Always convergent methods for nonlinear equations of several variables

Abstract

We develop always convergent methods for solving nonlinear equations of the form $f\left (x\right ) =0$ ( $f:\mathbb {R}^{n}\rightarrow \mathbb {R}^{m}$ , $x\in B=\times _{i=1}^{n}\left [ a_{i},b_{i}\right ] $ ) under the assumption that f is continuous on B. The suggested methods use continuous space curves lying in the rectangle B and have a kind of monotone convergence to the nearest zero on the given curve, if it exists, or the iterations leave the region in a finite number of steps. The selection of space curves is also investigated. The numerical test results indicate the feasibility and limitations of the suggested methods.