The Condition Number for Nonlinear Problems

Abstract

The goal of this chapter is to describe a measure of condition for problems that search for a solution of nonlinear systems of equations. Eventually a condition number μ is defined as a bound on the infinitesimal error of a solution caused by an infinitesimal error in the defining system of equations. With our emphasis on polynomial systems, we impose a norm on the space of such systems that reflects an important computational invariant, the distance between the zeros. To avoid the distortion caused by very large zeros, the analysis and metrics are defined in a projective space setting. The result is a unitarily invariant theory.