  We present ECLIPSE (Efficient Cmb poLarization and Intensity Power Spectra
Estimator), an optimized implementation of the Quadratic Maximum Likelihood
(QML) method for the estimation of the power spectra of the Cosmic Microwave
Background (CMB). This approach allows one to reduce significantly the
computational costs associated to this technique, allowing to estimate the
power spectra up to higher multipoles than previous implementations. In
particular, for a resolution of $N_\mathrm{side}=64$, $\ell_{\mathrm{max}}=192$
and a typical Galactic mask, the number of operations can be reduced by
approximately a factor of 1000 in a full analysis including intensity and
polarization with respect to an efficient direct implementation of the method.
In addition, if one is interested in studying only polarization, it is possible
to obtain the power spectra of the E and B modes with a further reduction of
computational resources without degrading the results. We also show that for
experiments observing a small fraction of the sky, the Fisher matrix becomes
singular and, in this case, the standard QML can not be applied. To solve this
problem, we have developed a binned version of the method that is unbiased and
of minimum variance. We also test the robustness of the QML estimator when the
assumed fiducial model differs from that of the sky and show the performance of
an iterative approach. Finally, we present a comparison of the results obtained
by QML and a pseudo-$C_{\ell}$ estimator (NaMaster) for a next-generation
satellite, showing that, as expected, QML produces significantly smaller errors
at low multipoles. The ECLIPSE fast QML code developed in this work will be
made publicly available.
