We describe Ccluster, a software for computing natural ε-clusters of complex roots in a given box of the complex plane. This algorithm from Becker et al. (2016) is near-optimal when applied to the benchmark problem of isolating all complex roots of an integer polynomial. It is one 4 of the first implementations of a near-optimal algorithm for complex roots. We describe some low level techniques for speeding up the algorithm. Its performance is compared with the well-known MPSolve library and Maple.

The recent advanced sub-division algorithm is nearly optimal for the approximation of the roots of a dense polynomial given in monomial basis; moreover, it works locally and slightly outperforms the user's choice MPSolve when the initial region of interest contains a small number of roots. Its basic and bottleneck block is counting the roots in a given disc on the complex plain based on Pellet's theorem, which requires the coefficients of the polynomial and expensive shift of the variable. We implement a novel method for both root-counting and exclusion test, which is faster, avoids the above requirements, and remains efficient for sparse input polynomials. It relies on approximation of the power sums of the roots lying in the disc rather than on Pellet's theorem. Such approximation was used by Schönhage in 1982 for the different task of deflation of a factor of a polynomial provided that the boundary circle of the disc is sufficiently well isolated from the roots. We implement a faster version of root-counting and exclusion test where we do not verify isolation and significantly improve performance of subdivision algorithms, particularly strongly in the case of sparse inputs. We present our implementation as heuristic and cite some relevant results on its formal support presented elsewhere.

Decomposition of constraint systems is a key component of geometric constraint solving in CAD. On the other hand, some authors have introduced the notion of reparameterization which aims at helping the solving of indecomposable systems by replacing some geometric constraints by other ones. In previous works, the minimal change of the initial system is a main criterion. We propose to marry these two ingredients, decomposition and reparameterization, in a method able to reparameterize and to decompose a constraint system according to this reparameterization. As a result, we do not aim at minimizing the number of added constraints during the reparameterization, but we want to decompose the system such that each component owns a minimal number of such added constraints.

We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial p of degree d with real or complex coefficients. As in their previous best subdivision algorithms our rootfinders are robust even for multiple roots of a polynomial given by a black box for the approximation of its coefficients, and their complexity decreases at least proportionally to the number of roots in a region of interest (ROI) on the complex plane, such as a disc or a square, but we greatly strengthen the main ingredient of the previous algorithms. Namely our new counting test essentially amounts to the evaluation of a polynomial p and its derivative p , which is a major benefit, e.g., for sparse polynomials p. Moreover with evaluation at about log(d) points (versus the previous record of order d) we output correct number of roots in a disc whose contour has no roots of p nearby. Moreover we greatly soften the latter requirement versus the known subdivision algorithms. Our second and less significant contribution concerns subdivision algorithms for polynomials with real coefficients. Our tests demonstrate the power of the proposed algorithms.