Kepler equation and accelerated Newton method

We study the efficiency of the accelerated Newton method (Garlach, SIAM Rev. 36 (1994) 272–276) for several orders of convergence versus Danbyʹs method for the resolution of Keplerʹs equation; we find that the cited method of order three is competitive with Danbyʹs method and the classical Newtonʹs method. We also generalize the accelerated Newton method for the resolution of system of algebraic equations, obtaining a formula of order three and a proof of its convergence; its application to several examples shows that its efficiency is greater than Newtonʹs method.