3 ms·
This is nice and pretty, but it still requires evaluating the function in infinitely many points. Is there an easy fix for that?
by thomasahle 9y ago
This is nice and pretty, but it still requires evaluating the function in infinitely many points. Is there an easy fix for that?
- kmill 9y ago1. If you can calculate rough bounds for the terms and their derivatives in the winding number integral, you can figure out how densely you need to sample the curve to calculate the integral exactly. 2. If the winding number is zero, there still might be a root in the region because poles and zeros contribute opposite winding number, and they might exactly cancel. 3. Polynomials are great because the bounds are easy to compute and because you don't have to worry about cancelation since the only pole is at infinity.
- EtDybNuvCu 9y agoYou'd think so, but check out the 'smooth surprise' in Kahan's great writeup: https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf We can approximate quite a bit, but ultimately further analysis is required to show that there are no such surprises on any particular root-finding problem.