[Minuit2] Fix analytical Hessian/G2 transform for parameters with limits

[guitargeek]

When a parameter has limits, Minuit2 minimizes in an internal coordinate q that is a non-linear function of the external parameter. Converting a user-provided external Hessian (or G2) to internal coordinates therefore requires, on the diagonal, an extra term involving the second derivative of the transformation:

H_int(i,i) = (dx/dq)^2 * H_ext(i,i) + (d^2x/dq^2) * g_ext(i)

AnalyticalGradientCalculator::Hessian() and ::G2() only applied the first-order Jacobian factor (dx/dq)^2 and dropped the (d^2x/dq^2) * g_ext(i) term. The numerical path does not have this problem because it differentiates directly in internal coordinates, so the term is included implicitly.

This was introduced in 888a767 ("[math][minuit2] First implementation in Minuit2 of mNHesse using external Hessian calculator"), which added the external->internal Hessian/G2 transform.

At the minimum the external gradient is ~0, so the missing term vanishes and the final errors are correct. But it is non-zero everywhere else, in particular at the seeding point. When a parameter starts close to a limit (dx/dq small) and far from the solution (g_ext large), the dropped term dominates: the seed G2/covariance built by MnSeedGenerator is corrupted and Migrad can take a catastrophic first step, silently converging to a wrong result that is still reported as valid. Relying on the numerical Hessian for the same fit works fine.

Fix this by adding the missing curvature term:

• add D2Int2Ext() to Sin/SqrtLow/SqrtUp parameter transformations and a dispatching MnUserTransformation::D2Int2Ext() (returns 0 without limits, so unlimited fits are unchanged)
• add (d^2x/dq^2) * g_ext(i) to the diagonal in AnalyticalGradientCalculator::Hessian() and both branches of ::G2()

The off-diagonal entries are unchanged: the transformation is per-parameter (diagonal), so mixed second derivatives only get the (dx/dq_i)(dx/dq_j) factor.

Add a regression test (Minuit2.AnalyticalHessianLimitTransformation). It does not rely on the end-to-end Migrad convergence behaviour, which is chaotic and compiler/optimization dependent for the reproducer, but instead checks the analytical internal Hessian/G2 against a central finite difference of the internal gradient at a near-limit, non-minimum point. Without the fix the G2 value is off by several orders of magnitude and has the wrong sign.

Closes #22692.

🤖 Done with the help of AI.