The Gaia core solution aims to solve the astrometric parameters for more than 1
billion sources mainly in our Galaxy. This clearly presents an enormous computational
challenge as the size of the data set, and the large number of parameters, cannot be
processed sequentially. The difficulty is caused by the strong connectivity among the
unknowns: each source is effectively observed relative to a large number of other
sources simultaneously in the field of view, or in the complementary field of view
some $106{.}^{\circ}5$ away on the sky, linked together by the attitude and
calibration models. The complexity of the astrometric solution in terms
of the connectivity between the sources provided by the attitude modelling
was analysed by Bombrun et al. (2010), who concluded that a direct solution
is infeasible, by many orders of magnitude, with today’s computational
capabilities. The study neglected the additional connectivity due to
the calibration model, which makes the problem even more unrealistic to
attack by a direct method. However, the mathematical system of equations under consideration
guarantees that a unique (apart for the six degrees of freedom discussed in the reference frame alignment
in Section 3.3.2), coherent and completely independent
global solution for the whole sky can be derived.
To overcome this difficulty an iterative method has been developed over a number of years
using increasingly complex and efficient algorithms. This approach became known as the
Astrometric Global Iterative Solution (AGIS) and now relies on a Conjugate Gradient (CG)
algorithm to converge the solution efficiently (Bombrun et al. 2012).
The numerical approach to AGIS is a blockiterative leastsquares
solution. In its simplest form, four blocks are evaluated in a cyclic sequence until
convergence. The blocks map to the four different kinds of unknowns outlined
in Section 3.1.1, namely:

S:
the source (star) update, in which the astrometric parameters $\bm{s}$
of the primary sources are improved;

A:
the attitude update, in which the attitude parameters $\bm{a}$ are
improved;

C:
the calibration update, in which the calibration parameters $\bm{c}$
are improved;

G:
the global update, in which the global parameters $\bm{g}$ are improved.
The G block is optional, and will perhaps only be used in some of the final solutions,
since the global parameters can normally be assumed to be known a priori to high
accuracy. The blocks must be iterated because each one of them needs data from the three other
processes. For example, when computing the astrometric parameters in the S block,
the attitude, calibration and global parameters are taken from the previous iteration.
The resulting (updated) astrometric parameters are used the next
time the A block is run, and so on. The mathematical description of the AGIS blockiterative
leastsquares solution and the updating of each block has been outlined in detail in Sections 4
and 5 respectively of Lindegren et al. (2012).
In addition to these blocks, separate processes are required
for the alignment of the astrometric solution with the ICRS (see also Section 3.3.2),
the selection of primary sources, and the calculation of standard uncertainties; these auxiliary
processes are discussed in (Section 6 of Lindegren et al. 2012).
Additionally, it is not necessary for the AGIS solution to include all one billion sources.
Instead, it is done using a selection of about 10% of the astrometrically well
behaved single sources and this is sufficient to converge the attitude and calibration solutions.
The other sources can then be solved for in a secondary solution (see Section 3.4.3)
using the converged parameters found in the primary AGIS solution. The primary solution will
consist of about ${10}^{8}$ sources so the number of unknowns in the global minimization problem
is about $5\times {10}^{8}$ for the sources ($\bm{s}$), $4\times {10}^{7}$ for the
attitude ($\bm{a}$, assuming a knot interval of 15 s for the 5 yr mission;
${10}^{6}$ for the calibration $\bm{c}$, and less than
100 global parameters ($\bm{g}$). The number of elementary observations
($l$) considered is about $8\times {10}^{10}$.