Fix analytic Jacobian for tied parameters in non-linear modeling fitters - #20711

Description

This pull request is to address the analytic Jacobian being wrong for models with tied parameters in the non-linear fitters (LevMarLSQFitter, TRFLSQFitter, DogBoxLSQFitter, LMLSQFitter).

_NonLinearLSQFitter._wrap_deriv computed the full analytic derivative and then deleted the columns of both the fixed and the tied parameters. That is correct for fixed parameters, but for a tied parameter p_t = f(p_free) it drops the ∂y/∂p_t · ∂f/∂p_free term from the derivative with respect to p_free, so fits could stop away from the least-squares minimum (see #20664 for the impact on [O III] doublet fits in specutils).

The full derivative is now multiplied by the Jacobian of the mapping from free to full model parameters: identity rows for free parameters, zero rows for fixed ones, and rows for tied parameters estimated by central differences of the tied callables (which can be arbitrary functions of the model, so their derivatives are not known in advance). Models with only fixed parameters take the same path as before. The cost is two evaluations of the tie expressions per free parameter per Jacobian call.

This is the same factorization used by the emission-line fitting in FastSpecFit (J = W · M · J_I · J_S, see py/fastspecfit/emline_fit/sparse_rep.py), where J_S is known exactly because ties there are restricted to fixed multiplicative factors. No code from it is used here.

Tests added:

• test_tied_par_analytic_deriv: the [O III] 4959/5007 doublet case from the issue, comparing each fitter (with the analytic Jacobian) against scipy.optimize.leastsq on the equivalent function of the free parameters only. Fails on main.
• test_wrap_deriv_tied_chain_rule: compares the Jacobian returned by _wrap_deriv against finite differences for a model with several ties and a fixed parameter, for both col_fit_deriv=True and False, and checks the model parameters are left unchanged. Fails on main.

While working on this I found that tied expressions depending on other tied parameters see values from the previous call; that is reported separately in #20710 and is not changed here.

Fixes #20664

AI Disclosure

This pull request was developed with Claude Code using Claude Opus 5.5 (claude-opus-5-5). It was used to investigate the issue (including comparing with the approach in FastSpecFit), reproduce the bug, and draft the code change, tests, changelog entry, and this description. I reviewed the changes and verified the tests fail on main and pass with the fix, and that the full astropy.modeling test suite passes.

• I certify that I am human and that I take full responsibility for this pull request including all interactions with reviewers.

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