Toeplitz band matrices with exponentially growing condition numbers

Electronic Journal of Linear Algebra

Volumn 5, Issue , 1999, Pages 104-125

  Bottcher A ^a   Grudsky, S ^b  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

^b SOUTHERN FEDERAL UNIVERSITY   (Russian Federation)

Author keywords

Band matrix; Condition number; Matrix norm; Toeplitz matrix

Indexed keywords

EID: 0000210040     PISSN: 10813810     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (23)

1. L. Berg, Lineare Gleichungssysteme mit Bandstruktur, Deutscher Verlag der Wissenschaften, Berlin 1986.
2. R. Bhatia, Matrix Analysis, Springer-Verlag, New York 1996.
3. A. Böttcher, Status report on rationally generated block Toeplitz and Wiener-Hopf determinants, Unpublished manuscript, 34 pages, 1989 (available from the author on request).
4. A. Böttcher, On the approximation numbers of large Toeplitz matrices, Doc. Math., 2:1-29, 1997.
5. A. Böttcher and S. Grudsky, On the condition numbers of large semi-definite Toeplitz matrices, Linear Algebra Appl., 279:285-301, 1998.
6. A. Böttcher, S. Grudsky, and B. Silbermann, Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices, New York J. Math., 3:1-31, 1997.
7. A. Böttcher and B. Silbermann, The asymptotic behavior of Toeplitz determinants for generating functions with zeros of integral orders, Math. Nachr., 102: 79-105, 1981.
8. A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag, Berlin 1989 and Springer Verlag, Berlin, Heidelberg, New York 1990.
9. A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer-Verlag, New York 1998.
10. K.M. Day, Toeplitz matrices generated by the Laurent series expansion of an arbitrary rational function, Trans. Amer. Math. Soc., 206:224-245, 1975.
11. K.M. Day, Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions, Trans. Amer. Math. Soc., 209:175-183, 1975.
12. I. Gohberg and I.A. Feldman, Convolution Equations and Projection Methods for Their Solution, Amer. Math. Soc., Providence, RI, 1974 [Russian original: Nauka, Moscow 1971].
13. U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, University of California Press, Berkeley and Los Angeles 1958.
14. I.I. Hirschman, Jr., The spectra of certain Toeplitz matrices, Illinois J. Math., 11:145-159, 1967.
15. E. Hlawka, J. Schoissengaier, and R. Taschner, Geometric and Analytic Number Theory, Springer-Verlag, Berlin, Heidelberg, New York 1991.
16. M. Kac, W.L. Murdock, and G. Szegö, On the eigenvalues of certain Hermitian forms, J. Rational Mech. Anal., 2:767-800, 1953.
17. S.V. Parter, Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations, Trans. Amer. Math. Soc., 99:153-192, 1966.
18. S. Roch and B. Silbermann, Index calculus for approximation methods and singular value decomposition, J. Math. Anal. Appl., 225:401-426, 1998.
19. M. Rosenblatt, Some purely deterministic processes, J. Math. Mech., 6:801-810, 1957.
20. P. Schmidt and P. Spitzer, The Toeplitz matrices of an arbitrary Laurent polynomial, Math. Scand., 8:15-38, 1960.
21. S. Serra, On the extreme eigenvalues of Hermitian (block) Toeplitz matrices, Linear Algebra Appl., 270:109-129, 1997.
22. S. Serra, How bad can positive definite Toeplitz matrices be?, Talk given at the conference "Fourier Analysis and Applications", Kuwait University, May 1998.
23. H. Widom, On the eigenvalues of certain Hermitian operators, Trans. Amer. Math. Soc., 88:491-522, 1958.