Optimization

The top image is the machine-precision trajectory of 0.1 under the quadratic map 4x (1 − x). It suffers from roundoff error and the terms beyond the 60th are not the same as the values in the true trajectory of $$\frac{1}{{10}}$$ shown just below it. But it turns out that the noisy trajectory can be shadowed, meaning that there is a value near $$\frac{1}{{10}}$$ — it turns out to be 0.09999999999999998884314845320503 — whose true trajectory under f matches the noisy trajectory very closely (to within 10−15). Finding this value is a true needle-in-a-gigantic-haystack problem, but sophisticated optimization algorithms are capable of getting it in under one second, and with only a few lines of code. The third image shows the absolute difference between the noisy trajectory and true trajectories in the vicinity of the shadow value; this gives an indication of the difficulty of finding this value, since the spike that defines it is very narrow.