Difference between revisions of "ApCoCoA-1:NC.TruncatedGB"
| Line 2: | Line 2: | ||
<title>NC.TruncatedGB</title> | <title>NC.TruncatedGB</title> | ||
<short_description> | <short_description> | ||
| − | + | D-truncated Groebner basis of a finitely generated two-sided ideal in a free associative <tt>K</tt>-algebra. | |
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
| − | NC.TruncatedGB(G:LIST, | + | NC.TruncatedGB(G:LIST, D:INT):LIST |
| − | NC.TruncatedGB(G:LIST, | + | NC.TruncatedGB(G:LIST, D:INT, LoopBound:INT, IFlag:INT):LIST |
</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/> | ||
| − | Please set ring environment coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and ordering through the functions <ref>NC.SetFp</ref>(Prime), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(Ordering), respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering | + | Please set ring environment coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and ordering <tt>Ordering</tt> through the functions <ref>NC.SetFp</ref>(Prime), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(Ordering), respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering <quotes>LLEX</quotes>. For more information, please check the relevant functions. |
<itemize> | <itemize> | ||
| − | <item>@param <em>G</em>: a LIST of non-zero homogeneous polynomials generating a two-sided ideal in <tt>K<X></tt>. Each polynomial is represented as a LIST of LISTs, | + | <item>@param <em>G</em>: a LIST of non-zero homogeneous polynomials generating a two-sided ideal in <tt>K<X></tt>. Each polynomial is represented as a LIST of LISTs, i.e. as [[C1,W1],...,[Cs,Ws]] where C1,...,Cs are coefficients and W1,...,Ws are terms. Each term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Polynomial <tt>F=xy-yx</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>yx</quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST [].</item> |
| − | <item>@param <em> | + | <item>@param <em>D:</em> a positive integer which is the truncated degree of Groebner basis.</item> |
</itemize> | </itemize> | ||
| − | About the optional parameters: | + | About the optional parameters: in some situations, D-truncated Groebner basis is huge. Thus instead of computing the whole D-truncated Groebner basis, the function has 2 optional parameters to control the computation. Note that at the moment <em>all</em> of the following 2 additional optional parameters must be used at the same time. |
<itemize> | <itemize> | ||
| − | <item>@param <em>LoopBound:</em> a positive integer which gives a bound for the main loop of Groebner basis computation. When it runs through the main loop LoopBound times, the function stops | + | <item>@param <em>LoopBound:</em> a positive integer which gives a bound for the main loop of Groebner basis computation. When it runs through the main loop LoopBound times, the function stops and returns a partial Groebner basis.</item> |
| − | <item>@param <em> | + | <item>@param <em>IFlag:</em> a non-negative integer which is a multi-switch for the output of ApCoCoAServer. If Flag=0, the server prints nothing on the screen. If Flag=1, the server prints basic information about computing procedure, such as number of S-elements has been checked and to be checked. If Flag=2, the server additionally prints current partial Groebner basis before each loop as well as the information when Flag=1. Note that the initial idea of Flag is to use it as a tool for debugging and tracing the computing process.</item> |
| − | <item>@return: a LIST of polynomials, which is a | + | <item>@return: a LIST of polynomials, which is a D-truncated Groebner basis if (1)finite Groebner basis of degree no more than DegreeBound exists or (2)the computation reaches the degree D, and which is a partial Groebner basis otherwise.</item> |
</itemize> | </itemize> | ||
<example> | <example> | ||
Revision as of 09:05, 18 April 2011
NC.TruncatedGB
D-truncated Groebner basis of a finitely generated two-sided ideal in a free associative K-algebra.
Syntax
NC.TruncatedGB(G:LIST, D:INT):LIST NC.TruncatedGB(G:LIST, D:INT, LoopBound:INT, IFlag:INT):LIST
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.
Please set ring environment coefficient field K, alphabet (or indeterminates) X and ordering Ordering through the functions NC.SetFp(Prime), NC.SetX(X) and NC.SetOrdering(Ordering), respectively, before calling the function. Default coefficient field is Q. Default ordering is length-lexicographic ordering "LLEX". For more information, please check the relevant functions.
@param G: a LIST of non-zero homogeneous polynomials generating a two-sided ideal in K<X>. Each polynomial is represented as a LIST of LISTs, i.e. as [[C1,W1],...,[Cs,Ws]] where C1,...,Cs are coefficients and W1,...,Ws are terms. Each term is represented as a STRING. For example, xy^2x is represented as "xyyx", unit is represented as an empty string "". Polynomial F=xy-yx is represented as F:=[[1,"xy"], [-1, "yx"]]. 0 polynomial is represented as an empty LIST [].
