We will denote by the matrix of the derivatives of order of with respect to the variable. Let us consider a point and define as

and . If is strictly lower than , then has a single root in a ball centered at with radius and the Newton scheme with initial guess will converge to the solution.

The most difficult part for using this theorem is to determine
. For algebraic equations it is easy to determine a value
, that we will call the *order* of Rouche theorem, such that
and consequently may be
obtained by computing

for all in and taking as the Sup of all .

For non algebraic finding requires an analysis of the system.

Rouche theorem may be more efficient than Moore or Kantorovitvh theorems. For example when combined with a polynomial deflation (see section 5.9.6) it allows one to solve Wilkinson polynomial of order up to 18 with the C++ arithmetic on a PC, while stand solving procedure fails for order 13.