Numerical certification is the process of verifying the correctness of a candidate solution to a system of equations. In (numerical) computational mathematics, such as numerical algebraic geometry, candidate solutions are computed algorithmically, but there is the possibility that errors have corrupted the candidates. For instance, in addition to the inexactness of input data and candidate solutions, numerical errors or errors in the discretization of the problem may result in corrupted candidate solutions. The goal of numerical certification is to provide a certificate which proves which of these candidates are, indeed, approximate solutions.
Methods for certification can be divided into two flavors: a priori certification and a posteriori certification. A posteriori certification confirms the correctness of the final answers (regardless of how they are generated), while a priori certification confirms the correctness of each step of a specific computation. A typical example of a posteriori certification is Smale's alpha theory, while a typical example of a priori certification is interval arithmetic.
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