Newton's method converges linearly from sufficiently close to a root of finite multiplicity greater than one
Suppose is a function of one variable that is at least one time continuously differentiable at a root . Further, suppose , so that is root of multiplicity greater than 1. Then, there exists such that for any , the sequence obtained by applying Newton's method either reaches the root in finitely many steps or has linear convergence to the root . The convergence rate is where is the multiplicity of the root (i.e., the order of as a zero).
Based on the information about the order of the root being , there exists and constants such that: