Difference between revisions of "ApCoCoA-1:Weyl.WRedGB"

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(New page: <command> <title>Weyl.WRedGB</title> <short_description>Computes reduced Groebner basis of a D-ideal in Weyl algebra <tt>A_n</tt>.</short_description> <syntax> Weyl.WRedGB(GB:LIS...)
 
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   <command>
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   {{Version|1}}
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<command>
 
     <title>Weyl.WRedGB</title>
 
     <title>Weyl.WRedGB</title>
 
     <short_description>Computes reduced Groebner basis of a D-ideal in Weyl algebra <tt>A_n</tt>.</short_description>
 
     <short_description>Computes reduced Groebner basis of a D-ideal in Weyl algebra <tt>A_n</tt>.</short_description>
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</syntax>
 
</syntax>
 
     <description>
 
     <description>
 +
This function converts Groebner basis <tt>GB</tt> computed by ApCoCoAServer into the reduced Groebner Basis. If <tt>GB</tt> is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list <tt>GB</tt> of Weyl polynomials using <ref>ApCoCoA-1:Weyl.WNR|Weyl.WNR</ref> into a new list <tt>L</tt> such that <tt>Ideal(L) = Ideal(GB)</tt>.
 
<par/>
 
<par/>
This function converts Groebner basis <tt>GB</tt> computed by ApCoCoAServer into the reduced Groebner Basis. If <tt>GB</tt> is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list <tt>GB</tt> of Weyl polynomials using <ref>Weyl.WNR</ref> into a new list L such that Ideal(L) = Ideal(GB).
+
<em>Note:</em> This function is faster than <ref>ApCoCoA-1:Weyl.WRGB|Weyl.WRGB</ref> for a list <tt>GB</tt> of large size.
This function is replaced by the function <ref>Weyl.WRGB</ref> inside the function <ref>Weyl.WGB</ref> to get a list of minimal Groebner basis elements for the ideal I.
 
Note: This function is faster than <ref>Weyl.WRGB</ref> for a list GB of large size.
 
 
<itemize>
 
<itemize>
 
<item>@param <em>GB</em> Groebner Basis of an ideal in the Weyl algebra.</item>
 
<item>@param <em>GB</em> Groebner Basis of an ideal in the Weyl algebra.</item>
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Weyl.WRedGB(L);
 
Weyl.WRedGB(L);
 
[1]
 
[1]
 +
-------------------------------
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-- Done.
 +
-------------------------------
 +
</example>
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<example>
 +
A2::=ZZ/(7)[x[1..2],y[1..2]]; -- define appropriate ring
 +
Use A2;
 +
I:=Ideal(2x[1]^14y[1]^7,x[1]^2y[1]^3+x[1]^2-1,y[2]^7-1,x[2]^3y[2]^2-x[2]y[2]-3x[2]-1);
 +
GbI:=Weyl.WGB(I,0);Len(GbI);
 +
 +
-------------------------------
 +
-- CoCoAServer: computing Cpu Time = 0.485
 +
-------------------------------
 +
42  -- size of complete GB of the ideal I
 +
-------------------------------
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Time GbI:=Weyl.WRedGB(GbI);
 +
 +
Cpu time = 10.89, User time = 11
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-------------------------------
 +
11  -- GbI is now reduced Groebner Basis of the ideal I.
 +
-------------------------------
 +
-- Done.
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
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   </description>
 
   </description>
 
     <seealso>
 
     <seealso>
       <see>Weyl.WNormalForm</see>
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       <see>ApCoCoA-1:Weyl.WNormalForm|Weyl.WNormalForm</see>
       <see>Weyl.WGB</see>
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       <see>ApCoCoA-1:Weyl.WGB|Weyl.WGB</see>
       <see>Weyl.WRGB</see>
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       <see>ApCoCoA-1:Weyl.WRGB|Weyl.WRGB</see>
       <see>Introduction to Groebner Basis in CoCoA</see>
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      <see>ApCoCoA-1:Weyl.WRGBS|Weyl.WRGBS</see>
       <see>Introduction to CoCoAServer</see>
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       <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see>
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       <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
     </seealso>
 
     </seealso>
 
     <types>
 
     <types>
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       <type>groebner</type>
 
       <type>groebner</type>
 
     </types>
 
     </types>
     <key>weyl.wrgb</key>
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     <key>weyl.wredgb</key>
     <key>wrgb</key>
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     <key>wredgb</key>
     <wiki-category>Package_weyl</wiki-category>
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     <wiki-category>ApCoCoA-1:Package_weyl</wiki-category>
 
   </command>
 
   </command>

Latest revision as of 10:40, 7 October 2020

This article is about a function from ApCoCoA-1.

Weyl.WRedGB

Computes reduced Groebner basis of a D-ideal in Weyl algebra A_n.

Syntax

Weyl.WRedGB(GB:LIST):LIST

Description

This function converts Groebner basis GB computed by ApCoCoAServer into the reduced Groebner Basis. If GB is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list GB of Weyl polynomials using Weyl.WNR into a new list L such that Ideal(L) = Ideal(GB).

Note: This function is faster than Weyl.WRGB for a list GB of large size.

  • @param GB Groebner Basis of an ideal in the Weyl algebra.

  • @result The reduced Groebner Basis of the given ideal.

Example

A1::=QQ[x,d];	--Define appropriate ring
Use A1;
L:=[x,d,1];
Weyl.WRedGB(L);
[1]
-------------------------------
-- Done.
-------------------------------

Example

A2::=ZZ/(7)[x[1..2],y[1..2]]; -- define appropriate ring
Use A2;
I:=Ideal(2x[1]^14y[1]^7,x[1]^2y[1]^3+x[1]^2-1,y[2]^7-1,x[2]^3y[2]^2-x[2]y[2]-3x[2]-1);
GbI:=Weyl.WGB(I,0);Len(GbI);

-------------------------------
-- CoCoAServer: computing Cpu Time = 0.485
-------------------------------
42   -- size of complete GB of the ideal I
-------------------------------
Time GbI:=Weyl.WRedGB(GbI);

Cpu time = 10.89, User time = 11
-------------------------------
11  -- GbI is now reduced Groebner Basis of the ideal I.
-------------------------------
-- Done.
-------------------------------


See also

Weyl.WNormalForm

Weyl.WGB

Weyl.WRGB

Weyl.WRGBS

Introduction to Groebner Basis in CoCoA

Introduction to CoCoAServer