Abstract
A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form \(A=T(a)+E\) where T(a) is the Toeplitz matrix with entries \((T(a))_{i,j}=a_{j-i}\), for \(a_{j-i}\in \mathbb {C}\), \(i,j\ge 1\), while E is a matrix representing a compact operator in \(\ell ^2\). The matrix A is finitely representable if \(a_k=0\) for \(k<-m\) and for \(k>n\), given \(m,n>0\), and if E has a finite number of nonzero entries. The problem of numerically computing eigenpairs of a finitely representable QT matrix is investigated, i.e., pairs \((\lambda ,\varvec{v})\) such that \(A\varvec{v}=\lambda \varvec{v}\), with \(\lambda \in \mathbb {C}\), \(\varvec{v}=(v_j)_{j\in \mathbb {Z}^+}\), \(\varvec{v}\ne 0\), and \({\sum }_{j=1}^\infty |v_j|^2<\infty\). It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind \(WU(\lambda )\varvec{\beta }=0\), where W is a constant matrix and U depends on \(\lambda\) and can be given in terms of either a Vandermonde matrix or a companion matrix. Algorithms relying on Newton’s method applied to the equation det \(WU(\lambda )=0\) are analyzed. Numerical experiments show the effectiveness of this approach. The algorithms have been included in the CQT-Toolbox [Numer. Algorithms 81 (2019), no. 2, 741–769].
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References
Barnett, S.: Polynomials and linear control systems, volume 77 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York (1983)
Bini, D.A., Fiorentino, G., Gemignani, L., Meini, B.: Effective fast algorithms for polynomial spectral factorization. Numer. Algorithms 34(2–4), 217–227 (2003)
Bini, D.A., Iannazzo, B., Meng, J.: Algorithms for approximating means of semi-infinite quasi-Toeplitz matrices. In: Nielsen, B.F. (ed) Geometric Science of Information, GSI 2021, volume 12829 of Lecture Notes in Computer Science, pp. 405–414. Springer
Bini, D.A., Iannazzo, B., Meng, J.: Geometric means of quasi-Toeplitz matrices. arXiv preprint. (2021)
Bini, D.A., Latouche, G., Meini, B.: Numerical methods for structured Markov chains. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2005)
Bini, D.A., Massei, S., Meini, B.: Semi-infinite quasi-Toeplitz matrices with applications to QBD stochastic processes. Math. Comp. 87(314), 2811–2830 (2018)
Bini, D.A., Massei, S., Meini, B., Robol, L.: On quadratic matrix equations with infinite size coefficients encountered in QBD stochastic processes. Numer. Linear Algebra Appl. 25(6), 2128, 12 (2018)
Bini, D.A., Massei, S., Meini, B., Robol, L.: A computational framework for two-dimensional random walks with restarts. SIAM J. Sci. Comput. 42(4), A2108–A2133 (2020)
Bini, D.A., Massei, S., Robol, L.: Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox. Numerical Algorithms 81(2), 741–769 (2019)
Bini, D.A., Meini, B.: On the exponential of semi-infinite quasi-Toeplitz matrices. Numer. Math. 141(2), 319–351 (2019)
Bini, D.A., Meini, B., Meng, J.: Solving quadratic matrix equations arising in random walks in the quarter plane. SIAM J. Matrix Anal. Appl. 41(2), 691–714 (2020)
Böttcher, A., Embree, M., Sokolov, V.I.: Infinite Toeplitz and Laurent matrices with localized impurities. Linear Algebra Appl. 343—-344, 101–118 (2002)
Böttcher, A., Embree, M., Sokolov, V.I.: On large Toeplitz band matrices with an uncertain block. Linear Algebra Appl 366, 87–97 (2003)
Böttcher, A., Grudsky, S.M.: Toeplitz matrices, asymptotic linear algebra, and functional analysis. Birkhäuser Verlag, Basel (2000)
Böttcher, A., Grudsky, S.M.: Spectral properties of banded Toeplitz matrices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2005)
Böttcher, A., Halwass, M.: A Newton method for canonical Wiener-Hopf and spectral factorization of matrix polynomials. Electron. J. Linear Algebra 26, 873–897 (2013)
