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Numerics of Gram-Schmidt orthogonalization. (English) Zbl 0801.65039
The paper surveys the numerical properties of the classical and the modified Gram-Schmidt (MGS) orthogonalization procedures. The key observation is the numerical equivalence of the modified Gram-Schmidt procedure to the Householder QR factorization of the matrix $$A$$ augmented by an $$n \times n$$ zero matrix on top. This result is used to derive bounds on the loss of orthogonality in MGS. A backward-stable algorithm based on MGS is developed. The use of reorthogonalization and iteration is also investigated. Block Gram-Schmidt algorithms are presented.

##### MSC:
 65F25 Orthogonalization in numerical linear algebra 65F05 Direct numerical methods for linear systems and matrix inversion
ALGOL 60
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##### References:
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