Enclosures for the solution set of parametric interval linear systems

Abstract

We investigate parametric interval linear systems of equations. The main result is a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for this case, comparing and refinement of both. We show that the latter bounds are not provable better, and that they are also sometimes too pessimistic. The presented form of both methods is suitable for combining them into one to get a more efficient algorithm. Some numerical experiments are carried out to illustrate performances of the methods.

Bracketing the solutions of an ordinary differential equation with uncertain initial conditions

“…In this context, we define the positive graph as the set [9]. Some existing approaches use guaranteed integration [6,20,23] to bracket those sets [7].…”

“…Some existing approaches use guaranteed integration [6,20,23] to bracket those sets [7]. For efficiency reasons we will propose in this paper, a guaranteed approach based on interval computation [13,9] and constraint networking [14] that do not use guaranteed integration. The main difference with existing approaches is that bisections will take place both in the time space and the state space, which makes the method both Eulerian and Lagrangian [15].…”

Enclosure for the Solution Set of Parametric Linear Systems with Non-affine Dependencies

“…The results of the M1 method are compared to the results obtained using the following methods: Rump's parametric fixed-point iteration (RPFPI) ( [2], [12]), Bauer-Skeel (BS) method [5], Hansen-Bliek-Rohn method [5] and Interval-Affine Gaussian Elimination [1]. All of them are polynomial complexity methods for solving parametric interval linear systems.…”

Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

“…Linear parametric forms are also used to model linear dependencies between parameters in interval linear equation solving [10,30,31,41]. Linear dependencies cause not only the problem to be more difficult from the computational viewpoint, but it is also hard to describe the corresponding solution set; see Mayer [23].…”