filtering with secure eigenvalue calculation and precise integration

Based on the analogy between structural mechanics and optimal control, interval mixed variable energy and the corresponding variational method are introduced to transform the induced norm c of an H= "lter into the fundamental eigenvalue c~2 #3 of a self-adjoint operator, which can itself be expressed as a generalized Rayleigh quotient of the variational principle with two kinds of variable, i.e. the state vector x and the co-state vector k. These vectors correspond, respectively, to the displacement and internal force vectors in the theory of structural mechanics. The interval matrices Q, G and F of the mixed variable energy are introduced in a natural way, based upon which the precise integration of a time invariant system is proposed and then combined with the extended Wittrick}Williams algorithm to solve the eigenvalue problem, enabling the optimal eigenvalue parameter c~2 #3 to be computed with high precision.