Translates a LAPACK info return into a failedAt position.

A positive info from dgetrf or dsytrf is the 1-based index of the pivot that came out exactly zero, so it becomes that index 0-based; anything else means the factorization succeeded. Public because any LAPACK-backed com.eignex.koblas.dense.Lapack implementation needs the same translation, including ones outside this repository.

Negative info is an illegal-argument report rather than a singularity, and callers check for it separately — it maps to NOT_SINGULAR here so a backend that forgot the check reports "not singular" rather than a nonsense position.

The com.eignex.koblas.dense.LuDecomposition.failedAt / com.eignex.koblas.sparse.SparseFactorization.failedAt value of a factorization that succeeded.

Every factorization koblas produces, dense or sparse, reports singularity the same way: the position of the pivot that had no acceptable candidate, or this when there wasn't one. Deriving singular from it rather than carrying a separate flag means the two cannot disagree - which they could when the flag was the source of truth and each backend set it independently.

It lives in the root package because both the dense and the sparse factorization types need it and neither owns the other. Whether they should share a Factorization interface outright is a separate question, deliberately still open: the sparse factors are opaque handles while the dense ones are a shared LAPACK format, and the two have different reasons to exist.

A failedAt meaning "singular, but this backend cannot say where".

koblas's own factorizations always know the position: they pick pivots themselves, so the step that ran out of candidates is exactly where they stopped. A host solver need not tell you. UMFPACK reports UMFPACK_WARNING_singular_matrix and, in Info, how many zero pivots there were — but not which, and recovering it would mean extracting U to hunt for a zero diagonal, which is absurd for a diagnostic.

So this exists rather than having a host backend invent a position or, worse, report NOT_SINGULAR for a matrix it just called singular. singular stays true because the value is not NOT_SINGULAR, and a caller that switches on the position has one more case to handle honestly.