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

Abstract. 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 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 optimal in a certain sense); and computation shows that sometimes it can be much smaller. These results also hold for the matrix polynomial case, when we are not looking for polynomial roots but rather for eigenvalues where the matrix polynomial is singular. This current paper explores the influence of the geometry of the interpolation nodes on the conditioning of the rootfinding and eigenvalue problems.