  We reexamine Smale's alpha theory as a way to certify a numerical solution to
an analytic system. For a given point and a system, Smale's alpha theory
determines whether Newton's method applied to this point shows the quadratic
convergence to an exact solution. We introduce the alpha theory computation
using interval arithmetic to avoid costly exact arithmetic. As a
straightforward variation of the alpha theory, our work improves computational
efficiency compared to software employing the traditional alpha theory.
