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

Hermitian plus possibly non-Hermitian 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 $\mO{n}$ floating point operations instead of the standard $\mO{n^2}$ operations needed for performing a $QR$-step on a Hessenberg matrix. The algorithm is based on a suitable$\mO{n}$ 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.