  In this paper the static, spherically symmetric and electrically charged
black hole solutions in Einstein-Born-Infeld gravity with massive dilaton are
investigated numerically.
  The Continuous Analog of Newton Method (CANM) is used to solve the
corresponding nonlinear multipoint boundary value problems (BVPs). The
linearized BVPs are solved numerically by means of collocation scheme of fourth
order.
  A special class of solutions are the extremal ones. We show that the extremal
horizons within the framework of the model satisfy some nonlinear system of
algebraic equations. Depending on the charge $q$ and dilaton mass $\gamma$, the
black holes can have no more than three horizons. This allows us to construct
some Hermite polynomial of third order. Its real roots describe the number, the
type and other characteristics of the horizons.
