Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

Abstract

In this article, we have constructed an iterative method of third order for solving polynomial equations with multiple polynomial zeros. We have combined two well known third order methods one is Halley and another is Super-Halley for this construction purpose. This constructed method is basically the mean of the methods Halley and Super-Halley, so we name the method as H-S Combined Mean Method. We have proposed some local convergence theorems of this H-S Combined Mean Method to establish the computation of a polynomial with known multiple zeros. For the establishment of this local convergence theorem, the key role is performed by a function (Real valued) termed as the function of initial conditions. Function of initial conditions \(I\) is a mapping from the set \(D\) into the set \(X\), where \(D\) (subset of \(X\)) is the domain of the H-S Combined Mean iterative scheme. Here the initial conditions uses the information only at the initial point and are given in the form \(I(w_0)\) which belongs to \(J\), where \(J\) is an in interval on the positive real line which also contains zero and \(w_0\) is the starting point. We have used the notion of gauge function which also plays very important role in establishing the convergence theorem. Here we have used two types of initial conditions over an arbitrary normed field and established local convergence theorems of the constructed H-S Combined Mean Method. The error estimations are also found in our convergence analysis. For simple zero, the method as well as the results hold good.