Basic Concepts

In this chapter, we give some basic concepts and properties, necessary in our investigation of convergence characteristics of root finding methods. Most of these methods are reviewed in Sect. 1.1, together with some historical notes and various principles for their construction. Section 1.2 contains several results concerning the localization of polynomial zeros. We restrict ourselves to inclusion disks in the complex plane that contain complex zeros of a given polynomial. In Sect. 1.3, we give the basic properties and operations of circular complex interval arithmetic, prerequisite to a careful analysis of the bounds of complex quantities that appear in our study and the construction of inclusion methods described in Sect. 5.3.