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

(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...) |
m (insert version info) |
||

(7 intermediate revisions by 3 users not shown) | |||

Line 1: | Line 1: | ||

+ | {{Version|1}} | ||

<command> | <command> | ||

− | <title>CharP. | + | <title>CharP.MNLASolve</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.MNLASolve(F:LIST):LIST | CharP.MNLASolve(F:LIST):LIST | ||

Line 9: | Line 10: | ||

<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 Mutant NLA | + | This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant NLA-Algorithm to find the unique zero. The Mutant NLA-Algorithm generates a sequence of linear systems to solve the given system. The Mutant NLA-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 Mutant NLA-Algorithm is the NLA-Algorithm with mutant strategy. It uses <ref>ApCoCoA-1:LinAlg.EF|LinAlg.EF</ref> for gaussian elimination. |

Line 34: | Line 35: | ||

---------------------------------------- | ---------------------------------------- | ||

− | + | The size of Matrix is: | |

− | + | No. of Rows=4 | |

− | + | No. of Columns=11 | |

− | + | Applying Gaussian Elimination finding Muatants... | |

− | |||

− | |||

− | |||

Gaussian Elimination Compeleted | Gaussian Elimination Compeleted | ||

− | + | Finding Variable: x[4] | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=11 | |

− | + | No. of Columns=5 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

Gaussian Elimination Completed. | Gaussian Elimination Completed. | ||

− | + | The size of Matrix is: | |

− | + | No. of Rows=11 | |

− | + | No. of Columns=5 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | |||

− | |||

Gaussian Elimination Completed. | Gaussian Elimination Completed. | ||

− | + | The size of Matrix is: | |

− | + | No. of Rows=4 | |

− | + | No. of Columns=11 | |

− | + | Applying Gaussian Elimination finding Muatants... | |

− | |||

− | |||

− | |||

Gaussian Elimination Compeleted | Gaussian Elimination Compeleted | ||

− | + | The No. of Mutants found = 0 | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=11 | |

− | + | No. of Columns=9 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | |||

Gaussian Elimination Completed. | Gaussian Elimination Completed. | ||

− | + | The size of Matrix is: | |

− | + | No. of Rows=11 | |

− | + | No. of Columns=9 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | |||

− | |||

Gaussian Elimination Completed. | Gaussian Elimination Completed. | ||

− | + | The size of Matrix is: | |

− | + | No. of Rows=8 | |

− | + | No. of Columns=11 | |

− | + | Applying Gaussian Elimination finding Muatants... | |

− | |||

− | |||

− | |||

Gaussian Elimination Compeleted | Gaussian Elimination Compeleted | ||

− | + | The No. of Mutants found = 1 | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=11 | |

− | + | No. of Columns=12 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | |||

Gaussian Elimination Completed. | Gaussian Elimination Completed. | ||

− | + | The size of Matrix is: | |

− | + | No. of Rows=11 | |

− | + | No. of Columns=12 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | |||

− | |||

Gaussian Elimination Completed. | Gaussian Elimination Completed. | ||

x[4] = 1 | x[4] = 1 | ||

− | + | Finding Variable: x[3] | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=7 | |

− | + | No. of Columns=10 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

Gaussian Elimination Completed. | Gaussian Elimination Completed. | ||

x[3] = 0 | x[3] = 0 | ||

− | + | Finding Variable: x[2] | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=4 | |

− | + | No. of Columns=5 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

Gaussian Elimination Completed. | Gaussian Elimination Completed. | ||

− | + | The size of Matrix is: | |

− | + | No. of Rows=4 | |

− | + | No. of Columns=5 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | |||

− | |||

Gaussian Elimination Completed. | Gaussian Elimination Completed. | ||

x[2] = 1 | x[2] = 1 | ||

+ | |||

[0, 1, 0, 1] | [0, 1, 0, 1] | ||

− | |||

</example> | </example> | ||

Line 205: | Line 125: | ||

---------------------------------------- | ---------------------------------------- | ||

