ApCoCoA-1:Num.SubEXTABM
Num.SubEXTABM
Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the Num.EXTABM algorithm.
Syntax
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
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 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.
Example
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]); Dec(-Eval(R[1],Points[1]),3); Dec(-Eval(R[1],Points[2]),3); Dec(-Eval(R[1],Points[3]),3); -- CoCoAServer: computing Cpu Time = 0 ------------------------------- [<quotes>1.000</quotes>, <quotes>0.999</quotes>, <quotes>1.000</quotes>, <quotes>0.999</quotes>, <quotes>0.999</quotes>, <quotes>1.000</quotes>] ------------------------------- [<quotes>0.099</quotes>, <quotes>0.100</quotes>, <quotes>0.100</quotes>, <quotes>0.099</quotes>, <quotes>0.099</quotes>, <quotes>0.100</quotes>] ------------------------------- [<quotes>0.200</quotes>, <quotes>0.200</quotes>, <quotes>0.200</quotes>, <quotes>0.199</quotes>, <quotes>0.199</quotes>, <quotes>0.199</quotes>] -------------------------------
See also
