Iterative Methods for Roots of Polynomials

We describe iterative methods for polynomial zero finding and, specifically, the Laguerre method and how it is used in the NAG subroutine C02AFF. We also investigate a bug that has been in this subroutine for ten years. In chapter two, we give a brief survey of some zero finding methods. These include Bairstow's method, Bernoulli's method, Graeffe's root-squaring method, Muller's method, the Newton-Raphson method and the Jenkins-Traub and Laguerre methods. In chapter three, we look at the Laguerre method as used in C02AFF in further detail, describe the behaviour of the bug and how the problem has been solved. We also describe general tests for zero finding algorithms and results of comparisons between NAG's C02AFF and other zero finding programs. Chapter 4 involves comparisons of C02AFF with other methods and a note on error bounds. Finally, we make our proposals and conclusions in chapter 5.

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