Difference between revisions of "ApCoCoA-1:CharP.MXLSolve"

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(New page: <command> <title>CharP.GBasisF2</title> <short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description> <synt...)
 
Line 3: Line 3:
 
     <short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description>
 
     <short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description>
 
<syntax>
 
<syntax>
−
CharP.XLSolve(F:LIST):LIST
+
CharP.MXLSolve(F:LIST):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.
 +
 
<par/>
 
<par/>
−
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses XL-Algorithm to find the unique zero. The XL-Algorithm is impelemented only to find a unique solution. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound.
+
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant XL-Algorithm to find the unique zero. The Mutant XL-Algorithm is impelemented only to find a unique solution. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound.  
−
 
 
−
<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.
 
  
  
 
<itemize>
 
<itemize>
−
<item>@param <em>F</em> A system of polynomial over <tt>F_2</tt> having a unique zero in <tt>F_2^n</tt>. </item>
+
<item>@param <em>F:</em> List of polynomials of given system.</item>
 
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item>
 
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item>
 
</itemize>
 
</itemize>
  
 
<example>
 
<example>
−
Use R::=QQ[x,y,z];
+
Use Z/(2)[x[1..4]];
−
I:=Ideal(x-y^2,x^2+xy,y^3);
+
F:=[
−
GBasis(I);
+
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1,  
−
[x^2 + xy, -y^2 + x, -xy]
+
    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.MXLSolve(F);
 +
 
 +
-- 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
 
-------------------------------
 
-------------------------------
−
Use Z::=ZZ[x,y,z];
+
Gaussian Elimination Completed.
−
-- WARNING: Coeffs are not in a field
+
The variables found till now, if any are:
−
-- GBasis-related computations could fail to terminate or be wrong
+
[0, 1, 0, 1]
 +
[0, 1, 0, 1]
 +
 
 +
</example>
 +
 
 +
 
 +
<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.MXLSolve(F);
  
 +
-- 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
 
-------------------------------
 
-------------------------------
−
I:=Ideal(x-y^2,x^2+xy,y^3);
+
Gaussian Elimination Completed.
−
CharP.GBasisF2(I);
+
The variables found till now, if any are:
−
-- WARNING: Coeffs are not in a field
+
[x[1], x[2], x[3], x[4]]
−
-- GBasis-related computations could fail to terminate or be wrong
+
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
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
-------------------------------
−
[y^2 + x, x^2, xy]
+
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>
 +
  
 
     </description>
 
     </description>
 
     <seealso>
 
     <seealso>
−
       <see>GBasis</see>
+
       <see>CharP.XLSolve</see>
 
     <see>Introduction to CoCoAServer</see>
 
     <see>Introduction to CoCoAServer</see>
 
     <see>Introduction to Groebner Basis in CoCoA</see>
 
     <see>Introduction to Groebner Basis in CoCoA</see>
Line 46: Line 169:
 
     <see>CharP.GBasisF8</see>
 
     <see>CharP.GBasisF8</see>
 
     <see>CharP.GBasisF16</see>
 
     <see>CharP.GBasisF16</see>
−
     <see>CharP.GBasisF32</see>
+
     <see>CharP.IMXLSolve</see>
 
      
 
      
 
   </seealso>
 
   </seealso>

Revision as of 08:43, 7 December 2010

CharP.GBasisF2

Computing the unique F_2-rational zero of a given polynomial system over F_2.

Syntax

CharP.MXLSolve(F:LIST):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.


This function computes the unique zero in F_2^n of a polynomial system over F_2 . It uses Mutant XL-Algorithm to find the unique zero. The Mutant XL-Algorithm is impelemented only to find a unique solution. If the given polynomial system has more than one zeros in F_2^n then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound.


  • @param F: List of polynomials of given system.

  • @return The unique solution of the given system in F_2^n.

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.MXLSolve(F);

-- 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

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.MXLSolve(F);

-- 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.


See also

CharP.XLSolve

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.GBasisF4

CharP.GBasisF8

CharP.GBasisF16

CharP.IMXLSolve






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