# Can RK5 be considered as part of Predictor–corrector method?

It seems that the 5th order is the corrector and the 4th order is the predictor. Can RK5 be considered as part of Predictor–corrector method?

No. The 4th order is not a predictor for the 5th order. Instead, what is done is that the $k_i$ are constructed in such a way that one linear combination $b_i k_i$ gives a 5th order method and $\hat{b}_i k_i$ gives a 4th order method. They are calculated independently, but constructed in a way such that the exact same intermediate steps ($k_i$) build both methods, saving the computation of actually computing two RK methods at the same time. This is called method embedding.

Predictor-corrector is when one method is used as the input to the second (usually implicit method). That requires non-trivial calculation to compute the second.