[SciPy-dev] implement reorthogonalization method in gmres

nicky van foreest vanforeest@gmail....
Wed Feb 24 16:43:44 CST 2010


I am not at all a specialist on this, but I recall that Numerical
Recipes states that the Gram-Schmidt process gives terrible results,
and should be avoided nearly always. Instead, singular value
decomposition should be used. To avoid the problem below, would is be
an idea to replace the GM algo by SVD?


On 24 February 2010 13:07, Pauli Virtanen <pav+sp@iki.fi> wrote:
> Tue, 23 Feb 2010 20:26:02 -0500, Darcoux Christine wrote:
>> It seems that the vectors produced by the Gram-Schmidt process may not
>> be orthogonal (probably due to the presence of roundoff errors or when
>> the matrix is ill conditionned). I would suggest to add an option to
>> test for loss of orthogonality and reorthogonalize if needed. See
>> http://www4.ncsu.edu/~ctk/newton/SOLVERS/nsoli.m for an example.
> From the literature, it appears that orthogonality of the Krylov vectors
> is not very important for GMRES convergence, see e.g.
>        A. Greenbaum, M. Rozložník, Z. Strakoš
>        BIT Numerical Mathematics, 37, 706 (1997).
>        http://www.springerlink.com/content/277614q38kj6q376/
> The GMRES implementations in Scipy use the modified Gram-Schmid
> procedure, so they should be OK in this respect.
> Do you have some evidence suggesting that reorthogonalization is a
> significant improvement for some problems?  I'll accept patches providing
> such a flag in any case, though.
> Best regards,
> Pauli
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