Abstract
We show how to stabilize the generalized Schur algorithm for the Cholesky factorization of positive-definite structured matrices R that satisfy R-FRF T = GJG T, where J is a signature matrix, F is a stable lower-triangular matrix, and G is a generator matrix. We use a perturbation analysis to indicate the best accuracy that can be expected from any finite precision algorithm that uses the generator matrix as the input data. We then show that the modified Schur algorithm proposed in this work essentially achieves this bound for a large class of structured matrices.
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References
Kailath, T., “A theorem of I. Schur and its impact on modern signal processing,”, Operator Theory: Advances and Applications, vol. 18, pp. 9–30, Birkhäuser, Boston, 1986.
Golub, G. H. and Van Loan, C. F., Matrix Computations, The Johns Hopkins University Press, Baltimore, second ed., 1989.
Cybenko, G., “The numerical stability of the Levinson-Durbin algorithm for Toeplitz system of equations,” SIAM J. Sci. Statis. Comput., vol. 1, pp. 303–319, 1980.
Sweet, D. R., Numerical Methods for Toeplitz Matrices, PhD thesis, University of Adelaide, Adelaide, Australia, 1982.
Bojanczyk, A. W., Brent, R. P., de Hoog, F. R., and Sweet, D. R., “On the stability of the Bareiss and related Toeplitz factorization algorithms,” SIAM J. Matrix Anal. Appl., vol. 16, pp. 40–57, 1995.
Bojanczyk, A. W., Brent, R. P., van Dooren, P., and de Hoog, F. R., “A note on downdating the Cholesky factorization,” SIAM J. Sci. Stat. Comput., vol. 8, pp. 210–221, 1987.
Kailath, T. and Sayed, A. H., “Displacement structure: Theory and applications,” SIAM Review, vol. 37, no. 3, Sep. 1995.
Kailath, T., Kung, S. Y., and Morf, M., “Displacement ranks of a matrix,” Bulletin of the American Mathematical Society, vol. 1, no. 5, pp. 769–773, Sep. 1979.
Gu, M., “Stable and efficient algorithms for structured systems of linear equations,” in preparation.
Heinig, G., “Inversion of generalized Cauchy matrices and other classes of structured matrices,” in Linear Algebra for Signal Processing, IMA volumes in Mathematics and its Applications, vol. 69, pp. 63–81, 1995.
Gohberg, I., Kailath, T., and Olshevsky, V., “Fast Gaussian elimination with partial pivoting for matrices with displacement structure,” to appear in Math, of Computation.
Chandrasekaran, S. and Sayed, A. H., “Stabilizing the fast generalized Schur algorithm,” submitted for publication.
Sayed, A. H., Kailath, T., and Lev-Ari, H., “Generalized Chandrasekhar recursions from the generalized Schur algorithm,” IEEE Transactions on Automatic Control, vol. 39, no. 11, pp. 2265–2269, Nov. 1994.
Sayed, A. H. and Kailath, T., “A state-space approach to adaptive RLS filtering,” IEEE Signal Processing Magazine, vol. 11, no. 3, pp. 18–60, Jul. 1994.
Chandrasekaran, S. and Sayed, A. H., “A fast and stable solver for non-symmetric structured systems,” in preparation.
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Dedicated with admiration and gratitude to Prof. Thomas Kailath on the occasion of his 60th birthday.
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Chandrasekaran, S., Sayed, A.H. (1997). Improving the Accuracy of the Generalized Schur Algorithm. In: Paulraj, A., Roychowdhury, V., Schaper, C.D. (eds) Communications, Computation, Control, and Signal Processing. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6281-8_11
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DOI: https://doi.org/10.1007/978-1-4615-6281-8_11
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