Proceedings of the American Mathematical Society
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 × 2 matrices. Here we develop an analogous theory based on 3 × 3 matrices which yields a two-variable generally convergent algorithm for cubics.
Northshield, S. (2013). A root-finding algorithm for cubics. Proceedings of the American Mathematical Society, 141 (2), 645-649.