Penalty Function Based Critical Point Approach to Compute Real Witness Solution Points of Polynomial Systems

Abstract

We present a critical point method based on a penalty function for finding certain solution (witness) points on real solutions components of general real polynomial systems. Unlike other existing numerical methods, the new method does not require the input polynomial system to have pure dimension or satisfy certain regularity conditions. This method has two stages. In the first stage it finds approximate solution points of the input system such that there is at least one real point on each connected solution component. In the second stage it refines the points by a homotopy continuation or traditional Newton iteration. The singularities of the original system are removed by embedding it in a higher dimensional space. In this paper we also analyze the convergence rate and give an error analysis of the method. Experimental results are also given and shown to be in close agreement with the theory.