Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 5 / b

Apply Newton's method to $F(t)=1-3\sin t/2-{\sin }^{3}t/3$. The derivative in part (a) givesChoose a starting point near a desired root, such as $t_0=0.6$ for the root in $(0,\pi /2)$; do not divide when $\cos t_k=0$. On $(0,\pi /2)$, $F$ is strictly decreasing and changes sign between its endpoints, so the intermediate value theorem gives exactly one root there. Its derivative is nonzero, giving the usual local convergence of Newton's method. The periodic function has other roots; the iteration does not impose global uniqueness or convergence from every starting value.