Minimization and uncertainty¶

Independent parameters and Jacobian¶

For the weighted residual vector \(\mathbf r\) defined in Rietveld method, let \(J_{ij}=\partial r_i/\partial p_j\). The Jacobian columns correspond to independent free parameters; fixed, symmetry-related and constrained dependent values are not extra columns.

The scientific models and their derivatives share their declared parameter conventions. A parameter that is only stored has no nonzero model derivative merely because a numerical value is present in the file.

Levenberg–Marquardt step¶

A scaled damped least-squares step has the form

\[(J^{\mathsf T}J+\lambda D)\Delta\mathbf p=-J^{\mathsf T}\mathbf r.\]

\(D\) supplies the diagonal scaling/damping convention; \(\lambda\) controls damping. The implemented strategy controls the acceptance and damping search, while the scientific objective remains the weighted residual sum.

Descent selector	Meaning
`ladder`	Damping-ladder LM search
`fast_descent`	Additional in-iteration damping re-search
`fast_descent_linear_snap`	Re-search with a full linear-block solve
`fast_descent_guarded_linear_snap`	Accept a linear snap only when it improves the objective
`multistart_prune`	A two-probe multi-start/pruning strategy using one progress stream

The project records its descent, positive relative objective tolerance and positive iteration budget. A budget stop is not the same as convergence. Some strategies have restrictions when bounds are declared; inspect the reported diagnostic rather than assuming every combination is supported.

Covariance and standard uncertainties¶

At the final state, the current covariance convention is the scaled inverse curvature

\[C=\chi^2_{\rm red}(J^{\mathsf T}J)^+,\qquad u(p_j)=\sqrt{\max(C_{jj},0)},\qquad \rho_{jk}=C_{jk}/\sqrt{C_{jj}C_{kk}}.\]

The Jacobian here is already weighted. The implementation computes a column-scaled pseudoinverse and transforms back to physical parameter units, using the undamped final curvature. The + denotes that numerical pseudoinverse; it is not an inverse of the damped LM step matrix.

The reduced-χ² scaling is an explicit convention. Do not silently substitute an unscaled covariance from another program. A zero-sensitivity direction or numerical degeneracy requires inspecting the associated diagnostics; a zero reported uncertainty does not make an unmeasured parameter exact.

Constraints and bounds¶

A constraint relates quantities and reduces independent freedom. Physical admissibility limits, editable optimizer bounds and symmetry restrictions serve different purposes. The application's displayed minimum and maximum are model-admissibility information; they are not a promise of controls for setting optimizer search bounds.

Use Parameters for practical interpretation. The fit's observation set and parameter count also determine the reliability factors.