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Theses

Formalisations d’analyses d’erreurs en analyse numérique et en arithmétique à virgule flottante

Abstract : This thesis consists of three contributions related to the Coq formalization of error analysis in numerical analysis and floating-point arithmetic.First, we have exhibited an algorithm computing the average of two decimal floating-point numbers and have proved that this algorithm computes the correct rounding. We have formalized the algorithm and its correctness proof in the Coq proof assistant.The second contribution of the thesis is the analysis and the formalization of rounding error bounds associated to the implementation of Runge-Kutta methods applied to linear systems. We have proposed a generic methodology to build a bound on the error accumulated over the iterations, taking potential underflow and overflow into account. We have then instantiated this methodology forclassic Runge-Kutta methods, e.g. the Euler and RK2 methods. We have proposed a formalization of the results, including the definition of matrix norms, theproof of rounding error bounds for matrix operations and the formalization of the generic results and their instantiations.Finally, we have proposed a formalization of functional analysis results that serve as foundations for the finite element method. This formalization is based on the Coquelicot library and includes the theory of Hilbert spaces, the formal proof of the Lax--Milgram Theorem and the proof of completeness of finite dimensional subspaces of Hilbert spaces.
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Florian Faissole. Formalisations d’analyses d’erreurs en analyse numérique et en arithmétique à virgule flottante. Logique en informatique [cs.LO]. Université Paris Saclay (COmUE), 2019. Français. ⟨NNT : 2019SACLS594⟩. ⟨tel-02470728⟩

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