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

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compatible term-ordering.  The matrix Points contains the points: each
 
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
 
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 >0
+
number of indeterminates in the current ring.  Epsilon is a rational <formula>&gt;0</formula>
 
describing which singular values should be treated as 0 (smaller values for
 
describing which singular values should be treated as 0 (smaller values for
 
Epsilon lead to bigger errors of the polynomials evaluated at the point
 
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,  
+
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  
 
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  
 
be either True or False.  If it is true, the command returns a list  
 
of two values: the first contains the border basis, the
 
of two values: the first contains the border basis, the
second one a vector space basis of P/I comprising those power products
+
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
 
lying outside the leading term ideal of I.  If GetO is false, the function
 
returns only the border basis (not in a list).
 
returns only the border basis (not in a list).

Revision as of 01:08, 4 November 2007

Numerical.BBasisOfPoints

Border basis of almost vanishing ideal for a set of points

Syntax

$numerical.BBasisOfPoints(Points, Epsilon, GetO):Object

Description

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

 D. Heldt, M. Kreuzer, H. Poulisse, S.Pokutta: Approximate Computation
 of Zero-Dimensional Ideals Submitted: August 2006

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).

Example

Points := Mat([[1,0,0],[0,0,1],[0,2,0]]);
$numerical.BBasisOfPoints(Points,0.001,True);
-------------------------------
[[x + 9007199254740991/18014398509481984y + z - 1, z^2 - 9007199254740991/9007199254740992z, 1/2yz, xz, 1/4y^2 - 9007199254740991/18014398509481984y, 1/2xy], [y, z, 1]]

See also

Introduction to CoCoAServer

Numerical.GBasisOfPoints

Numerical.HBasisOfPoints

Numerical.GBasisOfPointsInIdeal

Numerical.BBasisOfPointsInIdeal

Numerical.HBasisOfPointsInIdeal

Numerical.FirstVanishingRelations

Numerical.FirstVanishingRelationsInIdeal