Actually, I think that the test in the Cholesky decomposition should be
three-pronged:
lt[i] > threshold proceed normally, use sqrt(lt[i])
lt[i] >= -threshold treat as zero. Either return rank-deficient result
if pivoting or
signal that original argument is not positive
definite.
lt[i] < -threshold treat as indication that original argument is not
semi positive definite.
This is different from being simply rank deficient
These tests, as framed above, are obviously sequential. This can be avoided
if you replace the second test with abs(lt[i]) <= threshold
On Fri, Sep 9, 2011 at 4:13 AM, Gilles Sadowski <
gil...@harfang.homelinux.org> wrote:
> > > Replace that with the correct value. I meant it to be an index.
> > >
> > > That doesn't change my other points. There is an inherent problem with
> > > "less than" comments when you have subtracted several other elements
> > > previously and only now notices that the remainder is less than some
> other
> > > adjusted value. The user cannot relate either value back to anything
> that
> > > they have and they may well look at their own matrix and see the
> opposite
> > > condition. It is better to give a slightly more abstract message such
> as
> > > "Non-positive definiteness in input detected at diagonal element number
> > > xxx"
> > > where you will have to fill in the xxx since you know which
> place-holders
> > > are which. NPD-ness is the condition that the decomposition talks
> about to
> > > the user but that condition doesn't occur at any single point in the
> > > matrix... it is a property of the whole. All the algorithm can do is
> say
> > > "Badness happened, I found it this far along".
>
> OK, I understand. But do you agree that the test should be changed from
>
> ltI[i] < absolutePositivityThreshold
>
> to
>
> ltI[i] <= absolutePositivityThreshold
>
> ?
