ABSTRACT
One of the basic applications of iteration is Newton’s method—a classical algorithm for finding roots of a function. Newton’s method, unfortunately, does not always converge. That is, a given initial guess need not lead to an orbit that tends to one of the attracting fixed points of N which are, by the Newton Fixed Point Theorem, the roots of F. One of the reasons for the importance of Newton’s method is the speed with which it converges. There are many methods for finding roots of functions, and many of them involve iteration. Most methods suffer from the deficiency that they do not always work. Some methods, when they do converge, do so more rapidly than others. The trade-off is always how often an algorithm works (its efficiency) versus how quickly it converges (its speed). A full study of the speed and efficiency of algorithms to find roots is one of the topics of the field of mathematics known as numerical analysis.
