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Note about the validity check.

When the computation for the current $ \rho $ is complete, we check the model (see step 14 of the algorithm) around $ \boldsymbol {x}_{(k)}$, then one or both of the conditions:

$\displaystyle \Vert\boldsymbol{x}_{(j)}-\boldsymbol{x}_{(k)}\Vert \leq 2
 \rho$ (6.5)

$\displaystyle \frac{1}{6}M \Vert \boldsymbol{x}_{(j)}- \boldsymbol{x}_{(k)} \Ve...
...ert P_j(\boldsymbol{x}_{(k)}+d)\vert : \Vert d \Vert \leq \rho \} \leq \epsilon$ (6.6)

must hold for every points in the dataset. When $ \rho $ is reduced by formula 6.4, the equation 6.5 is very often NOT verified. Only Equation 6.6, prevents the algorithm from sampling the model at $ N=(n+1)(n+2)/2$ new points. Numerical experiments indicate that the algorithm is highly successful in that it computes less then $ \frac{1}{2}n^2$ new points in most cases.

Frank Vanden Berghen 2004-04-19