On the stability of the Bareiss and related Toeplitz factorization algorithms
Abstract
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.
Keywords
Cite
@article{arxiv.1004.5510,
title = {On the stability of the Bareiss and related Toeplitz factorization algorithms},
author = {Adam W. Bojanczyk and Richard P. Brent and Frank R. de Hoog and Douglas R. Sweet},
journal= {arXiv preprint arXiv:1004.5510},
year = {2021}
}
Comments
18 pages. An old Technical Report, submitted for archival purposes. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub144.html