ApCoCoA-1:CharP.IMNLASolve

CharP.GBasisF2

Computing 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 Improved Mutant 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 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

CharP.MXLSolve

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.GBasisF2

CharP.XLSolve

CharP.IMXLSolve

CharP.MNLASolve