− | + | The size of Matrix is: | |

− | + | No. of Rows=4 | |

− | + | No. of Columns=9 | |

− | + | Applying Gaussian Elimination finding Muatants... | |

− | + | Gaussian Elimination Compeleted | |

− | + | Finding Variable: x[4] | |

− | + | The size of Matrix is: | |

− | Gaussian Elimination Compeleted | + | No. of Rows=9 |

− | + | No. of Columns=4 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | + | Gaussian Elimination Completed. | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=9 | |

− | + | No. of Columns=4 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | + | Gaussian Elimination Completed. | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=3 | |

− | + | No. of Columns=9 | |

− | + | Applying Gaussian Elimination finding Muatants... | |

− | + | Gaussian Elimination Compeleted | |

− | Gaussian Elimination Completed. | + | The No. of Mutants found = 0 |

− | + | The size of Matrix is: | |

− | + | No. of Rows=14 | |

− | + | No. of Columns=16 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | + | Gaussian Elimination Completed. | |

− | + | The size of Matrix is: | |

− | Gaussian Elimination Completed. | + | No. of Rows=14 |

− | + | No. of Columns=16 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | + | Gaussian Elimination Completed. | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=15 | |

− | + | No. of Columns=14 | |

− | + | Applying Gaussian Elimination finding Muatants... | |

− | Gaussian Elimination Compeleted | + | Gaussian Elimination Compeleted |

− | + | The No. of Mutants found = 4 | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=14 | |

− | + | No. of Columns=28 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | + | Gaussian Elimination Completed. | |

− | Gaussian Elimination Completed. | + | The size of Matrix is: |

− | + | No. of Rows=14 | |

− | + | No. of Columns=28 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | + | Gaussian Elimination Completed. | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=27 | |

− | Gaussian Elimination Completed. | + | No. of Columns=14 |

− | + | Applying Gaussian Elimination finding Muatants... | |

− | + | Gaussian Elimination Compeleted | |

− | + | The No. of Mutants found = 0 | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=14 | |

− | + | No. of Columns=13 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | Gaussian Elimination Compeleted | + | Gaussian Elimination Completed. |

− | + | The size of Matrix is: | |

− | + | No. of Rows=14 | |

− | + | No. of Columns=13 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | + | Gaussian Elimination Completed. | |

− | + | The size of Matrix is: | |

− | Gaussian Elimination Completed. | + | No. of Rows=12 |

− | + | No. of Columns=14 | |

− | + | Applying Gaussian Elimination finding Muatants... | |

− | + | Gaussian Elimination Compeleted | |

− | + | The No. of Mutants found = 0 | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=15 | |

− | Gaussian Elimination Completed. | + | No. of Columns=20 |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | + | Gaussian Elimination Completed. | |

− | + | The size of Matrix is: | |

− | + | No. of Rows=15 | |

− | + | No. of Columns=20 | |

− | + | Applying Gaussian Elimination to check solution coordinate... | |

− | + | Gaussian Elimination Completed. | |

− | Gaussian Elimination Compeleted | + | x[4] = NA |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | Gaussian Elimination Completed. | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | Gaussian Elimination Completed. | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | Gaussian Elimination | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | Gaussian Elimination Completed. | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | Gaussian Elimination Completed. | ||

− | x[4] = NA | ||

Please Check the uniqueness of solution. | Please Check the uniqueness of solution. | ||

The Given system of polynomials does not | The Given system of polynomials does not | ||

seem to have a unique solution or it has | seem to have a unique solution or it has | ||

no solution over the finite field F2. | no solution over the finite field F2. | ||

− | |||

</example> | </example> | ||

Line 356: | Line 216: | ||

</description> | </description> | ||

<seealso> | <seealso> | ||

− | <see>CharP.MXLSolve</see> | + | <see>ApCoCoA-1:CharP.MXLSolve|CharP.MXLSolve</see> |

− | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |

− | <see>Introduction to Groebner Basis in CoCoA</see> | + | <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see> |

