metal-linalg

The band thresholds’ tie-break

Status: done in 2.15.0 (2026-10-07), both options. The eigh grid gains N = 2560 and 3584, the SVD’s k = 1280 and 1792; and in stages 3b and 4b a threshold counts as near-optimal only if, on the points where it chooses differently from the best, it is within 3% of the best there (near_on_disagreement in tuning/tune_eigh.py). Re-analysed, run 20261004-06bc11 now chooses 3072 for eigvalsh, as it should have. See the study, section 6, and reading-reports.md.

What follows is the proposal as written on 2026-10-04.

What

On the M5 Pro the routing sweep sends eigenvalues alone to the band backend from N = 4096, although at the one point measured between 3072 and 4095, N = 3072, band is 7% faster than tridiag. The fit is working as designed; the design picks the higher threshold here because there is too little data in the band region. Give the fit more points there, so that the conservative tie-break is not deciding on one.

Why: the measurements

Stage 4b of run 20261004-06bc11 (eigh), scored on the 28 points where band_vals was timed (N ≥ 512), geometric-mean regret by threshold:

threshold 1536 2048 3072 4096 never
geomean 1.0344 1.0261 1.0187 1.0212 1.0308

3072 is the best, 4096 is within the fit’s 0.5% tolerance, and the tie-break (lowest worst case, then the highest threshold, as the tridiag and bidiag thresholds’ stages also do) takes 4096. The difference is one point of 28: (3072, batch 1), tridiag 96.2 ms, band 89.9 ms. The SVD’s stage 3b had more separation and chose 1536, the best.

Plan

Options, cheapest first:

  1. More points where it matters. Add N = 2560 and 3584 to the eigh grid’s large sizes (N_EXTRA, which --max-n adds; batches 1, 2 and 4 up to HUGE_N, 1 above), so that the band region has several points. Costs a minute or two of sweep. The tie-break then has data rather than a single point.
  2. Score each threshold on the points it changes. A threshold of 3072 against 4096 only differs at 3072 ≤ N < 4096; scoring both over all 28 points dilutes the difference by the other 27. Comparing candidates on the points where they disagree (with the same tolerance) would choose 3072. This changes the method, so apply it to stages 3b and 4b only and say so in docs/reading-reports.md.
  3. Leave it: the cost is 7% for N between 3072 and 4095, one or two matrices, eigenvalues alone.

Option 1 is the least intrusive; fold it into the next eigh re-measure.

Effort

About two hours for option 1 (the grid in tuning/tune_eigh.py, check the cost model’s estimate for the new points, re-run or reuse the next eigh re-measure, 28 minutes unattended); half a day for option 2, with its tests and docs.

Where to start

tuning/tune_eigh.py: the grid (N_EXTRA, LARGE_BATCHES, HUGE_N, BAND_MIN_GRID_N) and stage 4b (band_choice, near_b, the min(..., key=...) tie-break); the SVD’s counterparts in tuning/tune_svd.py (stage 3b) if option 2 is taken.

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