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/lib.rs

@@ -13,7 +13,9 @@
 //! 4. **Solution** ([`cholesky`]): sparse Cholesky; reactions K u − f at the prescribed degrees
 //!    of freedom; stresses at the integration points, extrapolated and averaged at the nodes, and
 //!    von Mises. Frequency analysis ([`eigen`]) runs shift-invert block Lanczos on the same
-//!    kind of factorization.
+//!    kind of factorization. Indefinite matrices (tangents past a limit point, strained mixed
+//!    elements, shifted operators) are factored by the symmetric indefinite LDLᵀ ([`ldlt`]) on
+//!    the same symbolic factorization, which also counts their negative eigenvalues.
 //!
 //! Element matrices and stresses, the factorization's subtrees and dense tiles, the Lanczos
 //! block solves and matrix products run on threads ([`fieldcraft_parallel`]), but every number
@@ -31,6 +33,7 @@ pub mod constraints;
 pub mod discrete;
 pub mod eigen;
 pub mod heat;
+pub mod ldlt;
 pub mod nonlinear;
 pub mod ordering;
 pub mod sparse;