Re: [eigen] Checking if a dense linear system is solvable?

[Christoph Hertzberg <chtz@xxxxxxxxxxxxxxxxxxxxxxxx>]

On 05.10.2012 14:36, Helmut Jarausch wrote:
> Hi,
>
> unfortunately, the dense linear solvers don't have an .info() method.
> Invoked with a singular matrix I just get "+-inf".
> Checking the determinant isn't an option since, e.g.,
> 0.1*IdentyMatrix(100,100) has a determinant of  1E-100 although it has
> only a condition number of 100.

It should have a condition of 1, if you define cond(A) = 
norm(A)*norm(A^-1), shouldn't it?

> I suggest one either adds an info-method or one returns a vector
> (matrix) with zero rows.
>
> What do think about that.
>
> Probably, the solve method should have an additional parameter with a
> sensible default,
> giving a (relative) tolerance for deciding if a pivot element has to be
> considered as zero.


FullPivLU, ColPivHouseholderQR and FullPivHouseholderQR have a 
setThreshold() and a rank() method, which might be what you need.
http://eigen.tuxfamily.org/dox-devel/classEigen_1_1FullPivLU.html#ae47399e3cf7943075ce18ef89fe17f21

Another thing you can always do is
> SomeDecomposition D(A);
> VectorXd x = D.solve(b);
> double error = (b-A*x).norm()/b.norm(); // or squaredNorm


hth,
Christoph



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Dipl.-Inf. Christoph Hertzberg
Cartesium 0.049
Universität Bremen
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