This paper is concerned with the reduction of a unitary matrix U 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 U, i.e., of the matrix U+UH. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U 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. © 2014 Elsevier Inc. All rights reserved.

Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding

BEVILACQUA, ROBERTO;DEL CORSO, GIANNA MARIA;GEMIGNANI, LUCA
2015

Abstract

This paper is concerned with the reduction of a unitary matrix U 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 U, i.e., of the matrix U+UH. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U 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. © 2014 Elsevier Inc. All rights reserved.
Bevilacqua, Roberto; DEL CORSO, GIANNA MARIA; Gemignani, Luca
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/565867
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