  The inference and the verification of numerical relationships among variables
of a program is one of the main goals of static analysis. In this paper, we
propose an Abstract Interpretation framework based on higher-dimensional
ellipsoids to automatically discover symbolic quadratic invariants within
loops, using loop counters as implicit parameters. In order to obtain
non-trivial invariants, the diameter of the set of values taken by the
numerical variables of the program has to evolve (sub-)linearly during loop
iterations. These invariants are called ellipsoidal cones and can be seen as an
extension of constructs used in the static analysis of digital filters.
Semidefinite programming is used to both compute the numerical results of the
domain operations and provide proofs (witnesses) of their correctness.
