  The elliptic Monge-Amp\`ere equation is a fully nonlinear Partial
Differential Equation that originated in geometric surface theory and has been
applied in dynamic meteorology, elasticity, geometric optics, image processing
and image registration. Solutions can be singular, in which case standard
numerical approaches fail. Novel solution methods are required for stability
and convergence to the weak (viscosity) solution.
  In this article we build a wide stencil finite difference discretization for
the \MA equation. The scheme is monotone, so the Barles-Souganidis theory
allows us to prove that the solution of the scheme converges to the unique
viscosity solution of the equation.
  Solutions of the scheme are found using a damped Newton's method. We prove
convergence of Newton's method and provide a systematic method to determine a
starting point for the Newton iteration.
  Computational results are presented in two and three dimensions, which
demonstrates the speed and accuracy of the method on a number of exact
solutions, which range in regularity from smooth to non-differentiable.
