|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. Cheers, Benoit > > Gael. >
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