Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices

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

2010/2/24 Jitse Niesen <jitse@xxxxxxxxxxxxxxxxx>:
> On Wed, 24 Feb 2010, Gael Guennebaud wrote:
>
>> note that the LDLT decomp does pivoting, so it is normal that you don't
>> get the same diagonal matrix. If you reconstruct the matrix from the decomp
>> you will see that in this case the decomp is correct.
>
> To further clarify, because this also threw me off for a while: I think the
> important thing here is that the matrix A is singular.

Great observation :)

For a detailed analysis of the stability of LDLt with pivoting for
singular matrices, see
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.3360

Formulas are given on page 12, suggesting that it's not very stable, AFAIU.

In particular, LDLt, even with pivoting, is not a rank-revealing
decomposition, which means that the data "how many zeros in vectorD()"
should be considered meaningless.

Benoit

> The diagonal D is
> unique (up to reordering) for positive-definite matrices. However, there is
> no such uniqueness for singular matrices. Simple example
>
> A = [ 0  0;  0  1 ]      (these are 2-by-2 matrices in Matlab notation)
> L = [ 0  0;  sqrt(3)  1 ]
> D = [ 1/4  0;  0  1/4 ]
>
> The matrix A has two LDL^T decompositions, A = A * I * A^T = L * D * L^T,
> but the diagonal parts are not the same.
>
> Cheers,
> Jitse
>
>
>