By Raf Vandebril
In fresh years numerous new sessions of matrices were found and their constitution exploited to layout speedy and exact algorithms. during this new reference paintings, Raf Vandebril, Marc Van Barel, and Nicola Mastronardi current the 1st finished evaluate of the mathematical and numerical homes of the family's latest member: semiseparable matrices.
The textual content is split into 3 components. the 1st presents a few ancient history and introduces innovations and definitions relating established rank matrices. the second one bargains a few conventional equipment for fixing structures of equations related to the fundamental subclasses of those matrices. The 3rd part discusses established rank matrices in a broader context, provides algorithms for fixing higher-order dependent rank matrices, and examines hybrid editions corresponding to block quasiseparable matrices. An obtainable case research sincerely demonstrates the overall subject of every new suggestion mentioned. a number of the workouts featured are carried out in Matlab and will be downloaded from the internet for additional exploration.
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Additional info for Matrix Computations and Semiseparable Matrices, Volume 1: Linear Systems
L. Wang and W. B. Gragg. Convergence of the unitary QR algorithm with unitary Wilkinson shift. Mathematics of Computation, 72(241):375– 385, 2003. As already mentioned, unitary Hessenberg matrices have the upper triangular part in the matrix of semiseparable form. In the following paper by Gemignani the quasiseparable structure of the unitary Hessenberg matrix is exploited to develop a QR-algorithm. Moreover the presented algorithm is also valid for unitary Hessenberg plus rank 1 matrices. The presented method transforms the matrix into a hermitian semiseparable plus diagonal matrix via the M¨ obius transform, then a QR-method is applied for computing its eigenvalues.
T. Heath, editor, Hypercube multiprocessors 1987, pages 639–647, Philadelphia, 1987. SIAM. ☞ T. L. Wang and W. B. Gragg. Convergence of the shifted QR algorithm, for unitary Hessenberg matrices. Mathematics of Computation, 71(240):1473– 1496, 2002. ☞ T. L. Wang and W. B. Gragg. Convergence of the unitary QR algorithm with unitary Wilkinson shift. Mathematics of Computation, 72(241):375– 385, 2003. As already mentioned, unitary Hessenberg matrices have the upper triangular part in the matrix of semiseparable form.