Volume 44, pp. 73-82, 2015.

Revisiting the stability of computing the roots of a quadratic polynomial

Nicola Mastronardi and Paul Van Dooren

Abstract

We show in this paper that the roots $x_1$ and $x_2$ of a scalar quadratic polynomial $a{x}^2+bx+c=0$ 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.

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Key words

quadratic polynomial, roots, numerical stability

AMS subject classifications

65G30, 65G50, 65H04