# Difference between revisions of "ApCoCoA-1:CharP.IMNLASolve"

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<command> | <command> | ||

<title>CharP.GBasisF2</title> | <title>CharP.GBasisF2</title> | ||

− | <short_description> | + | <short_description>Computes the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description> |

<syntax> | <syntax> | ||

CharP.IMNLASolve(F:LIST):LIST | CharP.IMNLASolve(F:LIST):LIST | ||

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− | This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses | + | This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2</tt>. It uses <tt>I</tt>mproved <tt>M</tt>utant <tt>NLA</tt>-Algorithm to find the unique zero. The Improved Mutant <tt>NLA</tt>-Algorithm generates a sequence of linear systems to solve the given system. The Improved Mutant <tt>NLA</tt>-Algorithm can find the unique zero only. 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. In fact Improved Mutant NLA-Algorithm is the NLA-Algorithm with improved mutant strategy. It uses <ref>LinAlg.EF</ref> for gaussian elimination. |

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<itemize> | <itemize> | ||

<item>@param <em>F:</em> List of polynomials of given system.</item> | <item>@param <em>F:</em> List of polynomials of given system.</item> | ||

− | <item>@return | + | <item>@return Possibly the unique solution of the given system in <tt>F_2^n</tt>. </item> |

</itemize> | </itemize> | ||

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<see>CharP.IMXLSolve</see> | <see>CharP.IMXLSolve</see> | ||

<see>CharP.MNLASolve</see> | <see>CharP.MNLASolve</see> | ||

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</seealso> | </seealso> | ||

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− | <key>charP. | + | <key>charP.imnlasolve</key> |

− | <key> | + | <key>imnlasolve</key> |

<key>finite field</key> | <key>finite field</key> | ||

<wiki-category>Package_charP</wiki-category> | <wiki-category>Package_charP</wiki-category> | ||

</command> | </command> |

## Revision as of 14:08, 14 December 2010

## CharP.GBasisF2

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

### Syntax

CharP.IMNLASolve(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 `I`mproved `M`utant `NLA`-Algorithm to find the unique zero. The Improved Mutant `NLA`-Algorithm generates a sequence of linear systems to solve the given system. The Improved Mutant `NLA`-Algorithm can find the unique zero only. 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. In fact Improved Mutant NLA-Algorithm is the NLA-Algorithm with improved mutant strategy. It uses LinAlg.EF for gaussian elimination.

@param

*F:*List of polynomials of given system.@return Possibly 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.IMNLASolve(F); -- And we achieve the following information on the screen together with the solution at the end. ---------------------------------------- Finding Variable: x[4] The size of Matrix is: No. of Rows=4 No. of Columns=11 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=4 No. of Columns=11 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=11 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=11 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=8 No. of Columns=11 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=11 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=11 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 1 The total No. of Mutants found are = 1 The No. of Mutants of Minimum degree (Mutants used) are = 1 The size of Matrix is: No. of Rows=11 No. of Columns=11 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=11 No. of Columns=10 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=11 No. of Columns=10 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. x[4] = 1 Finding Variable: x[3] The size of Matrix is: No. of Rows=4 No. of Columns=7 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=4 No. of Columns=7 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=7 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=7 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=4 No. of Columns=7 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=7 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=7 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=7 No. of Columns=7 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=7 No. of Columns=7 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. x[3] = 0 Finding Variable: x[2] The size of Matrix is: No. of Rows=3 No. of Columns=4 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=3 No. of Columns=4 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=4 No. of Columns=4 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=4 No. of Columns=4 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. x[2] = 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.IMNLASolve(F); -- And we achieve the following information on the screen. ---------------------------------------- Finding Variable: x[4] The size of Matrix is: No. of Rows=4 No. of Columns=9 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=3 No. of Columns=9 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=9 No. of Columns=4 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=9 No. of Columns=4 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=7 No. of Columns=14 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=14 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=14 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 2 The total No. of Mutants found are = 2 The No. of Mutants of Minimum degree (Mutants used) are = 2 The size of Matrix is: No. of Rows=9 No. of Columns=10 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=10 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=10 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=7 No. of Columns=10 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=10 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=10 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=22 No. of Columns=14 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=14 No. of Columns=13 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=14 No. of Columns=13 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. x[4] = NA Please Check the uniqueness of solution. The Given system of polynomials does not seem to have a unique solution or it has no solution over the finite field F2.

### See also

Introduction to Groebner Basis in CoCoA