Revisiting the stability of computing the roots of a quadratic polynomial

TL;DR

This paper investigates the stability of algorithms for computing roots of quadratic polynomials, demonstrating that element-wise mixed stability is achievable but element-wise backward stability is impossible, supported by numerical experiments.

Contribution

It establishes the feasibility of element-wise mixed stability for quadratic root computation and proves the impossibility of element-wise backward stability for such methods.

Findings

Element-wise mixed stability can be achieved in quadratic root computations.

Element-wise backward stability is proven to be impossible for these methods.

Numerical experiments support the theoretical results.

Abstract

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 a 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.