Solution of systems of nonlinear equations – a semi-implicit approach

An iterative method for globally convergent solution of nonlinear equations and systems of nonlinear equations is presented. Convergence is quasi-monotonous and approaches second order in the proximity of the real roots. The algorithm is related to semi-implicit methods, earlier being applied to partial differential equations. It is shown that the Newton–Raphson and Newton methods are special cases of the method. The degrees of freedom introduced by the semi-implicit parameters are used to control convergence. When applied to a single equation, efficient global convergence and convergence to a nearby root makes the method attractive in comparison with methods as those of Newton–Raphson and van Wijngaarden–Dekker–Brent. An extensive standard set of systems of equations is solved and convergence diagrams are introduced, showing the robustness, efficiency and simplicity of the method as compared to Newton's method using linesearch.