Numerical Software Verification 

Abstract: We present in this paper a process which is suitable for the complete analysis of the numerical quality of a large industrial scientific computing code. Random rounding, using the Verrou diagnostics tool, is first used to evaluate the numerical stability, and locate the origin of errors in the source code. Once a small code part is identified as unstable, it can be isolated and studied using higher precision computations and interval arithmetic to compute guaranteed reference results. An alternative implementation of this unstable algorithm is then proposed and experimentally evaluated. Finally, error bounds are given for the proposed algorithm, and the effectiveness of the proposed corrections is assessed in the computing code.

Abstract: We present a framework for the verification of the numerical algorithms used in Ariadne, a tool for analysis of nonlinear hybrid system. In particular, in Ariadne, smooth functions are approximated by Taylor models based on sparse polynomials. We use the Coq theorem prover for developing Taylor models as sparse polynomials with floating-point coefficients. This development is based on the formalisation of an abstract data type of basic floating-point arithmetic . We show how to devise a type of continuous function models and thereby parametrise the framework with respect to the used approximation, which will allow us to plug in alternatives to Taylor models.

Abstract: We present mathematically rigorous computational methods for the transport of large domains through ODEs with the goal of making rigorous statements about their long term evolution. Of particular interest are determination of locations of attractors, reachability of certain sets, and proof of non-reachability of others. The methods are based on Taylor model verified integrators for the propagation of large domains, and heavily rely on automatic domain decomposition for accuracy. We illustrate the behavior and performance of these methods using several commonly studied dynamical systems.

Abstract: Semantic static analysis allows sound verification of program properties, that is, to prove that a given property holds for all possible executions. However, modern object-oriented applications make heavy use of third-party frameworks. These provide various functionalities (like libraries), as well as an extension of the execution model of the program. Applying standard models to statically analyze software relying on such frameworks could be potentially unsound and imprecise.