We derive upper and lower bounds on the smallest and largest eigenvalues, respectively, of real symmetric Toeplitz matrices. The bounds are rst obtained for positive-deenite matrices and then extended to the general real symmetric case. Our bounds are computed as the roots of rational and polynomial approximations to spectral, or secular, equations. The decomposition of the spectrum into even a...

In this article various existence results for the LDU-factorization of semi-innnite and bi-innnite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonor-malization of splines is indicated. Both banded and non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the a...

ABSTllACT-In this article various existence results for the LDU-factorization of semi-infinite and bi-infinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormal-ization of splines is indicated. Both banded and non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Resu...

A general scheme is proposed for computing the QR-factorization of certain displacement structured matrices, including Cauchy-like, Vandermonde-like, Toeplitz-like and Hankel-like matrices, hereby extending some earlier work for the QR-factorization of the Cauchy matrix. The algorithm employs a chasing scheme for the recursive construction of a diagonal plus semiseparable matrix of semiseparabi...

Matrix-vector multiplication is one of the most fundamental computing primitives that has been studied extensively. Given a matrix A ∈ FN×N and a vector b ∈ FN , it is known that in the worst-caseΘ(N 2) operations over F are needed to compute Ab. Many classes of structured dense matrices have been investigated which can be represented with O(N ) parameters, and for which matrix-vector multiplic...

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that RR is close to AA. Thus, when the algorithm is used to solve the semi-normal equations RRx = A b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear systemAx = b. The algorithm also applies to the solution of the full-rank Toeplitz ...

We propose a method for computing any Gelfand-Dickey τ function living in SegalWilson Grassmannian as the asymptotics of block Toeplitz determinant associated to a certain class of symbols W(t; z). Also truncated block Toeplitz determinants associated to the same symbols are shown to be τ function for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from ...

We show that every n × n matrix is generically a product of n/2 + 1 Toeplitz matrices and always a product of at most 2n + 5 Toeplitz matrices. The same result holds true if the word ‘Toeplitz’ is replaced by ‘Hankel,’ and the generic bound n/2 + 1 is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not, in general, replace the subspace of Toeplit...

In this paper, we consider the approximation of dense block Toeplitz-plus-Hankel matrices by sums of Kronecker products. We present an algorithm for efficiently computing the matrix approximation that requires the factorization of matrices of much smaller dimension than that of the original. The main results are described for block Toeplitz matrices with Toeplitz-plus-Hankel blocks (BTTHB), but...

We consider the solution of a class of complex symmetric block Toeplitz linear systems, arising from integral equations problems. Algorithms that exploit the Toeplitz structure provide considerable savings on the number of arithmetic operations, compared to the classical Cholesky factorization. We propose a fast Schur algorithm adapted to the complex symmetric case. We detail blocked variants, ...