From a first guess of the
parameters 
, a sequence 
is generated
and is intended to converge to a local minimum of 
.
At each iteration, one computes
where 
is a certain descent direction and 
is a real
coefficient which is chosen such that
is approximately minimum. The
direction 
is ideally the solution of the Newton equation
which can also be rewritten
Neglecting the second derivatives matrix 
, we obtain
the ``normal equations'' and the Gauss--Newton direction
This so--called Gauss--Newton method is intended for problems where
is small. If the Jacobian
is singular or near singular or if 
is very large (the
so--called large residuals problem), the Gauss--Newton equation is not a
good approximation of the normal equations and the convergence is not
guaranteed.
The algorithm implemented here is a modification
of that Gauss--Newton method, that allows convergence even for
rank deficient Jacobians or for large residuals. The Gauss--Newton
direction is computed in 
,
the invariant space corresponding to the non--null eigenvalues.
A correction is taken in 
, the orthogonal of 
,
according to the second derivatives if
the decrease of the objective function at the last
iteration is considered too small.
The Hessian matrix is estimated using finite
differences of the gradient.
This method requires the availability of the derivatives and as
the number of gradient evaluations is almost p at each iteration, it
is recommended for problems with a small number of parameters,
let us say 