Compression of unitary rank-structured matrices to CMV-like shape with an application to polynomial rootfinding
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Roberto Bevilacqua [1] ; Gianna M. Del Corso [1] ; Luca Gemignani [1]
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[1] University of Pisa
University of Pisa
Pisa, Italia
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- Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 278, Nº 1 (15 April 2015), 2015, págs. 326-335
- Idioma: inglés
- DOI: 10.1016/j.cam.2014.09.023
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Resumen
This paper is concerned with the reduction of a unitary matrix UU to CMV-like shape. A Lanczos-type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of UU, i.e., of the matrix U+UHU+UH. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If UU is rank-structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank-structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration.
