Stability of rootfinding for barycentric Lagrange interpolants

Abstract

Computing the roots of a univariate polynomial can be reduced to computing the eigenvalues of an associated companion matrix. For the monomial basis, these computations have been shown to be numerically stable under certain conditions. However, for certain applications, polynomials are more naturally expressed in other bases, such as the Lagrange basis or orthogonal polynomial bases. For the Lagrange basis, the equivalent stability results have not been published. We show that computing the roots of a polynomial expressed in barycentric form via the eigenvalues of an associated companion matrix pair is numerically stable, and give a bound for the backward error. Numerical experiments show that the error bound is approximately an order of magnitude larger than the backward error. We also discuss the matter of scaling and balancing the companion matrix to bring it closer to a normal pair. With balancing, we are able to produce roots with small backward error.