  High precision measurements of the Cosmic Microwave Background (CMB)
anisotropies, as can be expected from the Planck satellite, will require
high-accuracy theoretical predictions as well. One possible source of
theoretical uncertainty is the numerical error in the output of the Boltzmann
codes used to calculate angular power spectra. In this work, we carry out an
extensive study of the numerical accuracy of the public Boltzmann code CAMB,
and identify a set of parameters which determine the error of its output. We
show that at the current default settings, the cosmological parameters
extracted from data of future experiments like Planck can be biased by several
tenths of a standard deviation for the six parameters of the standard
Lambda-CDM model, and potentially more seriously for extended models. We
perform an optimisation procedure that leads the code to achieve sufficient
precision while at the same time keeping the computation time within reasonable
limits. Our conclusion is that the contribution of numerical errors to the
theoretical uncertainty of model predictions is well under control -- the main
challenges for more accurate calculations of CMB spectra will be of an
astrophysical nature instead.
