Block Toeplitz matrices and preconditioning |
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Authors: | M Miranda P Tilli |
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Affiliation: | (1) Scuola Normale Superiore, Piazza Cavalieri 7, 56100 Pisa, Italy |
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Abstract: | We study the asymptotic behaviour of the eigenvalues of Hermitiann×n block Topelitz matricesT
n
, withk×k blocks, asn tends to infinity. No hypothesis is made concerning the structure of the blocks. Such matrices{T
n
} are generated by the Fourier coefficients of a Hermitian matrix valued functionf∈L
2, and we study the distribution of their eigenvalues for largen, relating their behaviour to some properties of the functionf. We also study the eigenvalues of the preconditioned matrices{P
n
−1
Tn}, where the sequence{P
n
} is generated by a positive definite matrix valued functionp. We show that the spectrum of anyP
n
−1
T
n
is contained in the interval r, R], wherer is the smallest andR the largest eigenvalue ofp
−1
f. We also prove that the firstm eigenvalues ofP
n
−1
Tn tend tor and the lastm tend toR, for anym fixed. Finally, exact limit values for both the condition number and the conjugate gradient convergence factor for the preconditioned
matricesP
n
−1
Tn are computed. |
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Keywords: | |
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