[pmocz]

Fixes #1064

The inverse quadratic interpolation branch of eval_brent_safe_zero (num/private/mod_brent.f90) used the left bracket end a, which the routine never updates, where Brent's algorithm has q = fa/fc:

-           p = s * ( 2.0D+00 * m * a * ( q - r ) - ( sb - sa ) * ( r - 1.0D+00 ) )
+           p = s * ( 2.0D+00 * m * q * ( q - r ) - ( sb - sa ) * ( r - 1.0D+00 ) )

This matches Burkardt's own 2013 fix to zero and FMM zeroin (2.0d0*xm*q*(q - r)).

The bad step was still screened by Brent's acceptance tests, so roots stayed bracketed and within the requested tolerance. Convergence was just slower, and how much slower depended on where the bracket sat on the x axis. This affects every safe_root* / brent_safe_zero caller, including EOS inversions, pre_ms_model, rsp_build and pgbinary_orbit.

Testing

I built the real routine standalone with gfortran, from main and from this branch, and counted function evaluations (bracket, t = 1e-12):

f(x)	bracket	evals before	evals after	|Δroot|
x³ − 2x − 5	[2, 3]	17	6	4e-13
sin x − x/2	[1.5708, 3.14159]	22	7	6e-15
eˣ − 2	[0, 1]	8	6	3e-16
e^(x−1000) − 2	[1000, 1001]	16	7	0
x¹⁰ − 1	[0, 1.3]	10	8	2e-13

The test_brent_zero cases in the num unit tests still converge to |f(z)| < 1e-14, so they stay silent in the output.

Expected follow-ups

• num/test/test_output: the test_root / safe_root* lines print roots to 16 digits, and they change in the last digits. All results remain within the tests' epsx/epsy. The golden file was made with crmath, which I don't have locally. I'll update it from CI's make check output.
• Test-suite checksums: EOS and other callers now converge along a different iteration path, so some results may differ at round-off level.

🤖 Generated with Claude Code

[pmocz]

Updated num/test/test_output from CI's make check diff. The diff was identical on all four build jobs (Linux and macOS, OpenMP on and off), and num was the only module that failed. Three test_root lines change:

test	old	new	tolerance
root (expected 0.74800611)	7.4800610880593488D-01	7.4800609950637120D-01	epsx = 1e-6
safe_root (exact 0.5)	5.0000000000000011D-01	5.0000000000026457D-01	epsx = 1e-10
second safe_root_with_guess (exact 0.5)	5.0000000000000377D-01	5.0000000000003031D-01	epsx = 1e-10

The new roots are slightly less over-converged because Brent now reaches the tolerance in fewer steps. All of them are still well within each test's tolerance. first safe_root_with_guess is unchanged, since it converges in the Newton phase.

[pmocz]

CI is now green on all four SDK builds. Besides num, the eos unit test also needed a test_output update. That test calls eosDT_get_T to invert logP, and its found logT and got logP lines shifted. Linux and macOS gave identical results, and the result still meets the test's tol = 1d-5. The logP error went from 4.7e-6 to 9.5e-6, both under 1e-5. No other module's make check changed.

I haven't run the star test suite: it's workflow_dispatch only. Since EOS inversions and other safe_root callers now converge along a different path, a TestHub run before merging would be worthwhile to check checksums.

[Debraheem]

Closing as a duplicate of #1065, which contains the same solver correction. We can keep the remaining TestHub/EOS investigation there, however the testing/timing in this pr is still useful to that discussion.