# ApCoCoA-1:Num.CABM

This article is about a function from ApCoCoA-1. |

## Num.CABM

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

### Syntax

Num.CABM(PointsReal:MAT, PointsComp:MAT, Epsilon:RAT):Object Num.CABM(PointsReal:MAT, PointsComp: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.

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

compatible term-ordering. The matrix `PointsReal` and `PointsComp` contain 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

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

*PointsComp*The imaginary part of the points for which a border basis is computed.@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).`Epsilon`should be in the interval (0,1). As a rule of thumb,`Epsilon`is the expected percentage of error on the input points.@return A list of two lists. First the border basis as a list of polynomials. Two polynomials always belong together containing the real and the complex part. 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*An 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 euclidian norm of the column.

#### Example

Use P ::= Q[x]; PointsReal := Mat([[0],[0]]); PointsComp := Mat([[1],[-1]]); Res := Num.CABM(PointsReal, PointsComp, 0.1); Dec(Res,3); -- CoCoAServer: computing Cpu Time = 0 ------------------------------- [["-0.707 x^2 -0.707 ", "0"], ["1", " x "]] -------------------------------

### See also