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Mathematics of Computation

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Equivalent forms of multistep formulas

Author: Robert D. Skeel
Journal: Math. Comp. 33 (1979), 1229-1250
MSC: Primary 65L05
Corrigendum: Math. Comp. 47 (1986), 769.
MathSciNet review: 537967
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Abstract: 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$, which can be used as predictor formulas. Different polynomial bases lead to different equivalent forms of multistep formulas. In particular, local monomial bases yield Nordsieck formulas. An explicit one-to-one correspondence is established between Nordsieck formulas and k-step formulas of order at least k, and a strong equivalence result is proved for all but certain pathological cases. Equivalence is also shown for P(EC)$ ^\ast$ formulas but not for P(EC)$ ^\ast$E formulas.

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Keywords: Linear multistep formula, multistep formula, linear multistep method, multistep method, Nordsieck method, multivalue method, predictor-corrector method
Article copyright: © Copyright 1979 American Mathematical Society

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