Ball Comparison Between Four Fourth Convergence Order Methods Under the Same Set of Hypotheses for Solving Equations

Abstract

There is a plethora of techniques used to generate iterative methods. But the convergence order is determined by assuming the existence of higher order derivative for the operator involved. Moreover, these techniques do not provide estimates on error distances or results on the uniqueness of the solution based Lipschitz or Hölder type conditions. Hence, the use-fullness of these schemes is very restricted. We deal with these challenges using only the first derivative which is only actually appearing on these schemes and under the same set of conditions. Moreover, we provide a computable ball comparison between these schemes. That is how we extend these methods under weaker conditions. Numerical experiments are conducted to find the convergence balls and test the criteria of convergence. Moreover, different set of criteria usually based on the fifth derivative are needed in the ball convergence of fourth order methods. Then, these methods are compared using numerical examples. But we do not know: if the results of those comparisons are true if the examples change; the largest radii of convergence; error estimates on $\Vert x_n-\alpha \Vert$ and uniqueness results that are computable. We conclude that DSNM is the best among these four methods. Examples are used to compare the results. Our ideas can be utilized to make comparisons between other methods.