BLAS: Account for subnormal rounding in NRM2 tests

[rmlarsen]

Stop the NRM2 tests from rejecting correctly rounded subnormal results: floor the reference magnitude at SAFMIN in the error denominator, which takes a false-failure rate of 6.1e-5 per draw to zero over 2e7 draws, and add deterministic subnormal cases so the regression no longer rides on a lucky random draw.

The tests measure only relative error, so a subnormal result whose absolute error is one subnormal ulp is rejected outright. In #1382 an ARM64 OpenMP shared build reported ten DNRM2 failures at N=3 and INCX=0 while the tested BLAS sources were unchanged from master.

That count is the signature of the mechanism rather than a coincidence: at INCX=0 every IV row shares X(1), so a single unlucky draw of the random tail produces exactly ten reported failures. Sampling the same configuration on x86-64 — x = 2*SAFMIN*w with w uniform on (-1,1), compared against the reference DNRM2 — over 2e7 draws gives

SNRM != ZNRM	32% of draws, an ordinary one-subnormal-ulp disagreement
rejected by the current comparison	1229, i.e. 6.1e-5 per draw
rejected by the comparison in this PR	0, largest ratio 0.385 against THRESH = 10

So this is neither ARM-specific nor flush-to-zero; it is a rare lottery any platform can lose. It does gate CI: #1371's LAPACK_Test_Summary parses the .out files, and #1382's own rerun failed it with failing to pass the threshold: 10.

Fix. Use max(reference, SAFMIN) in the error denominator for all four real and complex norm tests. This is bit-identical to the current comparison for every normal result and supplies an absolute tolerance of order SAFMIN*ULP near underflow. It also replaces the overly permissive zero-reference branch: ZNRM == 0 happens only when the whole vector is exactly zero, in which case the SNRM == ZNRM fast path already claims it, so that branch was reachable only from a broken implementation. NaN and infinity checks remain in place.

Coverage. Add a fixed subnormal value and an unrandomized tail alongside every existing randomized case. The value has to sit in a narrow window: deep enough that one subnormal ulp is a relative error above THRESH*N*ULP, and shallow enough that SQRT(N)*x and NRM2(N,x,0) still round differently. Of 2*SAFMIN/k for k = 2, ..., 200000, only 13 values in double and 9 in single satisfy both, so the constants are commented rather than left bare, and a named parameter selects the unrandomized case instead of keying it to the end of the value list. The new cases fail the old comparison with test ratios of 3.2E+02 to 2.7E+03 in all four precisions and both index widths; each norm test now runs 1946 checks instead of 1610. No production BLAS or LAPACK routine changes.

Validation on a 13th Gen Intel(R) Core(TM) i7-13700HX (x86-64, WSL2) with GNU Fortran 13.3.0:

• All eight Level 1 BLAS executables pass, 1946 norm checks each, in Release static, Release shared with -fopenmp -frecursive -fcheck=all, -O3 -march=native, and -O0 -g -fcheck=all, using both the default and _64 APIs. Output summaries were checked for numerical failures, since executable exit status alone does not detect them; LAPACK_Test_Summary is what turns them into a CI failure.
• 200 repetitions per precision with zero failures, and 8e7 focused subnormal draws across the four routines whose largest test ratio is 0.385.
• Mutation checks: reverting the comparison to the relative-only form fails 14, 14, 14 and 20 checks in S, D, C and Z; flushing nonzero subnormal norms to zero fails 104 to 128; perturbing normal norms by 1% fails about 1230 per precision.
• git diff --check passes. Added fixed-form source lines fit within 72 columns.

Under a flush-to-zero build the new value itself flushes, so the deterministic case degenerates into the existing zero case and tests nothing rather than failing; -ffast-math shows only the pre-existing flush-to-zero failures, at twice the count because the new row duplicates the zero row.

ARM64/GNU 15.2 and other compiler configurations remain for upstream CI.

Related: #1382

Checklist

• The documentation has been updated. (Test comments describe the subnormal cases, the window the fixed value has to sit in, and the error scale; public behavior is unchanged.)
• If the PR solves a specific issue, it is set to be closed on merge. (No tracking issue; related CI failure is in PR Return INFO = 1 from xLALSD and xBDSDC instead of stopping in XERBLA when the bidiagonal contains a NaN #1382.)

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 69.36%. Comparing base (a6c6e74) to head (0b0f51b).
⚠️ Report is 2 commits behind head on master.
✅ All tests successful. No failed tests found.

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1423   +/-   ##
=======================================
  Coverage   69.36%   69.36%           
=======================================
  Files        6122     6122           
  Lines      486711   486723   +12     
  Branches    23268    23268           
=======================================
+ Hits       337584   337600   +16     
+ Misses     148689   148685    -4     
  Partials      438      438           

Components	Coverage Δ
BLAS	97.94% <ø> (ø)
CBLAS	96.98% <ø> (ø)
LAPACK	82.38% <ø> (ø)
LAPACKE	2.17% <ø> (ø)
TMGLIB	55.69% <ø> (ø)
BLAS testing	88.36% <100.00%> (+0.03%)	⬆️
CBLAS testing	89.63% <ø> (ø)
LAPACK testing	82.20% <ø> (ø)
LAPACKE testing	∅ <ø> (∅)

Files with missing lines	Coverage Δ
BLAS/TESTING/cblat1.f	89.22% <100.00%> (+0.45%)	⬆️
BLAS/TESTING/dblat1.f	91.12% <100.00%> (+0.27%)	⬆️
BLAS/TESTING/sblat1.f	92.23% <100.00%> (+0.27%)	⬆️
BLAS/TESTING/zblat1.f	89.22% <100.00%> (+0.45%)	⬆️

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