Some Observations on Khovanskii's Matrix Methods for Extracting Roots of Polynomials

In this article we apply a formula for the n-th power of a 3 × 3 matrix (found previously by the authors) to investigate a procedure of Khovanskii’s for finding the cube root of a positive integer. We show, for each positive integer α, how to construct certain families of integer sequences such that a certain rational expression, involving the ratio of successive terms in each family, tends to α1/3. We also show how to choose the optimal value of a free parameter to get maximum speed of convergence. We apply a similar method, also due to Khovanskii, to a more general class of cubic equations, and, for each such cubic, obtain a sequence of rationals that converge to the real root of the cubic. We prove that Khovanskii’s method for finding the m-th (m ≥ 4) root of a positive integer works, provided a free parameter is chosen to satisfy a very simple condition. Finally, we briefly consider another procedure of Khovanskii’s, which also involves m × m matrices, for approximating the root of an arbitrary polynomial of degree m.