Linearizations for interpolation bases -- a comparison I

Abstract

One strategy to solve a nonlinear eigenvalue problem $T(\lambda )x=0$ is to solve a polynomial eigenvalue problem (PEP) $P(\lambda )x=0$ that approximates the original problem through interpolation. Then, this PEP is usually solved by linearization. Most of the literature about linearizations assumes that $P(\lambda )$ is expressed in the monomial basis but, because of the polynomial approximation techniques, in this context, $P(\lambda )$ is expressed in a non-monomial basis. The bases used with most frequency are the Chebyshev basis, the Newton basis and the Lagrange basis. In this paper we construct a family of linearizations of $P(\lambda )$ that is easy to construct from the matrix coefficients of $P(\lambda )$ when this polynomial is expressed in any of those three bases. We also provide recovery formulas of eigenvectors (when $P(\lambda )$ is regular) and recovery formulas of minimal bases and minimal indices (when $P(\lambda )$ is singular). Our ultimate goal is to compare the numerical behavior of these linearizations, within the same family (to select the best one) and with the linearizations of other families based on the location of the eigenvalues with respect to the interpolation nodes.