Proof of the Cordero–Torregrosa Conjecture and an Improved Order Bound for Paired Jacobian Evaluations

In solving systems of nonlinear equations, one of the main objectives is to construct iterative methods that attain high orders of convergence using as few evaluations as possible. However, the information a method obtains during each iteration imposes limitations on the order it can guarantee. The Cordero–Torregrosa conjecture proposes a bound on this order in terms of the number of evaluations of the function and its Jacobian. In this work, we prove this bound for the universal local order of one iteration, without requiring the number of Jacobian evaluations to be less than or equal to the number of function evaluations. In particular, our result also covers the case in which more evaluations of the Jacobian than of the function are performed. The proof encompasses deterministic, causal, memoryless methods whose rules may be arbitrary, without imposing rationality, continuity, or computability conditions. Moreover, we prove that when each Jacobian evaluation is performed at a point where the value of the function is already known, a more restrictive bound on the order of convergence can be obtained. Finally, we propose a new conjecture that establishes a refined bound through a product of factors determined by the distribution of evaluations, under additional dimensional conditions. These results provide a better understanding of the limitations imposed by the available information and raise new possibilities for studying high-order iterative methods.