\(\star\)E6.1. Show that the iteration \[x_{n+1} = \frac{1 + 9x_n -x_n^3}8\] has a fixed point in the interval \([1,2]\), and that this fixed-point iteration is convergent.
E6.2. By verifying the conditions in the fixed-point theorem, show that the iteration \[x_{n+1} = g(x_n), \qquad% g(x):=\frac{x+1}{x+2}\] converges to a fixed point in the interval \([0,1]\) for any initial condition \(x_0\in[0,1]\).
\(\star\)E6.3. By considering a suitable equation of the form \(f(x)=0\), write down Newton’s method for computing the cube root of a given number \(a\). Use this technique to find an approximation of \(25^{1/3}\) correct to four-significant figures. How many iterations do you require?
E6.4. Determine a numerical value (correct to 12 decimal places) of \(1/3\) without division by use of Newton’s method applied to \(f(x) = 1/x - a\) with \(a = 3\), \(x_{0} = 0.3\). Examine closely the notion of quadratic converge. (Before computer hardware was fully developed, this was the standard way of performing division on early computers).