Speed up Thiele continuum approximation; pluggable evaluation strategy

[tobydriscoll]

Summary

Memory- and type-stability work on the Thiele continuum approximate path, plus a cleaner mechanism for selecting evaluation strategies. All 925 tests pass; no behavioral changes to results (the two recurrences agree to ~1e-15).

Performance (vs. main, median via benchmark/compare.jl)

benchmark	change
Thiele continuum exp_interval / tanh_steep / abs_circle	−62% / −72% / −53%
Barycentric continuum exp_interval / tanh_steep / abs_circle	−21% / −10% / −6%
thiele_vector (scalar r.(z))	−23% (3010 → 3 allocations)
poles	−5%
discrete approximate, bary_vector	invariant

What changed

• evaluate! scratch reuse: new evaluate!(t, r, z, a, b) overload lets the hot loop reuse the recurrence's work arrays instead of allocating per call; the 3-arg form delegates to it.
• Allocation-light index management: the live test-point set is tracked as a linear integer index vector (active), grown incrementally and rebuilt only on path reset, supporting a non-rectangular active set. Test points are gathered into a dense reusable buffer so evaluate! runs on contiguous vectors.
• Function barriers (_sweep!): the per-point work is routed through a single barrier function (two methods) so it specializes on concrete element types. This is necessary because ComplexRegions' point(d, ·) infers as Any, making σ/fσ/fτ/buffers abstract inside approximate.
• DiscretizedPath.points parameterized (Matrix{P}) — fixes an abstract field type and benefits all consumers (Barycentric, parfrac, spacing, collect).
• Evaluation-strategy mechanism: replaced the non-const-global _evaluator/_new_weight toggles (which forced a dynamic dispatch + heap box on every scalar evaluation) with strategy singletons OneDiv/Classic <: ThieleMethod. The default path is now a static, allocation-free call; evaluate(r, z, method) / add_node!(r, z, y, method) take an optional explicit strategy, and set_eval_method / set_weight_method retain the global-switch capability for tests.

API additions (exported)

ThieleMethod, OneDiv, Classic, set_eval_method, set_weight_method.

Housekeeping

• Version bumped to 0.3.7.
• Tightened ComplexRegions compat to 0.3.9 (earlier 0.3 allowed 0.3.7, which lacks points used by the Barycentric path).
• Added benchmark/compare.jl, a before/after benchmark harness.

[codecov]

Codecov Report

❌ Patch coverage is 94.36620% with 4 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/thiele.jl	94.28%	4 Missing ⚠️

📢 Thoughts on this report? Let us know!

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Benchmark Results

Ratios above 1 mean this PR is slower than main.

	main	12e9781...	main / 12e9781...
approximate/aaa/abs_circle	0.0451 ± 0.0021 s	0.0448 ± 0.0026 s	1.01 ± 0.075
approximate/aaa/exp_interval	0.114 ± 0.027 ms	0.0998 ± 0.025 ms	1.15 ± 0.39
approximate/aaa/tanh_steep	1.21 ± 0.055 ms	1.12 ± 0.068 ms	1.08 ± 0.082
approximate/thiele/abs_circle	9.23 ± 0.35 ms	3.35 ± 0.1 ms	2.76 ± 0.13
approximate/thiele/exp_interval	0.175 ± 0.022 ms	0.0659 ± 0.0055 ms	2.65 ± 0.4
approximate/thiele/tanh_steep	0.694 ± 0.027 ms	0.166 ± 0.013 ms	4.18 ± 0.36
approximate_discrete/aaa/abs_shift	3.37 ± 0.26 ms	3.41 ± 0.36 ms	0.988 ± 0.13
approximate_discrete/aaa/sin_recip	0.882 ± 0.11 ms	0.881 ± 0.13 ms	1 ± 0.19
approximate_discrete/aaa/tanh_steep	12.1 ± 0.61 ms	12.2 ± 0.98 ms	0.995 ± 0.095
approximate_discrete/thiele/abs_shift	0.206 ± 0.016 ms	0.206 ± 0.016 ms	1 ± 0.11
approximate_discrete/thiele/sin_recip	0.12 ± 0.0082 ms	0.119 ± 0.008 ms	1 ± 0.097
approximate_discrete/thiele/tanh_steep	0.626 ± 0.026 ms	0.623 ± 0.023 ms	1 ± 0.056
evaluate/bary_vector	0.0872 ± 0.004 ms	0.0876 ± 0.0046 ms	0.995 ± 0.069
evaluate/thiele_vector	0.0847 ± 0.0041 ms	0.0584 ± 0.00077 ms	1.45 ± 0.073
poles/thiele	0.356 ± 0.011 ms	0.354 ± 0.011 ms	1.01 ± 0.044
time_to_load	0.66 ± 0.0052 s	0.667 ± 0.0021 s	0.989 ± 0.0084

Benchmark Plots

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("Actions" -> this run -> "Artifacts" at the bottom).

[tobydriscoll]

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Release notes:

• Improved Thiele construction speed by eliminating type instabilities and dynamic dispatches.
• Changed and exposed the undocumented mechanism for switching Thiele evaluation methods.

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