perf: cooperative pivot search in the device LDLᵀ kernels (#75)

PERFORMANCE.md

@@ -8,9 +8,9 @@ Performance issues carry the GitHub label `performance` ([list](https://github.c
 
 | issue | what | experiment | status |
 | --- | --- | --- | --- |
-| #81 (PR) | regime-A subtrees ran a whole KKT tree on one workgroup; flop limit `subtree_parallelism` | 0 | open PR |
+| #81 (PR) | regime-A subtrees ran a whole KKT tree on one workgroup; flop limit `subtree_parallelism` | 0 | merged |
 | #82 | KKT refactorization and solve are level-bound after #81: 54–59 launches per refactorization, 62–85 per solve | 5, 6 | open |
-| #75 | device LDLᵀ: serial pivot search, one workgroup per regime-B/C front, no vendor `sytrf` | 1, 2 | open, triaged |
+| #75 | device LDLᵀ: serial pivot search, one workgroup per regime-B/C front, no vendor `sytrf` | 1, 2 | open, triaged; step 1 (cooperative pivot search) in a PR from `perf/exp1-pivot-search` |
 | #60 | regime-A follow-ups: CUDA timings (partly answered by experiment 0) and per-backend local-memory caps | 7 | open, triaged |
 | #25 | T25 performance pass (task) | 5, 6, 7 | open |
 
@@ -109,6 +109,44 @@ A side effect: the default solve (`deterministic_mode = 0`) now uses its atomic
 
 Revised order: experiments 1–2 (LDLᵀ kernels, #75) and 5–6 (launches and solve) now have the largest measured payoff; experiment 3 (matching and scaling) remains the accuracy blocker on K2 (relres unchanged by this fix).
 
+## Experiments 1–2 results (2026-10-05, RTX 4080)
+
+Issue #75 in three steps, one PR each, measured on the same machine with nothing else running. Refactorization: best of 5 warm runs of `bench/profile_phases.jl` (`*`: the `bench/compare.jl` median, for the matrices the profile skips); cuDSS from `bench/comparison/cudss.csv`; SDS Cholesky from the profile of `main`. The contract does not move: `piv` and `pivot_kind` equal `ref_ldlt!` on the test matrices on the CPU backend and CUDA, D and the panels within `panel_tol`.
+
+### Step 1: cooperative pivot search
+
+The pivot search of `front_ldlt_kernel!` and `subtree_ldlt_kernel!` ran on work item 1. Its column maxima are now workgroup reductions in up to three passes (λ and r with the column maximum; σ and the column-r maxima of Bunch–Kaufman; the fallback scan, one lane per candidate column), with the reference's tie-breaking (first maximum), so every decision is the serial one. Work item 1 takes the pivot from the values before the interchange, which keeps five barriers per step, and folds only the lanes that can hold data. The reference's fallback (`_best_1x1`) skips columns that cannot win and stops the threshold test at the first violation; the result is the same. The regime-A LDLᵀ kernel reserves 1024 B of local memory (`SUBTREE_LOCAL_RESERVE_LDLT`) for the reduction slots.
+
+LDLᵀ refactorization, `main` → step 1:
+
+| matrix | main ms | step 1 ms | step 1 / cuDSS | step 1 / SDS Cholesky |
+| --- | --- | --- | --- | --- |
+| lap2d_300 | 228 | 192 | 37.4× | 6.12× |
+| lap3d_40 | 14,777* | 14,494* | 248× | 64.7× |
+| HB/bcsstk17 | 206 | 167 | 49.5× | 6× |
+| Boeing/bcsstk38 | 137 | 106 | 32.1× | 4.69× |
+| GHS_psdef/apache2 | 92,576* | 97,296* | 199× | 94.5× |
+| kkt_pglib_opf_case118_ieee_condensed_1 | 1.77 | 1.82 | 4.66× | 1.99× |
+| kkt_pglib_opf_case118_ieee_condensed_10 | 1.72 | 1.79 | 5.13× | 2.04× |
+| kkt_pglib_opf_case118_ieee_condensed_20 | 1.87 | 1.82 | 5.18× | 2.08× |
+| kkt_pglib_opf_case118_ieee_k2_1 | 1.69 | 1.69 | 4.66× |  |
+| kkt_pglib_opf_case118_ieee_k2_10 | 1.62 | 1.68 | 4.74× |  |
+| kkt_pglib_opf_case118_ieee_k2_20 | 1.6 | 1.66 | 4.16× |  |
+| kkt_pglib_opf_case1354_pegase_condensed_1 | 10.9 | 7.2 | 10.5× | 2.46× |
+| kkt_pglib_opf_case1354_pegase_condensed_10 | 10.9 | 7.3 | 11.4× | 2.5× |
+| kkt_pglib_opf_case1354_pegase_condensed_20 | 11 | 7.32 | 10.4× | 2.51× |
+| kkt_pglib_opf_case1354_pegase_k2_1 | 8.31 | 6.6 | 8.08× |  |
+| kkt_pglib_opf_case1354_pegase_k2_10 | 7.15 | 6.05 | 7.32× |  |
+| kkt_pglib_opf_case1354_pegase_k2_20 | 9.1 | 6.71 | 8.32× |  |
+| kkt_pglib_opf_case14_ieee_condensed_1 | 0.446 | 0.503 | 2.61× | 1.38× |
+| kkt_pglib_opf_case14_ieee_condensed_10 | 0.443 | 0.52 | 2.26× | 1.53× |
+| kkt_pglib_opf_case14_ieee_condensed_11 | 0.448 | 0.463 | 1.59× | 1.41× |
+| kkt_pglib_opf_case14_ieee_k2_1 | 0.669 | 0.697 | 2.49× |  |
+| kkt_pglib_opf_case14_ieee_k2_10 | 0.722 | 0.708 | 2.98× |  |
+| kkt_pglib_opf_case14_ieee_k2_15 | 0.97 | 1.03 | 3.2× |  |
+
+On the K2 dumps the inertia and `nperturbed` are those of `main` on all nine. On the T15 matrix `kkt_matrix(3000, 1000, 1e-8)` (fallback scans on most steps of its two large root fronts) CUDA goes from 33.9 s to 8.3 s; the KA CPU backend stays at 7.0 s against 2.6 s for `ref_ldlt!` (it runs the work items of a workgroup one after another, so a parallel search saves nothing there). The medium matrices gain 16–34%; the smallest KKT dumps (case14, w ≤ 16) lose up to 17% to the reductions' barriers; the two matrices with very large regime-C fronts (apache2, lap3d_40) are within ±5%, still one workgroup per front.
+
 ## Recommendations: Prioritized Experiment Plan
 
 | # | Experiment | Payoff | Effort | Key measurement | Success criterion |