Question $2$

$50$ marks

$($a$)$ Write a function that implements the following pseudo$-$code:

Input: $f,f^{\text{'}},x,lon,N.$

Output: $x^{**}.$

Repeat $N$ times:

$($a$)$ Set $y_1=x.$

$($b$)$ Take one Newton step, starting from $y_1.$ Call the result $y_2.$

$($c$)$ Take one Newton step, starting from $y_2.$ Call the result $y_3.$

$($d$)$ Set

$x=y_1-\frac{(y_2-y_1{)}^2}{y_3-2y_2+y_1}$

$($e$)$ Display $|f(x)|.$