  It is well known that the Newton method may not converge when the initial
guess does not belong to a specific quadratic convergence region. We propose a
family of new variants of the Newton method with the potential advantage of
having a larger convergence region as well as more desirable properties near a
solution. We prove quadratic convergence of the new family, and provide
specific bounds for the asymptotic error constant. We illustrate the advantages
of the new methods by means of test problems, including two and six variable
polynomial systems, as well as a challenging signal processing example. We
present a numerical experimental methodology which uses a large number of
randomized initial guesses for a number of methods from the new family, in turn
providing advice as to which of the methods employed is preferable to use in a
particular search domain.
