Guard the power-mean relaxation factor estimate

[timholy]

The asymmetric power-mean sweeps overrelax by a factor ω estimated from plain sweeps and raised from the rates of later windows. Two defects in that rule made some solves far slower than plain sweeps.

An ω set on a plateau was never lowered. Far from the fixed point the imbalance can stay nearly constant for several sweeps before it contracts quickly. An ω estimated there is near its cap, and above the optimum the relaxed rate is ω - 1 whatever the plain rate is, so the windows read it as steady progress. The plain estimate now waits while the contraction is still improving from one sweep to the next, and sets ω no higher than 1.5, where a relaxed sweep contracts by at least half even above the optimum. Windows raise it from there.

A window was judged against the smallest imbalance of all earlier windows. The imbalance is the size of the plain step, which a relaxed iterate overshoots by ω - 1 times, so its level rises with ω, and windows at a raised ω could show no progress while the iteration converged. Those windows tripped the stall rule, which halves ω - 1 when it should rise. Each window is now judged against the window before it. The first window after ω changes is still judged against one at the old ω and reads as slow, so one window shrinks 2 - ω by at most a factor of 8.

The handoff to Newton after the plain estimate is predicted at the rate of the factor that estimate implies, as before, not at the capped factor the first relaxed sweeps use.

On 1200 random matrices of 2 to 8 rows and columns (lognormal entries with σ of 1, 3 or 6, up to half zeroed), with p of 1 and 2 and the default and flat starts, 4800 runs in all. With newton = false, on the 4663 runs that plain sweeps finish in 50,000:

	before	after
sweeps	552,513	254,336
runs more than 2 sweeps slower than plain	213	9
runs where the stall rule fires (of 4800)	55	3

The 3 remaining stalls are runs that neither rule finishes in 10^5 sweeps. With newton = true:

	before	after
runs handed to Newton	949	974
passes over the support	317,998	302,284
factorizations	5130	5227

Sweeps with newton = false on larger problems, plain / before / after:

problem	plain	before	after
bidiagonal n=100, flat start, p=1	97,194	6114	6466

Assisted-by: Claude Opus 5.5 noreply@anthropic.com

[codecov]

Codecov Report

❌ Patch coverage is 92.85714% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 98.84%. Comparing base (cb4bc5e) to head (bb6cf5a).

Files with missing lines	Patch %	Lines
src/powermean.jl	92.85%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #88      +/-   ##
==========================================
+ Coverage   98.79%   98.84%   +0.04%     
==========================================
  Files          23       23              
  Lines        6302     6303       +1     
==========================================
+ Hits         6226     6230       +4     
+ Misses         76       73       -3     

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