TY - JOUR

T1 - Eigenvalue condition numbers and pseudospectra of Fiedler matrices

AU - De Terán, Fernando

AU - Dopico, Froilán M.

AU - Pérez, Javier

N1 - Funding Information:
This work was partially supported by the Ministerio de Economía 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).
Funding Information:
This work was partially supported by the Ministerio de Econom??a 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 P??rez).
Publisher Copyright:
© 2016, Springer-Verlag Italia.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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).

AB - 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).

KW - Companion matrices

KW - Conditioning

KW - Eigenvalues

KW - Fiedler matrices

KW - Pseudospectrum

KW - Pseudozero sets of polynomials

KW - QR algorithm

KW - Roots of polynomials

UR - http://www.scopus.com/inward/record.url?scp=84964380977&partnerID=8YFLogxK

U2 - 10.1007/s10092-016-0189-9

DO - 10.1007/s10092-016-0189-9

M3 - Article

AN - SCOPUS:84964380977

SN - 0008-0624

VL - 54

SP - 319

EP - 365

JO - Calcolo

JF - Calcolo

IS - 1

ER -