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We develop two fast algorithms for Hessenberg reduction of a structuredmatrix $A = D + UV^H$ where $D$ is a real or unitary $n \times n$ diagonalmatrix and $U, V \in\mathbb{C}^{n \times k}$. The proposed algorithm for thereal case exploits a two--stage approach by first reducing the matrix to ageneralized Hessenberg form and then completing the reduction by annihilationof the unwanted sub-diagonals. It is shown that the novel method requires$O(n^2k)$ arithmetic operations and it is significantly faster than otherreduction algorithms for rank structured matrices. The method is then extendedto the unitary plus low rank case by using a block analogue of the CMV form ofunitary matrices. It is shown that a block Lanczos-type procedure for the blocktridiagonalization of $\Re(D)$ induces a structured reduction on $A$ in a blockstaircase CMV--type shape. Then, we present a numerically stable method forperforming this reduction using unitary transformations and we show how togeneralize the sub-diagonal elimination to this shape, while still being ableto provide a condensed representation for the reduced matrix. In this way thecomplexity still remains linear in $k$ and, moreover, the resulting algorithmcan be adapted to deal efficiently with block companion matrices.

Fast Hessenberg reduction of some rank structured matrices