On Existence and Uniqueness of Solutions of Linear Algebraic Equations in Kaucher’s Interval Arithmetic

Abstract

Presently there is a growing interest to the investigation of linear algebraic equations in Kaucher’s arithmetic [1]. This interest is mainly substantiated by the fact that with the use of solutions of linear algebraic equations in Kaucher’s arithmetic allows in some cases to obtain both external and internal estimates of various sets of solutions of linear interval equations. First such results appeared rather long ago and were concerned with the external estimation of the joint set of solutions for the system of linear interval equations by algebraic solving of this system (see, e.g., G. Alefeld and J. Herzberger [2], A. Neumaier [3], and the references in these books). Later, in L.V. Kupriyanova’s [4] and S.P. Shary’s [5] works, it was shown that with the use of algebraic solutions (but already in an extended Kaucher’s interval arithmetic) it is possible to obtain both internal and external estimates for generalized sets of solutions for the systems of linear interval equations.