(Extended Abstract) Integral Analysis of Evaluation-Based Root Isolation

Let f be a univariate polynomial with real coefficients, f ∈ R[X]. Subdivision methods are widely used for isolating the roots of f in a given interval. In this paper we consider evaluation-based subdivision algorithms that use using simpler primitives than well-known subdivision methods such as Sturm or Descartes methods. In particular, we study the EVAL algorithm based on an algorithm by Mitchell, which can be seen as a 1-dimensional analogue of Plantinga-Vegter’s meshing algorithm. (1) First we give a general framework for performing an adaptive analysis of the EVAL algorithm. This leads to an upper bound, defined via an integral, for the complexity of the EVAL algorithm. This novel technique for complexity analysis can be viewed as a continuous amortization technique. In addition, this framework is quite general and promises to be applicable for analyzing the complexity of other evaluation-based algorithms. (2) Next we consider the benchmark case of a square-free integer polynomial f of degree d and logarithmic height L. We give a priori worst-case upper bounds of the form O(d(log d+L)) for the size of the subdivision tree. This bound exploits the Mahler-Davenport root bound as well as new evaluation analogues of such bounds.