We survey and extend the recent progress in polynomial root-finding based on reducing the problem to eigen-solving for highly structured generalized companion matrices. In particular we cover the selection of the eigen-solvers and the matrices and the resulting benefits based on exploiting matrix structure. No good estimates for global convergence of the basic eigen-solvers have been proved, bu...

We show that a generally convergent root-finding algorithm for cubic polynomials defined by C. McMullen is of order 3, and we give generally convergent algorithms of order 5 and higher for cubic polynomials. We study the Julia sets for these algorithms and give a universal rational map and Julia set to explain the dynamics.

A set of simple stopping criteria is presented, which improve the efficiency of iterative root finding by terminating the iterations immediately when no further improvement of the roots is possible. The criteria use only the function evaluations already needed by the root finding procedure to which they are applied. The improved efficiency is achieved by formulating the stopping criteria in ter...

We present two algorithms for computing the root, or equivalently the xed point, of a function in exact real arithmetic. The rst algorithm uses the iteration of the expression tree representing the function in real arithmetic based on linear fractional transformations and exact oating point. The second and more general algorithm is based on a trisection of intervals and can be compared with the...

The stochastic root finding problem (SRFP) involves finding points in a region where a function attains a prespecified target value, using only a consistent estimator of the function. Due to the properties that the SRFP contexts entail, the development of good solutions to SRFPs has proven difficult, at least in the multi-dimensional setting. This paper discusses certain key issues, insights an...

Introduction. The problem of devising optimal methods for numerically approximating the roots of a polynomial has been of interest for several centuries, and is far from solved. There are numerous recent works on root-finding algorithms and their cost, for example, the work of Jenkins and Traub [JT70], Renegar [Ren87], Schönhage [Sch82], and Shub and Smale [SS85, SS86, Sma85]. This list is far ...

We consider growing random networks {Gn}n≥1 where, at each time, a new vertex attaches itself to collection of existing vertices via fixed number m≥1 edges, with probability proportional function f (called attachment function) their degree. It was shown in [BB21] that such network models exhibit two distinct regimes: (i) the persistent regime, corresponding ∑i=1∞f(i)−2<∞, where top K maximal de...