A new preconditioner is proposed for the solution of an N x N Toeplitz system TNX = b, where TN can be symmetric indefinite or nonsymmetric, by preconditioned iterative methods. The preconditioner FN is obtained based on factorizing the generating function T(z) into the product of two terms corresponding, respectively, to minimum-phase causal and anticausal systems and therefore called the mini...

In this paper we will give a survey on iterative methods for solving linear equations with Toeplitz matrices. We introduce a new class of Toeplitz matrices for which clustering of eigenvalues and singular values can be proved. We consider optimal (ω)circulant preconditioners as a generalization of the circulant preconditioner. For positive definite Toeplitz matrices, especially in the real case...

This paper studies optimization for inner products of real vectors assuming monotonicity properties for the entries in one of the vectors. Resulting inequalities have been useful recently in bounding reciprocals of power series with rapidly decaying coefficients and in proving that all symmetric Toeplitz matrices generated by monotone convex sequences have offdiagonal decay preserved through tr...

This paper is concerned with the influence of frequency modulation on the semiFredholm properties of Toeplitz operators with oscillating matrix symbols. The main results give conditions on an orientation-preserving homeomorphism α of the real line that ensure the following: if b belongs to a certain class of oscillating matrix functions (periodic, almost periodic, or semi-almost periodic matrix...

The Fredholm properties of Toeplitz operators on the Bergman space A have been well-known for continuous symbols since the 1970s. We investigate the case p = 1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on A that ari...

In this paper we study the condition number of linear systems, the condition number of matrix inversion, and the distance to the nearest singular matrix, all problems with respect to normwise structured perturbations. The structures under investigation are symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Hankel matrices (some resu...

This paper provides an error analysis of the generalized Schur algorithm of Kailath and Chun|a class of algorithms which can be used to factorize Toeplitz-like matrices, including block-Toeplitz matrices, and matrices of the form T T T , where T is Toeplitz. The conclusion drawn is that if this algorithm is implemented with hyperbolic transformations in the factored form which is well known to ...

This paper provides an error analysis of the generalized Schur algorithm of Kailath and Chun|a class of algorithms which can be used to factorize Toeplitz-like matrices , including block-Toeplitz matrices, and matrices of the form T T T, where T is Toeplitz. The conclusion drawn is that if this algorithm is implemented with hyperbolic transformations in the factored form which is well known to ...

In this paper, we parallelize a new algorithm for solving non– symmetric Toeplitz linear systems. This algorithm embeds the Toeplitz matrix in a larger structured matrix, then transforms it into an embedded Cauchy–like matrix by means of trigonometric modifications. Finally, the algorithm applies a modified QR transformation to triangularize the augmented matrix. The algorithm combines efficien...

This paper is concerned with the construction of circulant preconditioners for Toeplitz systems arising from a piecewise continuous generating function with sign changes. If the generating function is given, we prove that for any " > 0, only O(log N) eigenvalues of our preconditioned Toeplitz systems of size N N are not contained in ?1?"; ?1+"] 1?"; 1+"]. The result can be modiied for trigonome...