Formal Verifcation for Numerical Methods

Ioana Pasca 1
1 MARELLE - Mathematical, Reasoning and Software
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : This thesis deals with the formalization of mathematics in the proof assistant Coq with the purpose of verifying numerical methods. We focus in particular on formalizing concepts involved in solving systems of equations, both linear and non-linear.

We analyzed Newton's method which is a numerical method widely used for approximating solutions of equations or systems of equations. The goal was to formalize Kantorovitch's theorem which gives the convergence of Newton's method to a solution, the speed of the convergence and the local stability of the method. The formal study of this theorem also demanded a formalization of concepts of multivariate analysis. Based on these classic results on Newton's method, we showed that rounding at each step in Newton's method still yields a convergent process with an accurate correlation between the precision of the input ant that of the result. In a joint work with Nicolas Julien, we studied formally computations with Newton's method in a library of exact real arithmetic.

For linear systems of equations, we analyzed the case where the associated matrix has interval coefficients. For solving such systems, an important issue is to establish whether the associated matrix is regular. We provide a collection of formally verified criteria for regularity of interval matrices.

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Submitted on : Wednesday, January 12, 2011 - 2:50:40 PM
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Ioana Pasca. Formal Verifcation for Numerical Methods. Other [cs.OH]. Université Nice Sophia Antipolis, 2010. English. ⟨tel-00555158⟩

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