The use of exponential sums in step by step integration

customary to assume that F(t) is a polynomial of degree n — 1 whose coefficients are determined by the given values of F, i.e., the values at t, t — h, • • •, t — (n — \)h.x If the polynomial F(t) is specified by these conditions, then the desired integral is a linear combination of the values F{t),F{t-h), ■■■,F{t-{n-\)h). Greenwood2 has pointed out that the polynomial assumption for F(t) may not always be the most desirable one. In particular it might be assumed that F{t) is a linear sum of exponentials and still obtain that the integral is a linear combination of the values F(t), F(t — h), • ■ •, F(t — (n — \)h) with coefficients independent of t. The authors also were led to this conclusion by their own previous work.3 Sections 1-5 constitute a discussion of practical procedures and contain a description of the results obtained. Sections 6-9 contain the proofs of the results. The present discussion is for an individual step. The final total error effect on the solution and stability considerations is not given. 1. In an open integration procedure, the expression