The Initial Value Problem y' = f(x, y); y(c) = d 103A simple single-step, predictor-corrector method, for instance, might em-
ploy Euler's formula for the predictor and the trapezoidal scheme (Adams-Moulton formula with m = 0) for the corrector. Hence, one would have the
following iteration procedure:(24c)
<k> _ + h(j(Xn+i, Y~!;:-^1 >) + f(xn, Yn))
Yn+l - Yn
2, k = 1, 2, ...
The lo cal discretization error for the predictor formula (24p) is h^2 y(^2 ) (~) /2
and the local error of the correction formula (24c) is h^3 yC^3 l(~)/12.
The following multistep, predictor-corrector, iteration procedure was de-
rived and published in 1926 by the American mathematician William E. Milne
(1890-1971). The explicit predictor formula was derived by choosing r = n,
p = 3, q = 1, and m = 3 and the implicit corrector was derived by choosing
r = n + 1, p = 2, q = 1, and m = 2 or m = 3 (which yield the same formula).
Hence, the Milne predictor-corrector iteration is(25p) Yn+l <0> = Yn-3 + 4h(2j n - J n-1 + 2fn-2)
3(25c) h(f(xn+1, Y~!;:-^1 >) + 4fn + fn-i)
Yn+l = Yn-1 +
3, k = 1, 2, ...
The local discretization error of each of these formulas is of order h^5.
Another commonly used predictor-corrector method for which each formula
has local error of order h^5 but a slightly smaller error coefficient than Milne's
method employs the Adams-Bashforth formula with m = 3 as the predictor
and the Adams-Moulton formula with m = 2 as the corrector. This iteration
procedure is
(26p)
<0> h(55fn - 59fn-l + 37fn-2 - 9fn-3)
Yn+l = Yn + 24(26c)
<k> h(9j(xn+l, Y~!;:-
1
>) + 19fn - 5fn-l + fn-2)
Yn+l = Yn + 24 ' k = 1, 2, ...In order to approximate the solution of an initial value problem using a
predictor-corrector a lgorithm, one needs to specify (1) the stepsize, h, to be
taken; (2) the maximum absolute iteration error, E = IY~!r -Y~!;:-^1 > I, or the
maximum relative iteration error, E = IY~!r -y~;_-^1 > l/IY~!?i, to be a llowed