| Summary: | We introduce a new domain for finding precise numerical invariants of
programs by abstract interpretation. This domain, which consists of level sets
of non-linear functions, generalizes the domain of linear "templates"
introduced by Manna, Sankaranarayanan, and Sipma. In the case of quadratic
templates, we use Shor's semi-definite relaxation to derive computable yet
precise abstractions of semantic functionals, and we show that the abstract
fixpoint equation can be solved accurately by coupling policy iteration and
semi-definite programming. We demonstrate the interest of our approach on a
series of examples (filters, integration schemes) including a degenerate one
(symplectic scheme).
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