Can anyone explain this problem? I am wondering what it means for evaluate $g'(x^∗)$, isn't that zero?
Newton's method can be viewed as a way of transforming a root-finding problem $f(x)=0$ into a fixed-point problem $x=g(x)$, where $$g(x)=x−f(x)/f′(x).$$ Recall that for simple roots, Newton's method has a quadratic convergence rate since $g'(x^∗)=0$, where $$g′(x)=f(x)f''(x)/(f'(x))^2.$$
When there is a root with multiplicity, $f'(x^∗)= 0$ which leads to a division by $0$ in $g'(x^∗)$. We will study how this affects the convergence rate of Newton's Method.
Consider the function $$f(x)=(x−x^∗)^m * h(x),$$ where $m$ is an integer greater than $1$ and $h$ is arbitrary function such that $h(x^∗)≠0$
Please answer the following:
What is the fixed point problem obtained by applying Newton's Method? Please give the formula for $g(x)$.
Evaluate $g'(x^∗)$. Based on that value, what can you conclude about the convergence rate of Newton's Method for a root of multiplicity $m$?
Based on the previous result, can you propose a modified Newton's method that recovers quadratic convergence rate? What is the formula for $g(x)$ in this modified version? Prove that your new method achieves quadratic convergence.
