Coefficients for
Linear multistep methods
for ordinary differential equations $y' = f(y)$
can be found by making an approximation like
$\qquad y_{t+1} \approx y_t + \sum a_i y'_{t-i} $
good for polynomials ${ 1,\ t,\ t^2 ... }$ .
Why polynomials -- how about fitting $e^{\ z_i t}$ for a few problem-specific $z_i$ ? (If $z_i$ is complex, fit a sin-cos pair, to keep everything real.) Least-squares can easily fit polynomial + exponential terms, and is robust and flexible -- can fit more terms, or down-weight noisier terms. Exponentials $e^{\ z_i t}$ should be good for nearly linear $y' \approx z_i y$ . My questions:
- Do such custom-fit-exponential methods work ? Are they used in practice ? Examples would be welcome.
- Given a trajectory $y(t)$ in $\mathbb{R}^n$ for a particular ODE, how can one fit a matrix $Z: y(t) \approx e^{Z t} y_0$ ?
Added: the question is not how to derive coefficients that fit exponentials
(least squares is one way to do that),
but whether the flow
$\qquad$ specific $f(y), y(t) \to $ fit $e^{\ Z t} \to$ coefficients $\to$ custom-fit solver
is much better than than just using off-the-shelf coefficients.
You can do that. However, what you want is to have in $$ e^h\approx 1+h\sum_{k=0}^s a_ke^{-kh} $$ an error $O(h^{p+1})$ of maximal order. Using $z=1-e^{-h}$ this leads to the identity $$ z-(1-z)\ln(1-z)\sum_{k=0}^s a_k(1-z)^k=O(z^{p+1}) $$ and since $\ln(1-z)=O(z)$, $$ \sum_{k=0}^s a_k(1-z)^k=\frac{z}{(1-z)\ln(1-z)}+O(z^p) $$ which can be evaluated in truncated Taylor series arithmetics.