Computer-Assisted Proofs in Solving Linear Parametric Problems

Abstract

Consider a linear system A(p)x = b(p) whose input data depend on a number of uncertain parameters p = (p1,...,pk) varying within given intervals [p]. The objective is to verify by numerical computations monotonic (and convexity/concavity) dependence of a solution component xi(p) with respect to a parameter pj over the interval box [p], or more general, to prove if some boundary inf / sup xi(p) for all p isin [p] is attained at the end-points of [p]. Such knowledge is useful in many applications in order to facilitate the solution of some underlying linear parametric problem involving uncertainties. In this paper we present a technique, for proving the desired properties of the parametric solution, which is alternative to the approaches based on extreme point computations. The proposed computer-aided proof is based on guaranteed interval enclosures for the partial derivatives of the parametric solution for all p isin [p]. The availability of self-validated methods providing guaranteed enclosure of a parametric solution set by floating-point computations is a key for the efficiency and the expanded scope of applicability of the proposed approach. Linear systems involving nonlinear parameter dependencies, and dependencies between A(p) and b(p), as well as non-square linear parametric systems can be handled successfully. Presented are details of the algorithm design and mathematica tools implementing the proposed approach. Numerical examples from structural mechanics illustrate its application.