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

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2009/1/28 Jitse Niesen <jitse@xxxxxxxxxxxxxxxxx>:
> On Wed, 28 Jan 2009, Benoit Jacob wrote:
>>> When solving a linear system with an invertible matrix, partial pivoting
>>> performs as well as complete pivoting, doesn't it? LAPACK uses partial
>>> pivoting, as does GSL.
>> No, partial pivoting fails even for invertible matrices of size
>> 200x200, and fails consistently at sizes like 1000x1000.
> How did you arrive at this conclusion?

When I implemented LU, I was initially hoping that partial pivoting
would be enough. What I say here is the results of my experiments. So
maybe I got something wrong, but aside from that possibility, I was
talking first-hand experience when saying that.

>I'm by no means an expert in
> numerical linear algebra, but the books I have (including Golub & Van Loan,
> Matrix Computations, 3e and Higham, Accuracy and Stability ..., 2e) say that
> partial pivoting should work fine.

At least Golub & van Loan do mention the numerical stability problem
of partial pivoting...
(plus I don't consider this book to be a solid reference on numerical
stability, e.g. with Keir we found a numerical stability issue in
their householder vector computation).

>> 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:

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.

> 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.

Here we're talking about a cubic complexity algorithm. So while a 50%
speedup sounds big, it just means that in the same time you handle a
matrix with 14% more rows and columns ( 1.5^(1/3) = 1.14). I want
Eigen to be easy to use and that implies prefering safeness over

Now let me retry my experiment with partial pivoting to see if it's
really unstable...


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