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

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On 07/18/10 00:32:26, Benoit Jacob wrote:
> 2010/7/17 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,
> If (lhs * solution - rhs) is small compared to rhs, then what more
> could you ask for?

A lot more!
A linear equation  A x = b is a mathematical problem.
If it has a solution the users wants an approximation to that.
If I find a vector x which satisfies  norm(A*x-b) = tiny in finite 
arithmetic, that doesn't say much about  norm(x_computed - x_true)
unless I have a bound on the condition number. And in some cases the
mathematical matrix A is singular while the numerical matrix is not 
(just has a huge condition number).
So, the users needs an estimate of norm(x_computed - x_true) .


Helmut Jarausch

Lehrstuhl fuer Numerische Mathematik
RWTH - Aachen University
D 52056 Aachen, Germany

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