On the stability of the Bareiss and related Toeplitz factorization algorithms

144. A. W. Bojanczyk, R. P. Brent, F. R. de Hoog and D. R. Sweet, On the stability of the Bareiss and related Toeplitz factorization algorithms, SIAM J. Matrix Analysis and Applications 16 (1995), 40-57. MR 95k:65030.

Also appeared as Technical Report TR-CS-93-14, Computer Sciences Laboratory, ANU, November 1993, 18 pp. arXiv:1004.5510v1

Abstract: dvi (3K), pdf (67K), ps (23K).

Paper: dvi (32K), pdf (225K), ps (91K).

Technical Report: dvi (31K), pdf (201K), ps (95K).


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 reflection coefficients are not all positive) the Levinson algorithm is not stable; certainly it can give much larger residuals than the Bareiss algorithm.


For a preliminary version, see [126].

The stability results were later generalised by Chandrasekharan and Sayed [SIAM J. Matrix Anal. Appl. 17 (1996), 950-983] and (independently) by M. Stewart and Van Dooren [SIAM J. Matrix Anal. Appl. 18 (1997), 104-118]; see [177, Sec. 5.2] for a discussion.

Go to next publication

Return to Richard Brent's index page