− | <see>CharP.GBasisF2</see> | + | <see>ApCoCoA-1:CharP.GBasisF2|CharP.GBasisF2</see> |

− | <see>CharP.XLSolve</see> | + | <see>ApCoCoA-1:CharP.XLSolve|CharP.XLSolve</see> |

− | <see>CharP.IMXLSolve</see> | + | <see>ApCoCoA-1:CharP.IMXLSolve|CharP.IMXLSolve</see> |

− | <see>CharP.IMNLASolve | + | <see>ApCoCoA-1:CharP.IMNLASolve|CharP.IMNLASolve</see> |

− | |||

− | |||

− | |||

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

<types> | <types> | ||

<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||

− | <type> | + | <type>poly_system</type> |

− | |||

</types> | </types> | ||

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

− | <key> | + | <key>mnlasolve</key> |

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

− | <wiki-category>Package_charP</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_charP</wiki-category> |

</command> | </command> |

## Latest revision as of 09:56, 7 October 2020

This article is about a function from ApCoCoA-1. |

## CharP.MNLASolve

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

### Syntax

CharP.MNLASolve(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 NLA-Algorithm to find the unique zero. The Mutant NLA-Algorithm generates a sequence of linear systems to solve the given system. The 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 Mutant NLA-Algorithm is the NLA-Algorithm with mutant strategy. It uses LinAlg.EF for gaussian elimination.

@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.MNLASolve(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 Applying Gaussian Elimination finding Muatants... Gaussian Elimination Compeleted Finding Variable: x[4] The size of Matrix is: No. of Rows=11 No. of Columns=5 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=11 No. of Columns=5 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=4 No. of Columns=11 Applying Gaussian Elimination finding Muatants... Gaussian Elimination Compeleted The No. of Mutants found = 0 The size of Matrix is: No. of Rows=11 No. of Columns=9 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=11 No. of Columns=9 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=8 No. of Columns=11 Applying Gaussian Elimination finding Muatants... Gaussian Elimination Compeleted The No. of Mutants found = 1 The size of Matrix is: No. of Rows=11 No. of Columns=12 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=11 No. of Columns=12 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. x[4] = 1 Finding Variable: x[3] The size of Matrix is: No. of Rows=7 No. of Columns=10 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. x[3] = 0 Finding Variable: x[2] The size of Matrix is: No. of Rows=4 No. of Columns=5 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=4 No. of Columns=5 Applying Gaussian Elimination to check solution coordinate... 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.MNLASolve(F); -- And we achieve the following information on the screen. ---------------------------------------- The size of Matrix is: No. of Rows=4 No. of Columns=9 Applying Gaussian Elimination finding Muatants... Gaussian Elimination Compeleted Finding Variable: x[4] The size of Matrix is: No. of Rows=9 No. of Columns=4 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=9 No. of Columns=4 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=3 No. of Columns=9 Applying Gaussian Elimination finding Muatants... Gaussian Elimination Compeleted The No. of Mutants found = 0 The size of Matrix is: No. of Rows=14 No. of Columns=16 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=14 No. of Columns=16 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=15 No. of Columns=14 Applying Gaussian Elimination finding Muatants... Gaussian Elimination Compeleted The No. of Mutants found = 4 The size of Matrix is: No. of Rows=14 No. of Columns=28 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=14 No. of Columns=28 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=27 No. of Columns=14 Applying Gaussian Elimination finding Muatants... Gaussian Elimination Compeleted The No. of Mutants found = 0 The size of Matrix is: No. of Rows=14 No. of Columns=13 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=14 No. of Columns=13 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=12 No. of Columns=14 Applying Gaussian Elimination finding Muatants... Gaussian Elimination Compeleted The No. of Mutants found = 0 The size of Matrix is: No. of Rows=15 No. of Columns=20 Applying Gaussian Elimination to check solution coordinate... Gaussian Elimination Completed. The size of Matrix is: No. of Rows=15 No. of Columns=20 Applying Gaussian Elimination to check solution coordinate... 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