Geodesic
Revisiting the stability of computing the roots of a quadratic polynomial
Electronic transactions on numerical analysis, Tome 44 (2015), pp. 73-82
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.
Classification : 65G30, 65G50, 65H04
Keywords: quadratic polynomial, roots, numerical stability 
@article{ETNA_2015__44__a26,
     author = {Mastronardi,  Nicola and Van Dooren,  Paul},
     title = {Revisiting the stability of computing the roots of a quadratic polynomial},
     journal = {Electronic transactions on numerical analysis},
     pages = {73--82},
     year = {2015},
     volume = {44},
     zbl = {1312.65074},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2015__44__a26/}
}
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Mastronardi,  Nicola; Van Dooren,  Paul. Revisiting the stability of computing the roots of a quadratic polynomial. Electronic transactions on numerical analysis, Tome 44 (2015), pp. 73-82. http://geodesic.mathdoc.fr/item/ETNA_2015__44__a26/