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For uniform meshes it is shown that any linear fc-step formula can be formulated so that only k values need to be saved between steps. By saving additional m values it is possible to construct local polynomial approximations of degree k + m — 1, which can be used as predictor formulas. Different polynomi- al bases lead to different equivalent forms of multistep formulas. In particular, local monomial bases yield Nordsieck formulas. An explicit one-to-one correspondence is established between Nordsieck formulas and fc-step formulas of order at least k, and a strong equivalence result is proved for all but certain pathological cases. Equivalence is also shown for P(EC)* formulas but not for P(EC)*E formulas. 1. Introduction. Multistep methods are the oldest and the most important class of numerical methods for solving systems of ordinary differential equations. Imple- mentations of these methods have become increasingly sophisticated in the last two decades. One paper having a considerable impact on the development of these algorithms is that of Nordsieck (1962), which introduces a class of formulas closely related to multistep formulas. It is the purpose of the present paper to explore thoroughly the relationship between multistep and Nordsieck formulas on a uniform mesh. The results apply to a limited extent to variable-order variable-stepsize algorithms which discourage order and stepsize changes, because such algorithms often produce sequences of values computed with the same formula and stepsize.

Equivalent forms of multistep formulas