Compare the efficiency of the standard newton method with

Question

Consider a function f (x) and a let p be a zero of multiplicity m of f, i.e. f(x) = (x - p)^mg(x), with g(p) ≠ 0.

Consider the expression μ(x) = f(x)/f'(x).

a) Show that p is a simple zero of μ(x). This is μ(p) = 0, but μ'(p) ≠ 0. Note in this case that any zero of μ(x) will be a zero of f(x).

b) Show that the Newton method applied to the function μ(x) will give the iteration:

x_n+1 = x_n - (f(x_n)f'(x_n))/(f'(x_n)² - f(x_n)f''(x_n))

c) Compare the efficiency of the standard Newton method with this new iteration in order to find the zero of multiplicity m = 3 of the polynomial:

p(x) = x⁵ -9/2x⁴ + 23/4x³ + 9/8 x² - 27/4 x + 27/8

located in the interval [0,2].