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Num.CEXTABM

Computes the border basis of an almost vanishing ideal for a set of points.
Syntax
          
Num.CEXTABM(PointsReal:MAT, PointsComp:MAT, ValReal:MAT, ValComp:MAT, Epsilon:RAT):Object
Num.CEXTABM(PointsReal:MAT, PointsComp:MAT, ValReal:MAT, ValComp:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT):Object

          

Description
Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

This command computes a border basis of an almost vanishing ideal for a set of complex points. A special property of the polynomials is, that when evaluated at the original set of points, one obtains approximately Val. To obtain an approximate ideal an additional indeterminate has to be added to each equation, which represents the coordinates of Val.

The current ring has to be a ring over the rational numbers with a standard-degree compatible term-ordering. The matrix PointsReal contains the real part of the points, the matrix PointsComp the complex part: each point is a row in the matrix, so the number of columns must equal the number of indeterminates in the current ring. The following parameters are optional:

Example
Use P::=Q[x,y];
PointsReal := Mat([[0,0],[1,1]]);
PointsComplex := Mat([[1,-1],[-1,1]]);

ValReal := Mat([[1],[1]]);
ValComp := Mat([[0],[1]]);
Res := Num.CEXTABM(PointsReal, PointsComplex, ValReal, ValComp, 0.1);
Dec(Res,3);

[[["-0.399 y -0.799  ", "-0.200 y -0.399  ", "0.400 x -1.199  ", "-0.199 x -0.399  "], [["1  "]]
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See Also