On the conditioning of some structured generalized eigenvalue problems

Authors

DOI:

https://doi.org/10.5206/mt.v3i3.16450

Keywords:

structured matrices, generalized eigenvalue problem, relative condition number

Abstract

This work continues the analysis of the conditioning of a Hankel structured generalized eigenvalue problem (GEP) started in [1]. The considered generalized eigenvalue problem appears in exponential analysis and sparse interpolation.

We generalize the proof in [1] and add expressions for the relative condition numbers of two reformulations of the GEP, a reformulation as a Loewner GEP valid for general complex data, and a compression to a Hankel+Toeplitz GEP in the case of real data. Both reformulations are compared to the original Hankel GEP. The analysis is concluded with ample numerical illustrations.

Author Biographies

Annie Cuyt

Annie Cuyt is emerita full professor at the Faculty of Science of the University of Antwerp. She received her Doctor Scientiae degree in 1982 from the same university, summa cum laude and with the felicitations of the jury. Subsequently she was a Research fellow with the Alexander von Humboldt Foundation (Germany), she obtained the Habilitation (1986) and she was honoured with a Masuda Research Grant (Japan).

Since 2014 she is a life-time member of the Royal Flemish Academy of Belgium for the Sciences and Arts. She is the author of more than 220 peer-reviewed publications in international journals and conference proceedings, the author or editor of several books, and the organizer of several larger and smaller international events.

Her main research interest is situated in the area of numerical approximation theory and its applications to a diversity of problems in scientific computing, computer science, engineering and bio-informatics. A lot of research has been carried out on rational approximation, in one as well as in many variables, and its relation to sparse interpolation and exponential analysis. In view of her expertise she served on many national and international science foundation boards and prestigious international award juries. In 2005 she stood at the cradle of a HPC center for Flanders, which grew into a very successful and ongoing project.

Naomi Flamand

Naomi Flamand is a mathematics student at the University of Antwerp. She obtained her secondary education diploma in 2019 at the Sint-Jan Berchmanscollege in Westmalle, magna cum laude and with the mathematics prize. In 2022, she received her bachelor’s degree from the University of Antwerp, summa cum laude, and is currently completing her master’s degree in fundamental mathematics, which she expects to obtain in 2024.

Ferre Knaepkens, University of Antwerp

Ferre Knaepkens received the Ph.D. degree in mathematics from the University of Antwerp, Belgium,
in May 2022. He started his academic carreer in 2017 as a doctoral student with the Computational Mathematics (CMA) research group of the University of Antwerp, and is now working as a Post-
Doctoral fellow with Cosys-Lab at the same institution.

His current research interests are numerical and applied mathematics, with a focus on impactful computational science and engineering problems. His work mainly centers on exponential analysis, sparse interpolation and signal processing.

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Published

2023-11-02

Issue

Vol. 3 No. 3 (2023): Autumn Issue

Section

Student Corner

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Copyright (c) 2023 Annie Cuyt, Naomi Flamand, Ferre Knaepkens

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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.