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Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations


Authors: Luis Miguel Anguas, Maria Isabel Bueno and Froilán M. Dopico
Journal: Math. Comp. 89 (2020), 767-805
MSC (2010): Primary 65F15, 65F35, 15A18, 15A22
DOI: https://doi.org/10.1090/mcom/3472
Published electronically: August 26, 2019
MathSciNet review: 4044450
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Abstract: We present the first general study on the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of polynomial eigenvalue problems (PEPs). By using the homogeneous formulation of PEPs, we are able to obtain two clear and simple results. First, we show that if the matrix inducing the Möbius transformation is well-conditioned, then such transformation approximately preserves the eigenvalue condition numbers and backward errors when they are defined with respect to perturbations of the matrix polynomial which are small relative to the norm of the whole polynomial. However, if the perturbations in each coefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backward errors are preserved approximately by the Möbius transformations induced by well-conditioned matrices only if a penalty factor, depending on the norms of those matrix coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation of the PEP is used.


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Additional Information

Luis Miguel Anguas
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911 Leganés, Spain
Address at time of publication: Universidad Pontificia Comillas, Departamento de Matematica Aplicada, C. Alberto Aguilera, 23, Madrid, 28015 Spain
Email: lmanguas@comillas.edu

Maria Isabel Bueno
Affiliation: Department of Mathematics and College of Creative Studies, South Hall 6607, University of California, Santa Barbara, California 93106
Email: mbueno@math.ucsb.edu

Froilán M. Dopico
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911 Leganés, Spain
Email: dopico@math.uc3m.es

DOI: https://doi.org/10.1090/mcom/3472
Keywords: Backward error, eigenvalue condition number, matrix polynomial, M\"obius transformation, polynomial eigenvalue problem
Received by editor(s): October 26, 2018
Received by editor(s) in revised form: April 17, 2019
Published electronically: August 26, 2019
Additional Notes: The research of the first author was funded by the “contrato predoctoral” BES-2013-065688 of MINECO
This work was partially supported by the Ministerio de Economía, Industria y Competitividad (MINECO) of Spain through grants MTM2012-32542, MTM2015-65798-P, and MTM2017-90682-REDT
Article copyright: © Copyright 2019 American Mathematical Society