Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 4 / d / Solution

Apply the mean value theorem, the first-order case of Taylor's theorem, to $f^{\prime }$ between $0$ and $x>1$. It yieldsAlso $f(3/2)\ge -5+9/2+9/4=7/4>0$, so the initial point is to the right of the root $x^{\ast }$. For an increasing strictly convex function, its tangent at a point to the right of a root meets the axis between that root and the point. Hence the exact Newton root-finding iteration remains in $[x^{\ast },3/2]$ and decreases towards $x^{\ast }$.

On every relevant interval, $|f^{\prime \prime }|<3$ and $f^{\prime }>5$, so the stated quadratic convergence bound gives, with $e_n=|x_n-x^{\ast }|$,Iterating the bound givesThus four Newton corrections suffice to reduce the exact-iteration error bound below the floating-point spacing scale. The quadratic convergence doubles the approximate number of correct digits at each late step. Beyond this point further corrections cannot reliably improve a generic floating-point answer. This is a bound on useful accuracy under the usual stable-evaluation model, not a claim that the data specify the exact iteration at which every implementation must stagnate.