feat: add NonlinearSolveHomotopyContinuation.jl

[AayushSabharwal]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

Add any other context about the problem here.

[ChrisRackauckas]

What is left here?

[AayushSabharwal]

JuliaHomotopyContinuation/HomotopyContinuation.jl#609

TL;DR: Despite testing the custom system/homotopy here the same way HomotopyContinuation.jl tests theirs, the solve fails because NaNs creep in somewhere somehow. I haven't yet been able to figure out why.

[ChrisRackauckas]

Add this to the CI

[AayushSabharwal]

This is a bit stochastic since whether HC considers a root "real" depends on if the imaginary part is less than the accuracy of their solve. Sometimes their results are more accurate, sometimes less.

[ChrisRackauckas]

That should be easily patchable on their end in like a day?

[AayushSabharwal]

JuliaHomotopyContinuation/HomotopyContinuation.jl#614

[ChrisRackauckas]

It would be nice to have a short tutorial that showcases a place where Newton methods fail but this gives an accurate answer. Do you have one?

[AayushSabharwal]

iirc bruss with some guesses can show this behavior. Will write something