An algorithm for the total, or partial, factorization of a polynomial

Abstract

A globally convergent algorithm is presented for the total, or partial, factorization of a polynomial. Firstly, a circle is found containing all the zeros. Secondly, a search procedure locates smaller circles, each containing a zero, and the multiplicities are then calculated. Thirdly, a simultaneous Iteration Function is used to accelerate convergence. The Iteration Function is chosen from a class of such functions derived herein to deal with the general case of multiple zeros; various properties of these functions are also discussed. Finally, sample numerical results are given which demon-strate the effectiveness of the algorithm.

Iteration Functions re-visited

“…In a later paper [FL77] we extended those results to multiple zeros in the context of a globally convergent algorithm [Far14]. This section presents a new class of IFs expressed in a polynomial format in preference to the rational format used previously by us and other authors, see [FL77], [Kis54], and [Tra82] for examples. If p(z), as defined in Equation (3), has a zero α ν with multiplicity m ν then the function P(z ν ) defined by…”

“…We think that this emphasises how the IFs in the different classes are generated as their order of convergence increases. To see the rational format IF corresponding to one of our current IFs, see [FL77, …”

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

“…The outcome approximation x k and the exact order of multiplicity m serve as the initial values for one iteration of the two-point method (30) in order to improve the accuracy of the wanted multiple zero. We used the six test functions, including f 2 (x) taken from [5] and f 4 from [29]. …”

Inclusion Weierstrass-like root-finders with corrections

“…, z ν be distinct approximations to the zeros ζ 1 , ..., ζ ν . Farmer and Loizou [4] constructed the following iterative formula for multiple zeros:…”