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# A note on unconditional fixed-h stability of linear multistep formulae

## Abstract

For a linear multistep method applied with a fixed integration step h to x=λx, λ constant, a concept of AD-stability is defined relative to any domain of the complex q=λh plane. The image of the unit circle, under the map q=q(z) induced by the characteristic equation, determines a "maximal" stability domain D, even if q (z) is not univalued or if the image of the unit circle is unbounded. This is proved via degree theory of analytic maps on Riemann surfaces. For a special class of formulae, easy-to-check, sufficient conditions are given for q to belong to D. © 1971 Springer-Verlag.