A control-based approach to the solution of nonlinear algebraic equations

A different approach to the solution of a nonlinear set of algebraic equations is presented. It is basically a revision of the Newton iterative algorithm from a digital control point of view. The Newton algorithm is considered like a digital control algorithm that acts on a set of nonlinear algebraic equations. Its target is to find a value x* that satisfies the algebraic equation set. This value can be considered as a particular "input" of the equation set which gives a zero "output" while the iteration index can be considered as the clock of the digital system. From this point of view some correlations between the stability of digital systems and the Newton algorithm can be shown. This approach allows us to understand the reasons behind the convergence failure of some modified Newton algorithms such as source stepping, damping, and limiting that the literature often reports as heuristic.