Fast computation of angular power spectra and covariances of high-resolution cosmic microwave background maps using the Toeplitz approximation

We present a simple approximation that can speed up the computation of the
mode-coupling matrices, which are usually the bottleneck for computing unbiased
angular power spectra, as well as their associated covariance matrices, of the
cosmic microwave background temperature and polarization anisotropies. The
approximation results in the speed up of the MASTER algorithm by more than an
order of magnitude with very little loss of precision. We demonstrate the
performance on simulations of forthcoming cosmic microwave background surveys
such as the Simons Observatory and CMB-S4 for a wide variety of survey window
functions.
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