Families of Rational Maps and Iterative Root-Finding Algorithms

Iterative algorithms for families of variational inequalities fixed points and equilibrium problems

Preperiodic points for families of rational maps

Let X be a smooth curve defined over Q̄, let a, b ∈ P(Q̄) and let fλ(x) ∈ Q̄(x) be an algebraic family of rational maps indexed by all λ ∈ X(C). We study whether there exist infinitely many λ ∈ X(C) such that both a and b are preperiodic for fλ. In particular, we show that if P,Q ∈ Q̄[x] such that deg(P ) 2 + deg(Q), and if a, b ∈ Q̄ such that a is periodic for P (x)/Q(x), but b is not preperiodic f...

Iterative Method for Families of Nonexpansive Maps

In this paper, we study an iterative method for approximating a common fixed point of a countably family of nonexpansive mappings in a Banach space. Our result is a generalization of the results of Bauschke [1], Maingé [12] and Chidume and Ali [4]. An asymptotic version of this class of operators is also studied and similar results obtained in a Banach space.

New stopping criteria for iterative root finding

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...

Polynomial Root-Finding Algorithms and Branched Covers

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 ...