On the stability of the Bareiss and related Toeplitz factorization algorithms

TL;DR

This paper analyzes the numerical stability of the Bareiss algorithm for Toeplitz matrix factorization, demonstrating its stability and comparing it with the Levinson algorithm through numerical experiments.

Contribution

It proves the stability of the Bareiss algorithm for symmetric positive definite Toeplitz matrices and compares its performance with the Levinson algorithm.

Findings

Bareiss algorithm is stable for Toeplitz matrices.

Levinson algorithm can be unstable when reflection coefficients are not positive.

Numerical experiments show Bareiss has smaller residuals than Levinson.

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.