Abstract
The theory of block locally Toeplitz (LT) sequences—along with its generalization known as the theory of block generalized locally Toeplitz (GLT) sequences—is a powerful apparatus for computing the spectral distribution of matrices arising from the discretization of differential problems. In this paper we develop the theory of block LT sequences, whereas the theory of block GLT sequences is the subject of the complementary paper (Chap. 3 of this book).
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References
Avram, F.: On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Theory Relat. Fields 79, 37–45 (1988)
Barbarino, G.: Equivalence between GLT sequences and measurable functions. Linear Algebra Appl. 529, 397–412 (2017)
Bhatia, R.: Matrix Analysis. Springer, New York (1997)
Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer, New York (1999)
Böttcher, A., Silbermann B.: Analysis of Toeplitz Operators, 2nd edn. Springer, Berlin (2006)
Donatelli, M., Garoni, C., Mazza, M., Serra-Capizzano, S., Sesana, D.: Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol. Appl. Math. Comput. 245, 158–173 (2014)
Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications (Volume I). Springer, Cham (2017)
Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications (Volume II). Springer, Cham (2018)
Garoni, C., Serra-Capizzano, S., Sesana, D.: Spectral analysis and spectral symbol of d-variate \(\mathbb Q_{\boldsymbol {p}}\) Lagrangian FEM stiffness matrices. SIAM J. Matrix Anal. Appl. 36, 1100–1128 (2015)
Garoni, C., Serra-Capizzano, S., Vassalos, P.: A general tool for determining the asymptotic spectral distribution of Hermitian matrix-sequences. Oper. Matrices 9, 549–561 (2015)
Garoni, C., Mazza, M., Serra-Capizzano, S.: Block generalized locally Toeplitz sequences: from the theory to the applications. Axioms 7, 49 (2018)
Garoni, C., Serra-Capizzano, S., Sesana, D.: Block generalized locally Toeplitz sequences: topological construction, spectral distribution results, and star-algebra structure. In: Bini, D.A., et al. (eds.) Structured Matrices in Numerical Linear Algebra. Springer INdAM Series, vol. 30, pp. 59–79. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-04088-8_3
Garoni, C., Speleers, H., Ekström, S.-E., Reali, A., Serra-Capizzano, S., Hughes, T.J.R.: Symbol-based analysis of finite element and isogeometric B-spline discretizations of eigenvalue problems: exposition and review. Arch. Comput. Meth. Eng. (in press). https://doi.org/10.1007/s11831-018-9295-y
Grenander, U., Szegő, G.: Toeplitz Forms and Their Applications, 2nd edn. AMS Chelsea Publishing, New York (1984)
Parter, S.V.: On the distribution of the singular values of Toeplitz matrices. Linear Algebra Appl. 80, 115–130 (1986)
Serra-Capizzano, S.: Distribution results on the algebra generated by Toeplitz sequences: a finite dimensional approach. Linear Algebra Appl. 328, 121–130 (2001)
Serra-Capizzano, S.: More inequalities and asymptotics for matrix valued linear positive operators: the noncommutative case. Oper. Theory Adv. Appl. 135, 293–315 (2002)
Serra-Capizzano, S.: Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl. 366, 371–402 (2003)
Serra-Capizzano, S.: The GLT class as a generalized Fourier analysis and applications. Linear Algebra Appl. 419, 180–233 (2006)
Serra-Capizzano, S., Tilli, P.: On unitarily invariant norms of matrix-valued linear positive operators. J. Inequal. Appl. 7, 309–330 (2002)
Tilli, P.: A note on the spectral distribution of Toeplitz matrices. Linear Multilinear Algebra 45, 147–159 (1998)
Tilli, P.: Locally Toeplitz sequences: spectral properties and applications. Linear Algebra Appl. 278, 91–120 (1998)
Tyrtyshnikov, E.E.: A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl. 232, 1–43 (1996)
Tyrtyshnikov, E.E., Zamarashkin, N.L.: Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships. Linear Algebra Appl. 270, 15–27 (1998)
Zamarashkin, N.L., Tyrtyshnikov, E.E.: Distribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the generating function. Sb. Math. 188, 1191–1201 (1997)
Acknowledgements
Carlo Garoni is a Marie-Curie fellow of the Italian INdAM under grant agreement PCOFUND-GA-2012-600198. The work of the authors has been supported by the INdAM GNCS (Gruppo Nazionale per il Calcolo Scientifico). The authors wish to thank Giovanni Barbarino for useful discussions.
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Garoni, C., Serra-Capizzano, S., Sesana, D. (2019). Block Locally Toeplitz Sequences: Construction and Properties. In: Bini, D., Di Benedetto, F., Tyrtyshnikov, E., Van Barel, M. (eds) Structured Matrices in Numerical Linear Algebra. Springer INdAM Series, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-030-04088-8_2
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