A multi-iterate method to solve systems of nonlinear equations

We propose an extension of secant methods for nonlinear equations using a population of previous iterates. Contrarily to classical secant methods, where exact interpolation is used, we prefer a least squares approach to calibrate the linear model. We propose an explicit control of the numerical stability of the method. We show that our approach can lead to an update formula. In that case, we prove the local convergence of the corresponding undamped quasi-Newton method. Finally, computational comparisons with classical quasi-Newton methods highlight a significant improvement in terms of robustness and number of function evaluations. We also present numerical tests showing the robust behavior of our method in the presence of noise.