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Iterative algorithms for Gram-Schmidt orthogonalization. (English) Zbl 0667.65037
A Gram-Schmidt algorithm to orthogonalize a sequence of vectors, where one vector at a time is orthogonalized to the set of already orthogonalized vectors is investigated. It is shown that if a reorthogonalization is performed whenever the norm of the vector is decreased more than a certain threshold, a prescribed accuracy of the whole sequence will be obtained both for the classical and the modified variants of the Gram-Schmidt algorithm.
Reviewer: A.Ruhe

65F25 Orthogonalization in numerical linear algebra
65G50 Roundoff error
15A23 Factorization of matrices
Full Text: DOI
[1] Björck, A.: Solving linear least squares problems by Gram-Schmidt orthogonalization. BIT7, 1–21 (1967). · Zbl 0183.17802 · doi:10.1007/BF01934122
[2] Businger, P., Golub, G. H.: Linear least squares solutions by Householder transformations. Numer. Math.1, 269–276 (1965). · Zbl 0142.11503 · doi:10.1007/BF01436084
[3] Chan, T. F.: Rank revealingQR factorizations. Linear Algebra Appl.88/89, 67–82 (1987). · Zbl 0624.65025 · doi:10.1016/0024-3795(87)90103-0
[4] Daniel, J. W., Gragg, W. B., Kaufman, L., Stewart, G. W.: Reorthogonalization and stable algorithms for updating the Gram-SchmidtQR factorization. Math. Comp.30, 772–795 (1976). · Zbl 0345.65021
[5] Golub, G. H., van Loan, C. F.: Matrix Computations. Oxford: North Oxford Academic 1983. · Zbl 0559.65011
[6] Parlett, B. N.: The Symmetric Eigenvalue Problem. Englewood Cliffs, N. J.: Prentice-Hall 1980. · Zbl 0431.65017
[7] Ruhe, A.: Numerical aspects of Gram-Schmidt orthogonalization of vectors. Linear Algebra Appl.52/53, 591–601 (1983). · Zbl 0515.65036
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