Practical considerations. Newton's method is a powerful technique—in general the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step.

Does Newton's method always converge?

If the initial value is too far from the true zero, Newton's method may fail to converge (has only local convergence). ... If the function is not continuously differentiable in a neighborhood of the root, it is possible that Newton's method will always diverge or fail. Solution: Try another initial point.

Why does Newton's method always converge in one iteration?

This theorem insure that Newton's method will always converge if the initial point is sufficiently close to the root and if this root if not singular (that is f^¢(x^*) is non zero). This process has the local convergence property.