A normal form for multistep companion matrices

Linear multistep methods, in particular Backward Differentiation Formulas ( BDF), are widely used for the numerical integration of differential equatio ns and in particular for stiff systems. Although the classical convergence theory for linear multistep methods is quite well-developed, there is no co mprehensive stiff convergence theory. Special classes of stiff problems hav e been covered so far, see e.g. Hairer, Wanner(11) for the analysis of stif f linear systems or Lubich(13) for the analysis of problems in standard sin gular perturbation form. We present a theoretical approach for the analysis of stiff linear systems which will be extended to nonlinear problems in fo rthcoming papers.