Editing Newton's method (section)

===Failure of the method to converge to the root===
It is important to review the [[#Proof of quadratic convergence for Newton's iterative method|proof of quadratic convergence]] of Newton's method before implementing it. Specifically, one should review the assumptions made in the proof. For [[#Failure analysis|situations where the method fails to converge]], it is because the assumptions made in this proof are not met.

For example, [[#Divergence even when initialization is close to the root|in some cases]], if the first derivative is not well behaved in the neighborhood of a particular root, then it is possible that Newton's method will fail to converge no matter where the initialization is set. In some cases, Newton's method can be stabilized by using [[successive over-relaxation#Other applications of the method|successive over-relaxation]], or the speed of convergence can be increased by using the same method.

In a robust implementation of Newton's method, it is common to place limits on the number of iterations, bound the solution to an interval known to contain the root, and combine the method with a more robust root finding method.