QR vs LU factorisation

[Jordi Gutiérrez Hermoso]

This isn't specifically an Octave question except tangentially.

I have some matrices that are as bad as can be: largish (1000x1000 or
so), full, unsymmetric, and ill-conditioned. I notice that Octave uses
LU factorisation with partial pivoting to invert these matrices as a
last resort, which has been giving me acceptable results. Someone
suggested to me that QR factorisation would be better suited. I'm
reading Golub & Van Loan, but see no clear indication of when to use
QR or LU.

Does anyone have any suggestions?

Thanks,
- Jordi G. H.