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

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homogeneous Groebner Basis in the current ring.  This basis defines the
 
homogeneous Groebner Basis in the current ring.  This basis defines the
 
ideal we compute the approximate vanishing ideal's basis in.  Warning: for
 
ideal we compute the approximate vanishing ideal's basis in.  Warning: for
reasons of efficiency the function does not check that the validity of
+
reasons of efficiency the function does not check the validity of
 
GBasis.
 
GBasis.
  

Revision as of 19:50, 20 May 2008

Numerical.BBasisOfPointsInIdeal

border basis of an almost vanishing sub-ideal for a set of points and ideal

Syntax

$numerical.BBasisOfPointsInIdeal(Points, Epsilon, GetO, GBasis):Object

Description

This command computes a border basis of an almost vanishing sub-ideal for a set of points and ideal using the algorithm described in the paper

 D. Heldt, M. Kreuzer, H. Poulisse: Computing Approximate 
 Vanishing Ideals (Work in progress)

The current ring has to be a ring over the rationals 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. Epsilon is a rational <formula>>0</formula>, describing which singular values should be treated as 0 (smaller values for epsilon lead to bigger errors of the polynomials evaluated at the point set). Epsilon should be in the interval <formula>(0,1)</formula>. As a rule of thumb, Epsilon is the expected percentage of error on the input points. GetO must be either True or False. If it is true, the command returns a list of two values: the first contains the border basis, the second one a vector space basis of <formula>P/I</formula> comprising those power products lying outside the leading term ideal of I. If GetO is false, the function returns only the border basis (not in a list). GBasis must be a homogeneous Groebner Basis in the current ring. This basis defines the ideal we compute the approximate vanishing ideal's basis in. Warning: for reasons of efficiency the function does not check the validity of GBasis.

Example

Points := Mat([[2,0,0],[0,3,0],[0,0,1]]);
$numerical.BBasisOfPointsInIdeal(Points, 0.001, False,[z,y]);
-------------------------------
[z^2 - z, 1/3yz, 1/2xz, 1/9y^2 - 9007199254740991/27021597764222976y, 1/6xy]
-------------------------------

See also

Introduction to CoCoAServer

Numerical.GBasisOfPoints

Numerical.BBasisOfPoints

Numerical.HBasisOfPoints

Numerical.GBasisOfPointsInIdeal

Numerical.HBasisOfPointsInIdeal

Numerical.FirstVanishingRelations

Numerical.FirstVanishingRelationsInIdeal