[SciPy-dev] Why does orth use svd instead of QR ?
Charles R Harris
charlesr.harris@gmail....
Fri Feb 5 02:37:43 CST 2010
On Fri, Feb 5, 2010 at 1:29 AM, David Cournapeau <david@silveregg.co.jp>wrote:
> Charles R Harris wrote:
> >
> >
> > On Fri, Feb 5, 2010 at 12:47 AM, Charles R Harris
> > <charlesr.harris@gmail.com <mailto:charlesr.harris@gmail.com>> wrote:
> >
> >
> >
> > On Fri, Feb 5, 2010 at 12:18 AM, David Cournapeau
> > <david@silveregg.co.jp <mailto:david@silveregg.co.jp>> wrote:
> >
> > Charles R Harris wrote:
> > >
> > >
> > > On Thu, Feb 4, 2010 at 8:45 PM, David Cournapeau
> > <david@silveregg.co.jp <mailto:david@silveregg.co.jp>
> > > <mailto:david@silveregg.co.jp
> > <mailto:david@silveregg.co.jp>>> wrote:
> > >
> > > Hi,
> > >
> > > I wanted to know if there was a rationale for using svd to
> > > orthonormalize the columns of a matrix (in scipy.linalg).
> > QR-based
> > > methods are likely to be much faster, and I thought this
> > was the
> > > standard, numerically-stable method to orthonormalize a
> > basis ? If the
> > > reason is to deal with rank-deficient matrices, maybe we
> > could add an
> > > option to choose between them ?
> > >
> > >
> > > QR with column rotation would deal with rank-deficient
> > matrices and
> > > routines for that are available in LAPACK
> > > <http://netlib.org/lapack/lug/node42.html>. The SVD was
> > probably used
> > > because it was available. The diagonal elements of the R
> > matrix can
> > > somewhat take the place of the singular values when column
> > rotation is used.
> >
> > So would be it ok to use this column-rotated QR in place of svd
> for
> > every case in orth ? I would have to check that QR with column
> > rotation
> > is still significantly faster than svd, but I would surprised if
> > if were
> > not the case. QR has also the advantage of being implemented in
> > PLASMA
> > already contrary to eigen/svd solvers,
> >
> >
> > I don't know how the two methods compare in practice. SVD algorithms
> > generally use iterated QR reductions in their implementation, so QR
> > reductions can't be worse numerically. But the SVD probably provides
> > a better metric for rank determination. A google search turns up
> > some literature on the subject that I can't access from home.
> >
> >
> > OK, here's a good reference
> > <http://www.math.sjsu.edu/%7Efoster/rank/rank_revealing_s.pdf>. A quick
> > look seems to indicate that the SVD is the way to go.
>
> AFAIK, SVD is indeed the way to go, but do we really need this for the
> orth function ? I am wrong to think that orthonormalizing a matrix of
> linearly independent vectors is the most common usage for orth ? The
> difference in terms of speed is really significant (for example, svd of
> a 2000x100 matrix takes ~1.9 second vs 0.1 s for QR).
>
>
This looks a bit like sorting: quicksort is almost always fastest, but no
quarantee. The other methods are safer but slower. Maybe the way to go is
use a keyword to choose between methods.
Chuck
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