Böttcher, A., Halwass, M.: Wiener-Hopf and spectral factorization of real polynomials by Newton’s method. Linear Algebra Appl. 438(12), 4760–4805 (2013)
Böttcher, A., Silbermann, B.: Introduction to large truncated Toeplitz matrices. Universitext. Springer-Verlag, New York (1999)
Breda, D., Liessi, D.: Approximation of Eigenvalues of Evolution Operators for Linear Renewal Equations. SIAM J. Numer. Anal. 56(3), 1456–1481 (2018)
Colbrook, M..J., Roman Bogdan, B., Hansen, A.C.: How to compute spectra with error control. Phys. Rev. Lett 122(25), 250201, 6 (2019)
Colbrook, M.J., Hansen, A.C.: On the infinite-dimensional QR algorithm. Numer. Math. 143(1), 17–83 (2019)
D’Angelo, J.P.: Several complex variables and the geometry of real hypersurfaces. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1993)
Gander, W.: New algorithms for solving nonlinear eigenvalue problems. Comput. Math. Math. Phys. 61(5), 761–773 (2021)
Garoni, C., Serra-Capizzano, S.: Generalized locally Toeplitz sequences: theory and applications, vol. I. Springer, Cham (2017)
Garoni, C., Serra-Capizzano, S.: Generalized locally Toeplitz sequences: theory and applications, vol. II. Springer, Cham (2018)
Gavin, B., Międlar, A., Polizzi, E.: FEAST eigensolver for nonlinear eigenvalue problems. J. Comput. Sci. 27, 107–117 (2018)
Güttel, S., Tisseur, F.: The nonlinear eigenvalue problem. Acta Numer. 26, 1–94 (2017)
Hochstenbach, M.E., Plestenjak, B.: Computing several eigenvalues of nonlinear eigenvalue problems by selection. Calcolo, 57(2), Paper No. 16, 25 (2020)
Jackson, J.R.: Networks of waiting lines. Operations Res. 5, 518–521 (1957)
Kim, H.-M., Meng, J.: Structured perturbation analysis for an infinite size quasi-Toeplitz matrix equation with applications. BIT Numerical Mathematics 61, 859–879 (2021)
Latouche, G., Ramaswami, V.: Introduction to matrix analytic methods in stochastic modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia, PA (1999)
Mignotte, M.: Some useful bounds. In: Computer algebra, pp. 259–263. Springer, Vienna (1983)
Neuts, M.F.: Matrix-geometric solutions in stochastic models: An algorithmic approach, volume 2 of Johns Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, Md. (1981)
Ostrowski, A.: Recherches sur la méthode de Graeffe et les zéros des polynomes et des séries de Laurent. Acta Mathematica 72, 99–155 (1940)
Ozawa, T.: Stability condition of a two-dimensional QBD process and its application to estimation of efficiency for two-queue models. Performance Evaluation 130, 101–118 (2019)
Ozawa, T.: Asymptotic properties of the occupation measure in a multidimensional skip-free Markov-modulated random walk. Queueing Syst. 97(1–2), 125–161 (2021)
Robol, L.: Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations. Linear Algebra Appl. 604, 210–235 (2020)
Schechter, M.: Basic theory of Fredholm operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 21(3)261–280 (1967)
Webb, M., Olver, S.: Spectra of Jacobi operators via connection coefficient matrices. Commun. Math. Phys. 382, 657–707 (2021)
Acknowledgements
The first author wishes to thank Matthew Colbrook and Mark Embree for helpful conversations and comments. The second author would like to thank Dimitri Breda for useful discussions.
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This work has been partially supported by University of Pisa’s project PRA_2020_61, and by GNCS of INdAM.
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Dedicated to Claude Brezinski on the occasion of his 80th birthday.
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Bini, D.A., Iannazzo, B., Meini, B. et al. Computing eigenvalues of semi-infinite quasi-Toeplitz matrices. Numer Algor 92, 89–118 (2023). https://doi.org/10.1007/s11075-022-01381-0
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DOI: https://doi.org/10.1007/s11075-022-01381-0