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Re: SingularValueDecomposition bug: fifty examples

• To: mathgroup at smc.vnet.net
• Subject: [mg60818] Re: SingularValueDecomposition bug: fifty examples
• From: John Sidles <sidles at u.washington.edu>
• Date: Thu, 29 Sep 2005 05:41:54 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

Hello Ssezi

Please give my regards to my friend, and your colleague
colleague at JEOL, namely, Yohsuke Yoshinari !!

Yohsuke will understand that the "terrain" that my SVD-based
algorithm is following is a *quantum* terrain (which is why the 
SVD matrices contain complex numbers).  Someday, perhaps
JEOL will build microscopes that track this complex terrain.

> it is quite clear that the algorithm can fail to compute an SVD 
> with a message that it did not converge

Hmmm ... twenty-five of the examples compute an invalid SVD,
but with *no* message that convergence failed.  So it seems most
likely that that the problem is a bug in the implementation, not in 
the algorithm or the matrices.

> I am unsure of the details as to what properties of a matrix 
> could cause a failure, but it seems to be related to issues of 
> numeric precision and not dependent on the condition number.

Yes, it can't depend on the condition number, because otherwise 
SVD could not be used as the basis of Mathematica's MatrixRank[].
So I am not sure exactly what the circumstances are that lead to
failure.  Most (but not all) failing matrices have dimension 100 to 
200, with rank roughly half the dimension.  I have now found
non-Hermitian matrices that fail, so the symmetry of the matrix
does not seem to be critical.  I will report back when I know more.

Sincerely ... John Sidles