Numerical certification
Numerical certification is the process of verifying the correctness of a candidate solution to a system of equations, often a result of algorithmic calculation. In numerical computational mathematics, such as numerical algebraic geometry, candidate solutions are computed, but there is the possibility that errors have been introduced. Beyond the inexactness of candidates, they may also simply be grossly incorrect as the result of any of a plethora of modes of failure. The goal of numerical certification is to provide a certificate which proves which of these candidates 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.
A posteriori certification methods
- Alpha theory (Smale)
- Krawczyk's method/Interval Newton (Moore)
- Miranda Test (Yap, Vegter, Sharma)
A priori certification methods
- Interval Arithmetic (Moore, Arb, Mezzarobba)
- Condition numbers (Beltran-Leykin)
Category:Algebraic geometry Category:Nonlinear algebra
- ^ Hauenstein, Jonathan; Sottile, Frank (2012). "Algorithm 921: alphaCertified: certifying solutions to polynomial systems". ACM Transactions on Mathematical Software (TOMS). 38 (4): 28. doi:10.1145/2331130.2331136.