TY - JOUR
T1 - Inside the eigenvalues of certain Hermitian Toeplitz band matrices
AU - Böttcher, A.
AU - Grudsky, S. M.
AU - Maksimenko, E. A.
N1 - Funding Information:
This work was partially supported by CONACYT project 80503, Mexico.
PY - 2010/3/1
Y1 - 2010/3/1
N2 - While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n tends to infinity and provides asymptotic formulas that are uniform in j for 1 ≤ j ≤ n. The real-valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum, and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum. The results, which are of interest in numerical analysis, probability theory, or statistical physics, for example, are illustrated and underpinned by numerical examples.
AB - While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n tends to infinity and provides asymptotic formulas that are uniform in j for 1 ≤ j ≤ n. The real-valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum, and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum. The results, which are of interest in numerical analysis, probability theory, or statistical physics, for example, are illustrated and underpinned by numerical examples.
KW - Asymptotic expansions
KW - Eigenvalue
KW - Toeplitz matrix
UR - http://www.scopus.com/inward/record.url?scp=73249127731&partnerID=8YFLogxK
U2 - 10.1016/j.cam.2009.10.010
DO - 10.1016/j.cam.2009.10.010
M3 - Artículo
SN - 0377-0427
VL - 233
SP - 2245
EP - 2264
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
IS - 9
ER -