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Topics in structured linear algebra - lecture 1

structured matrices

Appears in collection : 2021 - French computer algebra days / Journées nationales de calcul formel

Structure is a fundamental concept in linear algebra: matrices arising from applications often inherit a special form from the original problem, and this special form can be analysed and exploited to design efficient algorithms. In this short course we will present some examples of matrix structure and related applications. Here we are interested in data-sparse structure, that is, structure that allows us to represent an n × n matrix using only O(n) parameters. One notable example is provided by quasi separable matrices, a class of (generally dense) rank-structured matrices where off-diagonal blocks have low rank. We will give an overview of the properties of these structured classes and present a few examples of how algorithms that perform basic tasks – e.g., solving linear systems, computing eigenvalues, approximating matrix functions – can be tailored to specific structures.

Information about the video

Date of recording 02/03/2021
Date of publication 19/03/2021
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19718703
Cite this video Boito, Paola (02/03/2021). Topics in structured linear algebra - lecture 1. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19718703
URL https://dx.doi.org/10.24350/CIRM.V.19718703

Domain(s)

Numerical Analysis

Bibliography

AURENTZ, Jared L., MACH, Thomas, ROBOL, Leonardo, et al. Core-chasing algorithms for the eigenvalue problem. Society for Industrial and Applied Mathematics, 2018.
BOITO, Paola, EIDELMAN, Yuli, et GEMIGNANI, Luca. Implicit QR for companion-like pencils. Mathematics of Computation, 2016, vol. 85, no 300, p. 1753-1774. - https://doi.org/10.1090/mcom/3020
BOITO, Paola, EIDELMAN, Yuli, et GEMIGNANI, Luca. Efficient solution of parameter‐dependent quasiseparable systems and computation of meromorphic matrix functions. Numerical Linear Algebra with Applications, 2018, vol. 25, no 6, p. e2141. - https://doi.org/10.1002/nla.2141
EIDELMAN, Yuli, GOHBERG, Israel, et HAIMOVICI, Iulian. Separable type representations of matrices and fast algorithms. Birkhäuser, Vol 1& 2 , 2014. - http://dx.doi.org/10.1007/978-3-0348-0606-0
HIGHAM, Nicholas J. Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics, 2008. - https://doi.org/10.1137/1.9780898717778
VANDEBRIL, Raf, VAN BAREL, Marc, et MASTRONARDI, Nicola. Matrix computations and semiseparable matrices: linear systems. JHU Press, 2007. -

MSC codes

Document(s)

https://www.cirm-math.fr/RepOrga/2564/Slides/01_slidesBoito.pdf

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