Re: [eigen] Re: LU precision tuning

[Jitse Niesen <jitse@xxxxxxxxxxxxxxxxx>]

On Mon, 11 May 2009, Benoit Jacob wrote:

> I have no experience with QR so I don't realize how much pivoting is
> essential or not for QR. So I can't answer right away the following
> important questions:
> - what is the domain of application of a no-pivoting QR? Is it
> restricted to only invertible matrices or something like that?
> (the fact that our unittest passes doesn't say much on precision, as
> our tests use a very low precision)
> - how  would the domain of application be expanded if we added partial
> (column) pivoting?

Since nobody replied, I thought perhaps I could say what I know even 
though I don't have any practical experience.

One important application of QR is to solve least squares problems. That 
is, given an overdetermined system Ax = b, find the vector x which 
minimizes the Euclidean norm of the residual Ax - b. As I understand it 
(and the situation is probably more complicated), QR without pivoting 
works fine if A is full rank, but you need column pivoting if A is 
(nearly) rank deficient. Remember that A is rectangular, so full rank is 
not the same as invertible.

I remember there was a recent post about the least squares module, but I 
can't find it anymore. However, the SVN log shows that linearRegression() 
used to solve the least squares problem via normal equation A*Ax = A*b. 
That's another approach, which is faster than QR but less stable. It now 
calls fitHyperplane(), which actually solves another problem: total least 
squares instead of (ordinary) least squares. That seems a rather big 
change.

Cheers,
Jitse