Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 4 / a

Let $e_n=x_n-x_{\ast }$ for the limit $x_{\ast }$. Quadratic convergence means that eventually $|e_{n+1}|\le C|e_n{|}^2$ for a fixed constant $C$; exact asymptotic order two has $|e_{n+1}|/|e_n{|}^2\to c$ with $0<c<\infty$.

For this iteration and $x_{\ast }=3$, direct algebra givesUsing the stipulated convergence $x_n\to 3$, the ratio of successive squared errors isThus the convergence is quadratic. The iteration is well defined here: $x_1=7/2$, and thereafter $x_n>3$ implies $3<x_{n+1}<x_n$. The initial point is not already the limit, so the error ratio is meaningful.