Adam W. Bojanczyk

Numerically Stable Solution of Dense Systems of Linear Equations Using Mesh-Connected Processors

We propose a multiprocessor structure for solving a dense system of n linear equations. The solution is obtained in two stages. First, the matrix of coefficients is reduced to upper triangular form via Givens rotations. Second, a back substitution process is applied to the triangular system. A two-dimensional array of

A note on downdating the Cholesky factorization

\theta (n^2 )

QR factorization of toeplitz matrices

processors is employed to implement the first step, and (using a previously known scheme) a one-dimensional array of

Periodic Schur decomposition: algorithms and applications

\theta (n)

On the Stability of the Bareiss and Related Toeplitz Factorization Algorithms

processors is employed to implement the second step. These processor arrays allow both stages to be carried out in time

Transformation Techniques for Toeplitz and Toeplitz-plus-Hankel Matrices Part I. Transformations

O(n)

Solving the Indefinite Least Squares Problem by Hyperbolic QR Factorization

, and they are well suited for VLSI implementation as identical processors with a simple and regular interconnection pattern are required.

Existence of the hyperbolic singular value decomposition

We analyse and compare three algorithms for “downdating” the Cholesky factorization of a positive definite matrix. Although the algorithms are closely related, their numerical properties differ. Two algorithms are stable in a certain “mixed” sense while the other is unstable. In addition to comparing the numerical properties of the algorithms, we compare their computational complexity and their suitability for implementation on parallel or vector computers.

Stability analysis of a general toeplitz systems solver

SummaryThis 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.

Rank-k Modification Methods for Recursive Least Squares Problems

In this paper we derive a unitary eigendecomposition for a sequence of matrices which we call the periodic Schur decomposition. We prove its existence and discuss its application to the solution of periodic difference equations arising in control. We show how the classical QR algorithm can be extended to provide a stable algorithm for computing this generalized decomposition. We apply the decomposition also to cyclic matrices and two point boundary value problems.