Qiao, Sanzheng Hybrid algorithm for fast Toeplitz orthogonalization. (English) Zbl 0629.65031 Numer. Math. 53, No. 3, 351-366 (1988). New techniques for fast Toeplitz QR decomposition are presented. The methods are based on the shift invariance property of a Toeplitz matrix. The numerical properties of the algorithms are discussed and some comparisons are made with two other fast Toeplitz orthogonalization methods. Cited in 2 Documents MSC: 65F05 Direct numerical methods for linear systems and matrix inversion 65F20 Numerical solutions to overdetermined systems, pseudoinverses 65F25 Orthogonalization in numerical linear algebra 15A23 Factorization of matrices Keywords:fast Toeplitz QR decomposition; shift invariance; Toeplitz matrix; comparisons; fast Toeplitz orthogonalization methods; Givens rotation; Householder transformation; Cholesky factorization; least squares problem PDFBibTeX XMLCite \textit{S. Qiao}, Numer. Math. 53, No. 3, 351--366 (1988; Zbl 0629.65031) Full Text: DOI EuDML References: [1] Bojanczyk, A.W., Brent, R.P., Hoog, F.R. de: QR factorization of Toeplitz matrices. Numer. Math.49, 81-94 (1986) · Zbl 0574.65019 · doi:10.1007/BF01389431 [2] Cybenko, G.: A general orthogonalization technique with applications to time series analysis and signal processing. Math. Comput.40, 323-336 (1983) · Zbl 0539.93089 · doi:10.1090/S0025-5718-1983-0679449-6 [3] Cybenko, G.: Fast Toeplitz orthogonalization using inner products. SIAM J. Sci. Stat. Comput.8, 734-740 (1987) · Zbl 0689.65022 · doi:10.1137/0908063 [4] Gentleman, W.M.: Error analysis of QR decompositions by Givens Transformations. Linear Algebra Appl.10, 189-197 (1975) · Zbl 0308.65022 · doi:10.1016/0024-3795(75)90068-3 [5] Golub, G.H., Van Loan, C.F.: Matrix Computations. Baltimore MD: The Johns Hopkins University Press 1983 · Zbl 0559.65011 [6] Itakura, F., Saito, S.: Digital filtering techniques for speech analysis and synthesis. Proc. 7th Internat. Congr. Acousti., Budapest (1971), pp. 261-264 [7] Luk, F.T., Qiao, S.: A fast but unstable orthogonal triangularization technique for Toeplitz matrices. J. Linear Algebra Appl.88/89, 495-506 (1987) · Zbl 0617.65016 · doi:10.1016/0024-3795(87)90122-4 [8] Qiao, S.: Recursive least squares algorithm for linear prediction problems. SIAM J. Matrix Analysis Appl., July 1986 [9] Stewart, G.W.: Introduction to Matrix Computations. New York: Academic Press 1973 · Zbl 0302.65021 [10] Stewart, G.W.: The effects of rounding error on an algorithm for downdating a Cholesky factorization. J. Inst. Math. Appl.23, 203-213 (1979) · Zbl 0405.65019 · doi:10.1093/imamat/23.2.203 [11] Sweet, D.R.: Fast Toeplitz orthogonalization. Numer. Math.43, 1-21 (1984) · Zbl 0504.65017 · doi:10.1007/BF01389635 [12] Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press 1965 · Zbl 0258.65037 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.