problem arises for the temperature anisotropies because the $C_l$s do not go smoothly to zero as $l \rightarrow 2$. @efstathiou04 suggests replacing the block of the approximate covariance matrices with small $l$ and $l'$ with the exact expression, since the latter can be calculated quickly at small $l$ and $l'$. For diagonally-dominant matrices, this procedure returns an accurate covariance matrix; a similar procedure can be implemented for polarization using equations (\[eq:5-37\])–(\[eq:5-39\]) and the approximations developed in Section \[sec:approx-cov\].

Examples {#subsec:ex1}
--------

As our first application of equations (\[eq:5-37\])–(\[eq:5-39\]), we consider a full-sky survey from which we remove a $\pm 20\degr$ band around the Galactic plane. We adopt a north-south symmetric weighting which is uniform (which we expect to be nearly optimal in the noise-free limit) except for the edge of the retained survey which we apodize with the square of a cosine. In the northern hemisphere $$w(\theta,\phi) = \left\{ \begin{array}{ll} 1 \quad & \mbox{for
$\theta < 65\degr$} \\
\cos^2[36(70\degr-\theta)] & \mbox{for $65\degr \le \theta \le 70\degr$.}
\end{array} \right.
\label{eq:exact_cov1}$$ Since $w({\hat{\bmath{n}}})$ is azimuthally-symmetric, the matrices $\pm I_{(lm)(lm')}$ are block diagonal and the covariances can be evaluated exactly in $O(l^4)$ operations.

Figures \[fig:exactEE\_galcosine\]–\[fig:exactEB\_galcosine\] show the exact covariance matrices $\rmn{cov}(\tilde{C}_l^E,\tilde{C}_{l'}^E)$, $l(l+1)l'(l'+1)\rmn{cov}(\tilde{C}_l^B,\tilde{C}_{l'}^B)$ and 