Root finding methods: a dynamical approach

Abstract

One of the most classical problems in Mathematics is to find the zeroes of a given function $f$, or equivalently, to find the roots of the equation $f (z) = 0$. It has been studied this problem, from the simplest cases, like the case of $f$ being a polynomial of one or several real or complex variables, to a more general setting, like the case of $f$ being just a continuous function. Using algebraic and analytic methods it is possible to exactly solve the equation $f (x) = 0$ rarely. A part from these particular situations (like polynomials of degree less than 5) the unique approximation is to numerically find them; that is to construct root finding algorithms which allow us to find good approximations of the zeroes of $f$. The more well know root finding algorithms are defined by an iterative mechanism, and so, they can be thought and treated as dynamical systems defined in a certain space.