@param D: a positive integer which is the truncated degree of Groebner basis.
About the optional parameters: in some situations, D-truncated Groebner basis is huge. Thus instead of computing the whole D-truncated Groebner basis, the function has 2 optional parameters to control the computation. Note that at the moment all of the following 2 additional optional parameters must be used at the same time.
@param LoopBound: a positive integer which gives a bound for the main loop of Groebner basis computation. When it runs through the main loop LoopBound times, the function stops and returns a partial Groebner basis.
@param IFlag: a non-negative integer which is a multi-switch for the output of ApCoCoAServer. If Flag=0, the server prints nothing on the screen. If Flag=1, the server prints basic information about computing procedure, such as number of S-elements has been checked and to be checked. If Flag=2, the server additionally prints current partial Groebner basis before each loop as well as the information when Flag=1. Note that the initial idea of Flag is to use it as a tool for debugging and tracing the computing process.
@return: a LIST of polynomials, which is a D-truncated Groebner basis if (1)finite Groebner basis of degree no more than DegreeBound exists or (2)the computation reaches the degree D, and which is a partial Groebner basis otherwise.
Example
NC.SetX("xyz");
F1:=[[1,"yxy"],[-1,"zyz"]];
F2:=[[1,"xyx"],[-1,"zxy"]];
F3:=[[1,"zxz"],[-1,"yzx"]];
F4:=[[1,"xxx"],[1,"yyy"],[1,"zzz"],[1,"xyz"]];
G:=[F1,F2,F3,F4];
NC.TruncatedGB(G,6);
[[[1, "yzx"], [-1, "zxz"]], [[1, "yxy"], [-1, "zyz"]], [[1, "xyx"], [-1, "zxy"]], [[1, "xxx"], [1, "xyz"], [1, "yyy"], [1, "zzz"]],
[[1, "zxzy"], [-1, "zzxz"]], [[1, "xzyz"], [-1, "zxyy"]], [[1, "xxyz"], [1, "xyyy"], [-1, "xzxz"], [1, "xzzz"], [-1, "yyyx"],
[-1, "zzzx"]], [[1, "zzxyy"], [-1, "zzxzz"]], [[1, "yzzxz"], [-1, "zxzzy"]], [[1, "yzzxy"], [-1, "zzxzx"]], [[1, "yzyyy"],
[1, "yzzzz"], [1, "zxzxx"], [1, "zzxzz"]], [[1, "yxzxz"], [-1, "zyzzx"]], [[1, "xzzxz"], [-1, "zxyyx"]], [[1, "xyyyy"], [1, "xyzzz"],
[1, "zxyyz"], [1, "zzzxy"]], [[1, "xxzxz"], [1, "xyyyx"], [-1, "xzxzx"], [1, "xzzzx"], [-1, "yyyxx"], [-1, "zzzxx"]], [[1, "xxzxy"],
[1, "xyzyx"], [1, "yyyyx"], [1, "zzzyx"]], [[1, "xxyyy"], [1, "xxzzz"], [-1, "xyzyz"], [-1, "xzxzx"], [-1, "yyyxx"], [-1, "yyyyz"],
[-1, "zzzxx"], [-1, "zzzyz"]], [[1, "zxzzyz"], [-1, "zzxzxy"]], [[1, "yzzzxz"], [-1, "zxzzyy"]], [[1, "yzzzxy"], [-1, "zzxzxx"]],
[[1, "xzzzxz"], [-1, "zxyzyz"]], [[1, "xyyzyz"], [1, "xzxyyx"], [-1, "xzxzxy"], [1, "xzzzxy"], [-1, "yyyxxy"], [-1, "zzzxxy"]],
[[1, "xxzzzy"], [1, "xyyyzz"], [-1, "xyzyzy"], [-1, "xzxyyz"], [-1, "xzxzxy"], [-1, "xzxzzz"], [-1, "xzzzxy"], [1, "xzzzzz"],
[-1, "yyyxxy"], [-1, "yyyxzz"], [-1, "yyyyzy"], [-1, "zzzxxy"], [-1, "zzzxzz"], [-1, "zzzyzy"]], [[1, "xxzzxy"], [1, "xyzyxx"],
[1, "yyyyxx"], [1, "zzzyxx"]]]
-------------------------------
See also
