A root-finding algorithm for cubics
Newton's method applied to a quadratic polynomial converges rapidly to a root for almost all starting points and almost all coefficients. This can be understood in terms of an associative binary operation arising from 2 x 2 matrices. Here we develop an analogous theory based on 3 x 3 matrices which yields a two-variable generally convergent algorithm for cubics.
This article has been published in Proceedings of the American Mathematical Society: https://doi.org/10.1090/S0002-9939-2012-11324-3