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From "Tyler Ward (JIRA)" <j...@apache.org>
Subject [jira] Commented: (MATH-157) Add support for SVD.
Date Fri, 03 Nov 2006 03:05:17 GMT

Tyler Ward commented on MATH-157:
---------------------------------

Not bad. Looks like everything is in the right place, modulo a transpose or two perhaps, but
looks good. The eigenvector reduction is really the heart of this algorithm. Your QR iteration
will work (I think), but it's really inefficient. The trick is to do two different reductions.
The first should reduce to tridiagonal form (this can be done with only 2(N-2) matrix multiplications,
rather than the 200 or so you're using), and then use the givens rotations to reduce tridiagonal
to diagonal. Should take about 5N rotations, but each is only about 10N operations or so (rather
than N-cubed for a regular matrix multiply).

Congrats.

> --------------------
>
>                 Key: MATH-157
>                 URL: http://issues.apache.org/jira/browse/MATH-157
>             Project: Commons Math
>          Issue Type: New Feature
>            Reporter: Tyler Ward
>         Attachments: svd.tar.gz, svd2.tar.gz
>
>
> SVD is probably the most important feature in any linear algebra package, though also
one of the more difficult.
> In general, SVD is needed because very often real systems end up being singular (which
can be handled by QR), or nearly singular (which can't). A good example is a nonlinear root
finder. Often the jacobian will be nearly singular, but it is VERY rare for it to be exactly
singular. Consequently, LU or QR produces really bad results, because they are dominated by
rounding error. What is needed is a way to throw out the insignificant parts of the solution,
and take what improvements we can get. That is what SVD provides. The colt SVD algorithm has
a serious infinite loop bug, caused primarily by Double.NaN in the inputs, but also by underflow
and overflow, which really can't be prevented.
> If worried about patents and such, SVD can be derrived from first principals very easily
with the acceptance of two postulates.
> 1) That an SVD always exists.
> 2) That Jacobi reduction works.
> Both are very basic results from linear algebra, available in nearly any text book. Once
that's accepted, then the rest of the algorithm falls into place in a very simple manner.

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