Extension of the Lanczos and CGS methods to systems of nonlinear equations

In this work new methods for solving systems of nonlinear equations are considered: the first and second topological epsilon algorithms, the conjugate gradient squared (CGS), and some of their nonlinear variants. First, we point out the link between them and the Lanczos and the CGS methods, and we present the implementation of the algorithms. Then a sufficient condition for quadratic convergence of these nonlinear methods is proved. Finally, some examples for illustrating our purpose are given.