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/heat.rs

@@ -310,7 +310,7 @@ fn factor(a: &Csr, limits: &Limits, mesh: &HeatMesh, eqs: &Equations) -> Result<
         }
         FactorError::OutOfMemory { entries } => Error::Limit(format!("not enough memory for the factorization ({entries} entries)")),
         FactorError::Internal(what) => Error::Internal(what),
-        FactorError::NotPositiveDefinite { column } => {
+        FactorError::NotPositiveDefinite { column } | FactorError::Singular { column } => {
             let label = eqs.node.get(column).and_then(|n| mesh.labels.get(*n)).copied().unwrap_or(0);
             Error::Singular(format!(
                 "the temperature is undetermined at node {label}: no temperature boundary condition or convection reaches its part of the model"