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