Direct and iterative methods for interval parametric algebraic systems producing parametric solutions

Abstract

Summary This paper deals with interval parametric linear systems with general dependencies. Motivated by the so‐called parameterized solution introduced by Kolev, we consider the enclosures of the solution set in a revised affine form. This form is advantageous to a classical interval solution because it enables us to obtain both outer and inner bounds for the parametric solution set and, thus, intervals containing the endpoints of the hull solution, among others. We propose two solution methods, a direct method called the generalized expansion method and an iterative method based on interval‐affine Krawczyk iterations. For the iterative method, we discuss its convergence and show the respective sufficient criterion. For both methods, we perform theoretical and numerical comparisons with some other approaches. The numerical experiments, including also interval parametric linear systems arising in practical problems of structural and electrical engineering, indicate the great usefulness of the proposed methodology and its superiority over most of the existing approaches to solving interval parametric linear systems.

Complexity of computing interval matrix powers for special classes of matrices

“…Many problems that are tractable for interval matrices become problematic for parametric ones even when considering a linear parametric case, see [6], [23], [24]. A linear parametric matrix reads (4.1)…”

Utilizing Statistical Information for Interval Analysis: A Method for Analyzing the Interval Uncertainty of Line-of-Sight Measurement Error of Space-Borne Observation Platforms

“…At present, the methods to deal with the uncertainty include the probability method [5], [6], the fuzzy method [7], [8], and the interval method [9], [10], etc. Among them, the fuzzy method uses membership function to describe the uncertain quantity, and uses the fuzzy mathematics to solve the membership function of the state quantity; the probability method uses random variables to describe the uncertainty, and uses Monte Carlo simulation [11], point estimation [12], semi-invariant method [13] to obtain the probability distribution characteristics of state variables; the interval method uses interval number to describe the uncertainty, and uses interval analysis theory to solve the equation with interval number.…”

On unbounded directions of linear interval parametric systems and their extensions to AE solutions

“…A number of works were devoted to computation of a tight enclosure of the solution set; see book [19] for a survey and very recent papers [8,15,20]. However, to find a bounded enclosure, the solution set must be bounded.…”