`bit::gauss` — Echelon Form Access

If the gauss object was constructed from the system \(A \cdot x = b\) these methods provide read-only access to the reduced row echelon form of the bit-matrix \(A\) and also to the equivalently transformed bit-vector \(b\).

1const bit::matrix& lhs() const;
2const bit::vector& rhs() const;

1: Returns a read-only reference to the reduced row echelon form of the bit-matrix \(A\).
2: Returns a read-only reference to the equivalently transformed bit-vector \(b\).

On construction, a gauss object computes the reduced row echelon form of the input bit-matrix \(A\) using elementary row operations. It performs the same operations on a copy of the input bit-vector \(b\). The two methods here let you look at the transformed left-hand side bit-matrix and right-hand side bit-vector.

Example

#include <bit/bit.h>
int
main()
{
    std::size_t m = 12;

    auto A = bit::matrix<>::random(m);
    auto b = bit::vector<>::random(m);
    std::cout << "Solving the system A.x = b for the following A & b:\n";
    print(A, b);

    // Create a solver object for the system
    bit::gauss<> solver(A, b);

    // Print some general information
    std::cout << "Number of equations in system:   " << solver.equation_count() << '\n';
    std::cout << "Rank of the matrix A:            " << solver.rank()           << '\n';
    std::cout << "Number of free variables:        " << solver.free_count()     << '\n';
    std::cout << "Number of solutions to A.x = b:  " << solver.solution_count() << '\n';

    // Also have a look at the echelon form of A and the equivalently transformed b
    std::cout << "The echelon forms of A & b are:\n";
    print(solver.lhs(), solver.rhs());
}

Output (depends on the values of the random inputs)

Solving the system A.x = b for the following A & b:
011100100101    0
000111011100    1
111101000011    1
010000111110    1
110011110000    1
101100100100    1
011010110010    0
010010000111    1
101110110001    0
001100101110    1
100000011010    1
111111010100    1
Number of equations in system:   12
Rank of the matrix A:            11
Number of free variables:        1
Number of solutions to A.x = b:  2
The echelon forms of A & b are:
100000000000    1
010000000000    0
001000000000    1
000100000000    0
000010000100    0
000001000000    0
000000100100    1
000000010000    1
000000001000    0
000000000010    1
000000000001    0
000000000000    0

See Also