Against other codes

Magνs is not the fastest way to compute every oscillation probability, and this page says where it is not. It follows the cross-code comparison of the Magνs paper, whose source is in resources/paper/: eight codes on three setups, Magνs against NuOscProbExact at two to five flavors and the cost of an averaged solar probability, all computed in notebook 28. It ends with advice on which code to use for which problem. Every code is timed in one process on one machine.

Warning

Absolute timings are a property of the machine and are worth little on their own. Read the ratios within a row, and read a tolerance sweep only within one code’s own curve. A claim that one code is faster than another may not hold on different hardware; a claim that tightening one setting costs 20% and gains four orders of magnitude in accuracy usually does.

Eight codes, three setups

Error against time per probability for eight codes on three setups

Error against time per probability, for eight codes: constant density (top), a core-crossing Earth chord at \(\cos\theta_z = -0.9\) at three flavors (middle), and the same chord with one sterile state (bottom). Magνs and NuOscProbExact appear twice on the Earth panels because each has two settings, a slab count and a tolerance.

The codes are GLoBES, Prob3++, nuCraft, NuFast-LBL, NuFast-Earth, NuOscProbExact, nuSQuIDS and Magνs, with a second-order analytic expansion for reference. Each is scored against a 50-digit reference built in its own constants and conventions, chiefly the constant that fixes \(V_{\rm CC}\), so two codes can both reach \(10^{-14}\) here and still disagree with each other at \(10^{-4}\).

  • Constant density. Most codes land near round-off and differ only in cost, from about 0.06 µs per probability for NuFast-LBL to about 100 µs for nuSQuIDS. Magνs sits at \(3 \times 10^{-15}\) and 3.9 µs.

  • Earth, three flavors. To reach about \(3 \times 10^{-10}\), Magνs needs 0.43 ms (256 slabs), nuSQuIDS 1.3 ms, NuOscProbExact 26 ms, nuCraft 62 ms and GLoBES 0.5 s. NuOscProbExact starts twenty times cheaper, and the two cross near \(10^{-6}\): its error falls as the square of the slab width, Magνs’s as the fourth power. Asking Magνs for rtol=1e-8 instead of 256 slabs costs about three times as much, for the same accuracy; that is the price of the convergence check. NuFast-Earth is by far the cheapest when a scan varies only \(\delta_{\rm CP}\).

  • Earth, 3+1 flavors. NuFast-Earth, GLoBES and Prob3++ drop out. The crossing with NuOscProbExact moves up to about \(10^{-5}\), because its slab cost grows eightfold at four flavors against under threefold for Magνs.

Magνs against NuOscProbExact, two to five flavors

Accuracy against cost, Magνs against NuOscProbExact

Deviation from an extended-precision reference against time per probability, at two to five flavors, on a smooth exponential profile (left) and a core-crossing Earth chord (right). Magνs at orders 4, 6 and 8; NuOscProbExact through its own adaptive refinement. Both use \(Y_e = 0.5\) electrons per atomic mass unit (Conventions). NuOscProbExact stops at four flavors.

NuOscProbExact’s closed form is the cheaper code at loose tolerances. At tight tolerances it runs into an accuracy floor that more slabs cannot lower, and Magνs reaches past it.

The averaged solar probability

Cost of an averaged solar survival probability

Time for one averaged \(\nu_e\) survival probability at 5 MeV through the tabulated BS2005-AGS,OP model ('BS05-AGS-OP' in Magνs), two to five flavors, with and without NSI and LIV: Magνs against a DOP853 integration followed by an average over the last 1% of the path.

Magνs computes the average directly, from one eigendecomposition at production and one at detection, and is four or more orders of magnitude faster than integrating the \(10^5\) oscillations and averaging them away. The call is:

import magnus.globaldefs as gd
import magnus.oscprob as oscprob
from magnus import solarmodels

P = oscprob.osc_prob_3nu_sun(
    5.0*gd.UNIT_MEV, solarmodels.table_edge('BS05-AGS-OP'), 0.0,
    density_profile='BS05-AGS-OP', average=True, nu_i=gd.NUE, nu_f=gd.NUE)

When to use Magνs, and when not

Structurally, Magνs sits between NuOscProbExact and nuSQuIDS: it composes slab exponentials as the first does, while resolving the variation of the Hamiltonian inside each slab, as the second does by other means. Use Magνs when one of the following puts a problem beyond what constant-density slabs handle well:

  • The density varies continuously and fast against the oscillation length. A composition of constant-density slabs needs some \(10^4\) steps per resonance crossing in the Sun, whereas Magνs integrates across the slab.

  • The profile has structure at a known place. A shock front, a layer boundary or a kink is passed as t_breakpoints, which puts a slab edge on it at every refinement level.

  • The problem has more than three flavors. At 3+1, NuOscProbExact finds its eigenvalues numerically, at eight times the cost per slab; past four flavors the SU(N) closed forms stop.

  • The Hamiltonian has no closed form of its own. Nothing in Magνs assumes a form for \(H(l)\) beyond Hermiticity, so a non-standard interaction, a Lorentz-violating background or a new Hamiltonian is passed as a matrix, with no change to the solver.

  • The observable is an average. Phase-averaged probabilities and flavor compositions are returned directly rather than reconstructed from a scan (Phase-averaged probabilities).

  • The required accuracy is below the floor of a slab composition. On a core-crossing PREM chord, NuOscProbExact stops near \(2 \times 10^{-10}\); Magνs reaches further. Both are far below the percent level that matters in a data analysis.

Elsewhere another code may be cheaper:

Situation

Cheapest

Because

Constant density

NuFast-LBL (three flavors); closed-form codes

Every code reaches round-off; NuFast-LBL does it in about 0.06 µs, some sixty times faster than Magνs

Earth chord, three flavors, a fit that moves only \(\delta_{\rm CP}\)

NuFast-Earth

It reuses its layer solve across \(\delta_{\rm CP}\)

Earth chord, accuracy no better than about \(10^{-6}\)

NuOscProbExact

Twenty times cheaper per slab; Magνs overtakes it below about \(10^{-6}\) (\(10^{-5}\) at 3+1), because the error of Magνs falls as the fourth power of the slab width, against the second for NuOscProbExact

Decay, decoherence, collective effects

nuSQuIDS

They need a density matrix; no unitary solver here, Magνs included, represents them

Before comparing any two codes’ numbers

Check that they agree in vacuum first. If they do not, the disagreement is in the solvers. If they agree in vacuum and disagree in matter, it is in the conventions — a matter-potential factor, a \(Y_e\), a channel index — and no amount of tolerance will close it. Notebook 25 spends a whole section on a conventions trap for this reason: a 1% difference in \(V_{\rm CC}\) reads exactly like an accuracy difference until you look.

See also

Notebook 25 compares Magνs with NuOscProbExact and nuSQuIDS in more detail, from the frozen datasets in notebooks/external_*.json: batching, the compiled kernel, the solar average in nuSQuIDS, and a supernova shock at 3+1 and with NSI.