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1	The Matrix Sign Function Method and the Computation of Invariant Subspaces Byers, R., He, C., Mehrmann, V. 30 October 1998 (has links) (PDF) A perturbation analysis shows that if a numerically stable procedure is used to compute the matrix sign function, then it is competitive with conventional methods for computing invariant subspaces. Stability analysis of the Newton iteration improves an earlier result of Byers and confirms that ill-conditioned iterates may cause numerical instability. Numerical examples demonstrate the theoretical results. matrix sign function invariant subspaces stability Newton iteration numerical examples MSC 65F30 MSC 65F15 ddc:510
2	Newtons method with exact line search for solving the algebraic Riccati equation Benner, P., Byers, R. 30 October 1998 (has links) (PDF) This paper studies Newton's method for solving the algebraic Riccati equation combined with an exact line search. Based on these considerations we present a Newton{like method for solving algebraic Riccati equations. This method can improve the sometimes erratic convergence behavior of Newton's method. Newton method algebraic Riccati equation exact line search step size control MSC 65F30 MSC 15A24 ddc:510
3	The Matrix Sign Function Method and the Computation of Invariant Subspaces Byers, R., He, C., Mehrmann, V. 30 October 1998 (has links) A perturbation analysis shows that if a numerically stable procedure is used to compute the matrix sign function, then it is competitive with conventional methods for computing invariant subspaces. Stability analysis of the Newton iteration improves an earlier result of Byers and confirms that ill-conditioned iterates may cause numerical instability. Numerical examples demonstrate the theoretical results. info:eu-repo/classification/ddc/510 ddc:510 matrix sign function invariant subspaces stability Newton iteration numerical examples MSC 65F30 MSC 65F15
4	Newtons method with exact line search for solving the algebraic Riccati equation Benner, P., Byers, R. 30 October 1998 (has links) This paper studies Newton's method for solving the algebraic Riccati equation combined with an exact line search. Based on these considerations we present a Newton{like method for solving algebraic Riccati equations. This method can improve the sometimes erratic convergence behavior of Newton's method. info:eu-repo/classification/ddc/510 ddc:510
5	A cyclic low rank Smith method for large, sparse Lyapunov equations with applications in model reduction and optimal control Penzl, T. 30 October 1998 (has links) (PDF) We present a new method for the computation of low rank approximations to the solution of large, sparse, stable Lyapunov equations. It is based on a generalization of the classical Smith method and profits by the usual low rank property of the right hand side matrix. The requirements of the method are moderate with respect to both computational cost and memory. Hence, it provides a possibility to tackle large scale control problems. Besides the efficient solution of the matrix equation itself, a thorough integration of the method into several control algorithms can improve their performance to a high degree. This is demonstrated for algorithms for model reduction and optimal control. Furthermore, we propose a heuristic for determining a set of suboptimal ADI shift parameters. This heuristic, which is based on a pair of Arnoldi processes, does not require any a priori knowledge on the spectrum of the coefficient matrix of the Lyapunov equation. Numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the ADI parameters. ADI iteration Smith method iterative methods Lyapunov equation matrix equation model reduction balanced truncation optimal control Riccati equation Newton method MSC 65F30 MSC 65F10 MSC 15A24 MSC 93C05 ddc:510
6	A cyclic low rank Smith method for large, sparse Lyapunov equations with applications in model reduction and optimal control Penzl, T. 30 October 1998 (has links) We present a new method for the computation of low rank approximations to the solution of large, sparse, stable Lyapunov equations. It is based on a generalization of the classical Smith method and profits by the usual low rank property of the right hand side matrix. The requirements of the method are moderate with respect to both computational cost and memory. Hence, it provides a possibility to tackle large scale control problems. Besides the efficient solution of the matrix equation itself, a thorough integration of the method into several control algorithms can improve their performance to a high degree. This is demonstrated for algorithms for model reduction and optimal control. Furthermore, we propose a heuristic for determining a set of suboptimal ADI shift parameters. This heuristic, which is based on a pair of Arnoldi processes, does not require any a priori knowledge on the spectrum of the coefficient matrix of the Lyapunov equation. Numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the ADI parameters. info:eu-repo/classification/ddc/510 ddc:510 ADI iteration Smith method iterative methods Lyapunov equation matrix equation model reduction balanced truncation optimal control Riccati equation Newton method MSC 65F30 MSC 65F10 MSC 15A24 MSC 93C05

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