%0 Book Section %1 citeulike:13503605 %A Helmke, Uwe %A Wirth, Fabian %B Nonlinear Control in the Year 2000 %D 2001 %E Isidori, Alberto %E Lamnabhi-Lagarrigue, Francoise %E Respondek, Witold %I Springer London %K 65f15-numerical-eigenvalues-eigenvectors 65f10-iterative-methods-for-linear-systems %P 467--480 %R 10.1007/bfb0110234 %T Controllability properties of numerical eigenvalue algorithms %U http://dx.doi.org/10.1007/bfb0110234 %V 258 %X We analyze controllability properties of the inverse iteration and the QR-algorithm equipped with a shifting parameter as a control input. In the case of the inverse iteration with real shifts the theory of universally regular controls may be used to obtain necessary and sufficient conditions for complete controllability in terms of the solvability of a matrix equation. Partial results on conditions for the solvability of this matrix equation are given. We discuss an interpretation of the system in terms of control systems on rational functions. Finally, first results on the extension to inverse Rayleigh iteration on Grassmann manifolds using complex shifts is discussed.

@incollection{citeulike:13503605, abstract = {{We analyze controllability properties of the inverse iteration and the QR-algorithm equipped with a shifting parameter as a control input. In the case of the inverse iteration with real shifts the theory of universally regular controls may be used to obtain necessary and sufficient conditions for complete controllability in terms of the solvability of a matrix equation. Partial results on conditions for the solvability of this matrix equation are given. We discuss an interpretation of the system in terms of control systems on rational functions. Finally, first results on the extension to inverse Rayleigh iteration on Grassmann manifolds using complex shifts is discussed.}}, added-at = {2017-06-29T07:13:07.000+0200}, author = {Helmke, Uwe and Wirth, Fabian}, biburl = {https://www.bibsonomy.org/bibtex/2c3a7134dcca7583b9c8450bf7acad0a8/gdmcbain}, booktitle = {Nonlinear Control in the Year 2000}, citeulike-article-id = {13503605}, citeulike-attachment-1 = {helmke_01_controllability.pdf; /pdf/user/gdmcbain/article/13503605/1002899/helmke_01_controllability.pdf; 605290948f4068307eb0865bc89dde3ffb9091fa}, citeulike-linkout-0 = {http://dx.doi.org/10.1007/bfb0110234}, citeulike-linkout-1 = {http://link.springer.com/chapter/10.1007/BFb0110234}, doi = {10.1007/bfb0110234}, editor = {Isidori, Alberto and Lamnabhi-Lagarrigue, Fran\c{c}oise and Respondek, Witold}, file = {helmke_01_controllability.pdf}, interhash = {c15bf4c8851c11d45348508357542786}, intrahash = {c3a7134dcca7583b9c8450bf7acad0a8}, keywords = {65f15-numerical-eigenvalues-eigenvectors 65f10-iterative-methods-for-linear-systems}, pages = {467--480}, posted-at = {2015-01-29 05:42:43}, priority = {2}, publisher = {Springer London}, series = {Lecture Notes in Control and Information Sciences}, timestamp = {2019-02-28T23:43:44.000+0100}, title = {{Controllability properties of numerical eigenvalue algorithms}}, url = {http://dx.doi.org/10.1007/bfb0110234}, volume = 258, year = 2001 }