Abstract
We propose efficient methods for the numerical approximation of the field of values of the linear pencil \(A-\lambda B\), when one of the matrix coefficients A or B is Hermitian and \(\lambda \in \mathbbm {C}\). Our approach builds on the fact that the field of values can be reduced under compressions to the bidimensional case, for which these sets can be exactly determined. The presented algorithms hold for matrices both of small and large size. Furthermore, we investigate spectral inclusion regions for the pencil based on certain fields of values . The results are illustrated by numerical examples. We point out that the given procedures complement the known ones in the literature.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bebiano, N., da Providência, J., Nata, A., da Providência, J.P.: Computing the numerical range of Krein space operators. Open Math. 13, 2391–5455 (2014)
Bebiano, N., da Providência, J., Ismaeili, F.: The Characteristic Polynomial of Linear Pencils of Small Size and the Numerical Range, in this volume
Chien, M.-T., Nakazato, H.: The numerical range of linear pencils of 2-by-2 matrices. Linear Algebra Appl. 341, 69–100 (2002)
Davis, C.: The Toeplitz-Hausdorff theorem explained. Canad. Math. Bull. 14, 245–246 (1971)
Gustafson, K.E., Rao, D.K.M.: Numerical Range, the Field of Values of Linear Operators and Matrices. Springer, New York (1997)
Hochstenbach, M.E.: Fields of values and inclusion regions for matrix pencils. Electron. Trans. Numer. Anal. 38, 98–112 (2011)
Kippenhahn, R.: Über den wertevorrat einer matrix. Math. Nachr. 6, 193–228 (1951)
Li, C.-K., Rodman, L.: Numerical range of matrix polynomials. SIAM J. Matrix Anal. Appl. 15, 1256–1265 (1994)
Li, C.-K., Rodman, L.: Shapes and computer generation of numerical ranges of Krein space operators. Electron. J. Linear Algebra 3, 31–47 (1998)
Loghin, D., van Guzen, M., Jonkers, E.: Bounds on the eigenvalue range and on the field of values of non-Hermitian and indefinite finite element matrices. J. Comput. Appl. Mat. 189(1–2), 304–323 (2006)
Marcus, M., Pesce, C.: Computer generated numerical ranges and some resulting theorems. Linear Multilinear Algebra 20, 121–157 (1987)
Psarrakos, P.J.: Numerical range of linear pencils. Linear Algebra Appl. 317, 127–141 (2000)
The Matrix Market, a repository for test matrices. http://math.nist.gov/MatrixMarket
Acknowledgements
The authors wish to thank the Referees for most valuable comments. This work was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Bebiano, N., da Providência, J., Nata, A., da Providência, J.P. (2017). Fields of Values of Linear Pencils and Spectral Inclusion Regions. In: Bebiano, N. (eds) Applied and Computational Matrix Analysis. MAT-TRIAD 2015. Springer Proceedings in Mathematics & Statistics, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-49984-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-49984-0_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49982-6
Online ISBN: 978-3-319-49984-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)