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Mention lyapunov algorithms in doc

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Ian Jauslin 2025-01-31 10:55:41 -05:00
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docs/nstrophy_doc.tex
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@ -518,6 +518,7 @@ To do the computation numerically, we drop the limit, and compute the logarithm
\indent \indent
In practice, we approximate $\varphi_{t_{i-1},t_i}$ by running a Runge-Kutta algorithm for the tangent flow equation. In practice, we approximate $\varphi_{t_{i-1},t_i}$ by running a Runge-Kutta algorithm for the tangent flow equation.
Because the tangent flow equation depends on $u(t)$, we must run it at times for which $u$ has been computed, so the Runge-Kutta algorithm for the tangent flow cannot be an adaptive step method, and should be one of {\tt RK4} for the fourth order algorithm or {\tt RK2} for the second order algorithm.
To obtain the full matrix, we consider every element of the canonical basis as an initial condition $\delta_0$. To obtain the full matrix, we consider every element of the canonical basis as an initial condition $\delta_0$.
We then iterate the Runge-Kutta algorithm until the time $t_0$ (chosen in one of two ways, see below), at which point we perform a QR decomposition, save the diagonal entries of $R$, replace the family of initial conditions with the columns of $Q$, and continue the flow from there. We then iterate the Runge-Kutta algorithm until the time $t_0$ (chosen in one of two ways, see below), at which point we perform a QR decomposition, save the diagonal entries of $R$, replace the family of initial conditions with the columns of $Q$, and continue the flow from there.
The choice of the times $t_i$ can be done either by fixed-length intervals, specified with the option {\tt lyapunov\_reset}, or the QR decomposition can be triggered whenever $\|\delta\|_1$ exceeds a threshold, specified in {\tt lyapunov\_maxu} (after all, the intervals are used to prevent $\delta$ from becoming too large). The choice of the times $t_i$ can be done either by fixed-length intervals, specified with the option {\tt lyapunov\_reset}, or the QR decomposition can be triggered whenever $\|\delta\|_1$ exceeds a threshold, specified in {\tt lyapunov\_maxu} (after all, the intervals are used to prevent $\delta$ from becoming too large).