Section 1: Iteration Methods
There are various important factors involved in choosing an iteration method to approximate the roots of a function. These include:
- The Initial Value Problem. For what initial values will the method converge?
Will it converge to a root and, if so, which root?
- The Rate of Convergence. Will the convergence be quadratic or better near a root?
- The Complexity of the Calculation.
We show how complex dynamics can shed light on some of these problems when using a Newton type iteration for finding the real or complex roots of a polynomial in a single variable.
We can illustrate the basins of attraction of the roots and the set of initial points for which the method will not converge.
We can also determine some information about the order of convergence at a given root; in particular whether it is quadratic or not.
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