# ApCoCoA-1:Num.SubAVI

## Num.SubAVI

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

### Syntax

```Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST):Object
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST, Delta:RAT, NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType: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.ABM 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 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.

• @param GBasis A homogeneous Groebner Basis in the current ring. This basis defines the ideal in which we compute the basis of the approximate vanishing ideal. Warning: For reasons of efficiency the function does not check the validity of GBasis.

• @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 1. 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.

• @param RREFNormalizeType Describes, how in each RREF step the columns are normalized. The options correspond to the ones for NormalizeType and the default is 1 again.

• @param RREFUseEps A boolean value. If RREFUseEps=TRUE, the given Delta is used within the RREF to decide if a value equals 0 or not. If RREFUseEps=FALSE, a replacement value for Delta is used, which is based on the norm of the matrix.

• @param RREFType A integer of the set {1,2}. If RREFType=1, the RREF operates column-wise, otherwise it works row-wise. The default value is 1.

#### Example

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

Points := Mat([[2/3,0,0],[0,1,0],[0,0,1/3]]);
R:=Num.SubAVI(Points, 0.1, [x]);
Dec(R,2);
R;

-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[<quotes>0.83 x^2 -0.55 x </quotes>, <quotes>1 xy </quotes>, <quotes>1 xz </quotes>]
-------------------------------
[x]
-------------------------------

```