Solving Decidability Problems with Interval Arithmetic

An affine iterated function system (IFS, [4]) over the reals is given by a set of contractive affine transformations on a complete metric space R for some positive natural number k. Thus an interesting problem is to decide, whether a given matrix M describes a contraction mapping, i.e. whether there exists some s < 1 such that |Mz| ≤ s|z| for each z ∈ R, where | · | denotes the Euclidean norm. Using the characteristic polynomial and interval methods we present a simple algorithm deciding the contraction mapping property for square matrices over the rational numbers. Furthermore, we show that interval arithmetic can be used to enclose the roots of complex polynomials with coefficients having rational real and imaginary parts at arbitrary precision such that each computed interval can be annotated with the exact number of contained roots.