A comparison between two competing sixth convergence order algorithms under the same set of conditions

There is a plethora of algorithms of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them hard and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives not even appearing on these algorithms. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these algorithms and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other algorithms along the same lines.