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

¹ Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, Brazil
² Faculty of Sciences and Technologies in Engineering, University of Brasília, Brasília, DF, Brazil
³ Department of Applied Mathematics, Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, Brazil
⁴ Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific Computing, State University of Campinas, Campinas, SP, Brazil

^* Corresponding author: egbirgin@ime.usp.br

Received: 23 January 2024
Accepted: 28 December 2024

Abstract

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.