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W. Auzinger, W. Herfort:
"A uniform quantitative stiff stability estimate for BDF schemes";
Opuscula Mathematica, 26 (2006), 2; 203 - 227.

The concepts of stability regions, $A$- and $A(\alpha)$-stability --
albeit based on scalar models -- turned out to be
essential for the identification of implicit methods suitable
for the integration of stiff ODEs. However, for multistep methods,
knowledge of the stability region provides no information
about the quantitative stability behavior of the scheme.

In this paper we fill this gap for the important class of Backward
Differentiation Formulas (BDF). Quantitative stability bounds are
derived which are uniformly valid in the stability region of
the method. Our analysis is based on a study of the separation
of the characteristic roots and a special similarity decomposition
of the associated companion matrix.

BDF schemes, stiff ODEs, stability, companion matrix, univalence

http://www.math.tuwien.ac.at/~winfried/papers/BS.pdf