Difference between revisions of "ApCoCoA-1:CharP.XLSolve"
From ApCoCoAWiki
| Line 53: | Line 53: | ||
</example> | </example> | ||
| + | |||
| + | |||
| + | <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>@return The unique solution of the given system in <tt>F_2^n</tt>. </item> | ||
| + | </itemize> | ||
| + | |||
| + | <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.XLSolve(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 variables found till now, if any are: | ||
| + | [x[1], x[2], x[3], x[4]] | ||
| + | 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 size of Matrix is: | ||
| + | No. of Rows=18 | ||
| + | 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 size of Matrix is: | ||
| + | No. of Rows=13 | ||
| + | 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> | ||
| + | |||
</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see> | + | <see>MXLSolve</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> | ||
Revision as of 16:21, 6 December 2010
CharP.GBasisF2
Computing the unique F_2-rational zero of a given polynomial system over F_2.
Syntax
CharP.XLSolve(F:LIST):LIST
Description
This function computes the unique zero in F_2^n of a polynomial system over F_2 . 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 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.
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.
@param F A system of polynomial over F_2 having a unique zero in F_2^n.
@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.XLSolve(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 variables found till now, if any are:
[x[1], x[2], x[3], x[4]]
The size of Matrix is:
No. of Rows=16
No. of Columns=15
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]
@param F A system of polynomial over F_2 having a unique zero in F_2^n.
@return The unique solution of the given system in F_2^n.
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.XLSolve(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 variables found till now, if any are: [x[1], x[2], x[3], x[4]] 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 size of Matrix is: No. of Rows=18 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 size of Matrix is: No. of Rows=13 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
Introduction to Groebner Basis in CoCoA
