  The computer program SnapPea can approximate whether or not a three manifold
whose boundary consists of tori has a complete hyperbolic structure, but it can
not prove conclusively that this is so. This article provides a method for
proving that such a manifold has a complete hyperbolic structure based on the
approximations of SNAP, a program that includes the functionality of SnapPea
plus other features. The approximation is done by triangulating the manifold,
identifying consistency and completeness equations with respect to this
triangulation, and then trying to solve the system of equations using Newton's
Method. This produces an approximate, not actual solution. The method developed
here uses Kantorovich's theorem to prove that an actual solution exists,
thereby assuring that the manifold has a complete hyperbolic structure. Using
this, we can definitively prove that every manifold in the SnapPea cusped
census has a complete hyperbolic structure.
