Abstract
A variety of strategies are used to construct algorithms for solving equations. However, higher order derivatives are usually assumed to calculate the convergence order. More importantly, bounds on error and uniqueness regions for the solution are also not derived. Therefore, the benefits of these algorithms are limited. We simply use the first derivative to tackle all these issues and study the ball analysis for two sixth order algorithms under the same set of conditions. In addition, we present a calculable ball comparison between these algorithms. In this manner, we enhance the utility of these algorithms. Our idea is very general. That is why it can also be used to extend other algorithms as well in the same way.
Highlights
Academic Editor: Frank WernerReceived: 31 May 2021Accepted: 9 July 2021We consider two Banach spaces Y1 and Y2 with an open and convex subset Z (6= ∅)of Y1
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SM2 based on X (vii) do not work in this scenario, these algorithms may still converge with convergence order six
Summary
Sharma and Arora [18] constructed iterative algorithms of fourth and sixth convergence order for solving nonlinear systems. Conditions on derivatives of the seventh order and Taylor series expansion have been employed in [11,26] to determine their convergence rate. Because of such results needing derivatives of higher order, these algorithms are very difficult to implement, as their utility is reduced they may converge. The existing convergence results for methods SM1 and SM2 based on X (vii) do not work in this scenario, these algorithms may still converge with convergence order six This is the case, since the conditions in the aforementioned studies are only sufficient.
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