Balancing Algorithm for Three Matrices

Three versions of an algorithm for balancing three matrices simultaneously are proposed. The balancing is performed by premultiplication and postmultiplication with positive definite diagonal matrices, in order to reduce the magnitude range of all elements in the involved matrices. Some numerically stable algorithms, when applied to matrices with a wide range in the magnitude of elements, can produce results with a large error. As an application we present several problems from control theory. A reduction to the m-Hessenberg–triangular–triangular form is efficiently used for computing the frequency response G(σ) = C(σE − A)−1B + D of a descriptor system. The reduction algorithm can produce inaccurate results for badly scaled matrices. Numerical experiments confirmed that balancing matrices A, B and E before the m-Hessenberg–triangular–triangular reduction can produce an accurate frequency response matrix. Balancing three matrices can also improve the solution of the pole assignment problem for descriptor linear systems via state feedback, and the determination of the controllable part of the system. Other applications are: the quadratic eigenvalue problem λ2Ax+ λEx+Bx = 0, solution of the algebraic linear system (σ2A+ σB + C)x = b for several values of the parameter σ, and any other problems involving three matrices of same dimensions. These problems are not covered by the paper, but they can be balanced with a simplified version of the proposed algorithm.