Vanni Noferini

Dario A. Bini, Vanni Noferini, Meisam Sharify

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by A.E. Pellet [Bulletin des Sciences Mathématiques, (2), vol 5 (1881), pp.393-395], some results of D.A. Bini [Numer. Algorithms 13:179-200, 1996] based on the Newton polygon technique, and recent results of M. Akian, S. Gaubert and M. Sharify (see in particular [LNCIS, 389, Springer p.p.291-303] and [M. Sharify, Ph.D. thesis, École Polytechnique, ParisTech, 2011]). These extensions are applied for determining effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich-Aberth method. Numerical experiments that show the computational advantage of these results are presented.

Yuji Nakatsukasa, Vanni Noferini, Alex Townsend

We revisit the landmark paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, SIAM J. Matrix Anal. Appl., 28 (2006), pp.~971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials. We also show that every pencil in the double ansatz space is intrinsically connected to a Bézout matrix, which we use to prove the eigenvalue exclusion theorem. In addition our exposition allows for any polynomial basis and for any field. The new viewpoint also leads to new results. We generalize the double ansatz space by exploiting its algebraic interpretation as a space of Bézout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials. Moreover, we analyze the conditioning of double ansatz space linearizations in the important practical case of a Chebyshev basis.

Vanni Noferini, Lauri Nyman, Federico Poloni

Given a square complex matrix $A$, we tackle the problem of finding the nearest matrix with multiple eigenvalues or, equivalently when $A$ had distinct eigenvalues, the nearest defective matrix. To this goal, we extend the general framework described in [M. Gnazzo, V. Noferini, L. Nyman, F. Poloni, \emph{Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory}, Found. Comput. Math., To appear] and based on variable projection and Riemannian optimization, allowing the ambient manifold to simultaneously track left and right eigenvectors. Our method also allows us to impose arbitrary complex-linear constraints on either the perturbation or the perturbed matrix; this can be useful to study structured eigenvalue condition numbers. We present numerical experiments, comparing with preexisting algorithms.

Vanni Noferini

Given a polynomial matrix P(x) of grade g and a rational function $x(y) = n(y)/d(y)$, where $n(y)$ and $d(y)$ are coprime nonzero scalar polynomials, the polynomial matrix $Q(y) :=[d(y)]^gP(x(y))$ is defined. The complete eigenstructures of $P(x)$ and $Q(y)$ are related, including characteristic values, elementary divisors and minimal indices. A Theorem on the matter, valid in the most general hypotheses, is stated and proved.

Yuji Nakatsukasa, Vanni Noferini

Sylvester's law of inertia states that the number of positive, negative and zero eigenvalues of Hermitian matrices is preserved under congruence transformations. The same is true of generalized Hermitian definite eigenvalue problems, in which the two matrices are allowed to undergo different congruence transformations, but not for the indefinite case. In this paper we investigate the possible change in inertia under congruence for generalized Hermitian indefinite eigenproblems, and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil, especially when one of the matrices is almost definite. A prominent application of the original Sylvester's law is in finding the number of eigenvalues in an interval. Our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems.

Luca Gemignani, Vanni Noferini

An algorithm based on the Ehrlich-Aberth root-finding method is presented for the computation of the eigenvalues of a T-palindromic matrix polynomial. A structured linearization of the polynomial represented in the Dickson basis is introduced in order to exploit the symmetry of the roots by halving the total number of the required approximations. The rank structure properties of the linearization allow the design of a fast and numerically robust implementation of the root-finding iteration. Numerical experiments that confirm the effectiveness and the robustness of the approach are provided.