Relationships between order and efficiency of a class of methods for multiple zeros of polynomials
Newton's method is a well-known iterative method for solving polynomial equations. Let \(p(z)\) be a complex polynomial of degree \(n= \deg(p)\). Newton's method is given by the function \(N_1(z)= z- p(z)/p'(z)\). For an initial approximation \(z_0\in \mathbb{C}\) of some root \(w\) of \(p\) the iterative algorithm \(z_{\nu+ 1}= N_1(z_\nu)\) generates a sequence of successive approximations \(z_\nu\), \(\nu= 0, 1,\dots\), converging to \(w\). The authors study the efficiency of Newton's method and its higher order analogs. Let \(m\) denote the multiplicity of \(w\) and \(k\) the order of the iterative method. If \(k\leq m\) then convergence analysis gives \(|z_{\nu+ 1}- w|\approx (1- {k- 1\over m+ k- 2})|z_\nu- w|\). From this the authors derive that for \(k\geq 5\) the number of required iterations decreases as the multiplicity \(m\) increases. Furthermore, the analysis as well as practical experience indicate the following optimal choices for \(k: k\approx 1.5\sqrt m\) when \(z_0\) is close to a root of multiplicity \(m> 1\) and \(k\approx 1.5\sqrt n\) when far from any root of \(p\).
scientific article; zbMATH DE number 808365