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Communications of the ACM

Communications of the ACM

Connections between accuracy and stability properties of linear multistep formulas

This paper is concerned with stability and accuracy of families of linear k-step formulas depending on parameters, with particular emphasis on the numerical solution of stiff ordinary differential equations. An upper bound, p = k, is derived for the order of accuracy of A-stable formulas. Three criteria are given for A0-stability. It is shown that (1) for p = k, k arbitrary, A-stability implies certain necessary conditions for A0-stability and for strict stability (meaning that the extraneous roots of &rgr;(&zgr;) satisfy |&zgr;| < 1); (2) for p = k = 2, 3, 4, and 5, A-stability (for k = 5 together with another constraint) implies strict stability; and (3) for certain one-parameter classes of formulas with p = k = 3, 4, and/or 5, A-stability implies A0-stability.

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