Variable-step Adams predictor-evaluate-correct-evaluate procedure

Input an integer $k\ge 1$, a finite grid $t_0,\dots ,t_N$ with $N\ge k$, starting states $Y_0,\dots ,Y_{k-1}\in {\mathbb{R}}^d$, and a vector field $f:U\to {\mathbb{R}}^d$ on its evaluation domain $U$. Before evaluating $f$, if any starting state has dimension different from $d$, return invalid-input; otherwise, if the grid is not strictly increasing, return invalid-grid. Evaluate and store $F_j=f(t_j,Y_j)$ for $j=0,\dots ,k-1$; an undefined evaluation outside $U$ returns evaluation-failure with the valid prefix.

For $n=k-1,\dots ,N-1$, construct the predictor weights $b_j^n$ and corrector weights ${\widehat{b}}_j^n$ from the actual nodes as in Variable-step Adams weights are integrals of the actual-node Lagrange bases, then do the following:

1. Predict $Y_{n+1}^P=Y_n+{\sum }_{j=0}^{k-1}{b}_j^n{F}_{n-j}$.
2. Evaluate $F_{n+1}^P=f(t_{n+1},Y_{n+1}^P)$. If undefined, return evaluation-failure with phase predictor.
3. Correct once by setting $Y_{n+1}=Y_n+{\widehat{b}}_0^n{F}_{n+1}^P+{\sum }_{j=1}^k{\widehat{b}}_j^n{F}_{n+1-j}$.
4. Evaluate and store $F_{n+1}=f(t_{n+1},Y_{n+1})$ for the next predictor. If undefined, return evaluation-failure with phase post-corrector and the computed $Y_{n+1}$.

After all steps return completed with the states. This is one predictor and one corrector pass; it does not solve the fully implicit Adams corrector equation and asserts no convergence, stability, or error bound.