Table 1
Summary of computation methods to obtain the pseudo-power spectrum covariance matrix.
| Method | Equation numbers | Precision | Complexity | |
|---|---|---|---|---|
| Exact (this work) | Eq. (25) |  computed ∀ℓ′, ℓ1 ,ℓ2 |
N/A |  (Using HEALPix pixelation) |
| NKA | Eq. (26) |  |
4% | O(1) |
| FRI | Eq. (29) |  |
4% | O(1) |
| INKA | Eq. (31) |  |
4% |  or  with Louis et al. (2020) |
| ACC (this work) | Eq. (33) |  invariant for Δ ≡ |ℓ − ℓ′| =cst |
1% |  |
Notes. First column: name of approximation. Second column: equation to which they are referred. Third column: expression of 
in this approximation. Fourth column: precision determined by the maximum values of the relative difference of the EEEE binned covariance on diagonal for multipoles ℓcut ≤ ℓ ≤ ℓmax,ex in Fig. 6. For larger multipoles, the approximation are expected to be in this range of precision as shown in Fig. 12. Fifth column: summary of computing resources needed to obtain 
in each approximation. Let us here specify that for INKA, the kernel M is often already known, thus the practical complexity is O(1). ℓmax is the multipole range of the covariance, dmax is the number of diagonal computed in the ACC approximation, nside is the resolution chosen to compute the covariance coupling kernels in the ACC approximation (closest to ℓcut is sufficient).
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