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From "Tyler Ward (JIRA)" <>
Subject [jira] Commented: (MATH-157) Add support for SVD.
Date Thu, 02 Nov 2006 02:07:17 GMT
    [ ] 
Tyler Ward commented on MATH-157:

One more thing. I don't think you can just use the Q from the QR reduction (slipped my mind
earlier). Try it, you'll see that it doesn't come out tridiagonal. The trick is that you need
to do both sides, front and back on each iteration of the householder reduction. 

The household reduction reduces each column of the matrix my multiplying by a Q matrix that
has a single column of values, and the rest is the identity matrix. Like this A2 = Q1A1. You
need to do this ... A2 = Q1A1Q1`, on each iteration, so A3 = Q3Q2Q1A1Q1`Q2`Q3`, etc... 

Make sure it doesn't come out right before doing this, but I think the QR decomposition will
compute the matrix wrong, because it's assuming a one sided (forward only) transformation,
but check first. 

> Add support for SVD.
> --------------------
>                 Key: MATH-157
>                 URL:
>             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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