1966 Articles

A Class of Globally Convergent Iterations for the Solution of Polynomial Equations

Traub, Joseph F.

We introduce a class of new iteration functions which are ratios of polynomials of the same degree and hence defined at infinity. The poles of these rational functions occur at points which cause no difficulty. The classical iteration functions are given as explicit functions of P and its derivatives. The new iteration functions are constructed according to a certain algorithm. This construction requires only simple polynomial manipulation which may be performed on a computer. We shall treat here only the important case that the zeros of P are distinct and that the dominant zero is real. The extension to multiple zeros, dominant complex zeros, and sub-dominant zeros will be given in another paper. We shall restrict ourselves to questions relevant to the calculation of zeros. Certain aspects of our investigations which are of broader interest will be reported elsewhere.