Re: [eigen] on fuzzy comparisons

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On Thursday 05 June 2008 19:49:54 Gael Guennebaud wrote:
> hm...  after more thoughts maybe I could add the following:
> in the current implementation you assume a vector is a given entity, and a
> matrix is a collection of such entities. Assuming we support nested matrix
> with Array1D, Array2D, Vector and Matrix, the right type for this
> interpretation should be
> Array1D<Vector<scalar> >

Don't over interprete the current implementation of isMuchSmallerThan / 
isApprox for matrices !

It's just a compromise, between speed, simplicity, and retaining some 
geometric meaning. It gave me acceptable numeric stability in practice, and i 
remember the L2norm had more trouble. The reason is that it's much closer to 
the matrixNorm than the L2norm is. By closer I mean that the L/K ratio is 
closer to 1.

So, that a matrix was seen as a collection of vectors, was just an 
implementation detail. No ground for generalization!

> I don't know if that really helps, but this is an attempt to show that
> there is no single solution... IMHO.

Let me summarize:

The only single solution is matrixNorm and it might be too expensive and/or 
not an option to put it in QR.

All the rest is just an approximation to matrixNorm.

How good an approximation is, is measured by the L/K ratio. The closer to 1 
that ratio is, the better the approximation.

Unfortunately: ANY approximation will have the problem that L/K goes to 
infinity when n goes to infinity. Just some approximations are better in that 
L/K goes to infinity slower.


> Gael.

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