function reduce(k, ℓ)
        # BEFORE ================
        if verbose
            println("iteration $n_iter \t reduce \t k=$k \t ℓ=$ℓ \t [μ[k,l]] = $(round_ties_down(μ[k,ℓ]))")
            n_iter += 1
        end

        if abs(μ[k, ℓ]) > 1/2
            y[k, :] -= round_ties_down(μ[k, ℓ]) * y[ℓ, :]
            for j in 1:ℓ-1
                μ[k, j] -= round_ties_down(μ[k, ℓ]) * μ[ℓ, j]
            end
            μ[k, ℓ] -= round_ties_down(μ[k, ℓ])
        end
    end

    function reduce(k, ℓ)
        if abs(μ[k, ℓ]) > 1/2
            # AFTER =============
            if verbose
                println("iteration $n_iter \t reduce \t k=$k \t ℓ=$ℓ \t [μ[k,l]] = $(round_ties_down(μ[k,ℓ]))")
                n_iter += 1
            end

            y[k, :] -= round_ties_down(μ[k, ℓ]) * y[ℓ, :]
            for j in 1:ℓ-1
                μ[k, j] -= round_ties_down(μ[k, ℓ]) * μ[ℓ, j]
            end
            μ[k, ℓ] -= round_ties_down(μ[k, ℓ])
        end
    end