On the Solution of Parametrised Linear Systems

Abstract

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.

Parametric Interval Linear Solver

“…The arithmetic of proper and improper intervals possesses properties due to which inwardly rounded interval arithmetic can be applied at no additional cost [10]. Based on the methodology developed in [8] and the arithmetic supported by the package IntervalComputations'GeneralizedIntervals' [10], all the iterative solvers, supported by the package IntervalComputations'LinearSystems', provide rigorous inner estimations for the solution. The presented package is maybe the only that provides this feature.…”

“…Inner estimations allow to obtain the very important measure for the degree of sharpness of an outer solution set enclosure [14]. Computing inner approximations by the iterative solvers is based on generalized interval arithmetic (see [8]) and requires the package IntervalComputations'GeneralizedIntervals' [10] which, if available, is loaded automatically. Refinement is an option that, when set to True, implies the application of an iterative refinement procedure for the outer solution set approximation.…”

“…It was shown that the parametric Rump's residual iteration possesses an excellent performance for parameters with small tolerances and does not give very sharp enclosures for parameters with large tolerances [8], [15]. On another side, some practical problems require very sharp solution set enclosure.…”

“…Except for [8], by now there is no other work that would apply and study the results of this method. Remarkably, the parametric Rump's method was recently reinvented in slightly different notations [2].…”

Enclosures for the solution set of parametric interval linear systems

“…Kolev (2006) proposed a direct method and an iterative one (Kolev, 2004) for computing an enclosure of the solution set. Parametrized Gauss-Seidel iteration was employed by Popova (2001). A direct method was given by Skalna (2006), and a monotonicity approach by Popova (2006a), Rohn (2004), and Skalna (2008).…”