Vanni Noferini

LOCATING THE EIGENVALUES OF MATRIX POLYNOMIALS

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by Pellet [Bull. Sci. Math. (2), 5 (1881), pp. 393--395], some results from Bini [Numer. Algorithms, 13 (1996), pp. 179--200] based on the Newton polygon technique, and recent results from Gaubert and Sharify (see, in particular, [Tropical scaling of polynomial matrices, Lecture Notes in Control and Inform. Sci. 389, Springer, Berlin, 2009, pp. 291--303] and [Sharify, Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues, Ph.D. thesis, Ecole Polytechnique, Paris, 2011]). These extensions are applied to determine 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.

Vector Spaces of Linearizations for Matrix Polynomials: A Bivariate Polynomial Approach

We revisit the landmark paper [D. S. Mackey et al. 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 Bezout 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 Bezout 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.

Computing the common zeros of two bivariate functions via Bézout resultants

The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bézout matrices with polynomial entries. Using techniques including domain subdivision, Bézoutian regularization, and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (

Tropical roots as approximations to eigenvalues of matrix polynomials

\ge \!1{,}000

Fiedler-comrade and Fiedler--Chebyshev pencils

≥1,000). We analyze the resultant method and its conditioning by noting that the Bézout matrices are matrix polynomials. Two implementations are available: one on the Matlab Central File Exchange and another in the roots command in Chebfun2 that is adapted to suit Chebfun’s methodology.

The Ehrlich-Aberth method for palindromic matrix polynomials represented in the Dickson basis

Given a polynomial matrix P(x) of grade g and a rational function

An algorithm to compute the polar decomposition of a 3 × 3 matrix

x(y) = n(y)/d(y)

Modifications of Newton's method for even-grade palindromic polynomials and other twined polynomials

, where