An analytical model for solving generalized interval eigenvalue problem

Abstract

• Theorem and corollaries are derived for generalized interval eigenvalue problems . • An effective way to discriminate the sign invariance of eigenvector is presented. • The analytical model solves the exact bounds of eigenvalue with high efficiency. • Inapplicability of existing methods and superiority of proposed model are revealed. Previous algorithms on calculating the eigenvalue bounds of generalized interval eigenvalue problems usually require preconditions difficult to be met or their efficiency and accuracy are unguaranteed. To overcome these defects, an exact analytical model for obtaining the interval solution set of generalized eigenvalue problem is proposed by rigorous derivations. The eigenvalue set of generalized interval eigenvalue problem is firstly characterized according to the linear interval system and interval arithmetic. Then by introducing the Rayleigh quotient the formulation of mathematical programming on maximizing and minimizing the eigenvalue is derived for the generalized interval eigenvalue problem. The Kuhn-Tucker theorem is subsequently applied to deduce the theorem on computing the upper and lower bounds of interval solution set. By applying the proposed model in real engineering problems, it is demonstrated that the positive definiteness or non-negative decomposition of the matrix pair required by the existing analytical methods cannot be satisfied in some engineering situations where the proposed model is applicable.