Revisiting the stability of computing the roots of a quadratic polynomial

We show in this paper that the roots x1 and x2 of a scalar quadratic polynomial ax2+bx+c = 0 with real or complex coefficients a, b c can be computed in a elementwise 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.