Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {Tn(f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {Tn−1(g)Tn(f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators.

[1]  Albrecht Böttcher,et al.  Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols , 2015 .

[2]  Stefano Serra Capizzano,et al.  Preconditioning Strategies for Hermitian Indefinite Toeplitz Linear Systems , 2004, SIAM J. Sci. Comput..

[3]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[4]  Sergei M. Grudsky,et al.  Asymptotics of Eigenvalues for Pentadiagonal Symmetric Toeplitz Matrices , 2017 .

[5]  Stefano Serra Capizzano,et al.  Are the Eigenvalues of Banded Symmetric Toeplitz Matrices Known in Almost Closed Form? , 2018, Exp. Math..

[6]  Stefano Serra,et al.  On the extreme spectral properties of Toeplitz matrices generated byL1 functions with several minima/maxima , 1996 .

[7]  Albrecht Böttcher,et al.  Inside the eigenvalues of certain Hermitian Toeplitz band matrices , 2010, J. Comput. Appl. Math..

[8]  Gerry Leversha,et al.  Introduction to numerical analysis (3rd edn), by J. Stoer and R. Bulirsch. Pp. 744. £49. 2002. ISBN 0 387 95452 X (Springer-Verlag). , 2004, The Mathematical Gazette.

[9]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[10]  S. Serra.  New PCG based algorithms for the solution of Hermitian Toeplitz systems , 1995 .

[11]  Claude Brezinski,et al.  Extrapolation methods - theory and practice , 1993, Studies in computational mathematics.

[12]  Friedrich L. Bauer,et al.  New aspects in numerical quadrature , 1963 .

[13]  Stefano Serra Capizzano,et al.  Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols , 2018, Numer. Linear Algebra Appl..

[14]  Carlo Garoni,et al.  Generalized Locally Toeplitz Sequences , 2017 .

[15]  A. Böttcher,et al.  Introduction to Large Truncated Toeplitz Matrices , 1998 .

[16]  Albrecht Böttcher,et al.  Maximum norm versions of the Szegő and Avram-Parter theorems for Toeplitz matrices , 2015, J. Approx. Theory.

[17]  Stefano Serra,et al.  Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems , 1997 .

[18]  P. Davis.  Interpolation and approximation , 1965 .

[19]  Carlo Garoni,et al.  Generalized Locally Toeplitz Sequences: Theory and Applications: Volume I , 2017 .

[20]  Sergei M. Grudsky,et al.  Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness , 2017 .

[21]  Stefano Serra Capizzano,et al.  An ergodic theorem for classes of preconditioned matrices , 1998 .

[22]  J. M. Bogoya,et al.  From convergence in distribution to uniform convergence , 2015, 1509.01836.

[23]  Dario Bini,et al.  SPECTRAL AND COMPUTATIONAL PROPERTIES OF BAND SYMMETRIC TOEPLITZ MATRICES , 1983 .

[24]  Giuseppe Fiorentino,et al.  C. G. preconditioning for Toeplitz matrices , 1993 .

[25]  P. Revesz.  Interpolation and Approximation , 2010 .

[26]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[27]  Stefano Serra Capizzano,et al.  Preconditioning strategies for non‐Hermitian Toeplitz linear systems , 2005, Numer. Linear Algebra Appl..

[28]  Raymond H. Chan,et al.  Fast Band-Toeplitz Preconditioners for Hermitian Toeplitz Systems , 1994, SIAM J. Sci. Comput..

[29]  Paolo Tilli,et al.  A note on the spectral distribution of toeplitz matrices , 1998 .

[30]  Dario Andrea Bini,et al.  Metodi Numerici per l'Algebra Lineare. , 1989 .