feat: add implementations of CheckInit and OverrideInit

[AayushSabharwal]

@TorkelE @isaacsas this should make it so that your rebuild function is something along the lines of:

function rebuild(prob; u0, p)
   newprob = remake(prob; u0, p)
   u0, p, success = SciMLBase.get_initial_values(newprob, newprob, newprob.f, OverrideInit(), Val(SciMLBase.isinplace(newprob.f); nlsolve_alg = ...)
   @assert success
   newprob = remake(newprob; u0, p)
   return newprob
end

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.

[isaacsas]

OK, awesome! Thanks for sticking with this.

If you feel this is at a point where it would be helpful to test against Catalyst let me know. We have master capped right now to SciMLBase 2.57.1 as that was the last time tests passed (and we wanted to be able to work / merge other PRs).

[ChrisRackauckas]

This should use a cached one from the integrator?

[AayushSabharwal]

Cached what, exactly? 😅 I copied almost all of this from the existing implementation

[ChrisRackauckas]

This allocates

[AayushSabharwal]

Somehow changing this to tmp = algebraic_eqs .* _vec(tmp) allocates more

[AayushSabharwal]

At least, according to @allocated/@allocations