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Stable Border Basis of ideal of points
StableBBasis5(Pts: LIST, Toler: LIST): RECORD
StableBBasis5(Pts: LIST, Toler: LIST, Gamma: RAT): RECORD 
***** NOT YET IMPLEMENTED *****
See
ApproxPointsNBM
This function returns a record containing a
stable order ideal of
the ideal of points, and a list of
almost vanishing polynomials.
If the cardinality of the order ideal is equal to the number of points,
it is in fact a
quotient basis, and in this case a
stable border basis founded on it is also returned. The boolean field
StableBBasisFound
is set to
true
if a stable border basis was found,
otherwise
false
.
The first argument is a list of points in kdimensional space, and the
second argument is list of k positive tolerances (one for each dimension).
The function builds the stable order ideal stepwise by testing, from a
numerical point of view, the linear dependence of a set of vectors.
So that the answer can be represented, the current ring must have at
least k indeterminates; the term ordering is ignored as it plays no role
in determining the border basis.
There is a third, optional argument: it is a real non negative number
Gamma
which is used for scaling the threshold on the
admissible perturbation of the points.
A value of
Gamma
<< 1 should be used.
If no value is specified then by default
Gamma
= 0.1
For a full description of the algorithms we refer to the paper
J.Abbott, C.Fassino, L.Torrente
Stable Border Bases for Ideals of Points (to appear in JSC
or arXiv:07062316).
Pts := [[0.1,1],[1,1],[2,3]];
Toler := [0.1,0.1];
StableBBasis5(Pts, Toler);
record[
AlmostVanishing := [ (...) ],
BBasis := [
3602879701896397/288230376151711744y^2 + x 
32425917317067571/72057594037927936y 
154923827181545063/288230376151711744,
xy  140512308373959475/288230376151711744y^2 
39631676720860365/72057594037927936y +
10808639105689191/288230376151711744,
y^3  3y^2  y + 3,
xy^2  580063632005319885/288230376151711744y^2 
32425917317067571/72057594037927936y +
421536925121878425/288230376151711744],
SOI := [1, y, y^2],
StableBBasisFound := True]

Toler := [0.6, 0.6]:
StableBBasis5(Pts, Toler);
record[AlmostVanishing := [.....], SOI := [1, y], StableBBasisFound := False]

