New ecorr sqrtsolve

[vhaasteren]

Summary

• Added new non-Cholesky sqrtsolve implementation that computes N^{-1/2} X using a closed-form diagonal-plus-rank-1 inverse square-root identity, replacing the previous Cholesky-based approach as the default
• Preserved the original Cholesky-based whitening under cholsqrtsolve for all three classes (ShermanMorrison, FastShermanMorrison, and the test reference class)
• The new sqrtsolve is ~2.4x faster than cholsqrtsolve for the whitening step, and computing Z^T N^{-1} Z via sqrtsolve + W^T W is competitive with (or slightly faster than) the direct BLAS-accelerated Sherman-Morrison 2D solve

Details

Mathematical approach: For each ECORR block with covariance D + j 11^T, the new sqrtsolve uses the rank-1 identity

(I + u u^T)^{-1/2} x = x + alpha u (u^T x), alpha = -1 / (sqrt(1+v) (sqrt(1+v) + 1))

where v = ||u||^2. This avoids forming or solving a triangular Cholesky factor entirely. The alpha formula is the numerically stable equivalent of (1/sqrt(1+v) - 1)/v, avoiding cancellation for all values of v.

What changed:

• python_block_sqrtsolve_rank1 / python_idx_sqrtsolve_rank1: new pure-Python implementations using the closed-form formula
• cython_block_sqrtsolve_rank1 / cython_idx_sqrtsolve_rank1: new Cython implementations of the same
• The previous Cholesky-based implementations are renamed to *_cholsqrtsolve_* (all four variants: python block/idx, cython block/idx)
• ShermanMorrison and FastShermanMorrison classes now expose both sqrtsolve() (non-Cholesky, default) and cholsqrtsolve() (Cholesky-based) public methods
• _sqrtsolve_D2 on all classes now calls the non-Cholesky path; _cholsqrtsolve_D2 preserves the old behavior

Tests:

• Existing test_sqrtsolve_D12 updated to test the new non-Cholesky sqrtsolve (element-wise cross-checks between Python reference and Cython, plus end-to-end Z^T N^{-1} Z = (WZ)^T (WZ) verification)
• New test_cholsqrtsolve_D12 added for the Cholesky-based path, with full cross-validation across all three implementations (Python rank-1, scipy Cholesky, Cython rank-1) and the same end-to-end correctness check
• Redundant element-wise comparisons between classes that now share the same algorithm are commented out

Benchmark (not included in codebase)

48,000 TOAs (1500 epochs x 32 obs/epoch), 500 parameters:

Operation	Time
Direct Z^T N^{-1} Z (BLAS SM 2D)	319 ms
N^{-1/2} Z (sqrtsolve whitening)	193 ms
L^{-1} Z (cholsqrtsolve whitening)	513 ms
sqrtsolve + W^T W	277 ms

Numerical accuracy identical across all paths (~2e-15 relative error).

Test plan

• All existing tests pass (test_solve_D1, test_solve_1D1, test_solve_2D2, test_sqrtsolve_D12, test_errors, test_logdet)
• New test_cholsqrtsolve_D12 passes
• End-to-end correctness: Z^T N^{-1} Z == (WZ)^T (WZ) verified for all code paths (block, indexed, shuffled)****

[codecov-commenter]

Codecov Report

❌ Patch coverage is 93.10345% with 4 lines in your changes missing coverage. Please review.
✅ Project coverage is 91.76%. Comparing base (ff28e9e) to head (5e62e78).

Files with missing lines	Patch %	Lines
fastshermanmorrison/fastshermanmorrison.py	93.10%	4 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main       #8      +/-   ##
==========================================
- Coverage   92.42%   91.76%   -0.66%     
==========================================
  Files           2        2              
  Lines         198      255      +57     
==========================================
+ Hits          183      234      +51     
- Misses         15       21       +6     

Flag	Coverage Δ
unittests	91.76% <93.10%> (-0.66%)	⬇️

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Files with missing lines	Coverage Δ
fastshermanmorrison/fastshermanmorrison.py	92.40% <93.10%> (-0.87%)	⬇️

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