Solving systems of nonlinear equations via spectral residual methods

This thesis addresses the numerical solution of systems of nonlinear equations via spectral residual methods. Spectral residual methods are iterative procedures, they use the residual vector as search direction and a spectral steplength, i.e., a steplength that is related to the spectrum of the average matrices associated to the Jacobian matrix of the system. Such procedures are widely studied and employed since they are derivative-free and low-cost per iteration. The first aim of the work is to analyze the properties of the spectral residual steplengths and study how they affect the performance of the methods. This aim is addressed both from a theoretical and experimental point of view. The main contributions in this direction are: the theoretical analysis of the steplengths proposed in the literature and of their impact on the methods behaviour; the analysis of the performance of spectral methods with various rules for updating the steplengths. We propose and extensively test different steplength strategies. Rules based on adaptive strategies that suitably combine small and large steplengths result by far more effective than rules based on static choices of the steplength. Numerical experience is conducted on sequences of nonlinear systems arising from rolling contact models which play a central role in many important applications, such as rolling bearings and wheel-rail interaction. Solving these models gives rise to sequences which consist of a large number of medium-size nonlinear systems and represent a relevant benchmark test set for the purpose of the thesis. The second purpose of the thesis is to propose a variant of the derivative-free spectral residual method used in the first part and obtain a general scheme globally convergent under more general conditions. The robustness of the new method is potentially improved with respect to the previous version. Numerical experiments are conducted both on the problems arising in rolling contact models and on a set of problems commonly used for testing solvers for nonlinear systems.