Abstract.IntervalComputations'LinearSystems' is a Mathematica package supporting tools for solving parametric and non-parametric linear systems involving uncertainties. It includes a variety of functions, implementing different interval techniques, that help in producing sharp and rigorous results in validated interval arithmetic. The package is designed to be easy to use, versatile, to provide a necessary background for further exploration, comparisons and prototyping, and to provide some indispensable tools for solving parametric interval linear systems.This paper presents the functionality, provided by the current version of the package, and briefly discusses the underlying methodology. A new hybrid approach for sharp parametric enclosures, that combines parametric residual iteration, exact bounds, based on monotonicity properties, and refinement by interval subdivision, is outlined.

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Considered are parametrised linear systems which parameters are subject to tolerances. Rump's fixed-point iteration method for finding outer and inner approximations of the hull of the solution set is studied and applied to an electrical circuit problem. Interval Gauss-Seidel iteration for parametrised linear systems is introduced and used for improving the enclosures, obtained by the fixed-point method, whenever they are not good enough. Generalised interval arithmetic (on proper and improper intervals) is considered as a computational tool for efficient handling of proper interval problems (to obtain inner interval estimations without inward rounding and to eliminate the dependency problem in parametrised Gauss-Seidel iteration). Numerical results from the application of the above methods to an electrical circuit problem are discussed.

Consider a linear system A(p) · x = b(p), where the elements of the matrix and the right-hand side vector depend linearly on a m-tuple of parameters p = (p 1 , . . . , p m ), the exact values of which are unknown but bounded within given intervals. Apart from quantifier elimination, the only known general way of describing the solution set {x ∈ R n | ∃p ∈ [p], A(p)x = b(p)} is a lengthy and non-unique Fourier-Motzkin-type parameter elimination process that leads to a description of the solution set by exponentially many inequalities. In this work we modify the parameter elimination process in a way that has a significant impact on the representation of the inequalities describing the solution set and their number. An explicit minimal description of the solution set to 2D parametric linear systems is derived. It generalizes the Oettli-Prager theorem for non-parametric linear systems. The number of the inequalities describing the solution set grows linearly with the number of the parameters involved simultaneously in both equations of the system. The boundary of any 2D parametric solution set is described by polynomial equations of at most second degree. It is proven that when the general parameter elimination process is applied to two equations of a system in higher dimension, some inequalities become redundant.