up previous next
Computes the unique
F_2-rational zero of a given polynomial system over
F_2.
CharP.IMNLASolve(F:LIST):LIST
|
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 Possibly the unique solution of the given system in F_2^n.
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.
----------------------------------------
The size of Matrix is:
No. of Rows=4
No. of Columns=11
Applying Gaussian Elimination for finding Mutants...
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 for finding Mutants...
Gaussian Elimination Compeleted.
No. of New 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 for finding Mutants...
Gaussian Elimination Compeleted.
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=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]
|
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.
----------------------------------------
The size of Matrix is:
No. of Rows=4
No. of Columns=9
Applying Gaussian Elimination for finding Mutants...
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 for finding Mutants...
Gaussian Elimination Compeleted.
No. of New Mutants found = 0
The size of Matrix is:
No. of Rows=14
No. of Columns=8
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
The size of Matrix is:
No. of Rows=14
No. of Columns=8
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
The size of Matrix is:
No. of Rows=7
No. of Columns=14
Applying Gaussian Elimination for finding Mutants...
Gaussian Elimination Compeleted.
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=10
No. of Columns=14
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
The size of Matrix is:
No. of Rows=10
No. of Columns=14
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
The size of Matrix is:
No. of Rows=13
No. of Columns=10
Applying Gaussian Elimination for finding Mutants...
Gaussian Elimination Compeleted.
No. of New Mutants found = 0
The size of Matrix is:
No. of Rows=10
No. of Columns=9
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
The size of Matrix is:
No. of Rows=10
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=10
Applying Gaussian Elimination for finding Mutants...
Gaussian Elimination Compeleted.
No. of New Mutants found = 0
The size of Matrix is:
No. of Rows=14
No. of Columns=24
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
The size of Matrix is:
No. of Rows=14
No. of Columns=24
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.
|