Difference between revisions of "ApCoCoA:Num.SubAVI"

From CoCoAWiki
(typo)
(adding new parameters.)
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     <short_description>border basis of an almost vanishing sub-ideal for a set of points and ideal</short_description>
 
     <short_description>border basis of an almost vanishing sub-ideal for a set of points and ideal</short_description>
 
<syntax>
 
<syntax>
$numerical.BBasisOfPointsInIdeal(Points, Epsilon, GetO, GBasis):Object
+
$numerical.BBasisOfPointsInIdeal(Points, Tau, GetO, GBasis, Delta,  NormalizeType, RREFNormalizeType, RREFUseEps, RREFType):Object
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
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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 <formula>&gt;0</formula>,
+
number of indeterminates in the current ring.  
 +
 
 +
Tau 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
+
tau 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,
+
set).  Tau 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.  
+
tau is the expected percentage of error on the input points.  
 +
 
 +
 
 
GetO must be either True or False.  If it is true, the command
 
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
 
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
 
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). GBasis must be a
+
returns only the border basis (not in a list).  
 +
 
 +
GBasis must be a
 
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
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GBasis.
 
GBasis.
  
 +
 +
The last 5 parameters are optional.
 +
 +
 +
Delta must be a positiv rational. 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.
 +
 +
 +
NormalizeType must be one of 1,2,3 ,4. The default value is 1. This parameter describes, if / how the input points are normalized.
 +
If NormalizeType equals 1, each coordinate is divided by the maximal absolut value of the matrix's corresponding column. This ensures that all point's coordinates 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 column's euclidian norm.
 +
Due to backward compatibility, the default is 1, although 3 is in most cases a better choice.
 +
 +
 +
RREFNormalizeType describes, how in each RREF steps the columns are normalized. The options correspond to the ones for NormalizeType and the default is 1 again.
 +
 +
 +
RREFUseEps must be either true or false! If RREFUseEps is true, the given Delta is used within the RREF to decide if a value equals 0 or not. If this parameter is false,
 +
a replacement value for Delta is used, which is based on the matrix's norm.
 +
 +
 +
The last parameter is RREFType. This must be 1 or 2. If RREFType=1, the rref operates column-wise. Otherwise it works row-wise. The default is 1.
 
<example>
 
<example>
 
Points := Mat([[2,0,0],[0,3,0],[0,0,1]]);
 
Points := Mat([[2,0,0],[0,3,0],[0,0,1]]);

Revision as of 11:02, 27 May 2008

Numerical.BBasisOfPointsInIdeal

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

Syntax

$numerical.BBasisOfPointsInIdeal(Points, Tau, GetO, GBasis, Delta,  NormalizeType, RREFNormalizeType, RREFUseEps, RREFType):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.

Tau is a rational <formula>>0</formula>,

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 <formula>(0,1)</formula>. As a rule of thumb, tau 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.


The last 5 parameters are optional.


Delta must be a positiv rational. 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.


NormalizeType must be one of 1,2,3 ,4. The default value is 1. This parameter describes, if / how the input points are normalized.

If NormalizeType equals 1, each coordinate is divided by the maximal absolut value of the matrix's corresponding column. This ensures that all point's coordinates 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 column's euclidian norm. Due to backward compatibility, the default is 1, although 3 is in most cases a better choice.


RREFNormalizeType describes, how in each RREF steps the columns are normalized. The options correspond to the ones for NormalizeType and the default is 1 again.


RREFUseEps must be either true or false! If RREFUseEps is true, the given Delta is used within the RREF to decide if a value equals 0 or not. If this parameter is false,

a replacement value for Delta is used, which is based on the matrix's norm.


The last parameter is RREFType. This must be 1 or 2. If RREFType=1, the rref operates column-wise. Otherwise it works row-wise. The default is 1.

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

ApCoCoA:Introduction to CoCoAServer

ApCoCoA:Numerical.GBasisOfPoints

ApCoCoA:Numerical.BBasisOfPoints

ApCoCoA:Numerical.HBasisOfPoints

ApCoCoA:Numerical.GBasisOfPointsInIdeal

ApCoCoA:Numerical.HBasisOfPointsInIdeal

ApCoCoA:Numerical.FirstVanishingRelations

ApCoCoA:Numerical.FirstVanishingRelationsInIdeal