I was watching one of Brian Douglas’ videos on control theory (Discrete Control #5) and ran into a simple derivation of an approximation I presented earlier.
Back in April I wrote several post on simple approximations for log, exp, etc. In this post I gave an approximation for the exponential function:
The control theory video arrives at the same approximation as follows.
As I believe I’ve suggested before here, in a derivation like the one above, where you have mostly equalities and one or two approximations, pay special attention to the approximation steps. The approximation step above uses a first order Taylor approximation in the numerator and denominator.
The plot below shows that the approximation above (the bilinear approximation) is more accurate than doing a single Taylor approximation, approximating exp(x) by 1 + x (linear approximation).
Here’s a plot focusing on the error in the bilinear and linear approximations.
The bilinear approximation is hard to tell from 0 in the plot above for x up to 0.5.
The derivation above is simple, but why is the result so good? An explanation in terms of Padé approximation is given here.