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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

A rational SHIRA method for the Hamiltonian eigenvalue problem

Benner, Peter, Effenberger, Cedric 07 January 2009 (has links) (PDF)
The SHIRA method of Mehrmann and Watkins belongs among the structure preserving Krylov subspace methods for solving skew-Hamiltonian eigenvalue problems. It can also be applied to Hamiltonian eigenproblems by considering a suitable transformation. Structure induced shift-and-invert techniques are employed to steer the algorithm towards the interesting region of the spectrum. However, the shift cannot be altered in the middle of the computation without discarding the information that has been accumulated so far. This paper shows how SHIRA can be combined with ideas from Ruhe's Rational Krylov algorithm to yield a method that permits an adjustment of shift after every step of the computation, adding greatly to the flexibility of the algorithm. We call this new method rational SHIRA. A numerical example is presented to demonstrate its efficiency.
2

HAMEV and SQRED: Fortran 77 Subroutines for Computing the Eigenvalues of Hamiltonian Matrices Using Van Loanss Square Reduced Method

Benner, P., Byers, R., Barth, E. 30 October 1998 (has links) (PDF)
This paper describes LAPACK-based Fortran 77 subroutines for the reduction of a Hamiltonian matrix to square-reduced form and the approximation of all its eigenvalues using the implicit version of Van Loan's method. The transformation of the Hamilto- nian matrix to a square-reduced Hamiltonian uses only orthogonal symplectic similarity transformations. The eigenvalues can then be determined by applying the Hessenberg QR iteration to a matrix of half the order of the Hamiltonian matrix and taking the square roots of the computed values. Using scaling strategies similar to those suggested for algebraic Riccati equations can in some cases improve the accuracy of the computed eigenvalues. We demonstrate the performance of the subroutines for several examples and show how they can be used to solve some control-theoretic problems.
3

HAMEV and SQRED: Fortran 77 Subroutines for Computing the Eigenvalues of Hamiltonian Matrices Using Van Loanss Square Reduced Method

Benner, P., Byers, R., Barth, E. 30 October 1998 (has links)
This paper describes LAPACK-based Fortran 77 subroutines for the reduction of a Hamiltonian matrix to square-reduced form and the approximation of all its eigenvalues using the implicit version of Van Loan's method. The transformation of the Hamilto- nian matrix to a square-reduced Hamiltonian uses only orthogonal symplectic similarity transformations. The eigenvalues can then be determined by applying the Hessenberg QR iteration to a matrix of half the order of the Hamiltonian matrix and taking the square roots of the computed values. Using scaling strategies similar to those suggested for algebraic Riccati equations can in some cases improve the accuracy of the computed eigenvalues. We demonstrate the performance of the subroutines for several examples and show how they can be used to solve some control-theoretic problems.
4

A rational SHIRA method for the Hamiltonian eigenvalue problem

Benner, Peter, Effenberger, Cedric 07 January 2009 (has links)
The SHIRA method of Mehrmann and Watkins belongs among the structure preserving Krylov subspace methods for solving skew-Hamiltonian eigenvalue problems. It can also be applied to Hamiltonian eigenproblems by considering a suitable transformation. Structure induced shift-and-invert techniques are employed to steer the algorithm towards the interesting region of the spectrum. However, the shift cannot be altered in the middle of the computation without discarding the information that has been accumulated so far. This paper shows how SHIRA can be combined with ideas from Ruhe's Rational Krylov algorithm to yield a method that permits an adjustment of shift after every step of the computation, adding greatly to the flexibility of the algorithm. We call this new method rational SHIRA. A numerical example is presented to demonstrate its efficiency.
5

Lagrangian invariant subspaces of Hamiltonian matrices

Mehrmann, 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.
6

A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils

Benner, P., Mehrmann, V., Xu, H. 30 October 1998 (has links) (PDF)
A new method is presented for the numerical computation of the generalized eigen- values of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order sqrt(epsilon), where epsilon is the machine precision, the new method computes the eigenvalues to full possible accuracy.
7

A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils

Benner, P., Mehrmann, V., Xu, H. 30 October 1998 (has links)
A new method is presented for the numerical computation of the generalized eigen- values of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order sqrt(epsilon), where epsilon is the machine precision, the new method computes the eigenvalues to full possible accuracy.
8

Lagrangian invariant subspaces of Hamiltonian matrices

Mehrmann, 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.
9

$L_\infty$-Norm Computation for Descriptor Systems

Voigt, Matthias 15 July 2010 (has links) (PDF)
In many applications from industry and technology computer simulations are performed using models which can be formulated by systems of differential equations. Often the equations underlie additional algebraic constraints. In this context we speak of descriptor systems. Very important characteristic values of such systems are the $L_\infty$-norms of the corresponding transfer functions. The main goal of this thesis is to extend a numerical method for the computation of the $L_\infty$-norm for standard state space systems to descriptor systems. For this purpose we develop a numerical method to check whether the transfer function of a given descriptor system is proper or improper and additionally use this method to reduce the order of the system to decrease the costs of the $L_\infty$-norm computation. When computing the $L_\infty$-norm it is necessary to compute the eigenvalues of certain skew-Hamiltonian/Hamiltonian matrix pencils composed by the system matrices. We show how we extend these matrix pencils to skew-Hamiltonian/Hamiltonian matrix pencils of larger dimension to get more reliable and accurate results. We also consider discrete-time systems, apply the extension strategy to the arising symplectic matrix pencils and transform these to more convenient structures in order to apply structure-exploiting eigenvalue solvers to them. We also investige a new structure-preserving method for the computation of the eigenvalues of skew-Hamiltonian/Hamiltonian matrix pencils and use this to increase the accuracy of the computed eigenvalues even more. In particular we ensure the reliability of the $L_\infty$-norm algorithm by this new eigenvalue solver. Finally we describe the implementation of the algorithms in Fortran and test them using two real-world examples.
10

A restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem

Benner, P., Faßbender, H. 30 October 1998 (has links) (PDF)
A restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. Breakdowns and near-breakdowns are overcome by inexpensive implicit restarts. The method is used to compute eigenvalues, eigenvectors and invariant subspaces of large and sparse Hamiltonian matrices and low rank approximations to the solution of continuous-time algebraic Riccati equations with large and sparse coefficient matrices.

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