Consider the set of multistep formulas ∑<supscrpt><italic>l</italic>-1</supscrpt><subscrpt><italic>j</italic><subscrpt>mn</subscrpt>-<italic>k</italic></subscrpt> <italic>α<subscrpt>ij</subscrpt></italic><italic>x</italic><subscrpt><italic>mn</italic>+<italic>j</italic></subscrpt> - <italic>h</italic> ∑<supscrpt><italic>l</italic>-1</supscrpt><subscrpt><italic>j</italic><subscrpt>mn</subscrpt>-<italic>k</italic></subscrpt><italic>β<subscrpt>ij</subscrpt></italic><italic>x</italic><subscrpt><italic>mn</italic>+<italic>j</italic></subscrpt> = 0, <italic>i</italic> = 1, ···, <italic>l</italic>, where <italic>x</italic><subscrpt><italic>mn</italic>+<italic>j</italic></subscrpt> = <italic>y</italic><subscrpt><italic>mn</italic>+<italic>j</italic></subscrpt> for <italic>j</italic>= -<italic>k</italic>, ···, -1 and <italic>x<subscrpt>n</subscrpt></italic> = ƒ<subscrpt><italic>n</italic></subscrpt> = ƒ(<italic>x<subscrpt>n</subscrpt> , t<subscrpt>n</subscrpt></italic>). These formulas are solved simultaneously for the <italic>x</italic><subscrpt><italic>mn</italic>+<italic>j</italic></subscrpt> with <italic>j</italic> = 0, ···, <italic>l</italic> - 1 in terms of the <italic>x</italic><subscrpt><italic>mn</italic>+<italic>j</italic></subscrpt> with <italic>j</italic> = -<italic>k</italic>, ··· , - 1, which are assumed to be known.
Then <italic>y</italic><subscrpt><italic>mn</italic>+<italic>j</italic></subscrpt> is defined to be <italic>x</italic><subscrpt><italic>mn</italic>+<italic>j</italic></subscrpt> for <italic>j</italic> = 0, ··· , <italic>m</italic> - 1. For <italic>j</italic> = <italic>m</italic>, ··· , <italic>l</italic> - 1, <italic>x</italic><subscrpt><italic>mn</italic>+<italic>j</italic></subscrpt> is discarded. The set of <italic>y</italic>'s generated in this manner for successive values of <italic>n</italic> provide an approximate solution of the initial value problem: <italic>y</italic> = ƒ(<italic>y, t</italic>), <italic>y</italic>(<italic>t</italic><subscrpt>0</subscrpt>) = <italic>y</italic><subscrpt>0</subscrpt>. It is conjectured that if the method, which is referred to as the composite multistep method, is <italic>A</italic>-stable, then its maximum order is 2<italic>l</italic>. In addition to noting that the conjecture conforms to Dahlquist's bound of 2 for <italic>l</italic> = 1, the conjecture is verified for <italic>k</italic> = 1. A third-order <italic>A</italic>-stable method with <italic>m</italic> = <italic>l</italic> = 2 is given as an example, and numerical results established in applying a fourth-order <italic>A</italic>-stable method with <italic>m</italic> = 1 and <italic>l</italic> = 2 are described. <italic>A</italic>-stable methods with <italic>m</italic> = <italic>l</italic> offer the promise of high order and a minimum of function evaluations—evaluation of ƒ(<italic>y, t</italic>) at solution points. Furthermore, the prospect that such methods might exist with <italic>k</italic> = 1—only one past point—means that step-size control can be easily implemented
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