A modified Newton method for polynomials

A modified Newton method for polynomials is discussed. It is assumed one has approximations for all the roots of the polynomial. Three variations are described. If the roots are simple, it is shown that under appropriate conditions, two of the variations are cubically convergent, 1. ! ,l t r o d u e t io, l,et t:.here be given film polynomial J'(:C) = aMt(' @ a,:.-.tX"' @ "'" + aix-I-ao (1) ,i i,! where we assume the roots are distinct. Assume X~0), ... , ,~,, tare n distinct guesses for the roots of f(x).