Enclosing the solution set of parametric interval matrix equation A(p)X = B(p)

Abstract

Consider the parametric matrix equation A(p)X = B(p), where the elements of the matrices A(p) and B(p) depend linearly on a number of uncertain parameters varying within given intervals. We prove that the united parametric solution sets of the matrix equation and that of the corresponding linear system with multiple right-hand sides, although different as sets, have the same interval hull. A generalization of the parametric Krawczyk iteration with low computational complexity for the matrix equation is presented. Some details improving the implementation and the application of this method are discussed. An interval method, designed by A. Neumaier and A. Pownuk for enclosing the united solution set of parametric linear systems with particular dependency structure, is generalized for arbitrary linear dependencies between the parameters and for systems with multiple right-hand sides. A new, more powerful, sufficient condition for regularity of a parametric interval matrix is proven. An important application of the linear systems with multiple right-hand sides is presented as a key methodology for feasibility in computing the interval hull of a class of united parametric solution sets that appear in practical problems.