Add SpecializingFactorizations.jl package extension

[ChrisRackauckas-Claude]

Please ignore until reviewed by @ChrisRackauckas.

Adds an opt-in package extension wrapping SpecializingFactorizations.jl as two LinearSolve algorithms. This is a weakdep extension only — no hard dependency and no change to any default algorithm.

What SpecializingFactorizations.jl provides

It exposes two type-stable, single-workspace dense factorizations that detect the property a matrix actually has and dispatch to the specialized solve, all behind one concrete workspace type (the structure/rank choice is a runtime enum field, not a Julia type, so the pipeline infers and the warm path is allocation-free):

• specializinglu / SpecializedLU — a square solver that cheaply scans A for structure (diagonal, bidiagonal, tridiagonal, banded, triangular, symmetric PD, symmetric/Hermitian indefinite) and uses the matching specialized factorization instead of a general O(n^3) LU.
• specializingqr / SpecializedQR — a rectangular / rank-deficient least-squares solver. Column-pivoted rank-revealing QR (geqp3) that returns the least-squares / minimum-norm solution for any shape including singular/rank-deficient, never throwing.

Wrapped algorithms

• SpecializedLUFactorization — wraps specializinglu; for square systems.
• SpecializedQRFactorization — wraps specializingqr; for rectangular / rank-deficient least-squares.

Both wrapped because they cover complementary problem classes (square structured solves vs. rectangular/rank-deficient least-squares) and both expose the ldiv! + in-place re-factor (specializinglu! / specializingqr!) interface the LinearCache warm-solve path wants.

Implementation

• src/extension_algs.jl: two AbstractDenseFactorization structs with docstrings + examples; exported from src/LinearSolve.jl.
• ext/LinearSolveSpecializingFactorizationsExt.jl: init_cacheval builds the factorization; SciMLBase.solve! re-factors in place on cache.isfresh (reusing the workspace via specializing*!), guards issuccess, and returns build_linear_solution with the appropriate ReturnCode.
• Project.toml: SpecializingFactorizations added to [weakdeps], [extensions], [compat] ("0.1"), [extras], and the test target. No SparseArrays trigger — it operates on dense matrices only.
• test/specializing_factorizations.jl: focused test set (extension load, square/structured/complex/float32 LU, overdetermined and rank-deficient QR, cache reuse across re-factor for both), wired into the Core test group.

Registration

SpecializingFactorizations.jl is registered in the General registry (v0.1.0).

Caveats

• SpecializedLUFactorization is square-only by construction; rectangular systems should use SpecializedQRFactorization.
• Structure detection is structural (!iszero), not numerical/tolerant.
• Cache reuse re-factors into the existing workspace; a different matrix size on a warm cache regrows buffers (handled by specializing*!).

🤖 Generated with Claude Code

[ChrisRackauckas]

This should also get an opt-out #1023

[ChrisRackauckas-Claude]

Rebased onto current main (was conflicting after the test-suite restructure in #1022 and the test-target slimming).

Conflict resolution:

• Project.toml: kept main's slimmed compat/extras/test target and added SpecializingFactorizations to each (it stays a base test dep since the test runs in the Core group).
• test/runtests.jl: took main's new grouped structure and re-added the test under the Core group as core/specializing_factorizations.jl (file moved from test/specializing_factorizations.jl to match the new test/<group>/ layout).

Verified locally on Julia 1.12: the LinearSolveSpecializingFactorizationsExt extension precompiles and all 18 tests in test/core/specializing_factorizations.jl pass. Runic reports no formatting changes on the touched files.

[ChrisRackauckas-Claude]

Rebased on main and added the ForwardDiff opt-out per the review comment (same pattern as #1023):

• _use_direct_dual_solve in ext/LinearSolveForwardDiffExt.jl now returns true for SpecializedLUFactorization and SpecializedQRFactorization, so Dual-number problems are solved directly instead of through the primal/partials splitting path. This is valid because SpecializingFactorizations.jl has generic-element fallbacks for non-BlasFloat types (generic dense LU, generic column-pivoted QR). It also matters for correctness on the QR side: the split-path math (∂x = A⁻¹(∂b − ∂A·x)) is square-only, while the direct path handles Dual rectangular least-squares.
• Added tests in test/core/forwarddiff_overloads.jl: square Dual systems through both algorithms plus a rectangular Dual least-squares solve through SpecializedQRFactorization, all checked against A \\ b / qr(A) \\ b. Note the tests compare values and partials separately: when the primal diff is exactly zero but partials differ by an ulp, isapprox on Dual vectors returns false because norm(diff) evaluates to Dual(NaN,…) (sqrt has infinite slope at 0), so the plain ≈(sol, ref) pattern only passes for bitwise-identical results.

Ran locally on the rebased branch: test/core/forwarddiff_overloads.jl (full file) and test/core/specializing_factorizations.jl both pass; Runic reports no formatting changes.

Note for whichever of this and #1023 merges second: both touch the same line of _use_direct_dual_solve, so there will be a trivial conflict.

[ChrisRackauckas-Claude]

CI status on 3bd3d2a8: everything this PR touches is green — Core (julia 1 / lts / pre), QA, Spell Check, Runic, format-check, Sublibrary CI all pass (the earlier Core failure was resolve.jl's subtype sweep hitting the new algs without the extension loaded; fixed by loading SpecializingFactorizations there, which also gives both algorithms re-solve coverage).

Remaining red is not from this PR:

• tests/LinearSolveHYPRE — pre-existing on main, fix in Fix LinearSolveHYPRE CI: pass active project to MPI worker processes #1030
• tests/LinearSolvePETSc — pre-existing on main (PETSc.jl 0.4.10 API change), fix in Fix LinearSolvePETSc CI: adapt MatNullSpaceCreate to PETSc.jl 0.4.10 API #1031
• Documentation — pre-existing NonstructuralZeros @ref fatal error, fix in Fix fatal docs cross-reference: include NonstructuralZeros docstrings #1026
• Downgrade / Downgrade Sublibraries — pre-existing, fixes in Add LogExpFunctions lower-bound compat to fix Downgrade CI #1025 / Fix Downgrade Sublibraries CI: numeric julia-version and resolvable sublibrary lower bounds #1029
• tests/NoPre (julia lts) — new upstream Enzyme/LLVM.jl breakage (MethodError: LLVM.ConstantInt(::LLVM.VectorType, ::Int64)), reproduces on main run 27251663412; being investigated separately
• tests/NoPre (julia 1), Pardiso (julia 1), ParU (julia 1), GPU — self-hosted runner infra deaths (job ran 0 steps, no logs, died at ~10 min); need a rerun from someone with admin rights, or the next push will retry them