The cosine

The function $f:x\rightarrow\cos(x)$ provides a simple example of majorization, also used by Lange [2015]. We work out some additional details. Start with $$\cos(x)=\cos(y)-\sin(y)(x-y)-\frac12 \cos(y)(x-y)^2+\frac16 \sin(y)(x-y)^3+\frac{1}{24}\cos(y)(x-y)^4+...\tag{1}$$ Since $f''(x)=\cos(x)\leq 1$ we see that $$g(x,y):=\cos(y)-\sin(y)(x-y)+\frac12 (x-y)^2$$ provides a uniform quadratic majorizer. Thus the iteration map $$A(x)=x+\sin(x)$$ provides a uniform quadratic majorization algorithm.

Now $$\frac{A(x)-\pi}{x-\pi}=1-\frac{\sin(x-\pi)}{x-\pi},$$ and for $0<x<2\pi$ we have $0<1-\frac{\sin(x-\pi)}{x-\pi}<1,$ and thus the algorithm converges to $\pi.$ As Lange [2015] points out the algorithm has a cubic rate of convergence, because $$A(\pi+x)-\pi=x+\sin(\pi+x)=\frac16 x^3+o(x^3),$$ and thus $$\lim_{x\rightarrow 0}\frac{A(\pi+x)-\pi}{x^3}=\frac16.$$ As a consequence of cubic convergence, there is not much that can be done to improve the algorithm (which is of very limited practical usefulness anyway). Of course we could use the more precise majorizations $$\begin{align*} \cos(x)&\leq \cos(y)-\sin(y)(x-y)-\frac12 \cos(y) (x-y)^2+\frac16 |x-y|^3,\\ \cos(x)&\leq \cos(y)-\sin(y)(x-y)-\frac12 \cos(y)(x-y)^2+\frac16 \sin(y)(x-y)^3+\frac{1}{24}(x-y)^4, \end{align*}$$ but they mostly increase the amount of computation in an iteration and do not improve much. The quadratic majorization at $y=2$ has its minimum at 2.1974, the cubic majorization at 2.9952, and the quartic majorization at 2.9270. The three majorizations at $y=2$ are shown in Figure 1.

As a curiosity, we could also consider the sharp quadratic majorization $$g(x,y)=\cos(y)-\sin(y)(x-y)+\frac12\frac{\sin(y)}{\pi-y}(x-y)^2$$ which has support points at $x=y$ and $x=2\pi-y.$ Because of symmetry the correspondng majorization algorithm converges to $x=\pi$ in a single step in this case. In Figure 2 we show the uniform and sharp quadratic majorizations for $y=2$.