Hybrid Methods for Initial Value Problems in Ordinary Differential ...
Hybrid Methods for Initial Value Problems in Ordinary Differential ...
Hybrid Methods for Initial Value Problems in Ordinary Differential ...
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HYBRID METHODS FOR INITIAL VALUE PROBLEMS 75<br />
where the Pli ,P2i ,Pad, and P di are polynomials <strong>in</strong> h with bounded coefi-<br />
cients.<br />
Henrici [3, Lemma 5.51 shows that if<br />
is the polynomial of a stable method, and if<br />
then there exists a constant I? < w such that / do I 5 r, p = 0, 1, -.. .<br />
Multiply (2.7) by aN-n-l and sum from n = k to N -1to get<br />
N-1 k<br />
(2.10) = x6N-n-1Lh(~n, h) + x (bounded multiples of E; and ~i*)<br />
n-k i=O<br />
From (2.9), 60 = 1and terms like<br />
There<strong>for</strong>e, the left hand side of (2.10) equals EN. The last term of the<br />
right hand side <strong>in</strong>volves each ei and ei* no more than k + 1times. There-<br />
<strong>for</strong>e (2.10) gives the bound<br />
where M and C are bounded <strong>for</strong> all h less than or equal to some ho . Now<br />
N - k 5 (b - a)/h, and
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