On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis

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Experimental observations of univariate rootfinding by generalized companion matrix pencils expressed in the Lagrange basis show that the method can sometimes be numerically stable. It has recently been proved that a new rootfinding condition number, defined for points on a set containing the interpolation points, is never larger than the rootfinding condition number for the Bernstein polynomial (which is itself optimally small in a certain sense); and computation shows that sometimes it can be much smaller. These results extend to the matrix polynomial case, although in this case we are not computing polynomial roots but rather ‘polynomial eigenvalues’ (sometimes known as ‘latent roots’), i.e. finding the values of x where the matrix polynomial is singular. This paper gives two theorems explaining part of the influence of the geometry of the interpolation nodes on the conditioning of the rootfinding and eigenvalue problems.

In memoriam: Karin Gatermann (1961–2005)
This work was supported in part by a grant from the Natural Science and Engineering Research Council of Canada.