Benchmark: freqresp! optimization strategies

[baggepinnen]

Standalone benchmark (under benchmark_freqresp/) comparing strategies for maximizing freqresp! performance on the target use case: a 48-state SISO continuous state-space system over 150 frequencies. Nothing in the package source is modified — all candidate implementations live in benchmark_freqresp/variants.jl, with ControlSystemsBase.freqresp! as the correctness oracle.

Companion to #1062 (full write-up and discussion there).

Strategies benchmarked

• Baseline / serial — existing Hessenberg ldiv2! loop
• Threading — Threads.@spawn, Polyester.@batch, OhMyThreads over frequency chunks
• SIMD — @turbo and @tturbo ldiv2! (real/imag split, since @turbo lacks Complex support)
• Modal / pole-residue — eigendecompose once, then a pure broadcast+reduction (the Reactant/XLA/GPU-friendly form)

Key findings (24-core machine, loop-only µs)

Strategy	µs	speedup
serial Hessenberg (current)	~545	1×
@turbo ldiv2!	~370	1.5×
threaded (best, ~8 threads)	~90	6×
threaded + @turbo	~78	7×
modal, serial	~42	13×
modal, threaded	~10	54×

• The per-frequency loop dominates over the one-time factorization (~565 µs vs ~85 µs).
• @turbo is a solid ~1.5× free win (needs a manual real/imag split — bare @turbo won't compile on complex).
• Threading scales sub-linearly; ~8 threads beats 24 for this size.
• @tturbo is slower than @turbo — wrong axis to thread.
• Algorithm beats micro-optimization: the modal reformulation is 13–54×, but trades numerical robustness (depends on cond(V); degrades for non-normal/defective A), which is why the package defaults to the Hessenberg solve.
• Reactant.jl is not worth it at this scale — the modal form already runs in ~10 µs on CPU, below XLA dispatch / GPU launch latency.

Run

cd benchmark_freqresp
julia --project=. -e 'using Pkg; Pkg.develop(path="../lib/ControlSystemsBase"); Pkg.add(["BenchmarkTools","LoopVectorization","Polyester","OhMyThreads"]); Pkg.instantiate()'
JULIA_EXCLUSIVE=1 julia -t auto --project=. benchmark.jl

Draft — this is an exploratory benchmark, not a proposed package change. Opening for visibility/discussion of which strategy (likely the real/imag-split @turbo ldiv2!, and an opt-in modal fast path) to upstream.

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Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 91.64%. Comparing base (856eb2b) to head (76e9832).
⚠️ Report is 2 commits behind head on master.

Additional details and impacted files

@@             Coverage Diff             @@
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+ Coverage   19.68%   91.64%   +71.95%     
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  Files          42       42               
  Lines        5598     5660       +62     
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+ Misses       4496      473     -4023     

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