RT Journal Article
T1 Backward error and conditioning of Fiedler linearizations
A1 Terán Vergara, Fernando de
AB The standard way to solve polynomial eigenvalue problems isthrough linearizations. The family of Fiedler linearizations, which includesthe classical Frobenius companion forms, presents many interesting propertiesfrom both the theoretical and the applied point of view. These properties makethe Fiedler pencils a very attractive family of linearizations to be used in thesolution of polynomial eigenvalue problems. However, their numerical featuresfor general matrix polynomials had not yet been fully investigated. In thispaper, we analyze the backward error of eigenpairs and the condition numberof eigenvalues of Fiedler linearizations in the solution of polynomial eigenvalueproblems. We get bounds for: (a) the ratio between the backward error ofan eigenpair of the matrix polynomial and the backward error of the corresponding(computed) eigenpair of the linearization, and (b) the ratio betweenthe condition number of an eigenvalue in the linearization and the conditionnumber of the same eigenvalue in the matrix polynomial. A key quantity inthese bounds is ρ, the ratio between the maximum norm of the coefficients ofthe polynomial and the minimum norm of the leading and trailing coefficient.If the matrix polynomial is well scaled (i. e., all its coefficients have a similarnorm, which implies ρ ≈ 1), then solving the Polynomial Eigenvalue Problemwith any Fiedler linearization will give a good performance from the point ofview of backward error and conditioning. In the more general case of badlyscaled matrix polynomials, dividing the coefficients of the polynomial by themaximum norm of its coefficients allows us to get better bounds. In particular,after this scaling, the ratio between the eigenvalue condition number in anytwo Fiedler linearizations is bounded by a quantity that depends only on thesize and the degree of the polynomial. We also analyze the effect of parameterscaling in these linearizations, which improves significantly the backward errorand conditioning in some cases where ρ is large. Several numerical experimentsare provided to support our theoretical results.
PB American Mathematical Society
SN 0025-5718
YR 2020
FD 2020-05-01
LK https://hdl.handle.net/10016/32491
UL https://hdl.handle.net/10016/32491
LA eng
NO This work was partially supported by the Ministerio de Ciencia e Innovación of Spain through grant MTM-2009-09281, and by the Ministerio de Economía y Competitividad of Spain through grants MTM-2012-32542, MTM2015-68805-REDT, and MTM2015-65798-P.
DS e-Archivo
RD 27 jul. 2024