From ApCoCoAWiki
This article is about a function from ApCoCoA-1.


Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the Num.EXTABM algorithm.


Num.SubEXTABM(Points:MAT, Val:MAT, Epsilon:RAT, Basis:LIST):Object
Num.SubEXTABM(Points:MAT, Val:MAT, Epsilon:RAT, Basis:LIST, Delta:RAT, NormalizeType:INT):Object


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 sub-ideal for a set of points and an ideal using the Num.EXTABM algorithm.

The current ring has to be a ring over the rational numbers with a standard-degree

compatible term-ordering. Each row in the matrix Points represents one point, so the number of columns must equal the

number of indeterminates in the current ring.

  • @param Points The points for which a border basis is computed.

  • @param Val The time series we want to approximate using Points.

  • @param Epsilon A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for Epsilon lead to bigger errors of the polynomials evaluated at the point set).

  • @param Basis A set of polynomials in the current ring. This basis defines the ideal in which we compute the basis of the approximate vanishing ideal.

  • @return A list of two results. First the border basis as a list of polynomials, second the vector space basis of P/I as a list of terms.

The following parameters are optional:

  • @param Delta A positive rational number which describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in [-Delta, Delta] to be 0. The default value for Delta is 0.00000000001.

  • @param NormalizeType A integer of the set {1,2,3,4}. The default value is 2. This parameter describes, if and where required the input points are normalized. If NormalizeType equals 1, each coordinate of a point is divided by the maximal absolute value of all coordinates of this point. This ensures that all coordinates of the points are within [-1,1]. With NormalizeType=2 no normalization is done at all. NormalizeType=3 shifts each coordinate to [-1,1], i.e. the minimal coordinate of a point is mapped to -1 and the maximal coordinate to 1, which describes a unique affine mapping. The last option is NormalizeType=4. In this case, each point is normalized by its euclidean norm. Although NormalizeType=3 is in most cases a better choice, the default value is due to backward compatibility 1.


Use P::=QQ[x,y,z];

Points := Mat([[1,2,3],[4,5,6],[7,11,12]]);
Val := Mat([[1],[0.1],[0.2]]);
R:=Num.SubEXTABM(Points,Val, 0.1, [x]);

-- CoCoAServer: computing Cpu Time = 0
["1.000", "0.999", "1.000", "0.999", "0.999", "1.000"]
["0.099", "0.100", "0.100", "0.099", "0.099", "0.100"]
["0.200", "0.200", "0.200", "0.199", "0.199", "0.199"]

See also

Introduction to CoCoAServer