# ApCoCoA-1:Num.CEXTABM

## 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.

• @param PointsReal The real part of the points for which a border basis is computed.

• @param PointsComp The complex part of the points for which a border basis is computed.

• @param ValReal The real part of the time series we want to approximate using Points.

• @param ValComplex The complex part of 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).

• @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::=Q[x,y];
PointsReal := Mat([[0,0],[1,1]]);
PointsComplex := Mat([[1,-1],[-1,1]]);

ValReal := Mat([,]);
ValComp := Mat([,]);
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  "]]
-------------------------------
```