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A Multi-Grid Method for Generalized Lyapunov EquationsPenzl, Thilo 07 September 2005 (has links) (PDF)
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.
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DGRSVX and DMSRIC: Fortran 77 subroutines for solving continuous-time matrix algebraic Riccati equations with condition and accuracy estimatesPetkov, P. Hr., Konstantinov, M. M., Mehrmann, V. 12 September 2005 (has links) (PDF)
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.
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Compatible Lie and Jordan algebras and applications to structured matrices and pencils /Mehl, Christian, January 1900 (has links)
Diss.--Mathematik--Chemnitz--Technische Universität, 1998. / Bibliogr. p. 103-105.
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Lagrangian invariant subspaces of Hamiltonian matricesMehrmann, Volker, Xu, Hongguo 14 September 2005 (has links) (PDF)
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.
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Solving Linear-Quadratic Optimal Control Problems on Parallel ComputersBenner, Peter, Quintana-Ortí, Enrique S., Quintana-Ortí, Gregorio 11 September 2006 (has links) (PDF)
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.
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A Multi-Grid Method for Generalized Lyapunov EquationsPenzl, Thilo 07 September 2005 (has links)
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.
|
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DGRSVX and DMSRIC: Fortran 77 subroutines for solving continuous-time matrix algebraic Riccati equations with condition and accuracy estimatesPetkov, P. Hr., Konstantinov, M. M., Mehrmann, V. 12 September 2005 (has links)
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.
|
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Lagrangian invariant subspaces of Hamiltonian matricesMehrmann, Volker, Xu, Hongguo 14 September 2005 (has links)
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.
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Solving Linear-Quadratic Optimal Control Problems on Parallel ComputersBenner, Peter, Quintana-Ortí, Enrique S., Quintana-Ortí, Gregorio 11 September 2006 (has links)
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.
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Canonical forms for Hamiltonian and symplectic matrices and pencilsMehrmann, Volker, Xu, Hongguo 09 September 2005 (has links) (PDF)
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.
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