# ApCoCoA-1:Num.EXTABM

## Num.EXTABM

Computes the border basis of an almost vanishing ideal for a set of points.

### Syntax

Num.EXTABM(Points:MAT, Val:MAT, Tau:RAT):Object Num.EXTABM(Points:MAT, Val:MAT, Tau: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 points. A special property of polynomials is, that when evaluated at the original set of points, one obtains approximately `Val`. To obtain a "real" approximate ideal an additional indeterminate has to be added, 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 `Points` contains the points: each

point is a row in the matrix, 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

*Tau*A positive rational number describing which singular values should be treated as 0 (smaller values for`Tau`lead to bigger errors of the polynomials evaluated at the point set).`Tau`should be in the interval (0,1). As a rule of thumb,`Tau`is the expected percentage of error on the input points.@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 positiv rational number.`Delta`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 range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If`NormalizeType`equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With`NormalizeType=2`no normalization is done at all.`NormalizeType=3`shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is`NormalizeType=4`. In this case, each coordinate is normalized, using the column's euclidian norm. Due to backward compatibility, the default is 1, although 3 is in most cases a better choice.

#### 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.EXTABM(Points,Val, 0.1); 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>0.999</quotes> <quotes>0.999</quotes> <quotes>0.999</quotes> <quotes>1.000</quotes>] ------------------------------- [<quotes>0.099</quotes> <quotes>0.099</quotes> <quotes>0.099</quotes> <quotes>0.099</quotes> <quotes>0.099</quotes> <quotes>0.099</quotes>] ------------------------------- [<quotes>0.199</quotes> <quotes>0.200</quotes> <quotes>0.199</quotes> <quotes>0.200</quotes> <quotes>0.199</quotes> <quotes>0.199</quotes>] -------------------------------

### See also