fix: fix total degree with many parameters threaded (#599)

[oameye]

resolve #599

[PBrdng]

Awesome. Thanks!

You said this problem only appears when using :total_degree. Is that right?

[oameye]

I didn't test the other methods, but probably the rest is covered by the tests?

[oameye]

It seems that polyhedral is also broken with many parameters (threading doesn't matter this time) but for another reason (see #601).

[PBrdng]

Okay. I would like to fix #601 while we are at it before merging. Thanks for noticing.

[oameye]

I tried solve #601 but can't seem to solve it as I don't know enough of the internals. Could you take a look?

[PBrdng]

Yes, I will pick it up from here. Don't worry.

[PBrdng]

I added a warning message for when using total_degree and an error message for when using polyhedral on multiple parameters. I also changed your example the way it should best be solved.

[PBrdng]

@oameye I'm waiting for you to take a look at the PR and then merge it later today or tomorrow.

[oameye]

This is good to go for me :)

[oameye]

Can it be that this recommended way of computing the roots of many parameters sets does not always guarantee to find all solutions? Whereas doing a total degree does guarantee to find all solutions?

[PBrdng]

No. In theory this method finds all solutions (if the generic parameters are chosen complex).

[oameye]

Sorry one other question:

What if I want to avoid zero solutions? For one parameter set I can use :polyhedral with only_non_zero=true. But if I use the recommended way, I will just get one less solution which is not necessarily non zero. Do you never of a better way? Or is running :polyhedral in a loop the best way for the moment?

[PBrdng]

Sorry one other question:

What if I want to avoid zero solutions? For one parameter set I can use :polyhedral with only_non_zero=true. But if I use the recommended way, I will just get one less solution which is not necessarily non zero. Do you never of a better way? Or is running :polyhedral in a loop the best way for the moment?

The recommended method should, in theory, find all non-zero solutions (this doesnt mean that some solutions can't potentially be zero). Do you have an example that I can look at?

[saschatimme]

Just for context: The non-zero solutions option works by constructing a slightly different start system than in the general case. I assume in your case, you are actually sampling from a parameter space that is smaller than C^m (m = number of parameters). So by solving once for a general sample of your parameter space and then reusing these solutions you should have only the results you are interested in.

[PBrdng]

I think this is true in general (provided setting the parameters does not change the support). However, all solution nonzero is only true for generic coefficients, which is not necessarily the case here.

[oameye]

Thank you both for the answers. This is really useful. In general, the zero solutions are non-interesting solutions for my use case. So I was hoping it could give a speedup in finding the solutions. However, in my testing it doesn't seem to make much of a difference. Probably, this is because it just tracks one less solution of the generic system compared to only_non_zero=false case.