ApCoCoA-1:CharP.MBBasisF2: Difference between revisions

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New page: <command> <title>CharP.MBBasis</title> <short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description> <syntax> CharP.MBBasisF2(F:LIST):LIST </...
 
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<syntax>
<syntax>
CharP.MBBasisF2(F:LIST):LIST
CharP.MBBasisF2(F:LIST):LIST
CharP.MBBasisF2(F:LIST, NSol: 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/>
Let <tt>f_1</tt>, ... , <tt>f_m</tt> is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials.
Let <tt>f_1</tt>, ... , <tt>f_m</tt> is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials. Furthermore, it uses mutant strategy to compute a U-stable span.


If you want to use the second version with the parameter NSol, you need to provide the exact number of <tt>F_2</tt> rational solutions. The first version is safe to use if you do not know the exact number of <tt>F_2</tt> rational solutions.
<itemize>
<itemize>
<item>@param <em>F:</em> List of polynomials.</item>
<item>@param <em>F:</em> List of polynomials.</item>
<item>@param <em>NSol:</em> Number of <tt>F_2</tt> rational solutions. </item>
<item>@return A Border Basis of zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item>
<item>@return A Border Basis of zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item>
</itemize>
</itemize>
Line 26: Line 28:


-- Then we compute the solution with
-- Then we compute the solution with
CharP.MXLSolve(F);
CharP.MBBasisF2(F);


-- And we achieve the following information on the screen together with the solution at the end.
-- And we achieve the following information on the screen together with the solution at the end.
----------------------------------------
----------------------------------------
The size of Matrix is:
 
No. of Rows=4
No. of Columns=11
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The size of Matrix is:
No. of Rows=4
No. of Columns=11
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The variables found till now, if any are:
[x[1], x[2], x[3], x[4]]
The No. of Mutants found = 0
The size of Matrix is:
No. of Rows=8
No. of Columns=11
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The variables found till now, if any are:
[x[1], x[2], x[3], x[4]]
The No. of Mutants found = 1
The size of Matrix is:
No. of Rows=11
No. of Columns=11
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The variables found till now, if any are:
[0, 1, 0, 1]
[0, 1, 0, 1]


</example>
</example>
Line 83: Line 49:


-- Then we compute the solution with
-- Then we compute the solution with
CharP.MXLSolve(F);
CharP.MBBasisF2(F,NSol);


-- And we achieve the following information on the screen.
-- And we achieve the following information on the screen.
----------------------------------------
----------------------------------------
The size of Matrix is:
 
No. of Rows=4
No. of Columns=9
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The size of Matrix is:
No. of Rows=3
No. of Columns=9
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The variables found till now, if any are:
[x[1], x[2], x[3], x[4]]
The No. of Mutants found = 0
The size of Matrix is:
No. of Rows=14
No. of Columns=14
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The variables found till now, if any are:
[x[1], x[2], x[3], x[4]]
The No. of Mutants found = 4
The size of Matrix is:
No. of Rows=27
No. of Columns=14
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The variables found till now, if any are:
[x[1], x[2], x[3], x[4]]
The No. of Mutants found = 0
The size of Matrix is:
No. of Rows=12
No. of Columns=14
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The variables found till now, if any are:
[x[1], x[2], x[3], x[4]]
The No. of Mutants found = 0
The size of Matrix is:
No. of Rows=19
No. of Columns=15
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The variables found till now, if any are:
[x[1], x[2], x[3], x[4]]
The No. of Mutants found = 0
The size of Matrix is:
No. of Rows=14
No. of Columns=15
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
The variables found till now, if any are:
[x[1], x[2], x[3], x[4]]
Please Check the uniqueness of solution.
The Given system of polynomials does not
seem to have a unique solution.


</example>
</example>

Revision as of 13:18, 28 April 2011

CharP.MBBasis

Computing a Border Basis of a given ideal over F_2.

Syntax

CharP.MBBasisF2(F:LIST):LIST
CharP.MBBasisF2(F:LIST, NSol: 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.

Let f_1, ... , f_m is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by f_1, ... , f_m and the field polynomials. Furthermore, it uses mutant strategy to compute a U-stable span.

If you want to use the second version with the parameter NSol, you need to provide the exact number of F_2 rational solutions. The first version is safe to use if you do not know the exact number of F_2 rational solutions.

  • @param F: List of polynomials.

  • @param NSol: Number of F_2 rational solutions.

  • @return A Border Basis of zero-dimensional radical ideal generated by the polynomials in F and the field polynomials.

Example

Use Z/(2)[x[1..4]];
F:=[
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, 
    x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, 
    x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, 
    x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1
    ];

-- Then we compute the solution with
CharP.MBBasisF2(F);

-- And we achieve the following information on the screen together with the solution at the end.
----------------------------------------


Example

Use Z/(2)[x[1..4]];
F:=[ 
    x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], 
    x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4],  
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2],  
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]
   ];

-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions 

-- Then we compute the solution with
CharP.MBBasisF2(F,NSol);

-- And we achieve the following information on the screen.
----------------------------------------


See also

CharP.XLSolve

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.IMNLASolve

CharP.MNLASolve

CharP.NLASolve

CharP.IMXLSolve





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