In this expository paper, we survey some of the latest developments in using preconditioned conjugate gradient methods for solving Toeplitz systems. One of the main results is that the complexity of solving a large class of $n$-by-$n$ Toeplitz systems is reduced to $O(n \log n)$ operations as compared to $O(n \log ^2 n)$ operations required by fast direct Toeplitz solvers. Different preconditioners proposed for Toeplitz systems are reviewed. Applications to Toeplitz-related systems arising from partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis are given.

/pdf/conjugate-gradient-methods-for-toeplitz-systems-44re7adc4m.pdf

Conjugate Gradient Methods for Toeplitz Systems

Least squares methods

By deconvolution we mean the solution of a linear first-kind integral equation with a convolution-type kernel, i.e., a kernel that depends only on the difference between the two independent variables. Deconvolution problems are special cases of linear first-kind Fredholm integral equations, whose treatment requires the use of regularization methods. The corresponding computational problem takes the form of structured matrix problem with a Toeplitz or block Toeplitz coefficient matrix. The aim of this paper is to present a tutorial survey of numerical algorithms for the practical treatment of these discretized deconvolution problems, with emphasis on methods that take the special structure of the matrix into account. Wherever possible, analogies to classical DFT-based deconvolution problems are drawn. Among other things, we present direct methods for regularization with Toeplitz matrices, and we show how Toeplitz matrix–vector products are computed by means of FFT, being useful in iterative methods. We also introduce the Kronecker product and show how it is used in the discretization and solution of 2-D deconvolution problems whose variables separate.

/pdf/deconvolution-and-regularization-with-toeplitz-matrices-382ivzkkxq.pdf

Deconvolution and Regularization with Toeplitz Matrices

We present a family of new fast algorithms for QR factorization of certain structured matrices, including rectangular Toeplitz matrices and a variety of other Toeplitz-like matrices. It possesses a very regular structure, and appears to be very convenient for parallel implementation. Moreover it is shown that the same architecture can be used for either triangular factorization or QR factorization. Our approach separates the conceptual and implementational aspects of the problem. Our analysis reveals a variety of algorithmic implementations of the basic procedure, all with potentially different numerical properties that need further examination.Our approach is based on the observations that the matrix ${\bf R}$ in ${\bf A} = {\bf {QR}}$ is the Cholesky factor of ${\bf A}^T {\bf A}$ and that fast algorithms for Cholesky factorization of positive definite Toeplitz and certain “close-to-Toeplitz” matrices are readily obtained via a so-called displacement representation of matrices. In this paper we show that if ${\bf A}$ is Toeplitz or is expressed in suitable displacement form then ${\bf A}^T {\bf A}$ also has such a displacement representation, so that ${\bf R}$ can be obtained by a suitable fast Cholesky algorithm. The matrix ${\bf Q}$ is then found by extending a result, also presented in this paper, that ${\bf R}$ and ${\bf R}^{ - 1} $ can be found simultaneously by appropriate circular and hyperbolic rotations applied to a matrix (with Toeplitz blocks) constructed from the elements of ${\bf A}$.

Fast Parallel Algorithms for QR and Triangular Factorization

In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchy-like matrices using the FFT or other trigonometric transformations. These Cauchy-like matrices have a special property, that is, their off-diagonal blocks have small numerical ranks. This low-rank property plays a central role in our superfast Toeplitz solver. It enables us to quickly approximate the Cauchy-like matrices by structured matrices called sequentially semiseparable (SSS) matrices. The major work of the constructions of these SSS forms can be done in precomputations (independent of the Toeplitz matrix entries). These SSS representations are compact because of the low-rank property. The SSS Cauchy-like systems can be solved in linear time with linear storage. Excluding precomputations the main operations are the FFT and SSS system solve, which are both very efficient. Our new Toeplitz solver is stable in practice. Numerical examples are presented to illustrate the efficiency and the practical stability.

https://www.math.purdue.edu/~xiaj/work/toep.pdf

A Superfast Algorithm for Toeplitz Systems of Linear Equations

This paper presents a new algorithm for computing theQR factorization of anm×n Toeplitz matrix inO(mn) operations. The algorithm exploits the procedure for the rank-1 modification and the fact that both principal (m−1)×(n−1) submatrices of the Toeplitz matrix are identical. An efficient parallel implementation of the algorithm is possible.

QR factorization of toeplitz matrices

This report contains a numerical stability analysis of factorization algorithms for computing the Cholesky decomposition of symmetric positive definite matrices of displacement rank 2. The algorithms in the class can be expressed as sequences of elementary downdating steps. The stability of the factorization algorithms follows directly from the numerical properties of algorithms for realizing elementary downdating operations. It is shown that the Bareiss algorithm for factorizing a symmetric positive definite Toeplitz matrix is in the class and hence the Bareiss algorithm is stable. Some numerical experiments that compare behavior of the Bareiss algorithm and the Levinson algorithm are presented. These experiments indicate that in general (when the reflection coefficients are not all positive) the Levinson algorithm is not stable; certainly it can give much larger residuals than the Bareiss algorithm.

pdf/on-the-stability-of-the-bareiss-and-related-toeplitz-3463tx61ta.pdf

On the stability of the Bareiss and related Toeplitz factorization algorithms

Tridiagonalization of a symmetric matrix on a square array of mesh-connected processors

Parallel solution of certain Toeplitz least-squares problems

A systolic algorithm for extended GCD computation

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