T18 — FGMRES-IR extension (Krylov.jl) #18

Task T18 of TASKS.md — milestone M5 — Solve extras.

Depends on: #17 (T17) must be merged first; tasks are strictly sequential.

Summary (from TASKS.md): ext/SparseDirectSolverKrylovExt.jl: ir_mode = "fgmres" runs FGMRES with the factorization as right preconditioner; test: on an ill-conditioned Float64 system where plain IR stalls (relres not improving over 5 steps), FGMRES-IR reaches relres ≤ 1e-12 within 20 iterations on CPU and CUDA.

Scope and deliverables: the section T18 of TASKS.md on main is the single source of truth; it is not copied here so it cannot drift. Read it together with the PLAN.md sections it names and the Report blocks of the previous tasks.

Definition of done:

• Every test listed in the task passes on the KA CPU backend (GitHub runner) and on the CUDA/AMDGPU self-hosted runners (Run tests and Aqua checks green on the PR).
• The task's status marker in TASKS.md is set and its Report block is filled in honestly (CUDA counts come from CI).
• A PR titled T18: … from branch task/T18-<slug> with Closes #18 in its body, reviewed by the Claude reviewer with VERDICT: APPROVE.
• Out-of-scope bugs or follow-ups found on the way are opened as separate issues (label found-by-agent), not fixed silently.

How it starts: add the label claude:implement (the pipeline does this automatically after the previous task's PR merges; label on-hold to prevent that).

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