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 Danby's method for the resolution of Kepler's equation; we find that the cited method of order three is competitive with Danby's method and the classical Newton's 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 Newton's method.