Exercises on Newton’s Method#

Exercise 1#

a) Show that Newton’s method applied to

\[ f(x) = x^k - a \]

leads to fixed point iteration with function

\[ g(x) = \frac{(k-1) x + \displaystyle \frac{a}{x^{k-1}}}{k}. \]

b) Then verify mathematically that the iteration \(x_{k+1} = g(x_k)\) has super-linear convergence.