On Thu, Sep 8, 2011 at 12:42 PM, Luc Maisonobe <[email protected]>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.
