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I have a discrete time system of the form $$\mathbf{y}[k+1] = f(\mathbf{y}[k])$$ that I want to step until a specific condition $$g(\mathbf{y}[K]) < 0,$$ usually $K\sim1000$.

Evaluations of $f$ are expensive, but $f$ is pretty well-behaved: for example there are lots of cases where $f$ is constant, or close to linear in $k$, for $50 < k < K-50$. In that region I could easily extrapolate over several steps to avoid having to calculate $f$ at each step, but close to the ends I need to be more careful.

So my question is: are there any adaptive stepping algorithms for cases like this, which tell me how many steps I can extrapolate over? e.g. a discrete version of Dormand-Price?

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