The QR Decomposition of Toeplitz Matrices

We present a new algorithm for computing the QR factorization of an mxn Toeplitz matrix in O(mn) multiplications. The algorithm exploits the procedure for the rank-1 modification and the fact that successive columns of a Toeplitz matrix are related to each other. Both matrices Q and R are generated column by column, starting for their first columns. Each column is calculated from the previous column after rank-1 modification to the matrix R and a step of Gramm-Schmidt orthogonalization process applied to two auxiliary vectors

A Novel MVDR Beamforming Algorithm

We propose a novel algorithm and architecture for minimum variance distortions less response (MVDR) beamforming. Our approach extracts the least squares residual element directly, and requires only one systolic array for the many look directions

A Unified Systolic Array for Adaptive Beamforming

We present a new algorithm and systolic array for adaptive beamforming. Our approach improves on McWhirter\u27s pioneering work in two respects. First, our algorithm uses only orthogonal transformations and this should have better numerical properties. Second, the algorithms can be implemented on one single pxp triangular array of programmable processors that offers a throughput of one residual element per cycle

A Systolic Parsing Algorithm for a Visual Programming Language

In this paper we consider a problem of parsing a two-dimensional visual programming language Show and Tell on a two-dimensional array of processors. A program in Show and Tell is a bit-mapped, two-dimensional pattern satisfying a certain set of grammatical rules. The pattern consists of partially ordered set of rectilinear boxes and arrows distributed over the space of nxn pixel area. The corresponding directed graph, the box graph, where boxes are nodes and arrows are directed edges, may not have a cycle in a Show and Tell program. The cycle detection is the most computationally intensive stage of the parsing section of a Show and Tell program. We propose to exploit the concept of systolic array in parsing of Show and Tell programming language. A given bit pattern is mapped onto nxn array of mesh connected processors with one pixel assigned to one processing element. We show an algorithm for cycle detection which runs in time proportional to the size of the box graph. The complexity of any individual processor is independent on n, the parameter describing the size of the array

On the stability of the Bareiss and related Toeplitz factorization algorithms

This paper 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 reection coefficients are not all positive) the Levinson algorithm can give much larger residuals than the Bareiss algorithm

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

A Multi-step Algorithm for Hankel Matrices

AbstractIn this paper we discuss a Schur-type algorithm for solving indefinite Hankel systems of linear equations. The algorithm is derived using the language of linear algebra