MiniTensor: modernize the general log() and exp() algorithms

[lxmota]

@trilinos/minitensor

Motivation

The general log() used a transformation-free inverse scaling and squaring whose Denman–Villiers square-root iterations and determinant-based scaling are both expensive and fragile for ill-conditioned input (see #15513), and exp() used the 2005-era parameter selection with rounded threshold constants. This PR brings both up to the current published algorithms, specialized to the small dimensions MiniTensor serves.

Changes

log(), dimensions <= 3 (larger dimensions keep the previous path):

• Within the Padé trust region (norm_1(A - I) <= 0.72): direct Gauss–Legendre partial-fraction Padé evaluation on the exactly computed A − I. This preserves the relative accuracy of small logarithms, which any similarity-based method loses to the ~eps·norm(A) rounding of the transformation itself.
• Beyond it: Schur-based inverse scaling and squaring in the spirit of Al-Mohy & Higham, SIAM J. Sci. Comput. 34(4), 2012 — real Schur decomposition (Givens Hessenberg reduction, Francis double-shift QR with the classical difference-form shift vector and standardized 2×2 blocks), exact quasi-triangular square roots instead of Denman–Beavers iterations, Padé on the quasi-triangular factor, and exact recomputation of the diagonal blocks of the logarithm from the eigenvalues. Eigenvalues on the closed negative real axis now produce a clean error instead of non-convergence, and the determinant-scaling failure mode of MiniTensor: fix log() failure on ill-conditioned SPD tensors #15513 is structurally absent from this path.

exp(): parameter selection of Al-Mohy & Higham, SIAM J. Matrix Anal. Appl. 31(3), 2009. Their θ thresholds make the norm-based order choice provably exact short-circuit evaluation of the full algorithm whenever norm_1(A) <= θ13 = 4.25; the norm(A^p)^(1/p) overscaling analysis and the rounding-correction term (with a conservative early exit and binary powering) run only above that threshold. Also fixes a decimal-point typo (9,4e-0) in the scaling_squaring_theta table that silently shifted entries above order 16.

Related Issues

Stacked on #15513 and includes its commit; this PR will be rebased to a single commit once #15513 merges. Follows #15514.

Testing

• Full MiniTensor gtest suite passes (36 tests), including three new ones: a nonsymmetric logarithm with a complex-conjugate eigenvalue pair recovered through log(exp(L)), the analytic logarithm of a scaled 2×2 rotation, and agreement of exp() with the Taylor series on a nonsymmetric argument.
• Benchmarked on 500-sample suites per parameter point (35 parameter points for the log, 25 for the exp) against an independent 80-bit eigendecomposition reference:
  • log(): identical accuracy and speed near the identity; 1.8–4.3× more accurate and 1.4–4.7× faster away from it (condition numbers 10 to 1e14); zero failures across 17,500 samples.
  • exp(): median cost and accuracy unchanged; up to 1.8× more accurate at strong stretches; the extreme-norm cases (norm > 4.25) pay up to ~1.9× for the overscaling analysis.

[github-actions]

CDash for AT2 results

Reproducing CI builds locally

CI builds can be reproduced locally in a container.

Requirements: Python 3.9+, Podman, CMake

Steps:

• Check out the branch of the PR locally.
• Run commonTools/pr_reproducer/reproducer.sh.

[ikalash]

Looks good.