pybhpt.radial performance

Timing of the homogeneous radial Teukolsky solutions. See the Performance overview for the methodology and reference machine.

RadialTeukolsky(s, l, m, a, omega, r).solve(method) computes the In/Up homogeneous solutions on the supplied radial grid. Four things drive the cost, benchmarked separately: the solver method, the radial grid length, the mode frequency omega (more oscillatory solutions cost more, differently for each method), and the mode numbers and spin (l, m, a). The first three use a representative mode (s=-2, l=5, m=2, a=0.9); the last sweeps l, m, and a directly.

Method comparison

Median solve time by method, at omega = 0.3 on a 256-point grid:

method	median time
HBL	0.54 ms
AUTO	0.54 ms
GSN	4.9 ms

HBL and GSN are numerical ODE integrators, and AUTO dispatches between them. GSN is ~10× the cost of HBL at this mode; AUTO tracks HBL almost exactly, confirming it picks the cheap integrator for a generic mode.

Note

The analytic MST series solver is not benchmarked here. It was removed from AUTO’s dispatch in 6d214d5 as unreliable at high l, so AUTO will never select it. It remains callable explicitly via solve("MST"), but it is not part of the supported solve path these numbers describe.

Full data: benchmarks/data/radial_method.csv.

Grid-length scaling

Median solve time (method AUTO) versus the number of radial output points:

nsamples	64	256	1024	4096
median time	0.43 ms	0.54 ms	1.1 ms	3.5 ms

Cost grows sub-linearly at small grids (dominated by a fixed setup cost — boundary condition solve, method dispatch) and approaches linear in nsamples at large grids, where evaluating the solution on the output grid dominates.

Full data: benchmarks/data/radial_grid.csv.

Frequency sweep

Median solve time versus omega, for each method, on a fixed 256-point grid:

method	omega=0.02	0.1	0.6	2.5
HBL	0.49 ms	0.45 ms	0.60 ms	0.87 ms
AUTO	0.49 ms	0.42 ms	0.59 ms	0.92 ms
GSN	5.1 ms	4.9 ms	5.7 ms	16.0 ms

Both integrators grow with omega — more oscillatory solutions need more integration steps — but not at the same rate: HBL costs 1.8× more at omega = 2.5 than at 0.02, while GSN costs 3.1× more, so the gap between them widens with frequency. AUTO tracks HBL across the whole sweep.

Full data: benchmarks/data/radial_freq.csv.

Mode number and spin

Method AUTO, omega = 0.3, 256-point grid — so the mode (l, m) and the background spin a are the only things varying. Each cell is the median over five equally spaced m in [-l, l]:

l	a = 0	a = 0.3	a = 0.7	a = 0.9	a = 0.99
2	0.38 ms	0.42 ms	0.43 ms	0.46 ms	0.48 ms
5	0.43 ms	0.48 ms	0.49 ms	0.54 ms	0.61 ms
10	0.48 ms	0.55 ms	0.57 ms	0.64 ms	0.95 ms
15	0.57 ms	0.64 ms	0.69 ms	0.77 ms	1.32 ms
30	0.96 ms	1.06 ms	1.16 ms	1.37 ms	2.35 ms
50	1.87 ms	2.05 ms	2.16 ms	2.47 ms	3.94 ms

All 150 (l, m, a) combinations solved successfully — nothing failed, even at l = 50, a = 0.99.

The two parameters compound rather than acting independently. Growth from l=2 to l=50 is 4.9× at a = 0, but 8.1× at a = 0.99; equivalently, going from a = 0 to a = 0.99 costs 1.3× at l = 2 but 2.1× at l = 50. Spin is nearly free for slowly rotating black holes and only becomes a real cost driver in the near-extremal, high-l corner.

m matters mostly through its sign at high spin. At a = 0.9, l = 30 the spread across m is mild (1.22 ms at m = 0 to 2.01 ms at m = -30), but at a = 0.99, l = 15 the retrograde end costs 4.3× the m = 0 mode (3.33 ms vs 0.77 ms). Prograde modes stay cheaper than retrograde ones of the same |m| throughout.

Note

One reproducible spike. l = 50, m = -25, a = 0.99 takes ~18 ms — over 3× its immediate neighbours (m = -24: 5.8 ms, m = -26: 4.6 ms), breaking the otherwise smooth trend. It is not a method-dispatch artifact: timing the integrators directly gives HBL = 18.4 ms and AUTO = 17.9 ms at that mode versus 5.5 ms for both at m = -24, so the extra cost is inside the HBL integrator itself, not in AUTO falling back to something slower. Left in the data rather than smoothed away.

Full data: benchmarks/data/radial_mode.csv.

Random parameter-space sample

The sweeps above each hold two of the three cost drivers fixed, so none of them shows how l, a and omega interact. This one samples all of them at once: 600 random points with l uniform on [2, 50], one random m per point in [-l, l], a log-uniform in 1 - a over [0.01, 1], and omega uniform on [0.02, 5]. Method AUTO, 256-point grid. Every point solved — nothing failed anywhere in that box. Times run 0.42–4.81 ms, median 1.61 ms.

Spin is plotted as 1 - a on a log axis, near-extremal to the right: the cost structure bunches against a = 1, and a linear spin axis compresses the entire a = 0.9 to 0.99 range — where most of the variation lives — into the last tenth of the plot.

Cost climbs along both axes with no visible interaction between them; the cheap corner is low l, low omega.

The gradient is almost entirely vertical: l dominates, and spin only sharpens the tail in the near-extremal column on the right.

With l averaged out, the remaining structure is a mild climb in omega and an elevated near-extremal edge.

A log-log fit of log(time) against the four parameters (R² = 0.80) ranks them:

term	exponent
log l	+0.40
log omega	+0.20
−log(1 − a)	+0.09
m/l	−0.02

So cost scales roughly as l^0.4 ω^0.2, with l the leading driver by about 2× over frequency and 4× over spin. Rank correlations tell the same story: l +0.75, omega +0.51, 1 - a −0.29.

The m/l coefficient is near zero, but raw |m| correlates at +0.54 — that is |m| riding along with l, not an independent effect, consistent with the l versus l - |m| comparison above. The sign asymmetry does survive and is spin-dependent: near-extremal (1 - a < 0.05) retrograde modes run 2.04 ms against 1.83 ms prograde, narrowing to 1.37 vs 1.27 ms at slow spin.

Note

These exponents describe the box sampled here, not a universal scaling law. The earlier draft of this sample drew omega log-uniformly, which concentrated points at low frequency where the omega cost is flat, and reported log omega at +0.07 instead of +0.20 — the same physics weighted differently. Read them as a guide to relative cost within this range.

Full data: benchmarks/data/radial_random.csv.