The case n simultaneous nonlinear equations in n unknowns for n=2 is examined in detail. The basic ideas extend to the general case. Consider the system of equations
Or equivalently written in vector form
where f and z are 2-vectors.
To solve (1.3), Newton's method for finding the root of a single
nonlinear equation is generalized to two dimensions. The functions
and 
are expanded about the point
using Taylor's theorem for functions of two variables and
using only the first order terms. This process yields
the following equations.
In fact this can be written in matrix-vector form as:
where the matrix 
is the Jacobian matrix of the
function f.
Let 
be the zero of produced by this linear approximation.
It satisfies
In general 
is obtained from 
from
However, instead of computing 
,
we solve for 
from
and then compute
This algorithm can be formalize in the following.
Where 
.
This algorithm requires the matrix of Jacobians to be calculated.
