Let be a system of equations in the unknowns and be an estimate of the solution of the system.
Let be the Jacobian matrix of the system of equation. Then
the iterative scheme defined by:

(2.6) |

A *simplified Newton method* consist in using a constant matrix in
the classical Newton method, for example the inverse Jacobian matrix
at some point like . The iterative scheme become:

(2.7) |

Newton method has advantages and drawbacks that need to be known in order to use it in the best way:

- it may really be fast: this may be important, for example in real-time control
- it is very simple to use
**but**it does not necessarily converge toward the solution "closest" to the estimate (see the example in section 15.1.2)**but**it may not converge. Kantorovitch theorem (see section 3.1.2) enable to determine the size of the convergence ball but this size is usually small (but quite often in practice the size is greater than the size given by the theorem which however is exact in some cases)**but**a numerical implementation of Newton may overflow