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A unification of unitary similarity transforms to compressed representations

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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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Author information

Authors and Affiliations

  1. Department of Computer Science, K.U. Leuven, 3001, Louvain (Heverlee), Belgium

    Raf Vandebril

  2. Department of Computer Science, Università di Pisa, Largo Pontecorvo, 3, 56127, Pisa, Italy

    Gianna M. Del Corso

Authors
  1. Raf Vandebril
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  2. Gianna M. Del Corso
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Corresponding author

Correspondence to Gianna M. Del Corso.

Additional information

R. Vandebril has a grant as “Postdoctoraal Onderzoeker” from the Fund for Scientific Research—Flanders (Belgium). This research was also partially supported by the Research Council K.U. Leuven, project OT/11/055 (Spectral properties of (perturbed) normal matrices and their applications), and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization). The research of G. M. Del Corso is partially supported by the PRIN project “Analisi di strutture di matrici: aspetti teorici, computazionali e applicazioni” Prot. n. 20083KLJEZ by the Italian MIUR.

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Vandebril, R., Del Corso, G.M. A unification of unitary similarity transforms to compressed representations. Numer. Math. 119, 641–665 (2011). https://doi.org/10.1007/s00211-011-0400-5

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  • DOI: https://doi.org/10.1007/s00211-011-0400-5

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