Spectral functions for real symmetric Toeplitz matrices

Journal of Computational and Applied Mathematics

Volumn 98, Issue 2, 1998, Pages 233-243

  Melman, A ^a,b  

^a STANFORD UNIVERSITY   (United States)

^b BEN GURION UNIVERSITY OF THE NEGEV   (Israel)

Author keywords

Eigenvalues; Embedding; Odd and even spectra; Secular equation; Spectral function; Symmetric; Toeplitz matrix

Indexed keywords

COMPUTATIONAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; SPECTRUM ANALYSIS;

SECULAR EQUATIONS; SPECTRAL FUNCTIONS; TOEPLITZ MATRICES;

MATRIX ALGEBRA;

EID: 0032184328     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(98)00129-0     Document Type: Article

References (29)

1. N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, 1961.
2. G.S. Ammar, W.B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems, in: J. Gilewicz, M. Pindor, W. Siemaszko (Eds.), Rational Approximations and its Applications in Mathematics and Physics, Lecture Notes in Mathematics, vol. 1237, Springer Berlin, 1987, pp. 315-330.
3. G.S. Ammar, W.B. Gragg, Numerical experience with a superfast Toeplitz solver, Linear Algebra Appl. 121 (1989) pp. 185-206.
4. A.L. Andrew, Eigenvectors of certain matrices, Linear Algebra Appl. 7 (1973) 151-162.
5. C.F. Borges, W.B. Gragg, A parallel divide and conquer algorithm for the generalized real symmetric definite tridiagonal eigenproblem, in: L. Reichel, A. Rutan, R.S. Varga (Eds.), Numerical Linear Algebra and Scientific Computing, de Gruyter, Berlin, (1993) pp. 10-28.
6. J.R. Bunch, Stability of methods for solving Toeplitz systems of equations, SIAM J. Sci. Statist. Comput. 6 (1985) 349-364.
7. J.R. Bunch, C.P. Nielsen, D.C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math. 31 (1978) 31-48.
8. A. Cantoni, F. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra Appl. 13 (1976) 275-288.
9. J.J.M. Cuppen, A divide and conquer method for the symmetric tridiagonal eigenvalue problem, Numer. Math. 36 (1981) 177-195.
10. G. Cybenko, The numerical stability of the Levinson-Durbin algorithm for Toeplitz systems of equations, SIAM J. Sci. Statist. Comput. 1 (1980) 303-319.
11. G. Cybenko, On the eigenstructure of Toeplitz matrices, IEEE Trans. Acoustics Speech Signal Process. 32 (1984) 918-921.
12. G. Cybenko, C. Van Loan, Computing the minimum eigenvalue of a symmetric positive definite Toeplitz matrix, SIAM J. Sci. Statist. Comput. 7 (1) (1986) 123-131.
13. F. De Hoog, A new algorithm for solving Toeplitz systems of equations, Linear Algebra Appl. 88-89 (1987) 123-138.
14. P. Delsarte, Y. Genin, Spectral properties of finite Toeplitz matrices, in Mathematical Theory of Networks and Systems, Proc. MTNS-83 Int. Symp., Beer-Sheva, Israel, 1983, pp. 194-213.
15. A. Dembo, Bounds on the extreme eigenvalues of positive definite Toeplitz matrices, IEEE Trans. Inform. Theory 34 (1988) 352-355.
16. W. Gander, G.H. Golub, U. von Matt, A constrained eigenvalue problem, Linear Algebra Appl. 114-115 (1989) 815-839.
17. G.H. Golub, Some modified matrix eigenvalue problems, SIAM Rev. 15 (2) (1973) 318-334.
18. G. Golub, and C. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 1989.
19. D. Huang, Symmetric solutions and eigenvalue problems of Toeplitz systems, IEEE Trans. Signal Process. 40 (1992) 3069-3074.
20. I.S. Iohvidov, Hankel and Toeplitz Matrices and Forms, Birkhäuser, Boston, Basel, Stuttgart, 1982.
21. N. Levinson, The Wiener RMS (root mean square) error criterion in filter design and prediction, J. Math. Phys. 25 (1947) 261-278.
22. R.C. Li, Solving secular equations stably and efficiently, Technical Report UCB//CSD-94-851, Computer Science Division, University of California, Berkeley, CA, 1994. Also: LAPACK Working Notes 89.
23. W. Mackens, H. Voss, The minimum eigenvalue of a symmetric positive-definite Toeplitz matrix and rational Hermitian interpolation, SIAM J. Matrix Anal. Appl. 18 (3) (1997) 521-534.
24. A. Melman, A unifying convergence analysis of second-order methods for secular equations, Math. Comput. 66 (217) (1997) 333-344.
25. A. Melman, Analysis of third order methods for secular equations, Math. Comput. 67 (221) (1998) 271-286.
26. V.F. Pisarenko, The retrieval of harmonics from a covariance function, Geophys. J. Royal Astron. Soc. 33 (1973) 347-366.
27. W.F. Trench, Numerical solution of the eigenvalue problem for Hermitian Toeplitz matrices, SIAM J. Matrix Anal. Appl. 10 (2) (1989) 135-146.
28. W.F. Trench, Spectral evolution of a one-parameter extension of a real symmetric Toeplitz matrix, SIAM J. Matrix Anal. Appl. 11 (4) (1990) 601-611.
29. J.H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.