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

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   <command>
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   {{Version|1}}
 +
<command>
 
     <title>Num.SubAVI</title>
 
     <title>Num.SubAVI</title>
     <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>
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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>ApCoCoA-1:Num.AVI|Num.AVI</ref> algorithm.</short_description>
 
<syntax>
 
<syntax>
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST):Object
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Num.SubAVI(Points:MAT, Epsilon:RAT, Basis:LIST):Object
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST, Delta:RAT, NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType:INT):Object
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Num.SubAVI(Points:MAT, Epsilon:RAT, Basis:LIST, Delta:RAT, NormalizeType:INT):Object
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
 
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
<par/>
 
<par/>
This command computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.ABM</ref> algorithm.
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This command computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>ApCoCoA-1:Num.AVI|Num.AVI</ref> algorithm.
 
<par/>
 
<par/>
 
The current ring has to be a ring over the rational numbers with a standard-degree
 
The current ring has to be a ring over the rational numbers with a standard-degree
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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>Tau</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>
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<item>@param <em>Epsilon</em> A positive rational number describing which singular values should be treated as 0 (bigger values for 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, Tau is the expected percentage of error on the input points. </item>
  
<item>@param <em>GBasis</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.  <em>Warning:</em> For reasons of efficiency the function does not check the validity of <tt>GBasis</tt>.</item>
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<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>
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<item>@param <em>Delta</em> 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 <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item>
 
<item>@param <em>Delta</em> 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 <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item>
  
<item>@param <em>NormalizeType</em> A integer of the set <tt>{1,2,3,4}</tt>. The default value is 2. This parameter describes, if and where required the input points are normalized. If <tt>NormalizeType</tt> 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 <tt>[-1,1]</tt>. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to <tt>[-1,1]</tt>, 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 <tt>NormalizeType=4</tt>. In this case, each point is normalized by its euclidean norm. Although <tt>NormalizeType=3</tt> is in most cases a better choice, the default value is due to backward compatibility 1.</item>
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<item>@param <em>NormalizeType</em> A integer of the set <tt>{1,2,3,4}</tt>. The default value is 2. This parameter describes, if and where required the input points are normalized. If <tt>NormalizeType</tt> 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 <tt>[-1,1]</tt>. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to <tt>[-1,1]</tt>, 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 <tt>NormalizeType=4</tt>. In this case, each point is normalized by its euclidean norm.</item>
  
<item>@param <em>RREFNormalizeType</em> Describes, how in each RREF step the columns are normalized. The options correspond to the ones for <tt>NormalizeType</tt> and the default is 1 again.</item>
 
  
<item>@param <em>RREFUseEps</em> A boolean value. If <tt>RREFUseEps=TRUE</tt>, the given <tt>Delta</tt> is used within the RREF to decide if a value equals 0 or not. If <tt>RREFUseEps=FALSE</tt>, a replacement value for <tt>Delta</tt> is used, which is based on the norm of the matrix.</item>
 
 
<item>@param <em>RREFType</em> A integer of the set <tt>{1,2}</tt>. If <tt>RREFType=1</tt>, the RREF operates column-wise, otherwise it works row-wise. The default value is 1.</item>
 
 
</itemize>
 
</itemize>
  
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-- CoCoAServer: computing Cpu Time = 0
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
-------------------------------
[<quotes>0.83 x^2 -0.55 x </quotes>, <quotes>1 xy </quotes>, <quotes>1 xz </quotes>]
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["1 x^2 -0.66 x ", "1 xy ", "1 xz "]
 
-------------------------------
 
-------------------------------
 
[x]
 
[x]
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     </description>
 
     </description>
 
     <seealso>
 
     <seealso>
       <see>Introduction to CoCoAServer</see>
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       <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
       <see>Num.AVI</see>
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       <see>ApCoCoA-1:Num.AVI|Num.AVI</see>
 
     </seealso>
 
     </seealso>
 
     <types>
 
     <types>
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     <key>num.SubAVI</key>
 
     <key>num.SubAVI</key>
 
     <key>numerical.subavi</key>
 
     <key>numerical.subavi</key>
     <wiki-category>Package_numerical</wiki-category>
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     <wiki-category>ApCoCoA-1:Package_numerical</wiki-category>
 
   </command>
 
   </command>

Latest revision as of 13:49, 29 October 2020

This article is about a function from ApCoCoA-1.

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):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 (bigger values for 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, 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.


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
-------------------------------
["1 x^2 -0.66 x ", "1 xy ", "1 xz "]
-------------------------------
[x]
-------------------------------

See also

Introduction to CoCoAServer

Num.AVI