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