Abstract
The empty regions used in this paper are simple extensions of the empty regions that are used in [5]. The region around a real number α depends on f(α), f′(α) and ‖f‖, where ‖f‖ denotes the polynomial norm of f. We show that for almost all polynomials of degree greater than 1 with real coefficients, the Newton method is locally equivalent to a simple algorithm, which uses the empty regions. Moreover, analyzing the region, we obtained a modified Newton's formula such that
