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 number 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.

Certificates

A certificate is a computational proof of the correctness of a candidate solution.

A certificate in this context for a candidate solution is similar to a certificate for correctness in computer science. Whereas a computer science certificate proves that a function produces the correct result, numerical certification operates on solutions. Between the two flavors, a priori numerical certification bears the greatest similarity to certification of an algorithm, in that the results are mathematically guaranteed to be correct -- much like total correctness. On the other hand, a posteriori numerical certification operates on solution candidates, regardless of how they are computed, and hence is rather different from algorithmic correctness -- for an extreme example, one could randomly generate candidates and attempt to certify them a posteriori.

A posteriori certification methods

There are a variety of methods for a posteriori certification, including

Alpha theory (Smale)
Krawczyk's method/Interval Newton (Moore)
Miranda Test (Yap, Vegter, Sharma)

Alpha theory

The cornerstone of Smale's alpha theory is Newton's method. In Smale's 1986 work,^[1] named for the quantity $\alpha$ , a measurement on the distance to a true solution.

More precisely, let $f$ be a system of polynomials with variables $x$ , with $D$ the derivative operator, $N$ a Newton step. The quantities $\beta (f,x)=||x-N_{f}(x)||=||Df(x)^{-1}f(x)||$ $\gamma (f,x)=\sup _{k\geq 2}||{\frac {Df(x)^{-1}D^{k}f(x)}{k!}}||^{\frac {1}{k-1}}$ and $\alpha (f,x)=\beta (f,x)\gamma (f,x)$ can be computed and used to certify a candidate solution, since if $\alpha <{\frac {13-3{\sqrt {17}}}{4}}$ then $x$ is an approximate solution for $f$ .^[2]

Interval Newton

Miranda test

A priori certification methods

Interval arithmetic (Moore, Arb, Mezzarobba)
Condition numbers (Beltran-Leykin)

Interval arithmetic

Interval arithmetic can be used to provide an a priori numerical certificate by computing intervals containing unique solutions.

Condition numbers

By monitoring the condition number for a system as it is being numerically solved, and ensuring that no two solution paths ever intersect, one can compute, along with a solution, a numerical certificate. This scheme can be called a priori path tracking.