Difference between revisions of "ApCoCoA-1:Weyl.WRGB"
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| − | <command> | + | {{Version|1}} |
| + | <command> | ||
<title>Weyl.WRGB</title> | <title>Weyl.WRGB</title> | ||
<short_description>Reduced Groebner basis of an ideal <tt>I</tt> in Weyl algebra <tt>A_n</tt>.</short_description> | <short_description>Reduced Groebner basis of an ideal <tt>I</tt> in Weyl algebra <tt>A_n</tt>.</short_description> | ||
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<description> | <description> | ||
<par/> | <par/> | ||
| − | This function converts a Weyl Groebner basis <tt>GB</tt> computed by ApCoCoAServer into the reduced Weyl 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 <tt>L</tt> such that <tt>Ideal(L) = Ideal(GB)</tt>, every polynomial is reduced with respect to the remaining polynomials in the list <tt>L</tt> and leading coefficient of each polynomial in <tt>L</tt> is 1. | + | This function converts a Weyl Groebner basis <tt>GB</tt> computed by ApCoCoAServer into the reduced Weyl 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>, every polynomial is reduced with respect to the remaining polynomials in the list <tt>L</tt> and leading coefficient of each polynomial in <tt>L</tt> is 1. |
<itemize> | <itemize> | ||
| Line 20: | Line 21: | ||
Weyl.WRGB(L); | Weyl.WRGB(L); | ||
[1] | [1] | ||
| + | ------------------------------- | ||
| + | </example> | ||
| + | <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.WRGB(GbI);Len(GbI); | ||
| + | |||
| + | Cpu time = 9.61, User time = 10 | ||
| + | ------------------------------- | ||
| + | 11 | ||
| + | ------------------------------- | ||
| + | -- Done. | ||
| + | ------------------------------- | ||
| + | Time GbI:=Weyl.WRGBS(GbI);Len(GbI); -- Weyl.WRGBS() can now be used for calling same implementation in ApCoCoALib | ||
| + | -- note that this speeds up the computations | ||
| + | ------------------------------- | ||
| + | -- CoCoAServer: computing Cpu Time = 0 | ||
| + | ------------------------------- | ||
| + | Cpu time = 0.04, User time = 0 | ||
| + | ------------------------------- | ||
| + | 11 -- this is now size of reduced GB of the ideal I | ||
| + | ------------------------------- | ||
| + | -- Done. | ||
------------------------------- | ------------------------------- | ||
</example> | </example> | ||
</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>Weyl.WNormalForm</see> | + | <see>ApCoCoA-1:Weyl.WNormalForm|Weyl.WNormalForm</see> |
| − | <see>Weyl.WGB</see> | + | <see>ApCoCoA-1:Weyl.WGB|Weyl.WGB</see> |
| − | <see>Weyl.WRGBS</see> | + | <see>ApCoCoA-1:Weyl.WRGBS|Weyl.WRGBS</see> |
| − | <see>Weyl.WRedGB</see> | + | <see>ApCoCoA-1:Weyl.WRedGB|Weyl.WRedGB</see> |
| − | <see>Introduction to Groebner Basis in CoCoA</see> | + | <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see> |
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
| Line 38: | Line 70: | ||
<key>Weyl.WRGB</key> | <key>Weyl.WRGB</key> | ||
<key>wrgb</key> | <key>wrgb</key> | ||
| − | <wiki-category>Package_weyl</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_weyl</wiki-category> |
</command> | </command> | ||
Latest revision as of 10:39, 7 October 2020
| This article is about a function from ApCoCoA-1. |
Weyl.WRGB
Reduced Groebner basis of an ideal I in Weyl algebra A_n.
Syntax
Weyl.WRGB(GB:LIST):LIST
Description
This function converts a Weyl Groebner basis GB computed by ApCoCoAServer into the reduced Weyl 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), every polynomial is reduced with respect to the remaining polynomials in the list L and leading coefficient of each polynomial in L is 1.
@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.WRGB(L); [1] -------------------------------
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.WRGB(GbI);Len(GbI);
Cpu time = 9.61, User time = 10
-------------------------------
11
-------------------------------
-- Done.
-------------------------------
Time GbI:=Weyl.WRGBS(GbI);Len(GbI); -- Weyl.WRGBS() can now be used for calling same implementation in ApCoCoALib
-- note that this speeds up the computations
-------------------------------
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Cpu time = 0.04, User time = 0
-------------------------------
11 -- this is now size of reduced GB of the ideal I
-------------------------------
-- Done.
-------------------------------
See also
Introduction to Groebner Basis in CoCoA
