[shinaoka]

Summary

Closes #145. On strongly graded matrices D A D^-1, the real nonsymmetric EVD returned wrong eigenvectors for every complex pair (GET22 up to 2.6e9).

Stage-by-stage comparison with LAPACK 3.12.0

f64, n = 64, D = diag(10^linspace(-6, 6, n)):

stage	faer	LAPACK
Schur backward error (eps |A|)	8.0
eigenvectors of faer's own Schur form T (dtrevc3 on the same T)	3.4e7	2.2e-4
dgeevx on A, balanc = N / P / S / B		0.58 / 0.58 / 2.1e-10 / 2.1e-10
faer on A after this fix	0.45

• Where it breaks: in the eigenvector back substitution of the real Schur form, not in the reduction.
• Which columns: all 24 failing eigen-blocks are complex pairs. For each, the first wrong entry is a 2×2 block solve above the pair's eigenvalue.
• Balancing (?GEBAL) is not the cause: without it, dgeev gives 0.58 and the fixed faer 0.45, both well under the threshold. Balancing would bring both to about 1e-10. Adopting it is a separate decision.

Cause

• solve_cplx_shifted_upper_quasi_triangular_system solved each shifted 2×2 block through its inverse, and compared the determinant with the entry-sized clamp eps_norm.
• A determinant is a product of two entries. So every block with entries below sqrt(eps_norm) got det = eps_norm while keeping its original numerators (adjugate).
• scaled_into_range (LAPACK validation: the nonsymmetric EVD fixtures exceed the GET22-style residual threshold (4 dtypes) #112) shrinks the blocks of a graded matrix enough for that to happen to every one of them.

Fix

• Both 2×2 block solves now use solve_shifted_2x2_cplx, a port of ?LALN2's method:
  • if every entry is below eps_norm, the block is taken as eps_norm * I;
  • otherwise, Gaussian elimination with complete pivoting, with the second pivot raised to eps_norm when it is smaller.
• The two solves are the complex-shift one and the real-shift one above a real eigenvalue. The real-shift one inverted through a^2 - b c, which underflowed to inf for tiny blocks.
• Review history: the first version raised only the determinant. The independent review (gpt-6.1-sol) gave a 4×4 Schur form where that is not a solved system (residual 4.5e14). Its counterexample is now a test, and the fix was changed to the elimination above. The second review found no BLOCKER or IMPORTANT issues.

Tests (all four dtypes; the real ones reach the changed code)

test	before
lapack_nonsymmetric_evd_graded_pairs_keep_their_residual: graded n = 16, 64, 128 plus a coupled 4×4, left and right GET22	f64 5.9e8 at n = 16, 1.5e12 on the 4×4
lapack_nonsymmetric_evd_pair_solve_with_a_small_determinant	determinant-floor version: 4.5e14; main: f32 4.2e6
lapack_nonsymmetric_evd_real_shift_block_below_the_square_root_of_min_positive	main: NoConvergence

• linalg::evd with --include-ignored (release): 167 passed. The one failure is the known c32 kinds 6/7 case of LAPACK validation: the nonsymmetric EVD fixtures exceed the GET22-style residual threshold (4 dtypes) #112, on the complex path and untouched (diagnostics = 96 before and after).

Upstream (upstream-reports/faer-029-real-evd-pair-clamp)

• faer 0.24.4 has the same comparison (z < eps * norm, src/linalg/evd/mod.rs:828).
• It does not fail on the graded matrices, because upstream doesn't scale the Schur form first. It does fail on a 4×4 with two small, close complex pairs coupled by entries of size 1e8:

	residual ratio
faer 0.24.4	8.97e4
LAPACK dgeev	9.6e-14
0.24.4 with this 2×2 solve swapped in	6.6e-14
fork after this fix	6.0e-14

• The faer-028 number was taken by sparse lu: report an exact zero pivot as LuError::NumericSingular in both backends (#132) #147 in the meantime, so this entry is faer-029. The PATCHSET has a new row (KEEP) linking to it.

Coordination

#112 (primerose-faer-autopilot) has no open PR. What remains there is the c32 complex path (kinds 6/7), and this change does not touch it.

🤖 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).