A fast algorithm to solve systems of nonlinear equations

[EN] A new HSS-based algorithm for solving systems of nonlinear equations is presented and its semilocal convergence is proved. Spectral properties of the new method are investigated. Performance profile for the new scheme is computed and compared with HSS algorithm. Besides, by a numerical example in which a two-dimensional nonlinear convection diffusion equation is solved, we compare the new method and the Newton-HSS method. Numerical results show that the new scheme solves the problem faster than the NewtonHSS scheme in terms of CPU -time and number of iterations. Moreover, the application of the new method is found to be fast, reliable, flexible, accurate, and has small CPU time.This research was partially supported by Ministerio de Economia y Competitividad, Spain under grants MTM2014-52016-C2-2-P and Generalitat Valenciana, Spain PROMETEO/2016/089.Amiri, A.; Cordero Barbero, A.; Darvishi, M.; Torregrosa Sánchez, JR. (2019). A fast algorithm to solve systems of nonlinear equations. Journal of Computational and Applied Mathematics. 354:242-258. https://doi.org/10.1016/j.cam.2018.03.048S24225835

On some extensions of the accelerated overrelaxation (AOR) theory

This paper extends the convergence theory of the Accelerated Overrelaxation (AOR) method to cases analogous to those considered first by Ostrowski and then by Varga in connection with the Successive Overrelaxation (SOR) method. Among others, the Ostrowski Theorem, some of the theorems by Varga on the extensions of the SOR theory, and some recent results by Niethammer and by the authors are obtained as special cases of the work presented in this paper. In addition, several points are raised which suggest further research