Fix scaling-dependence of termination criterion#47
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The termination criterion for bisecting alpha implicitly assumes the optimal alpha to be in the order of magnitude of 1.0. For larger values of alpha, lower and upper can never get close enough, meaning the bisection never terminates successfully. For smaller values of alpha, the bisection may terminate much too early. This fixes that by checking the relative instead of the absolute difference using numpy.isclose. Also, this adds tests with a large and a tiny optimal alpha that would have failed with the previous termination criterion (they also wouldve failed for much more moderate scaling factors like 100).
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The termination criterion for bisecting alpha implicitly assumes the optimal alpha to be in the order of magnitude of
1.0. For larger values of alpha,loweranduppercan never get close enough, meaning the bisection never terminates successfully. This happens already with an example like this:For smaller values of alpha, the bisection may terminate much too early, returning a very suboptimal alpha.
This fixes that by checking the relative instead of the absolute difference using
numpy.isclose. Also, this adds tests with a large and a tiny optimal alpha that would have failed with the previous termination criterion.An alternative might be to check
np.nextafter(lower, upper) == upper, to ensure finding the exact threshold.See also #18 . A user-defined tolerance as suggested there might be a good future addition. But for now, this change would at least make the function usable for larger alphas.