rrefx.jl

export rrefx

"""
`rref(A)`  returns the row-reduced echelon form of the exact 
matrix `A`.
"""
function rrefx(A::AbstractMatrix{T}) where T<: IntegerX
    AA = big.(A//1)
    return rrefx(AA)
end

function rrefx(A::AbstractMatrix{T}) where T
    AA = copy(A)
    rrefx!(AA)
    return AA
end


function _pivot!(A::AbstractMatrix{T}, i::Int, j::Int) where T
    r,c = size(A)
    s = A[i,j]
    row_scale!(A,i,invx(s))

    for k=1:r
        if k != i
            s = -A[k,j]
            row_add_mult!(A,i,s,k)
        end
    end
end



function rrefx!(A::AbstractMatrix{T}) where T
    rn,nc = size(A)

    r = 1
    for j=1:nc
        # see if column j's first one is at level r or below
        col = A[r:end,j]
        if all(col .== 0)
            continue
        end
        k = findfirst(col .!= 0)
        row_swap!(A,r,k+r-1)
        _pivot!(A,r,j)
        r += 1
    end


end