Re: [eigen] Re: LU precision tuning

[Benoit Jacob <jacob.benoit.1@xxxxxxxxx>]

2009/5/11, Hauke Heibel <hauke.heibel@xxxxxxxxxxxxxx>:
>> ... but I rather think that what is invalidated is the QR precision
>> testing!
>> Indeed your patch introduced m_precision in QR but it was never used
>> in the computation of the QR decomposition, only in the rank() and
>> solve() methods. So what's really needed is to make eps2 honor the
>> m_precision.
>>
>> I can't tell whether the creation of random rank r matrices was
>> affected by that or not.
>
> It seems as if it was not really affected. After fixing the precision
> usage, the results did not improve.

Ah, ok.

> I took a short look at the code an briefly touched the literature
> about RRQR (rank revealing QR) decompositions. Since the current QR
> does (please correct me if I am wrong) no pivoting I assume it is
> pretty safe to assume that it is not rank revealing. There are a bunch
> of RRQR algorithms reviewed/described in Bischof & Quintana-Qrti,
> "Computing Rank-Revealing QR Factorizations of Dense Matrices" and all
> of them rely on pivoting.

Indeed, the current QR doesn't do any pivoting: that completely
escaped me (disclaimer -- i didn't write any of this code). So yes,
it's clearly not rank-revealing and i'm amazed then how well it
performed in your tests -- interesting.

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?

Just asking in case anyone here knows. I'm not saying we must have
sorted this completely for 2.1 but at least if we don't change the QR,
the API and documentation should be adapted to its domain of
applicability.

Thanks for the preliminary research on RRQR, it'll be useful if
someone decides to implement it.

Cheers,
Benoit