[SciPy-dev] implement reorthogonalization method in gmres

[Pauli Virtanen]

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