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I have a very sparse non-linear system $N(u) = 0$ that can be solved as a time-dependent ODE, $\frac{du}{dt} = N(u)$, and explicitly integrated until $\frac{du}{dt} = N(u) = 0$, e.g. by forward euler, $u^{n+1} = u^n+\Delta t N(u^n)$.

In the process of this numerical integration, is there any way to recover the eigenvalues of the system by examining something about the growth/response in $u$ and it's dependence on the timestep? Gershgorin's theorem provides a bound on the spectrum, but in practice this bound's range is too large to be useful. I don't need exact eigenvalues, but relatively tight approximations would be great. I suspect there's something about this in the literature of linear time-invariant (LTI) theory, but I don't know where to start looking.

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