a more reasonable sorting of complex numbers

[edgarcosta]

In the current sorting, we have z < conjugate(z) and conjugate(z) < z.

I propose sorting on the real part; if they overlap, use the imaginary part.

Note that in the library, _acb_vec_sort_pretty is used to sort polynomials' roots, which are multisets of non-overlapping balls.
Therefore, we will obtain a predictable and stable sorting with the new comparison function.

PS: this will break some tests at Sagemath, as the roots will now come in a different order, but this is how I got into this rabbit hole.

[edgarcosta]

@fredrik-johansson, I might have some tests to fix, but before I do that, please let me know if you agree with the change.

[fredrik-johansson]

I agree the comparison function is buggy. However...

Therefore, we will obtain a predictable and stable sorting with the new comparison function.

... I don't think this is true. Even if the roots are non-overlapping, real or imaginary parts can be overlapping. No matter how one defines the comparison function, the order of non-exact balls will be essentially random in case of componentwise overlap (though one can make such cases exceedingly rare by using a more artificial comparison function, e.g. mapping a+bi -> a+bc where c is some fixed, unattractive real number).

Now, there is also some logic to the current comparison: it intended to put real roots first and then group complex conjugates. A more robust version of the idea can be found in qqbar_cmp_root_order.

It would be useful to provide both "lexicographic" and "root order" comparisons/sorts for complex numbers.

This could also be implemented using generics with the option to error when the comparisons fail.

[edgarcosta]

I agree.
Honestly, I was thinking about this from the perspective that these roots arise from an irreducible polynomial defined over Q, as I am trying to order the embeddings of a number field into C.
In this scenario, one might have several roots with overlapping real parts $x$, but their imaginary parts will be distinguished.
I also agree that these might no longer overlap at higher precision, but while they overlap the order will not change, and it is predictable.