Abstract
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.
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Wagon, S. (2010). Optimization. In: Wagon, S. (eds) Mathematica in Action. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75477-2_14
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DOI: https://doi.org/10.1007/978-0-387-75477-2_14
Publisher Name: Springer, New York, NY
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