Re: [eigen] Problem inverting a Matrix4f with Eigen 2.0.0

[Jitse Niesen <jitse@xxxxxxxxxxxxxxxxx>]

On Sat, 17 Jul 2010, Benoit Jacob wrote:

> Another reason why we decided not to expose condition numbers in Eigen
> is that for the main purpose they're used for, namely checking if a
> result is reliable, there is a better approach which is: check the
> result itself. For example, if you want to check how accurate your
> matrix inverse is, just compute matrix*inverse and see how close it is
> to the identity matrix. Nothing beats that! When it comes to more
> general solving with potentially non full rank matrices, this is even
> better, because the condition number of the lhs matrix alone doesn't
> tell all you need to know (it also depends on your particular rhs), so
> the approach we're recommending in Eigen, to compute lhs*solution and
> compare with rhs, is the only way to know for sure how good your
> solution is.

I'm not sure I agree. The residual (lhs * solution - rhs) is certainly 
useful, but so is the condition number (norm of A multiplied by norm of 
A^{-1}, where I deliberately don't specify the norm used). If the residual 
is zero but the condition number is huge then you probably should not 
trust the solution, because even though Eigen may have found the exact 
solution to the problem you gave it, it is likely that the lhs and rhs 
that you gave to Eigen are not quite exact, and these errors are magnified 
hugely (in fact, if the condition number is huge then you probably cannot 
even reliably compute the residual).

This implies (if correct) that it is useful to have Eigen compute 
estimates for the condition number. The operative word here is 'estimate': 
I think the order of magnitude is what is usually important, not the exact 
value.

Cheers,
Jitse