We present a multi-grid method for a class of structured generalized Lyapunov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial differential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The efficiency of the method is demonstrated by numerical experiments.

We present new Fortran 77 subroutines which implement the Schur method and the matrix sign function method for the solution of the continuous­time matrix algebraic Riccati equation on the basis of LAPACK subroutines. In order to avoid some of the well­known difficulties with these methods due to a loss of accuracy, we combine the implementations with block scalings as well as condition estimates and forward error estimates. Results of numerical experiments comparing the performance of both methods for more than one hundred well­ and ill­conditioned Riccati equations of order up to 150 are given. It is demonstrated that there exist several classes of examples for which the matrix sign function approach performs more reliably and more accurately than the Schur method. In all cases the forward error estimates allow to obtain a reliable bound on the accuracy of the computed solution.

Diss.--Mathematik--Chemnitz--Technische Universität, 1998. / Bibliogr. p. 103-105.

The existence and uniqueness of Lagrangian invariant subspaces of Hamiltonian matrices is studied. Necessary and sufficient conditions are given in terms of the Jordan structure and certain sign characteristics that give uniqueness of these subspaces even in the presence of purely imaginary eigenvalues. These results are applied to obtain in special cases existence and uniqueness results for Hermitian solutions of continuous time algebraic Riccati equations.

We discuss a parallel library of efficient algorithms for the solution of linear-quadratic optimal control problems involving largescale systems with state-space dimension up to $O(10^4)$. We survey the numerical algorithms underlying the implementation of the chosen optimal control methods. The approaches considered here are based on invariant and deflating subspace techniques, and avoid the explicit solution of the associated algebraic Riccati equations in case of possible ill-conditioning. Still, our algorithms can also optionally compute the Riccati solution. The major computational task of finding spectral projectors onto the required invariant or deflating subspaces is implemented using iterative schemes for the sign and disk functions. Experimental results report the numerical accuracy and the parallel performance of our approach on a cluster of Intel Itanium-2 processors.

We present a multi-grid method for a class of structured generalized Lyapunov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial differential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The efficiency of the method is demonstrated by numerical experiments.

We present new Fortran 77 subroutines which implement the Schur method and the matrix sign function method for the solution of the continuous­time matrix algebraic Riccati equation on the basis of LAPACK subroutines. In order to avoid some of the well­known difficulties with these methods due to a loss of accuracy, we combine the implementations with block scalings as well as condition estimates and forward error estimates. Results of numerical experiments comparing the performance of both methods for more than one hundred well­ and ill­conditioned Riccati equations of order up to 150 are given. It is demonstrated that there exist several classes of examples for which the matrix sign function approach performs more reliably and more accurately than the Schur method. In all cases the forward error estimates allow to obtain a reliable bound on the accuracy of the computed solution.

The existence and uniqueness of Lagrangian invariant subspaces of Hamiltonian matrices is studied. Necessary and sufficient conditions are given in terms of the Jordan structure and certain sign characteristics that give uniqueness of these subspaces even in the presence of purely imaginary eigenvalues. These results are applied to obtain in special cases existence and uniqueness results for Hermitian solutions of continuous time algebraic Riccati equations.

We discuss a parallel library of efficient algorithms for the solution of linear-quadratic optimal control problems involving largescale systems with state-space dimension up to $O(10^4)$. We survey the numerical algorithms underlying the implementation of the chosen optimal control methods. The approaches considered here are based on invariant and deflating subspace techniques, and avoid the explicit solution of the associated algebraic Riccati equations in case of possible ill-conditioning. Still, our algorithms can also optionally compute the Riccati solution. The major computational task of finding spectral projectors onto the required invariant or deflating subspaces is implemented using iterative schemes for the sign and disk functions. Experimental results report the numerical accuracy and the parallel performance of our approach on a cluster of Intel Itanium-2 processors.

We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditions for the existence of Hamiltonian and symplectic triangular Jordan, Kronecker and Schur forms. The presented results generalize results of Lin and Ho [17] and simplify the proofs presented there.