Make length of Partials known at compile time in ForwardDiff overloads

[jClugstor]

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]

JET test it?

[jClugstor]

using LinearSolve
using ForwardDiff
using Test
using JET

function h(p)
    (A=[p[1] p[2]+1 p[2]^3;
            3*p[1] p[1]+5 p[2]*p[1]-4;
            p[2]^2 9*p[1] p[2]],
        b=[p[1] + 1, p[2] * 2, p[1]^2])
end

A, b = h([ForwardDiff.Dual(5.0, 1.0, 0.0), ForwardDiff.Dual(5.0, 0.0, 1.0)])

prob = LinearProblem(A, b)

cache = init(prob)

solve!(cache)

ext = Base.get_extension(LinearSolve, :LinearSolveForwardDiffExt)

@test_opt ext.linearsolve_dual_solution([1.0, 1.0, 1.0], [[1.0, 1.0, 1.0], [1.0, 1.0, 1.0]], cache)

This passes now.

Without this change there is indeed a runtime dispatch.

There are also a bunch of runtime dispatches for an overall solve!(cache), so I can try to work on those as well.

[ChrisRackauckas]

Okay, but you still didn't add that to the JET tests.

[hersle]

What is the state of this? I am currently hanging back with old versions of LinearSolve. It would be really nice to get back the old performance in SciML/OrdinaryDiffEq.jl#2837.

[ChrisRackauckas]

Test fails when rebased.

[jClugstor]

Tests pass now

[hersle]

Thanks!