An implicit multishift QR-algorithm for Hermitian plus low rank matrices

Vandebril, Raf and Del Corso, Gianna M. (2009) An implicit multishift QR-algorithm for Hermitian plus low rank matrices. Technical Report del Dipartimento di Informatica . Università di Pisa, Pisa, IT.

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Abstract

Hermitian plus possibly unhermitian low rank matrices can be efficiently reduced into Hessenberg form. The resulting Hessenberg matrix can still be written as the sum of a Hermitian plus low rank matrix. In this paper we develop a new implicit multishift QR-algorithm for Hessenberg matrices, which are the sum of a Hermitian plus a possibly non-Hermitian low rank correction.The proposed algorithm exploits both the symmetry and low rank structure to obtain a QR-step involving only O(n) floating point operations instead of the standard O(n^2) operations needed for performing a QR-step on a Hessenberg matrix. The algorithm is based on a suitable O(n) representation of the Hessenberg matrix. The low rank parts present in both the Hermitian and low rank part of the sum are compactly stored by a sequence of Givens transformations and few vectors. Due to the new representation, we cannot apply classical deflation techniques for Hessenberg matrices. A new, efficient technique is developed to overcome this problem. Some numerical experiments based on matrices arising in applications are performed.The experiments illustrate effectiveness and accuracy of both the $QR$-algorithm and the newly developed deflation technique.

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