Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program

Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring specific accuracy issues due to their massive use of floating-point computations. Yet, it is uncommon to guarantee their correctness. Indeed, we had to extend existing methods and tools for proving the correct behavior of programs to verify an existing numerical analysis program. This C program implements the second-order centered finite difference explicit scheme for solving the 1D wave equation. In fact, we have gone much further as we have mechanically verified the convergence of the numerical scheme in order to get a complete formal proof covering all aspects from partial differential equations to actual numerical results. To the best of our knowledge, this is the first time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with arXiv:1112.179

On periodic solutions of 2-periodic Lyness difference equations

We study the existence of periodic solutions of the non--autonomous periodic Lyness' recurrence u_{n+2}=(a_n+u_{n+1})/u_n, where {a_n} is a cycle with positive values a,b and with positive initial conditions. It is known that for a=b=1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a,b) different from (1,1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a is not equal to b, then any odd period, except 1, appears.Comment: 27 pages; 1 figur

Hyperbolic systems of conservation laws in one space dimension

Aim of this paper is to review some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. The main focus will be on the uniqueness and stability of entropy weak solutions and on the convergence of vanishing viscosity approximations

Network analysis of complex biological systems : boundedness of weakly reversible chemical reaction networks and conditions for synchronisation of coupled oscillators

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