guessdegree works when it should not work (from gforge #8865)
Imported issue: Initially reported by @chevilla in https://gforge.inria.fr/tracker/?group_id=1015&aid=8865
Currently, we observe the following behavior:
guessdegree(1,[-1,1],1b-20,1/sin(x));
[7;7]
p=remez(1,[|1,...,7|],[-1;1],1/sin(x)); round(-log2(abs(dirtyinfnorm(p/sin(x)-1,[-1;1]))),10,RD);
2.5375e1
p=remez(1,[|1,...,6|],[-1;1],1/sin(x)); round(-log2(abs(dirtyinfnorm(p/sin(x)-1,[-1;1]))),10,RD);
1.71875e1
Several remarks:
a) guessdegree has not been made for such purpose since it is supposed to search a polynomial in a basis of the form [|0,...,n|]. I would expect guessdegree to crash when looking for an approximation in relative error to a function that has a zero in the interval.
We should understand why it does not crash.
b) More surprising: it gives a correct answer. However, I do not know if this is by pure chance or not. Again, this should be investigated.
c) Independently of the first two items, it is an interesting feature that guessdegree manages to deal with such instances. We should think of doing something in this direction