# Difference between revisions of "ApCoCoA-1:Num.SubAVI"

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<short_description>Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.AVI</ref> algorithm.</short_description> | <short_description>Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.AVI</ref> algorithm.</short_description> | ||

<syntax> | <syntax> | ||

− | Num.SubAVI(Points:MAT, | + | Num.SubAVI(Points:MAT, Epsilon:RAT, Basis:LIST):Object |

− | Num.SubAVI(Points:MAT, | + | Num.SubAVI(Points:MAT, Epsilon:RAT, Basis:LIST, Delta:RAT, NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType:INT):Object |

</syntax> | </syntax> | ||

<description> | <description> | ||

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<item>@param <em>Points</em> The points for which a border basis is computed.</item> | <item>@param <em>Points</em> The points for which a border basis is computed.</item> | ||

− | <item>@param <em> | + | <item>@param <em>Epsilon</em> 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. </item> |

− | <item>@param <em> | + | <item>@param <em>Basis</em> A homogeneous Groebner Basis in the current ring. This basis defines the ideal in which we compute the basis of the approximate vanishing ideal.</item> |

<item>@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.</item> | <item>@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.</item> |

## Revision as of 15:11, 7 October 2009

## 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, Epsilon:RAT, Basis:LIST):Object Num.SubAVI(Points:MAT, Epsilon:RAT, Basis: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.AVI 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

*Epsilon*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

*Basis*A homogeneous Groebner Basis 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.@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[1],2); R[2]; -- CoCoAServer: computing Cpu Time = 0 ------------------------------- [<quotes>1 x^2 -0.66 x </quotes>, <quotes>1 xy </quotes>, <quotes>1 xz </quotes>] ------------------------------- [x] -------------------------------

### See also