[shinaoka]

Closes #112.

Summary

As the maintainer asked (2026-10-29), I compared faer's nonsymmetric EVD with LAPACK ?GEEV stage by stage. faer skipped ?GEEV's first step: when the largest entry is below smlnum = sqrt(safmin) / eps or above bignum = 1 / smlnum, LAPACK scales the matrix to that bound before the Hessenberg reduction and scales the eigenvalues back at the end. evd_imp now does the same with a power of two, which scales every entry exactly and does not change the eigenvectors.

Stage-by-stage comparison (c32 kinds 6/7, n = 63 to 65)

stage	as is	scaled to unit size first
Hessenberg backward error / eps	16 to 20	16 to 20
complex Schur backward error / eps	226 to 258	67 to 93
GET22	46 to 58	16 to 20 (cgeev: 29 to 40)

So the gap arises in the complex QR iteration on the unscaled matrix.

Where the gap grows

I swept 372 matrices over f32/f64/c32/c64, n = 16, 64, 128 and 257, and four families: random, Grcar, near-Jordan, and graded D A D^-1 with D spanning 1e±6. Scales went up to the range limits, and LAPACK ?geev (scipy/OpenBLAS) was run on the same files. Below the bound the gap was far worse than #112's factor of 1.5:

scale (n = 64; Grcar, random, near-Jordan; max of right and left)	faer before	LAPACK	faer after
f32/c32 2^-80	2.8 to 30	2.6 to 34	3.2 to 26
f32/c32 2^-90	27 to 1750; f32 near-Jordan NoConvergence	2.6 to 34	3.2 to 26
f32/c32 2^-100	2.4e4 to 3.5e5; f32 near-Jordan NoConvergence	2.6 to 34	3.0 to 29
f32/c32 2^-110, 2^-120	1.0e6 to 4.2e6	2.6 to 33	3.2 to 28
f64/c64 2^-900	3.9 to 31	3.1 to 37	3.9 to 44
f64/c64 2^-960	1.5e11 to 2.1e12	3.1 to 37	3.9 to 35
f64/c64 2^-1000, 2^-1010	5.4e14 to 2.3e15	3.1 to 41	3.9 to 44

At 2^-100, f32 near-Jordan n = 128 and 257 did not return at all. That hang is a separate defect, #144: with the scaling, Mat::eigen no longer reaches it.

After the change, every non-graded series is at most 1.74 times LAPACK, at every n, scale and type. The two cells above 1.5 are at n = 16, where the absolute values are small: f32 near-Jordan gives 9.15 against 5.26 (the same at scale 1), and c32 random at 2^-60 and 2^-100 gives 3.74 against 2.25.

The graded family has a different, serious defect in the real eigenvector back substitution. It is unrelated to scaling: GET22 reaches 2.6e9 against LAPACK below 5e-4, and upstream 0.24.4 also fails. It is filed as #145 and not touched here.

Changes

• faer/src/linalg/evd/mod.rs: geev_scaling and its use in evd_imp. The bound is tested on the largest |re| or |im|, not on faer's complex modulus, which is inf for 2^1023 + 0i and zero for 2^-1074 + 0i (post-review finding: a modulus scan picked a zero factor and returned a NaN eigenvalue with Ok; evd_scaling_at_the_complex_range_ends covers it and fails with the modulus scan). Also the test geev_scaling_tests::evd_scales_its_input_into_range. The test covers Grcar and pseudo-random n = 32, all four types, 2^-120 to 2^120 (32-bit) and 2^-1010 to 2^1015 (64-bit), right and left vectors, and the GET22 threshold 20 * max(1, sqrt(n / 16)).
• scripts/lapack-known-failures.json: the LAPACK validation: the nonsymmetric EVD fixtures exceed the GET22-style residual threshold (4 dtypes) #112 entry is removed, because the c32 group now passes.
• docs/lapack-validation.md, PATCHSET.md: one row and one paragraph.
• upstream-reports/faer-028-evd-input-not-scaled: reproduces the defect on faer 0.24.4: f64 2^-960 gives GET22 1.1e15, and f64 2^-1000 and f32 2^-100 return eigenvectors with NaN/inf entries, with Ok. Not posted upstream.

LAPACK sources were read only; no code was copied.

Verification (primerose, 4 cores)

• The new test fails with the scaling disabled (f32 2^-120, right: GET22 1.7e6) and passes with it.
• scripts/run-lapack-validation.py: every family exits 0, including nonsymmetric_evd 28/28, hessenberg, schur, balance and self_adjoint_evd. The known-failures list is now empty.
• cargo test -p t4a-faer --lib linalg::evd: 106 passed. The LAPACK nonsymmetric EVD tests (--include-ignored), rerun after the post-review fix: 158 passed.
• Independent post-review (gpt-6.1-sol): one BLOCKER (the complex modulus at the range ends) and one IMPORTANT (the test's GET22 helper dropped a NaN through f64::max), both fixed. The MINOR findings (the left side is checked as the right eigenvectors of A^H; universal wording in the docs) were addressed by narrowing the claims.
• faer-028 against this branch prints 17.7 (f64) and 15.2 (f32) at every scale.
• Performance: the change adds one pass over the input plus the eigenvalue unscaling. It was not timed.

🤖 Generated with Claude Code

[shinaoka]

Combined into #154 together with the other two nonsymmetric EVD fixes (#148, #151, #152), as the maintainer directed on 2026-10-29. The commits are carried over with their authorship; the upstream reports are renumbered faer-029 (#112), faer-030 (#144) and faer-031 (#145).