  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. 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.
