[ http://issues.apache.org/jira/browse/MATH-157?page=comments#action_12446421 ] Tyler Ward commented on MATH-157: ---------------------------------
I think I see the confusion. Take a look at my first comment here. You don't want to invert (or find eigenvectors of) the original matrix. Say the original matrix is A (and A` is A-transpose, with row and column labels flipped), which is NxM (N > M), you want to construct eigenvectors for... B = A`A [NxN square symmetric matrix] You can do most of this by performing a QR decomposition on B, and then producing T = QBQ`. Unfortunately, T is not diagonal, it will be tridiagonal (three rows of values), so you need to reduce it the rest of the way to diagonal form by multiplying by a matrix P. Finding which matrix does this however is a chore, it requires givens reductions, explained above. In the end, you want to produce a matrix P such that D = PQBQ`P` is diagonal. This should be reversible, B = Q`P`DPQ should be the B you started with. If you can do that, then you're home free, just a few matrix multiplications will finish the job. When you get to the step of needing givens reductions, let me know. I can probably give you some formulas and such that will make it really easy. If you let me know how you're doing with this, I can definitely help. I can't however contribute code (work forbids it), but I can help a lot with anything else, even formulas. So I'd say try this. Use the QR decomposition they already have to get to the point of being able to produce T, and reverse T to get the B that you started with. Do that, and I can give you a PDF that will make the givens step very easy, it's really only a couple dozen lines of code, but requires some care that I can walk you through. > 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. -- This message is automatically generated by JIRA. - If you think it was sent incorrectly contact one of the administrators: http://issues.apache.org/jira/secure/Administrators.jspa - For more information on JIRA, see: http://www.atlassian.com/software/jira --------------------------------------------------------------------- To unsubscribe, e-mail: [EMAIL PROTECTED] For additional commands, e-mail: [EMAIL PROTECTED]
