A unification of unitary similarity transforms to compressed representations |
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Authors: | Raf Vandebril Gianna M Del Corso |
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Institution: | (1) State Key Laboratory of Robotics, Shenyang Institute of Automation, Academia Sinica, 110016 Shenyang, China; |
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Abstract: | In this paper a new framework for transforming arbitrary matrices to compressed representations is presented. The framework
provides a generic way of transforming a matrix via unitary similarity transformations to, e.g., Hessenberg, Hessenberg-like
form and combinations of both. The new algorithms are deduced, based on the QR-factorization of the original matrix. Relying on manipulations with rotations, all the algorithms consist of eliminating
the correct set of rotations, resulting in a matrix obeying the desired structural constraints. Based on this new reduction
procedure we investigate further correspondences such as irreducibility, uniqueness of the reduction procedure and the link
with (rational) Krylov methods. The unitary similarity transform to Hessenberg-like form as presented here, differs significantly
from the one presented in earlier work. Not only does it use less rotations to obtain the desired structure, also the convergence
to rational Ritz-values is not observed in the conventional approach. |
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Keywords: | |
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