perf: fuse boundary predict_cov and parallelize backward smoother

[YuminosukeSato]

Summary

Optimize the Kalman filter for large seasonal models (s=168, T=1200). Two key changes:

• Fuse predict_state_covariance_flat boundary case from 4-pass (TP, TP*T', symmetrize, add Q) to 2-pass closed-form using the sparse structure of the seasonal transition matrix T. Reduces forward-filter memory traffic by ~58%.
• Split DK backward smoother into Phase 1 (sequential r recursion) and Phase 2 (Rayon parallel P*r correction via par_chunks_mut). Phase 2 is embarrassingly parallel since each smoothed state depends only on a_pred[i], P_pred[i], and r_store[i].
• Flatten alpha_plus and smoother output from Vec<Vec<f64>> to contiguous Vec<f64> buffers for better cache locality.

Benchmark

Config	Before (v1.3.1)	After
s=168, T=1200, niter=1000, nwarmup=500	~92s	~85s (8% improvement)
s=168, T=1200, niter=1000, nwarmup=0	~61s	~58s (5% improvement)

Note: The forward filter is memory-bound at this scale (p_pred_flat = 900 × 168² × 8B ≈ 203MB, exceeding L3 cache). The FLOPs reduction from 4→2 pass fusion has proportionally less effect under memory bandwidth saturation. Further optimization would require a p_pred recomputation strategy or blocked access pattern.

Changes

src/kalman.rs

• predict_state_covariance_flat: Replace 4-pass boundary logic with closed-form 2-pass using row_sums precomputation and lower-triangle + mirror write
• Add debug_assert for input P symmetry verification
• Backward pass: 2-phase split (Phase 1: sequential r_store, Phase 2: par_chunks_mut parallel)
• seasonal_kalman_smoother: Return flat Vec<f64> (t×s) instead of Vec<Vec<f64>>
• alpha_plus: Flatten to contiguous buffer with split_at_mut for borrow safety
• Add use rayon::prelude::*;

Tests (12 new)

• 9 PR-D tests: fused predict_cov vs naive reference (s=2,4,12,168 boundary, non-boundary, symmetry, diagonal positivity, identity P, zero P)
• 3 PR-E tests: Rayon backward vs sequential (DK vs RTS cross-validation at s=4/s=168, finite check)

Test plan

• cargo test -- -q — 91 Rust tests pass (79 existing + 12 new)
• .venv/bin/pytest tests/ -v — 236 Python integration tests pass
• Benchmark: python z-ai/bench_large_seasonal.py
• Verify on CI

Release

Version bump: 1.3.1 → 1.4.0 (minor: internal API change to seasonal_kalman_smoother return type + performance improvements)