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From "Remi Arntzen (JIRA)" <j...@apache.org>
Subject [jira] Commented: (MATH-157) Add support for SVD.
Date Wed, 01 Nov 2006 23:46:17 GMT
    [ http://issues.apache.org/jira/browse/MATH-157?page=comments#action_12446407 ] 
            
Remi Arntzen commented on MATH-157:
-----------------------------------

> You know you can invert that matrix (it's orthogonal, so just take the transpose), and
you can
> also invert the S matrix (it's diagonal, invert each value on the diagonal), so just
invert those two
> and multiply through the original matrix to get the other orthogonal matrix

however when the original matrix is not square that method fails to apply.  the inverse of
a non-square matrix does not exist, and as such you can not simply take the transpose of each
matrix product as they are not of equal size. (although this method does work splendidly on
square matrices)

> It just seems like you're kindof going in the wrong direction

I'd say I'm literally going in the wrong direction, some of the vectors seem to be randomly
inverted, and as I've said earlier I have never had any formal training on this subject (just
rudimentary high school training).  I'm basically coding as I am learning.

> Add support for SVD.
> --------------------
>
>                 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
>
>
> 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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