It should be obvious that, to find a root of F(x) = 0, we can generalize Newton's method to systems, writing
As before, we (would like to) believe that, if we start close enough to a root, the methods will converge. As we know, those hopes are not well founded.
Of course, we do not want to compute the inverse of the Jacobian in the formula above.
rather we proceed in two steps. Define 
. Then
There are two difficulties with the method as outlined. First, one has a
Jacobian to compute or evaluate at every step, as well as a
new linear system to solve. Second, we know from experience with
scalar equations that sometimes
the Newton step is too large a correction. There are several modifications
to the simple system outlined above that can be made.
First, we can evaluate
update the Jacobian not every step, but only after several iteration updates.
This will save on evaluation (either from a formula or by finite difference),
and on linear solves (with a direct solver). Second, one can take a multiple
of the correction step,
,
where 
can be chosen to minimize
.