1995/01/01 by Adam W. Bojanczyk, Adam W. Bojańczyk, Richard P. Brent +3 · 3 citations
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Advanced Topics in Algebra #Algebra over a field #Algorithm #Computer science #Factorization #Levinson recursion #Mathematics #Matrix Theory and Algorithms #Pure mathematics #Stability (learning theory) #Toeplitz matrix #acm:65F05 #acm:65G50 #cs.NA #math.NA #msc:65F05 #msc:65G50
paper · pdf · doi:10.1137/s0895479891221563
published as SIAM J. Matrix Analysis and Applications 16 (1995), 40-57 · 18 pages. An old Technical Report, submitted for archival purposes. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub144.html
openalex publication_date 1995/01/01 · arxiv created 2010/04/30 · arxiv updated 2021/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.