Qd-type methods for quasiseparable matrices

Bevilacqua, Roberto and Bozzo, Enrico and Del Corso, Gianna M. (2010) Qd-type methods for quasiseparable matrices. Technical Report del Dipartimento di Informatica . Università di Pisa, Pisa, IT.

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Abstract

In the last few years many numerical techniques for computing eigenvalues of structured rank matrices have been proposed. Most of them are<br />based on $QR$ iterations since, in the symmetric case, the rank structure is preserved and high accuracy is guaranteed. In the unsymmetric<br />case, however, the $QR$ algorithm destroys the rank structure, which is instead preserved if $LR$ iterations are used. We consider a wide<br />class of quasiseparable matrices which can be represented in terms of the same parameters involved in their Neville factorization. This<br />class, if assumptions are made to prevent possible breakdowns, is closed under $LR$ steps. Moreover, we propose an implicit shifted $LR$<br />method with a linear cost per step, which resembles the qd method for tridiagonal matrices. We show that for totally nonnegative<br />quasiseparable matrices the algorithm is stable and breakdowns cannot occur, if the Laguerre shift, or other shift strategy preserving<br />nonnegativity, is used. Computational evidence shows that good accuracy is obtained also when applied to symmetric positive definite<br />matrices.<br />

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