Equivalent Forms of Multistep Formulas

For uniform meshes it is shown that any linear k-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 polynomial 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 k-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 $\text {P(EC)}^\ast$ formulas but not for $\text {P(EC)}^\ast \text {E}$ formulas.