When should I use them, and how the relation of speed and precision changes?
Which are the advantages, and disadvantages of them?
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Did you look at http://en.wikipedia.org/wiki/Orthogonalization? there is a short advantage/disadvantage discussion there. – DKal May 05 '14 at 08:46
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Thank you. But they don't write about the spped. I'm looking for a site, where this metods are compared by their relation of speed. – Iter Ator May 05 '14 at 09:53
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For an $m\times n$ matrix, G. Stewart says that both the Householder and Gram-Schmidt cost $mn^2-n^2/3$ floating point additions and multiplications.
The orthogonal factor computed by the Householder factorization is generally more accurate than that of the (modified) Gram-Schmidt unless the GS is implemented with reorthogonalization. On the other hand, GS gives the orthogonal factor directly, while Householder provides it in a factored form (essentially as a product of elementary reflections).
The loss of orthogonality does not need to be a problem for the modified GS, e.g., when solving least squares problem, a careful implementation (without reorthogonalization) can give the solution with the same accuracy as the Householder variant.
Also, GS is easier to implement and computationally more efficient in parallel.
Algebraic Pavel
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1Thank you, just one question. On which page can I read about Householder and Gram-Schmidt cost ? – Iter Ator May 05 '14 at 18:21
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1Oh if you mean on which page in the book, there are two sections related to Householder and GS in the chapter on the QR factorization, look there, it cannot be missed:) – Algebraic Pavel May 06 '14 at 01:02