  Accurate covariance matrices are required for a reliable estimation of
cosmological parameters from pseudo-power spectrum estimators. In this work, we
focus on the analytical calculation of covariance matrices. We consider the
case of observations of the Cosmic Microwave Background in temperature and
polarization on a small footprint such as in the SPT-3G experiment, which
observes 4% of the sky. Power spectra evaluated on small footprints are
expected to have large correlations between modes, and these need to be
accurately modelled. We present, for the first time, an algorithm that allows
an efficient (but computationally expensive) exact calculation of analytic
covariance matrices. Using it as our reference, we test the accuracy of
existing fast approximations of the covariance matrix. We find that, when the
power spectrum is binned in wide bandpowers, current approaches are correct up
to the 5% level on the SPT-3G small sky footprint. Furthermore, we propose a
new approximation which improves over the previous ones reaching a precision of
1% in the wide bandpowers case and generally more than 4 times more accurate
than current approaches. Finally, we derive the covariance matrices for
mask-corrected power spectra estimated by the PolSpice code. In particular, we
include, in the case of a small sky fraction, the effect of the apodization of
the large scale modes. While we considered the specific case of the CMB, our
results are applicable to any other cosmological probe which requires the
calculation of pseudo-power spectrum covariance matrices.
