In this assignment we consider the polynomial $x^3+3x-4$.
This polynomial has a unique root, which is $x=1$, and can be rewritten as $\bigl(x-1\bigr)\bigl(x^2+x+4\bigr)=\bigl(x-1\bigr)\bigl((x+\tfrac{1}{2})^2+3\!\tfrac{3}{4}\bigr)$.

The zero point of the polynomial can be approximated via iteration of a function. In this assignment we will experimentally study the convergence behaviour of several functions of which $x=1$ is the fixed point.

Assignment

Write a computer program for function iteration and use the following functions:

$$\begin{aligned} f_1(x) &=\frac{4-x^3}{3x}\\ \\ f_2(x) &= \frac{4-3x}{x^2}\\ \\ f_3(x) &= x^3+4x-4\\ \\ f_4(x) &= x-\frac{1}{6}(x^3+3x-4)\\ \\ f_5(x) &= \frac{2x^3+4}{3x^2+3}\end{aligned}$$ Carry out experiments for each of these functions to approximate the zero point of $x^3+3x-4$. Vary the tolerance and check how many iterations are needed (note: sometimes there is no convergence!).
Also experiment with a few different starting values $x_0$. Along the way, write down any details you encounter (about convergence, speed of convergence, dependency on the starting value, etc.) and verify that the value of the derivative $f_k'(1)$ is the predominant factor in all experiments.