Eigenvalue condition numbers and Pseudospectra of Fiedler matrices

The aim of the present paper is to analyze the behavior of Fiedler companion matrices in the polynomial root-finding problem from the point of view of conditioning of eigenvalues. More precisely, we compare: (a) the condition number of a given root λ of a monic polynomial p(z) with the condition number of λ as an eigenvalue of any Fiedler matrix of p(z), (b) the condition number of λ as an eigenvalue of an arbitrary Fiedler matrix with the condition number of λ as an eigenvalue of the classical Frobenius companion matrices, and (c) the pseudozero sets of p(z) and the pseudospectra of any Fiedler matrix of p(z). We prove that, if the coefficients of the polynomial p(z) are not too large and not all close to zero, then the conditioning of any root λ of p(z) is similar to the conditioning of λ as an eigenvalue of any Fiedler matrix of p(z). On the contrary, when p(z) has some large coefficients, or they are all close to zero, the conditioning of λ as an eigenvalue of any Fiedler matrix can be arbitrarily much larger than its conditioning as a root of p(z) and, moreover, when p(z) has some large coefficients there can be two different Fiedler matrices such that the ratio between the condition numbers of λ as an eigenvalue of these two matrices can be arbitrarily large. Finally, we relate asymptotically the pseudozero sets of p(z) with the pseudospectra of any given Fiedler matrix of p(z), and the pseudospectra of any two Fiedler matrices of p(z). This work was partially supported by the Ministerio de Economı́a y Competitividad of Spain through grants MTM-2012-32542, MTM2015-68805-REDT and MTM2015-65798-P, and by the Engineering and Physical Sciences Research Council grant EP/I005293 (Javier Pérez). Fernando De Terán Department of Mathematics, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain. E-mail: [email protected] Froilán M. Dopico Department of Mathematics, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain. E-mail: [email protected] Javier Pérez School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK. E-mail: [email protected] 2 Fernando De Terán et al.