order multipoint methods. A few authors [12,13,14] have recently established optimal sixteenth-order methods, despite the lack of applicability to real-life nonlinear governing equations due to their algebraic complexities, to emphasize the theoretical importance of developing extremely high-order methods, and to apply them to root-finding of real-world nonlinear problems, we strongly desire to establish a new optimal family of sixteenth-order simple-root finders that are comparable to or competitive against the existing methods
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