[shinaoka]

The numeric sparse LU returned Ok for a numerically singular matrix, and the solve then returned NaN. Upstream reported the simplicial backend (sarah-quinones/faer#339); the supernodal backend has the same defect.

Closes #132.

What changes

• New error variant. LuError::NumericSingular { index }, the shape the upstream maintainer asked for in upstream PR sarah-quinones/faer#332 (closed unmerged). The maintainer approved it (option 1, 2026-10-29).
• Simplicial backend. Returns the error when the chosen pivot is exactly zero. The check reads the pivot itself (abs1), not the modulus used by the pivot search, because a complex modulus underflows to zero for a nonzero subnormal entry.
• Supernodal backend. Returns the error when a diagonal entry of a factored panel, which is U's pivot, is exactly zero.
• index. Counts elimination steps in the fill-reducing column order, the same way SymbolicSingular's does.
• Tolerance. Only an exact zero is reported. Tolerance-based detection is sparse LU: tolerance-based numeric singularity detection (follow-up to #132) #146 (an issue only).
• API change. This adds a variant to an exhaustive public enum. No tensor4all consumer references the sparse LuError; I checked tlinalg-rs, tcontract-rs and tenferro-rs.
• Records.
  • PATCHSET row.
  • upstream-reports/faer-028-sparse-lu-numeric-singular: a standalone program that reproduces the defect on faer 0.24.4 in both backends.

Provenance

I wrote the fix from the upstream report. No code comes from upstream PR #332, and nothing was posted to Codeberg.

Verification

• Tests, each run in both backends:
  • codeberg_339: the upstream reproduction at f32, f64, c32 and c64.
  • numeric_singular_after_an_elimination_step: a 4x4 singular/nonsingular pair.
  • nonzero_subnormal_pivot_is_not_singular.
• Before the fix: the first two fail without the checks. The third fails if the simplicial check reads the modulus instead of the pivot.
• Other checks:
  • All 31 sparse library tests and the sparse doctests pass.
  • cargo +1.89.0 check passes.
  • Clippy reports nothing in the changed files.
  • Nightly rustfmt leaves no diff in my edits; the only remaining diffs are three that already exist on main.
• The standalone report program prints Ok and then x = [NaN, NaN] for both backends on faer 0.24.4.
• Independent review (gpt-6.1-sol):
  • It found the subnormal complex modulus case, now fixed and pinned by a test.
  • It also corrected the report README's explanation: in this program the factors stay finite, and the NaN comes from the solve dividing by the zero pivot.

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