Certified Computations with UnCertainties`

Yves Papegay

INRIA Sophia Antipolis, COPRIN

Abstract

Taking into account uncertainties on data or parameters while certifying results of computations in terms of boundings of errors is a big challenge that has to be faced by next generation of software with numerical capabilities.

The COPRIN team at INRIA Sophia is one of the leading teams in research on interval analysis methods and their efficient implementation, and has developped several numerical library amongst which ALIAS is the most widely used. An interface of Mathematica to this library has been presented at the Wolfram Technology Conference in 2005. The 2007 edition of this conference gave me the opportunity to demonstrate how crucial is for their efficiency, the use of symbolic capabilities during the implementation and execution of interval analysis algorithms.

The first version of the UnCertainties` package that is described in details in this presentation will be released during the conference, as announced during the International Mathematica Symposium 2008. The functionnalities of this package includes a data representation for extending intervals to n-dimensionnal vectors and matrices, and the corresponding extension of the arithmetics and procedures based on symbolic pre-processing for sharp evaluation of expressions. Several solvers are available for finding roots of systems of non-necessarly algebraic equations. They are based on bisection and filtering through 2B- 3B- or hull-consistency and may use Jacobian or Hessian matrix of partial derivatives to improve their efficiency. For specific systems of equations, dedicated solvers have been implemented, namely Ridder or Brent method for solving one equation, Newton and Krawczyk method, etc... and a global optimization procedure based on intervals is also provided. Finally, the former Aliastica` package allowing to connect to the ALIAS library has been integrated as well.