Revisiting the stability of computing the roots of a quadratic polynomial

We show in this paper that the roots $x_1$ and $x_2$ of a scalar quadratic polynomial $ax^2 + bx + c = 0$ with real or complex coefficients $a, b, c$ can be computed in an element-wise mixed stable manner, measured in a relative sense. We also show that this is a stronger property than norm-wise backward stability but weaker than element-wise backward stability. We finally show that there does not exist any method that can compute the roots in an element-wise backward stable sense, which is also illustrated by some numerical experiments.