In this paper, we re-investigate the resolution of Toeplitz systems T u = g, from a new point of view, by correlating the solution of such problems with syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this module is generated by two elements of degree n an...

A compact scheme is a discretization scheme that is advantageous in obtaining highly accurate solutions. However, the resulting systems from compact schemes are tridiagonal systems that are di cult to solve e ciently on parallel computers. Considering the almost symmetric Toeplitz structure, a parallel algorithm, simple parallel pre x (SPP), is proposed. The SPP algorithm requires less memory t...

It can be shown directly from consideration of the Schur algorithm that any nn semi-deenite rank r Toeplitz matrix, T, has a factorization T = C r C T r with C r = C 11 C 12 0 0 where C 11 is r r and upper triangular. This paper explores the reliability of computing such a decomposition with O(nr) complexity using the Schur algorithm and truncating the Cholesky factor after computing the rst r ...

When computing an average of positive definite (PD) matrices, the preservation of additional matrix structure is desirable for interpretations in applications. An interesting and widely present structure is that of PD Toeplitz matrices, which we endow with a geometry originating in signal processing theory. As an averaging operation, we consider the barycenter, or minimizer of the sum of square...

Using the notion of displacement rank, we look for a unifying approach to representations of a matrix A as sums of products of matrices belonging to commutative matrix algebras. These representations are then considered in case A is the inverse of a Toeplitz or a Toeplitz plus Hankel matrix. Some well-known decomposition formulas for A (Gohberg-Semencul or Kailath et al., Gader, Bini-Pan, and G...

The index formula of Douglas is a formula which expresses the index of a Fredholm Toeplitz operator with a discontinuous symbol as the limit of the indices of a family of Fredholm Toeplitz operators with continuous symbols. This paper is concerned with Toeplitz operators on the Bergman space A of the unit ball of C. The symbols are supposed to be matrix functions with entries in C+H∞ or to be c...

We present a new superfast algorithm for solving Toeplitz systems. This algorithm is based on a relation between the solution of such problems and syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this module is generated by two elements and the solution of ...

The problem we are interested in is the best approximation of a given matrix by a positive semi–definite symmetric Toeplitz matrix. Toeplitz matrices appear naturally in a variety of problems in engineering. Since positive semi–definite Toeplitz matrices can be viewed as shift invariant autocorrelation matrices, considerable attention has been paid to them, especially in the areas of stochastic...

The di erential equation dX dt X k X where k is a Toeplitz annihilator has been suggested as a means to solve the inverse Toeplitz eigenvalue problem Starting with the diagonal matrix whose entries are the same as the given eigenvalues the solution ow has been observed numerically to always converge a symmetric Toeplitz matrix as t This paper is an attempt to understand the dynamics involved in...

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hami...