Proof and Refinement of the Cordero–Torregrosa Bound in a Class of Memoryless Rational Methods

In solving systems of nonlinear equations, one of the main objectives is to construct iterative methods that allow us to approach a solution rapidly. However, the order of convergence that a method can attain is related to the information it obtains from the function and its derivatives. A natural question then arises: what is the highest order of convergence that can be guaranteed when the number of available evaluations is limited? The Cordero–Torregrosa conjecture proposes a bound for this order in the case of nonlinear systems. In this work, we prove this bound for a class of memoryless methods defined by a finite number of rational operations. Our proof allows the function and its Jacobian to be evaluated at different points. Moreover, when Jacobian evaluations are performed at points where the function has previously been evaluated, we obtain a more restrictive bound on the order of convergence. Finally, we present some examples showing that certain orders of convergence reported in the literature do not hold for all nonlinear systems. These results help clarify the limitations imposed by the available information on the order of convergence of iterative methods.