Properties

One can show that has a zero of multiplicity when , and for but .

Thus for a zero of multiplicity 1, or a simple zero , and therefore the Newton-Raphson Method will converge quadratically.

On the other hand, quadratic convergence may no longer occur when looking for a zero of multiplicity larger then 1. Consider the following example:

> f:=x->exp(x)-x-1;

> solve(f(x)=0,x);

> D(f);D(f)(0);

> D(D(f));D(D(f))(0);

> p[0]:=1.0;

> for i from 1 to 10 do p[i]:=p[i-1]-f(p[i-1])/(D(f)(p[i-1])): print(i,p[i],abs(p[i]/p[i-1]),abs(p[i]/(p[i-1])^2)); od:

which suggests only linear convergence!