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From "Tyler Ward (JIRA)" <>
Subject [jira] Created: (MATH-157) Add support for SVD.
Date Tue, 29 Aug 2006 03:43:22 GMT
Add support for SVD.

                 Key: MATH-157
             Project: Commons Math
          Issue Type: New Feature
            Reporter: Tyler Ward

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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