Newton Backward

function NB = newton_backward(x, y, a)
    n = length(x);
    diff_table = zeros(n, n);

    for i = 1 : n
        diff_table(i, 1) = y(i);
    end

    for j = 2 : n
        for i = 1 : n - j + 1
            diff_table(i, j) = diff_table(i + 1, j - 1) - diff_table(i, j - 1);
        end
    end


    NB = diff_table(n, 1);
    s = (a - x(n)) / (x(2) - x(1));
    fact = 1;
    val = s;
    for j = 2 : n
        NB = NB + (val * diff_table(n - j + 1, j)) / fact;
        val = val * (s + j - 1);
        fact = fact * j;
    end