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

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(New page: <command> <title>Num.RatPoints</title> <short_description>Checks if a given set of polynomials vanishes at a given set of points</short_description> <syntax> Num.RatPoints(AppBB:...)
 
m (replaced <quotes> tag by real quotes)
 
(15 intermediate revisions by 5 users not shown)
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   <command>
+
   {{Version|1}}
 +
<command>
 
     <title>Num.RatPoints</title>
 
     <title>Num.RatPoints</title>
     <short_description>Checks if a given set of polynomials vanishes at a given set of points</short_description>
+
     <short_description>Computes the zero set of an exact zero dimensional border basis. The zeros are computed approximately using the eigenvalues of the transposed multiplication matrices.</short_description>
 
<syntax>
 
<syntax>
Num.RatPoints(AppBB:PolyList, OrderIdeal:PolyList)):[Matrix];
+
Num.RatPoints(BB:LIST, OrderIdeal:LIST)):LIST of MAT
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
{{ApCoCoAServer}} Please also note that you will have to use an ApCoCoAServer with enabled BLAS/LAPACK support.
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<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/>
Calculates a set of points which are the zeros of an exact border basis which is close to AppBB. Uses the eigenvalue methode.
+
This command computes a set of points, which are the zeros of an exact border basis. This border basis is close to the approximate border basis <tt>AppBB</tt>. The set of (complex) points is represented as two matrices. The <tt>j</tt>-th column of the first matrix gives the real part of a point and the <tt>j</tt>-th column of the second matrix gives the imaginary part. For computation the function is using the <ref>ApCoCoA-1:Num.EigenValues|Num.EigenValues</ref> command.
  
 
<itemize>
 
<itemize>
 
<item>@param <em>AppBB</em> An approximate border basis.</item>
 
<item>@param <em>AppBB</em> An approximate border basis.</item>
<item>@param <em>Points</em> The asscoiated order ideal</item>
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<item>@param <em>OrderIdeal</em> The associated order ideal</item>
<item>@return A set of points in matrix form.</item>
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<item>@return A set of points in matrix form described above.</item>
 
</itemize>
 
</itemize>
  
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Use P::=QQ[x,y,z];
 
Use P::=QQ[x,y,z];
  
Points := Mat([[2/3,0,0],[0,1,0],[0,0,1/3]]);
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Points := Mat([[2/3,0,0],[0,10,0],[0,0,1/3]]);
R:=Num.SubAVI(Points, 0.001, [1]);
+
R:=Num.ABM(Points, 0);
 
Dec(Num.RatPoints(R[1],R[2]),2);
 
Dec(Num.RatPoints(R[1],R[2]),2);
  
 
-- CoCoAServer: computing Cpu Time = 0
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
-------------------------------
-- CoCoAServer: computing Cpu Time = 0
+
-- CoCoAServer: computing Cpu Time = 0.016
 
-------------------------------
 
-------------------------------
 
[Mat([
 
[Mat([
   ["0.66", "0.00", "-0.00"],
+
   ["0.66", "0.00", "0"],
   ["0", "0", "1.00"],
+
   ["0", "0", "10"],
 
   ["0", "0.33", "0"]
 
   ["0", "0.33", "0"]
 
]), Mat([
 
]), Mat([
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     </description>
 
     </description>
 
     <seealso>
 
     <seealso>
       <see>Introduction to CoCoAServer</see>
+
       <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
     </seealso>
 
     </seealso>
 
     <types>
 
     <types>
 
       <type>apcocoaserver</type>
 
       <type>apcocoaserver</type>
 +
      <type>polynomial</type>
 +
      <type>points</type>
 
     </types>
 
     </types>
     <key>numerical.IsAvi</key>
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     <key>Num.RatPoints</key>
     <key>Num.IsAvi</key>
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     <key>RatPoints</key>
     <key>IsAvi</key>
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     <key>numerical.RatPoints</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 13:48, 29 October 2020

This article is about a function from ApCoCoA-1.

Num.RatPoints

Computes the zero set of an exact zero dimensional border basis. The zeros are computed approximately using the eigenvalues of the transposed multiplication matrices.

Syntax

Num.RatPoints(BB:LIST, OrderIdeal:LIST)):LIST of MAT

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 set of points, which are the zeros of an exact border basis. This border basis is close to the approximate border basis AppBB. The set of (complex) points is represented as two matrices. The j-th column of the first matrix gives the real part of a point and the j-th column of the second matrix gives the imaginary part. For computation the function is using the Num.EigenValues command.

  • @param AppBB An approximate border basis.

  • @param OrderIdeal The associated order ideal

  • @return A set of points in matrix form described above.

Example

Use P::=QQ[x,y,z];

Points := Mat([[2/3,0,0],[0,10,0],[0,0,1/3]]);
R:=Num.ABM(Points, 0);
Dec(Num.RatPoints(R[1],R[2]),2);

-- CoCoAServer: computing Cpu Time = 0
-------------------------------
-- CoCoAServer: computing Cpu Time = 0.016
-------------------------------
[Mat([
  ["0.66", "0.00", "0"],
  ["0", "0", "10"],
  ["0", "0.33", "0"]
]), Mat([
  ["0", "0", "0"],
  ["0", "0", "0"],
  ["0", "0", "0"]
])]
-------------------------------

See also

Introduction to CoCoAServer