A Class of Higher Order Algorithms for Computing Polynomial Zeros

A general framework for deriving multi-parameter classes of methods of any presribed order is presented. In particular, two classes of cubically and fourth order convergent iterative methods for simple zeros of polynomial equations are derived. As special cases, the cubically convergent class includes the one-parameter Hansen-Patricks family and other well-known Laguerre’s, Halley’s and Ostrowski’s methods. Additionally, methods of any order which are based on Taylor expansion of certain rational functions are developed. Modified methods for extracting multiple zeros are briefly discussed.