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Add a LinearSolve algorithm that wraps `lstsq` — the LinearSolve extension cannot solve rank-deficient/inconsistent A

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Assessment

Difficulty
4/5
Estimated time
3-5 days
Newbie friendliness
55/100
Issue type
Feature
Clarity
Mostly clear
Activity status
Quiet
Tech stack
julia
Domain
backend

Research direction

Read ext/SparseWithDenseRowColMatricesLinearSolveExt.jl alongside the existing SWDRCQRFactorizationAlg, then review src/factorize.jl and the LS refactor! work. Add the LinearSolve least-squares algorithm and public factorization described in the issue, and extend test/test_linearsolve.jl so the rank-deficient or inconsistent case is covered and cache.A updates reuse the analysis.

Written by the indexing model from the issue text.

Description

WHAT: ext/SparseWithDenseRowColMatricesLinearSolveExt.jl has zero references to lstsq/LeastSquares/pinv. Both existing algs (SWDRCFactorizationAlg, SWDRCQRFactorizationAlg) map a singular A to ReturnCode.Infeasible (lines 104-106, 166-168), so there is no way to get the min-norm A⁺b through the LinearSolve interface despite the full lstsq machinery existing. WHY IT MATTERS: LinearSolve is the standard SciML entry point and the natural home for the cached factorization; users in a LinearSolve pipeline cannot reach lstsq at all. FIX: Add SWDRCLeastSquaresAlg (mirroring the QR alg): init_cacheval builds a SparseWithDenseRowColLeastSquares (dense-COD fallback when structured doesn't apply), solve! does ldiv!(cache.u, F, cache.b), and once the LS refactor! lands its _refresh! reuses symbolic analysis across cache.A updates. Expose a public SparseWithDenseRowColLeastSquaresFactorization(; alg=:auto, tolC=...) in src/factorize.jl. Add a test/test_linearsolve.jl block (currently only LU+QR algs tested). Best done after or alongside the LS refactor! issue. EFFORT: M.

Priority: high. Filed from an automated next-steps audit of the QR/lstsq work (see PR #6).

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