Suppose that 
is either as in the nonlinear
equations Problem 1 or as in the Fritz-John
equations 12, suppose 
is
an interval vector, and suppose that 
. Then a
general form for multivariate interval Newton methods is
where 
is an interval vector that contains all solutions
to point systems 
, for
, where 
is an interval
extension to the Jacobi matrix of F over . Under
certain natural smoothness conditions,
must contain all solutions
with 
. (Consequently,
if 
,
then there are no solutions of F(X)=0 in .)
containing a solution of F(X)=0 and the
widths of the components of sufficiently small, the
width of
is roughly proportional to the
square of the widths of the components of .
,
where 
represents the interior of
, then there is a unique solution of F(x)=0 within
, and hence within
.
For details and further references, see [8, §1.5,], or see the preprint of the Encyclopedia of Optimization article at http://interval.louisiana.edu/preprints/EoO/interval_Newton.ps (Postscript, approx. 112KB) or http://interval.louisiana.edu/preprints/EoO/interval_Newton.dvi (TeX DVI, approx. 17KB).
In a multivariate interval Newton method, the interval vector
must be found in the iteration
formula (10), that is, the
solution set to the interval linear system
must be bounded. Within GlobSol, this is done in some contexts
with interval Gaussian elimination, and in other contexts with
the interval Gauss-Seidel method. In either case, the
system (11) must be preconditioned. For the
interval Gauss-Seidel method, GlobSol provides two options for
the preconditioner: the inverse midpoint preconditioner and the
width-optimal linear programming preconditioner. The width-optimal
linear programming preconditioner provides better box-reduction
properties for boxes with large widths, but is possibly more expensive
to compute, especially in higher dimensions. The choice of
preconditioner is controlled with parameter
PRECONDITIONER_TYPE in the configuration file
INTNEWT.CFG.
For more details on preconditioning, see [8, Chapter 3,].