Smale's α-theory uses estimates related to
Newton's method to certify solutions to polynomial systems.
The program alphaCertified implements algorithms based on α-theory to certify
solutions to polynomial systems using both exact rational arithmetic and
arbitrary precision floating point arithmetic.
It also implements an algorithm to certify whether a given point corresponds to a real
solution of a real polynomial system, as well as algorithms to heuristically validate
solutions to overdetermined systems. Examples are presented to demonstrate the
algorithms.