Accelerated derivative-free spectral residual method for nonlinear systems of equations

Many continuous models of natural phenomena require the solution of large-scale nonlinear systems of equations. For example, the discretization of many partial differential equations, which are widely used in physics, chemistry, and engineering, requires the solution of subproblems in which a nonlinear algebraic system has to be addressed, especially one in which stable implicit difference schemes are used. Spectral residual methods are powerful tools for solving nonlinear systems of equations without derivatives. In a recent paper [Birgin and Martínez, SIAM J. Numer. Anal . 60 (2022) 3145–3180], it was shown that an acceleration technique based on the Sequential Secant Method can greatly improve its efficiency and robustness. In the present work, an R implementation of the method is presented. Numerical experiments with a widely used test bed compare the presented approach with its plain ( i.e ., non-accelerated) version that is part of the R package BB. Additional numerical experiments compare the proposed method with NITSOL, a state-of-the-art solver for nonlinear systems. These comparisons show that the acceleration process greatly improves the robustness of its counterpart included in the existing R package. As a by-product, an interface is provided between R and the consolidated CUTEst collection, which contains over a thousand nonlinear programming problems of all types and represents a standard for evaluating the performance of optimization methods.