Eigenvalue condition numbers and pseudospectra of Fiedler matrices Articles uri icon

publication date

  • March 2017

start page

  • 319

end page

  • 365

issue

  • 1

volume

  • 54

International Standard Serial Number (ISSN)

  • 0008-0624

Electronic International Standard Serial Number (EISSN)

  • 1126-5434

abstract

  • 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 of a monic polynomial p(z) with the condition number of as an eigenvalue of any Fiedler matrix of p(z), (b) the condition number of as an eigenvalue of an arbitrary Fiedler matrix with the condition number of 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 of p(z) is similar to the conditioning of 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 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 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).

keywords

  • roots of polynomials; eigenvalues; companion matrices; qr algorithm; fiedler matrices; conditioning; pseudospectrum; pseudozero sets of polynomials; polynomials; linearizations; computation