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Real root finding for low rank linear matrices

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

We consider \(m \times s\) matrices (with \(m\ge s\)) in a real affine subspace of dimension n. The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, based on advanced methods for polynomial system solving, to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is polynomial in \(\left( {\begin{array}{c}n+m(s-r)\\ n\end{array}}\right) \). It improves on the state-of-the-art in computer algebra and effective real algebraic geometry. Moreover, computer experiments show the practical efficiency of our approach.

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Acknowledgements

The second author is supported by the FMJH Program PGMO and would like to thank LAAS-CNRS and the Mathematics Department of Technische Universität Dortmund where he was working when a significant part of this work was designed. The third author is supported by Institut Universitaire de France. We thank the reviewers for their help in improving the first version of this paper.

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Authors and Affiliations

  1. CNRS, LAAS, 7 Avenue du colonel Roche, 31400, Toulouse, France

    Didier Henrion

  2. Université de Toulouse; LAAS, 31400, Toulouse, France

    Didier Henrion

  3. Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic

    Didier Henrion

  4. CNRS, XLIM, UMR 7252, University of Limoges, 87000, Limoges, France

    Simone Naldi

  5. CNRS, Inria, Laboratoire d’Informatique de Paris 6, Sorbonne Université, LIP6, Equipe PolSys, 75252, Paris, France

    Mohab Safey El Din

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  1. Didier Henrion
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  2. Simone Naldi
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  3. Mohab Safey El Din
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Correspondence to Simone Naldi.

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Henrion, D., Naldi, S. & Din, M.S.E. Real root finding for low rank linear matrices. AAECC 31, 101–133 (2020). https://doi.org/10.1007/s00200-019-00396-w

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  • DOI: https://doi.org/10.1007/s00200-019-00396-w

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