Re: [eigen] Instability in LLT and LDLT methods.

[Gael Guennebaud <gael.guennebaud@xxxxxxxxx>]

>>> I think that LAPACK and GSL do a hybrid of partial and full pivoting
>>> that I have yet to understand.
>>
>> That's not what their documentation says:
>> http://netlib.org/lapack/lug/node38.html
>> http://www.gnu.org/software/gsl/manual/html_node/LU-Decomposition.html
>
> Ah OK. Here I was just repeating stuff I heared over IRC. So I don't
> know. Perhaps they still do kind of a hybrid but don't mention it in
> the docs. If they really do plain partial pivoting, then let's try
> inverting, say, a 2000x2000 matrix, it will fail. For this reason I
> doubt that they actually do plain partial pivoting.

yeah, GSL clearly only does partial pivoting. About LAPACK I thought I
read somewhere that the high level block LU algorithm was based on top
of the sgetc2 auxiliary routines (un-blocked full pivoting), but
searching over the net again, I found source code using the dgetf2
routine (un-blocked partial pivoting)... That would make sense because
I don't know how hybrid full/partial pivoting could work.


>> I think a 50% speedup is a worthwhile target. At least for vector solves,
>> you can compute the residual and redo the computation with complete pivoting
>> if necessary.

another argument in favor of having partial pivoting is that would
allow to implement a blocked version with only ~10 lines of code, and
for large matrix a blocked LU with partial pivoting is much much
faster than the current one.

Gael.