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From Ted Dunning <ted.dunn...@gmail.com>
Subject Re: [Math] "LUDecomposition" in "AbstractLeastSquaresOptimizer"
Date Thu, 08 Sep 2011 19:51:24 GMT
On Thu, Sep 8, 2011 at 12:42 PM, Luc Maisonobe <Luc.Maisonobe@free.fr>wrote:

> ... Luc - are there other reasons that QR would be better for cov
>> matrices?   I would have to play with a bunch of examples, but I
>> suspect with the defaults, Cholesky may do the best job detecting
>> singular problems.
>>
>
> I'm not sure about Cholesky, but I have always thought that at least QR was
> better than LU for near singular matrices, with only a factor 2 overhead in
> number of operations (but number of operations is not the main bottleneck in
> modern computers, cache behavior is more important).


This is my impression as well.

In addition, Cholesky very nearly is a QR decomposition through the back
door.  That is, if

     Q R = A

Then

     R' R = A' A

is the Cholesky decomposition.  The algorithms are quite similar when viewed
this way.  Cholesky does not produce as accurate a result for R if you are
given A, but if you are given A'A as in the discussion here, it is pretty
much just as good, I think.

I use this property for very large SVD's via stochastic projection because Q
is much larger than R in that context so computing just R can be done
in-core while the full QR would require out-of-core operations.

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