pybhpt.swsh performance

Timing of the spin-weighted spheroidal harmonic. See the Performance overview for the methodology and reference machine.

SpinWeightedHarmonic(s, l, m, gamma, theta) does two things at construction, which scale with different parameters and are benchmarked separately:

1. a spectral solve for the eigenvalue and the spherical–spheroidal coupling coefficients, whose cost is driven by the spheroidicity gamma = a·omega (it widens the coupling band and the eigenproblem);

2. evaluation on the theta grid of S, S', S'', whose cost is linear in the number of grid points on top of the fixed solve.

Spectral solve

Median time to build the eigenvalue + coupling representation (measured on a tiny grid so the solve dominates):

gamma	median solve time
0 (spherical)	3.6–15 µs
1	62–112 µs
4	73–250 µs
10	0.23–0.45 ms

At gamma = 0 the harmonic reduces to a spin-weighted spherical harmonic and no eigenvalue solve is needed, so it is ~50× cheaper. For nonzero gamma the cost is set by the coupling bandwidth rather than l alone, and rises with gamma (strong spheroidicity, e.g. high-frequency modes of a near-extremal black hole), where the truncated coupling matrix must be enlarged. The growth is smooth: the widest spread within a gamma column is a factor of ~3.4, across all (s, l) in the sweep.

Full data: benchmarks/data/swsh_solve.csv.

On-grid evaluation

Median total construction time (spectral solve plus on-grid evaluation) versus grid length, for three representative modes:

mode (s, l, m, gamma)	ns = 64	512	4096
(-2, 2, 2, 2)	0.18 ms	0.56 ms	3.8 ms
(-2, 8, 4, 4)	0.37 ms	0.99 ms	6.4 ms
(2, 13, 7, 8)	0.48 ms	1.5 ms	9.3 ms

The slope with nsamples is the on-grid evaluation cost; the intercept is the fixed per-mode spectral-solve offset, a few hundred µs even for the high-gamma mode, so from ns = 512 up the grid evaluation dominates. Once solved, an instance can be re-evaluated at arbitrary angles via .eval(theta, deriv=…) without repeating the solve.

Full data: benchmarks/data/swsh_grid.csv.

Random parameter-space sample

The same treatment as the radial random sample, with one continuous axis instead of two: the radial solver takes spin and frequency separately, while the harmonic sees only their product gamma = a·omega. 600 random points — l uniform on [2, 50], one random m per point in [-l, l], gamma uniform on [0.02, 5] — at s = -2 on a fixed 256-point theta grid, timing the full construction (spectral solve plus on-grid evaluation). Every point succeeded. Median 0.65 ms, p90 1.18 ms, p99 1.53 ms, slowest point 1.60 ms.

Cost rises with both, l far more steeply than gamma; the hottest points are the high-l end, at no particular gamma.

The expensive modes cluster near m = 0, not at |m| = l. This is the opposite of the radial solver, where large |m| is the costly end.

The m ≈ 0 band darkens as gamma grows: low |m| at high spheroidicity is the expensive combination.

Binning by |m|/l shows the effect cleanly, at fixed everything else:

abs(m)/l	n	median	p90
0.0–0.2	116	0.84 ms	1.45 ms
0.2–0.5	157	0.81 ms	1.24 ms
0.5–0.8	184	0.62 ms	1.07 ms
0.8–1.0	143	0.51 ms	0.86 ms

A log-log fit (R² = 0.69) gives log l +0.46, log gamma +0.15, and |m|/l −0.46 — the only strongly negative coefficient anywhere in these benchmarks. Nearly axisymmetric modes need the widest spherical–spheroidal coupling band, so they carry the largest eigenproblem.

Full data: benchmarks/data/swsh_random.csv.