I'm trying to find the Chebyshev interpolate for an ODE in a given interval.
That is, given an ODE that looks something like:
$$y'' = g(y) \ y'$$
I want to numerically integrate it inside the window $[0, t)$ and then form an interpolating polynomial for it (in a Chebyshev basis. An introduction to this idea is in this chapter in Numerical Recipes in C).
Forming the Chebyshev polynomial usually involves evaluating the function at Chebyshev points $cos(\Large{\frac{2 \pi i}{2^n}})$, where you actually increase $n$ over several iterations and reuse old evaluations of the function you're approximating. But I don't think that will work here since $g(y)$ is dependent on $y$. Presumably I'd have to start the integration fresh when I change $n$.
Are there any tricks I can use here beyond just using something like Runge-Kutta to integrate forward to each of the Chebyshev points? I'm thinking something like a modified Bulirsch-Stoer algorithm where you warp the domain so that the modified midpoint abscissas from Bulirsch-Stoer fall on the Chebyshev points. Hopefully if the integration is converging after the full step size $H$ it'll converge at all the other Chebyshev points, too?