Difference between revisions of "ApCoCoA-1:BB.BBasis"
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| + | {{Version|1}} | ||
<command> | <command> | ||
<title>BB.BBasis</title> | <title>BB.BBasis</title> | ||
| − | <short_description> | + | <short_description>Computes the border basis of a zero-dimensional ideal.</short_description> |
<syntax> | <syntax> | ||
BB.BBasis(I:IDEAL):LIST of POLY | BB.BBasis(I:IDEAL):LIST of POLY | ||
| + | BB.BBasis(I:IDEAL, Method:STRING):LIST of POLY | ||
</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/> | ||
| − | Let <tt>sigma</tt> be a degree compatible term ordering. The function <tt>BBasis</tt> calls the ApCoCoAServer to compute the O_sigma(I)-border basis of the zero-dimensional input ideal <tt>I</tt> and returns it as a list of polynomials. | + | Let <tt>sigma</tt> be a degree compatible term ordering. The function <tt>BBasis</tt> calls the ApCoCoAServer to compute the <tt>O_sigma(I)</tt>-border basis of the zero-dimensional input ideal <tt>I</tt> and returns it as a list of polynomials. |
| + | <par/> | ||
| + | You may specify a method parameter which indicates which variation of the Border Basis Algorithm should be used for the computation of the border basis. Possible method values are <tt>StandardBBA</tt>, <tt>MatrixBBA</tt> (polynomial reduction via transformation to matrices and row echelon form computation), and <tt>HeuristicBBA</tt> (uses a heuristic strategy for the computing universe enlargements). The default method is <tt>StandardBBA</tt>. | ||
<par/> | <par/> | ||
The return value will be the computed border basis. | The return value will be the computed border basis. | ||
| Line 15: | Line 19: | ||
<item>@param <em>I</em> A zero-dimensional ideal of which to compute a border basis.</item> | <item>@param <em>I</em> A zero-dimensional ideal of which to compute a border basis.</item> | ||
<item>@return A list of border basis polynomials.</item> | <item>@return A list of border basis polynomials.</item> | ||
| + | </itemize> | ||
| + | The following parameter is optional. | ||
| + | <itemize> | ||
| + | <item>@param <em>Method</em> A string that specifies which variation of the Border Basis Algorithm to use. Possible method values are "StandardBBA" (default), "MatrixBBA", and "HeuristicBBA".</item> | ||
</itemize> | </itemize> | ||
<example> | <example> | ||
Use QQ[x, y], DegLex; | Use QQ[x, y], DegLex; | ||
I := Ideal([x^2, xy + y^2]); | I := Ideal([x^2, xy + y^2]); | ||
| − | BB := BBasis(I); | + | BB := BB.BBasis(I); |
BB; | BB; | ||
------------------------------- | ------------------------------- | ||
[xy + y^2, x^2, y^3, xy^2] | [xy + y^2, x^2, y^3, xy^2] | ||
| + | ------------------------------- | ||
| + | </example> | ||
| + | <example> | ||
| + | Use QQ[x, y, z]; | ||
| + | I := Ideal([x^2, y^2, z^2]); | ||
| + | BB := BB.BBasis(I, "MatrixBBA"); | ||
| + | BB; | ||
| + | |||
| + | ------------------------------- | ||
| + | [z^2, y^2, x^2, yz^2, xz^2, y^2z, x^2z, xy^2, x^2y, xyz^2, xy^2z, x^2yz] | ||
------------------------------- | ------------------------------- | ||
</example> | </example> | ||
</description> | </description> | ||
<see>Introduction to CoCoAServer</see> | <see>Introduction to CoCoAServer</see> | ||
| − | <see>GBasis5, and more</see> | + | <see>CoCoA:GBasis5, and more</see> |
| − | <see>BB.BBasisForOI</see> | + | <see>ApCoCoA-1:BB.BBasisForOI</see> |
<types> | <types> | ||
<type>borderbasis</type> | <type>borderbasis</type> | ||
| Line 39: | Line 57: | ||
<key>borderbasis.BBasis</key> | <key>borderbasis.BBasis</key> | ||
<key>border basis</key> | <key>border basis</key> | ||
| − | <wiki-category>Package_borderbasis</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_borderbasis</wiki-category> |
</command> | </command> | ||
Latest revision as of 13:27, 29 October 2020
| This article is about a function from ApCoCoA-1. |
BB.BBasis
Computes the border basis of a zero-dimensional ideal.
Syntax
BB.BBasis(I:IDEAL):LIST of POLY BB.BBasis(I:IDEAL, Method:STRING):LIST of POLY
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.
Let sigma be a degree compatible term ordering. The function BBasis calls the ApCoCoAServer to compute the O_sigma(I)-border basis of the zero-dimensional input ideal I and returns it as a list of polynomials.
You may specify a method parameter which indicates which variation of the Border Basis Algorithm should be used for the computation of the border basis. Possible method values are StandardBBA, MatrixBBA (polynomial reduction via transformation to matrices and row echelon form computation), and HeuristicBBA (uses a heuristic strategy for the computing universe enlargements). The default method is StandardBBA.
The return value will be the computed border basis.
@param I A zero-dimensional ideal of which to compute a border basis.
@return A list of border basis polynomials.
The following parameter is optional.
@param Method A string that specifies which variation of the Border Basis Algorithm to use. Possible method values are "StandardBBA" (default), "MatrixBBA", and "HeuristicBBA".
Example
Use QQ[x, y], DegLex; I := Ideal([x^2, xy + y^2]); BB := BB.BBasis(I); BB; ------------------------------- [xy + y^2, x^2, y^3, xy^2] -------------------------------
Example
Use QQ[x, y, z]; I := Ideal([x^2, y^2, z^2]); BB := BB.BBasis(I, "MatrixBBA"); BB; ------------------------------- [z^2, y^2, x^2, yz^2, xz^2, y^2z, x^2z, xy^2, x^2y, xyz^2, xy^2z, x^2yz] -------------------------------
