2.1 Differential corrections
as an optimization algorithm

The principle of least squares assumes that a target function
has to
be minimized to find the nominal solution. The target function
is formed with the sum of squares of the residuals ,
with .
The residuals are normalized, as discussed in Section 3.3, thus
is dimensionless. In our case,
for astrometric
observations of two angular coordinates. The residuals are functions
of
the estimated parameters .
In the simplest problems of orbit determination
and is
some vector representing the orbital elements at some initial epoch :
in this paper
are the equinoctial elements as defined in Paper I, Sect. 4.1. Some of
the coordinates of the vector ,
e.g. in our case the mean longitude ,
are not real numbers, but are sometimes angles (defined ),
and this introduces some complications which will be noted later.

Thus the target function also depends upon ,
and the minimum of
is obtained by solving the nonlinear equations:

Now let the map between
and be
linearized in a neighborhood of some point :

where the target function is approximated by a quadratic form

The equations to be solved for the minimum are the normal equations

with normal matrix
and solution

computed with the covariance matrix ,
which exists whenever
is positive-definite, which is generically the case for .
We shall of course assume that the linearization is performed around the
solution
such that ;
in the standard differential corrections procedure
is obtained by iterating the solution of the normal equation until convergence
(pseudo-Newton method). For a standard reference on differential correction,
see [Cappellari et al., 1976].

Please note that to apply a single iteration of differential correction,
and even any fixed number of iterations, is not enough to guarantee convergence;
an iterative scheme with a tight convergence control needs to be used.
As an example, in our programs the convergence is controlled by requiring
that the correction norm

(1)

is below a small control value
to stop the iterative procedure;
is used in most cases.