On the stability of the Bareiss and related Toeplitz factorization algorithms

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 generally (when the reflection coefficients are not all of the same sign) the Levinson algorithm can give much larger residuals than the Bareiss algorithm.

[1]  H. Wozniakowski,et al.  Iterative refinement implies numerical stability , 1977 .

[2]  R. Brent,et al.  QR factorization of Toeplitz matrices , 1986 .

[3]  Adam W. Bojanczyk,et al.  Stability analysis of fast Toeplitz linear system solvers , 1991 .

[4]  Webb Miller,et al.  Software for Roundoff Analysis of Matrix Algorithms , 1980 .

[5]  M. Morf,et al.  Displacement ranks of matrices and linear equations , 1979 .

[6]  Israel Gohberg,et al.  Solution of the Yule-Walker equations , 1991, Optics & Photonics.

[7]  Jim M. Varah.  Backward Error Estimates for Toeplitz and Vandermonde Systems , 1991 .

[8]  S. Alexander,et al.  Analysis of a recursive least squares hyperbolic rotation algorithm for signal processing , 1988 .

[9]  Richard P. Brent,et al.  A Note on Downdating the Cholesky Factorization , 1987 .

[10]  N. Higham.  Optimization by Direct Search in Matrix Computations , 1993, SIAM J. Matrix Anal. Appl..

[11]  E. Bareiss.  Numerical solution of linear equations with Toeplitz and Vector Toeplitz matrices , 1969 .

[12]  M. Powell,et al.  On the Modification of LDL T Factorizations , 1974 .

[13]  Adam W. Bojanczyk,et al.  Matrix Downdating Techniques For Signal Processing , 1988, Optics & Photonics.

[14]  D. Slepian.  Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case , 1978, The Bell System Technical Journal.

[15]  J. Bunch.  The weak and strong stability of algorithms in numerical linear algebra , 1987 .

[16]  J. M. Bennett.  Triangular factors of modified matrices , 1965 .

[17]  Allan O. Steinhardt,et al.  Stabilized hyperbolic Householder transformations , 1989, IEEE Trans. Acoust. Speech Signal Process..

[18]  Charles M. Rader,et al.  Hyperbolic householder transformations , 1986, IEEE Trans. Acoust. Speech Signal Process..

[19]  Ilse C. F. Ipsen,et al.  From Bareiss' Algorithm to the Stable Computation of Partial Correlations , 1989 .

[20]  George Cybenko,et al.  The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations , 1980 .

[21]  Franklin T. Luk,et al.  Advanced Algorithms and Architectures for Signal Processing III , 1989 .