solver: symmetric indefinite LDLᵀ factorization with inertia

A supernodal LDLᵀ on the Cholesky factor's symbolic factorization (ordering, supernodes, fill,
cap and parallel subtree tasks shared through a Symbolic and a generic driver; the Cholesky
numerics are unchanged bit for bit). Pivots by the bounded Bunch–Kaufman (rook) search of
Ashcraft, Grimes & Lewis among each supernode's columns (Schenk & Gärtner's supernode
Bunch–Kaufman), 1 × 1 and 2 × 2, right-looking panels whose trailing update reuses the tiled
Cholesky kernels with L D formed on the fly. Reports the inertia; a column without a pivot above
round-off is a located Singular error; NaN is refused. Symmetric::new picks Cholesky, LDLᵀ, or
Cholesky with an LDLᵀ fallback; solve_refined adds iterative refinement. 1.1–1.2× the Cholesky
time on a 140 000-equation hex8 pattern.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

crates/solver/src/analysis.rs

@@ -265,6 +265,12 @@ pub(crate) fn factor_error(e: FactorError, mesh: &Mesh, reduction: &Reduction) -
                 "the stiffness is singular at {at}: the model can move without straining (missing or insufficient boundary conditions, a part not connected to the rest, or a reference point direction nothing holds)"
             ))
         }
+        FactorError::Singular { column } => {
+            let at = reduction.eq_dof.get(column).map_or_else(|| "an unknown degree of freedom".to_string(), |d| mesh.describe_dof(*d));
+            Error::Singular(format!(
+                "the matrix is singular at {at}: no pivot of its supernode is more than round-off (a mechanism, or a shift or load at an eigenvalue)"
+            ))
+        }
     }
 }