Reduce test time

[ohno]

Summary

The test suite is dominated by first-call compilation rather than steady-state numerical work. The largest bottleneck is the nested three-dimensional quadgk calculation in test/Basis.jl.

CI timings

These timings come from the same PR #56 CI run:

• Julia 1.10 job
• Julia 1.12 job

Test	Julia 1.10	Share	Julia 1.12	Share
Basis.jl	76.6 s	42.0%	595.9 s	80.5%
VNN.jl	34.2 s	18.8%	39.7 s	5.4%
QTT.jl	30.9 s	17.0%	45.8 s	6.2%
Rayleigh-Ritz.jl	12.7 s	7.0%	20.5 s	2.8%
GEM.jl	7.9 s	4.3%	14.1 s	1.9%
Total	182.2 s	—	740.7 s	—

Basis.jl is 7.8 times slower on Julia 1.12 and accounts for about 80% of that job's test time.

Detailed measurements

The current Fourier-transform test evaluates

4 basis functions × 3 momenta × 2 polar angles × 2 azimuthal angles
= 48 nested three-dimensional adaptive integrals

On Julia 1.10:

• complete Basis.jl: 76.18 s and 22.29 GB allocated
• first (a, k) group: 76.24 s and 22.13 GB allocated
• all remaining groups combined: about 0.84 s

This concentration in the first call identifies specialization of the nested quadgk closures as the dominant cost.

The other compile-heavy tests show the same first-call pattern:

Operation	First call	Repeated call
QTT solve	24.44 s, 2.32 GB	0.168 s
VNN solve	20.38 s, 2.20 GB	0.0007 s

The 100-iteration VNN training calculation takes 1.19 s after compilation.

Proposed change

For a spherical Gaussian, reduce the Fourier transform analytically to one radial integral,

$$(2\pi)^{-3/2}\int e^{i\boldsymbol{k}\cdot\boldsymbol{r}}\phi(r)\,d^3r = \sqrt{\frac{2}{\pi}}\int_0^\infty r^2\phi(r)j_0(kr)\,dr.$$

The replacement prototype evaluates 12 one-dimensional integrals using sphericalbesselj(0, k * r):

• 1.29 s and 172 MB allocated
• maximum absolute error: 7.9e-6
• approximately 59 times faster than the current Basis.jl test

Test expikr separately against cis(k * r * cosγ) so that plane-wave geometry remains covered without the nested adaptive integration.

After this change, remeasure QTT and VNN. Their steady-state calculations are already short, so further improvement should target precompilation or parallel CI jobs rather than reducing solver iterations.

Completion criteria

• Replace the nested three-dimensional Fourier-transform test with the radial spherical-Bessel integral.
• Add a separate direct test for expikr.
• Preserve the current numerical coverage and tolerances.
• Pass the complete suite on Julia 1.10 and Julia 1.12.
• Record the updated CI timing table.

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This issue was drafted with assistance from Codex using GPT-5.6 Sol with High reasoning effort.