A class of multi-iterate methods to solve systems of nonlinear equations

A new class of methods for solving systems of nonlinear equations is introduced. The main idea is to build a linear model 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 quasi-Newton method. Finally, computational comparisons with classical methods highlight a significant improvement in terms of robustness and number of function evaluations. We also present preliminary numerical tests showing the robust behavior of our methods in the presence of noise.