solver: Newton on indefinite tangents by LDLᵀ

A tangent that is not positive definite is factored by LDLᵀ (Cholesky first, one symbolic
factorization) instead of cutting the increment back; each iteration records the tangent's
negative eigenvalues and the summary warns when a converged state keeps some. A model that can
move freely is still an error at its first tangent, located by the undeformed Cholesky factor;
a singular tangent later is a cutback. Nearly incompressible hybrid hex8, hex20 and tet10 at
ν = 0.4999 under prescribed displacement now converge to the closed form.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

crates/solver/src/analysis.rs

@@ -14,6 +14,7 @@ use fieldcraft_parallel as parallel;
 use crate::cholesky::{Factor, FactorError, UpperColumns};
 use crate::constraints::{Budget, Reduction, coupling_rows};
 use crate::eigen::{Operator, Wanted, lanczos};
+use crate::ldlt::{Kind, Symmetric};
 use crate::ordering::nested_dissection;
 use crate::sparse::{Csr, NodeGraph, add_block, add_diagonal, add_scalar};
 use crate::system::{
@@ -250,6 +251,12 @@ pub(crate) fn factor_raw(a: &Csr, limits: &Limits) -> std::result::Result<Factor
     Factor::new(&UpperColumns::from_symmetric(a), limits.max_factor_entries)
 }
 
+/// Factor a free-dof matrix by Cholesky, LDLᵀ, or Cholesky with LDLᵀ when it is not positive
+/// definite ([`Kind`]).
+pub(crate) fn factor_symmetric(a: &Csr, limits: &Limits, kind: Kind) -> std::result::Result<Symmetric, FactorError> {
+    Symmetric::new(&UpperColumns::from_symmetric(a), limits.max_factor_entries, kind)
+}
+
 /// A factorization failure in the user's terms: the node (or reference point) and direction of
 /// a singular pivot.
 pub(crate) fn factor_error(e: FactorError, mesh: &Mesh, reduction: &Reduction) -> Error {