On two conjectures about Dennis-Moré conditions

Quasi-Newton methods for solving nonlinear systems of equations are generally defined in order to satisfy a “direct secant equation” or an “inverse secant equation” at every iteration. The classical Dennis-Moré condition, which, under suitable local assumptions, implies superlinear convergence, is related to the direct secant equation. A natural modification of the Dennis-Moré condition that evokes the inverse secant equation will be considered and two conjectures related to convergence will be formulated. In spite of the apparent plausibility of the conjectures, we will prove that both are false. Consequences with respect to the behavior of standard quasi-Newton methods will be derived.