A hallmark of the Runge-Kutta methods is that stepping to the next point, (tm+1, ym+1 ), uses information at (tm+1, ym+1 ) but at no other prior points. We must evaluate the slope function (derivative) at one or more subsequent points, depending on the order of the method. The fact that these methods do not use the accumulated information of prior points, plus the lack of a convenient error estimation procedure, suggest there might be value in devising other methods.
These newer methods turn out to be the Predictor-Corrector ones. As the name implies, a value for ym+1 is first predicted by one formula and it is then corrected by another. If required, the latter value can be re-corrected by iteration.
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