2017
Journal article  Open Access

Fast Hessenberg reduction of some rank structured matrices

Gemignani L., Robol L

Bulge chasing  Mathematics - Numerical Analysis  Analysis  65F15  Quasiseparable matrices  CMV matrix  Hessenberg reduction  Complexity 

We develop two fast algorithms for Hessenberg reduction of a structured matrix $A = D + UV^H$, where $D$ is a real or unitary n x n diagonal matrix and $U, V in mathbb{C}^{n times k}$. The proposed algorithm for the real case exploits a two-stage approach by first reducing the matrix to a generalized Hessenberg form and then completing the reduction by annihilation of the unwanted subdiagonals. It is shown that the novel method requires O(n^2 k) arithmetic operations and is significantly faster than other reduction algorithms for rank structured matrices. The method is then extended to the unitary plus low rank case by using a block analogue of the CMV form of unitary matrices. It is shown that a block Lanczos-type procedure for the block tridiagonalization of Re(D) induces a structured reduction on A in a block staircase CMV-type shape. Then, we present a numerically stable method for performing this reduction using unitary transformations and show how to generalize the subdiagonal elimination to this shape, while still being able to provide a condensed representation for the reduced matrix. In this way the complexity still remains linear in k and, moreover, the resulting algorithm can be adapted to deal efficiently with block companion matrices.

Source: SIAM journal on matrix analysis and applications (Print) 38 (2017): 574–598. doi:10.1137/16M1107851

Publisher: Society for Industrial and Applied Mathematics ,, Philadelphia, Pa. , Stati Uniti d'America


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BibTeX entry
@article{oai:it.cnr:prodotti:374248,
	title = {Fast Hessenberg reduction of some rank structured matrices},
	author = {Gemignani L. and Robol L},
	publisher = {Society for Industrial and Applied Mathematics ,, Philadelphia, Pa. , Stati Uniti d'America},
	doi = {10.1137/16m1107851},
	journal = {SIAM journal on matrix analysis and applications (Print)},
	volume = {38},
	pages = {574–598},
	year = {2017}
}