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Table of Contents.

>Z< Toeplitz and Circulant Matrices: A Review R.M. Gray

In this tutorial report the fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of ``finite section'' Toeplitz matrices are derived. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hopes of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

>Z< On the QR algorithm and updating the SVD and URV decomposition in parallel M. Moonen, P. Van Dooren, F. Vanpoucke

A Jacobi-type updating algorithm for the SVD or URV decomposition is developed, which is related to the QR algorithm for the symmetric eigenvalue problem. The algorithm employs one-sided transformations, and therefore provides a cheap alternative to earlier developed updating algorithms based on two-sided transformations. The present algorithm as well as the corresponding systolic implementation is therefore roughly twice as fast, compared to the former method, while the tracking properties are preserved. The algorithm is also extended to the 2-matrix QSVD or QURV case. Finally, the differences are discussed with a number of closely related algorithms that have been recently proposed.

>Z< Two-way bidiagonalization scheme for downdating the singular value decomposition H. Park and S. Van Huffel

We present a method that transforms the problem of downdating the singular value decomposition into a problem of diagonalizing a diagonal matrix bordered by one column. The first step in this diagonalization involves bidiagonalization of a diagonal matrix bordered by one column. For updating the singular value decomposition, a two-way chasing scheme has recently been introduced, which reduces the total number of rotations by 50% compared to previously developed one-way chasing schemes. Here, a two-way chasing scheme is introduced for the bidiagonalization step in downdating the singular value decomposition. We show how the matrix elements can be rearranged and how the nonzero elements can be chased away towards two corners of the matrix. The newly proposed scheme saves nearly 50% of the number of plane rotations compared to those required by one-way chasing schemes.

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>Z< An efficient total least squares algorithm based on a rank-revealing two-sided orthogonal decomposition S. Van Huffel and H. Zha

Solving Total Least Squares (TLS) problems $AX\approx B$ requires the computation of the noise subspace of the data matrix $[A;B]$. The widely used tool for doing this is the Singular Value Decomposition (SVD). However, the SVD has the drawback that it is computationally expensive. Therefore, we consider here a different so-called rank-revealing two-sided orthogonal decomposition which decomposes the matrix into a product of a unitary matrix, a triangular matrix and another unitary matrix in such a way that the effective rank of the matrix is obvious and at the same time the noise subspace is exhibited explicitly. We show how this decomposition leads to an efficient and reliable TLS algorithm that can be parallelized in an efficient way.

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