Laguere’s Method and a Circle which Contains at Least One Zero of a Polynomial

Abstract

Given an Nth degree polynomial $P(z)$ one may use Laguerre’s method to generate a sequence of complex numbers $x_0 ,x_1 ,x_2 , \ldots $ which usually converges to a, zero of $P(z)$. This note shows that each circle $\left| {z - x_n } \right| \leqq \sqrt N \left| {x_{n + 1} - x_n } \right|$ contains at least one zero of $P(z)$. If N is not a perfect square, then the constant $\sqrt N $ can be replaced by a slightly smaller constant $R_N $.