Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees

Abstract

We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion (10 9 ) so that the observed complexity to find all roots is between O(d ln d) and O(d ln 3 d) (measuring complexity in terms of number of Newton iterations or computing time). All computations were performed successfully on standard desktop computers built between 2007 and 2012.

Finding polynomial roots by dynamical systems -- a case study

“…In response to this problem, the "iterated refinement" Newton method has been developed in [SSt1,RaSS]: the idea is that initially, where the orbits are still close to the initial circles, there is a lot of control on the Newton dynamics so all orbits do just about the same thing. The iterated refinement approach thus starts with a small initial number of starting points on the original circles, perhaps 64 points, and iterates them while their orbits are "parallel" in a sense to be made precise (for instance, in the sense of cross ratios between three adjacent orbits, together with ∞).…”

“…These polynomials have degree 2 n but can be evaluated in complexity O(n) hence in logarithmic complexity with respect to the degree. For certain choices of c, all periodic points of periods up to 30 (i.e., degrees up to 2 30 > 10 9 ) were computed successfully by Newton's method in [SSt1,RaSS].…”

“…The results of the experiments are displayed in Figure 4. It may be interesting to note that these polynomials have been used as sample polynomials for evaluating the efficiency the Ehrlich-Abertmethod in its implementation in MPSolve (see [BF2]), and independently also for our experiments on Newton's method (see [SSt1,RaSS]). One has to keep in mind that formally speaking, [BF2] works with polynomials q n (c)/c of degree 2 n−1 − 1 but this difference is, of course, insignificant for root finding.…”