Algorithm for Predictor-Corrector Method

For the IVP (), set $x_0 = a, \; y(a) = y_0$ and choose a step size $h$ . Let $x_i = a + i h, \; i=0,1, 2, \ldots, n$ with $x_n = b$ . Let $M$ be the tolerance or the number of iteration for each step. Let $e_{\mbox{ref}}$ be the tolerance on the relative error at each step. For $k=0,1, 2, \ldots, n$

1. compute $y_{k+1}^0 = y_k + h f(x_k, y_k)$
2. for $m=0,1,2,\ldots, M$ , compute $y_{m+1}^{k+1} = y_k + \frac{h}{2} \bigl( f(x_k, y_k) + f(x_{k+1}, y_{k+1}^m) \bigr)$
3. Stop the iterations whenever the absolute of the relative error $\left\vert\displaystyle \frac{ y_1^{k+1} - y_1^k}{y_1^{k+1}}\right\vert < \epsilon$ for some previously prescribed positive real number $\epsilon$ .

Remark 14.4.1   The tolerance in the number of inner iteration has been incorporated in by setting $m=0,1,2,\ldots, M$ .

Figure: Flow-Chart of Predictor-Corrector Method