[skyon0522-ai]

The finite 2-by-2 matrix in MATH-1424 has two real eigenvalues, approximately 9,900,980.4766 and 20,099,019.5234. On current master, the unscaled input instead returns a complex pair with real parts 15,000,000 and imaginary parts approximately +/-5,099,019.5234; scaling the same input by 2^-20 works.

Schur convergence already identifies real pairs and negligible split boundaries, but their subdiagonal entries can retain rounding residuals. Eigenvalue extraction then interprets a residual above its absolute threshold as a complex block. Make those existing convergence decisions explicit by clearing accepted splits and the subdiagonal after a real-pair rotation. No convergence threshold, public API or dependency changes.

The regressions cover the analytic two-root result at both scales, a genuine 2 +/- i block beside 1e13, accepted split and one-root boundaries, and normalized Schur reconstruction and orthogonality residuals. All three new methods fail against the original Schur implementation; the mixed-scale complex control runs before the split assertion fails.

Validation on Linux, Temurin JDK 17.0.20.1+1 and Maven 3.9.11:

• The original EigenDecomposition test class reports 30 tests, 0 failures/errors and 2 existing skips.
• The fixed EigenDecomposition/Schur/Hessenberg classes report 50 tests, 0 failures/errors and 2 existing skips.
• Root clean verify with command-local JAVA_TOOL_OPTIONS=-Xmx6g: all 14 modules succeeded, exit 0. Retained reports contain 3,787 tests, 33 skips and zero final failures/errors. The existing retry policy recovered 41 cases after 79 failed attempts; this was not a first-pass-clean run.

The default-heap root run stopped at the existing FP64FieldDenseMatrixTest.testVeryLargeAllocation with heap exhaustion. The same allocation test also failed on pristine master at the default heap, then passed with the command-local 6 GiB setting before the fixed root run. Product code, POMs, JaCoCo and native retry settings were unchanged. I have not isolated the causes of the recovered flaky cases. The original-master comparison covered that allocation test, not a complete root build.

An isolated probe also reproduces the reported input failure in unmodified Commons Math 3.6.1. This patch and its regression tests target Math 4 master; it is not a tested Math 3 backport. Windows, macOS and new upstream GitHub CI have not been run for this patch.

AI assistance: OpenAI Codex assisted with source analysis, the implementation, regression tests and drafting.

[skyon0522-ai]

Using Temurin 17.0.20.1+1 and a direct JUnit4 Request.method launcher with no retries, CMAESOptimizerTest.testDiagonalRosen also fails on the PR's CI base commit 3d15331bf8b73ecb3f209fb165ad47d41dd13ef7 (expected 0, tolerance 1e-10, actual 1.416773781598585e-10). It fails on the tested PR merge 13b75826ecff8e0ded33baf51e2105da530a0a1c as well (4.08456980926037e-10); testMaximize passes once on each revision.

The test calls RandomSource.MT_64.create() without a fixed seed, so these are independent samples, not a deterministic same-seed comparison. Source inspection shows CMA-ES symmetrizes its covariance matrix before EigenDecomposition, selecting the symmetric/tridiagonal path; this check does not validate the modified SchurTransformer path. This check does not explain the other CI failures; I have not changed the patch or rerun CI.