We study the asymptotic behaviour of the eigenvalues of Hermitian n×n block Toeplitz matrices Tn, with k×k blocks, as n tends to infinity. No hypothesis is made concerning the structure of the blocks. Such matrices {Tn} are generated by the Fourier coefficients of a Hermitian matrix-valued function f∈L2, and we study the distribution of their eigenvalues for large n, relating their behaviour to some properties of the function f. We also study the eigenvalues of the preconditioned matrices {P−1nTn}, where the sequence {Pn} is generated by a positive-definite matrix-valued function p. We show that the spectrum of any P−1nTn is contained in the interval [r,R], where r is the smallest and R the largest eigenvalue of p−1f. We also prove that the first m eigenvalues of P−1nTn tend to r and the last m tend to R, for any fixed m. Finally, exact limit values for both the condition number and the conjugate gradient convergence factor for the preconditioned matrices P−1nTn are computed.
Block Toeplitz matrices and preconditioning
MIRANDA, Michele;
1996
Abstract
We study the asymptotic behaviour of the eigenvalues of Hermitian n×n block Toeplitz matrices Tn, with k×k blocks, as n tends to infinity. No hypothesis is made concerning the structure of the blocks. Such matrices {Tn} are generated by the Fourier coefficients of a Hermitian matrix-valued function f∈L2, and we study the distribution of their eigenvalues for large n, relating their behaviour to some properties of the function f. We also study the eigenvalues of the preconditioned matrices {P−1nTn}, where the sequence {Pn} is generated by a positive-definite matrix-valued function p. We show that the spectrum of any P−1nTn is contained in the interval [r,R], where r is the smallest and R the largest eigenvalue of p−1f. We also prove that the first m eigenvalues of P−1nTn tend to r and the last m tend to R, for any fixed m. Finally, exact limit values for both the condition number and the conjugate gradient convergence factor for the preconditioned matrices P−1nTn are computed.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
