Re: [eigen] Another LDLt issue

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

I had a closer look.

What happens is that the "positive" assumption allows to restrict the
pivot lookup to the diagonal. So we get full pivoting at the price of
partial pivoting.

What allows us to do that is that the norm of a positive matrix is
bounded by a multiple of the norm of its diagonal. In particular, if
the diagonal is zero, then the whole matrix is zero.

This is  really characteristic of positive matrices, it doesn't work
for general self-adjoint matrices, e.g.
0 1
1 0

So a generalization of LDLt handling non-positive matrices would have
to assume invertibility. It would work for self-adjoint invertible
matrices. Why not. Are you still interested in that?

The next question would then be: implement as separate dec or as a
variant of LDLt controlled by a template parameter?

By the way, Keir, at line 162:
m_matrix.block(j, j, size-j, size-j).fill(0);  // Zero unreliable data.
Unless this is backed by experiment, I kind of disagree with this
line, I don't see how zeroing this noise can be useful, and I can
think of disadvantages of zeroing: it makes the reconstruction of the
original matrix more inaccurate, not to mention that it generates more
code.

Cheers,
Benoit


2009/3/30 Benoit Jacob <jacob.benoit.1@xxxxxxxxx>:
> This is interesting. Indeed it seems that this works for all
> self-adjoint matrices. I'll try to have a  look at it.
>
> Cheers,
> Benoit
>
> 2009/3/29  <w.h.greene@xxxxxxxxx>:
>> I just finished reading a long recent thread on some issues with LDLt.
>>
>> In some numerical experiments I was doing today, I've come across a
>> related issue. It appears that the current implementation of LDLt requires
>> that the matrix be positive definite. This is not strictly necessary for
>> the LDLt factorization to succeed. It is necessary only that the diagonal
>> term not be zero during factorization. A negative diagonal term is not,
>> by itself, a problem.
>>
>> From reading the previous posts, it appears that a main reason for the
>> LDLt implementation compared with LLt was to avoid the performance
>> penalty of a square root. In fact, I think the main benefit of LDLt is that
>> it can factor both negative- and positive-definite symmetric matrices.
>> (As an aside, these routinely occur in dynamic analysis of mechanical
>> systems)
>>
>> I suggest the test in the LDLt factorization routine be changed to
>> check that abs(diag_term) > eps.
>>
>> Bill Greene
>