Is that possible for some function whose convergence rate is linear by using Newton's method?
I am solving the function $$f(x) = \sin^2(x) - x \sin(x) + \frac 14 x^2$$ by Newton's Method. I got the error with $0.11$, $0.05$, $0.024$, $0.012$ in the first few iterations, which looks like it has linear convergence rate with $1/2$. Is that possible?
As your function is a square, $f(x)=(\sin x-\frac12x)^2$, all roots will have even multiplicity. And indeed, Newton's method has linear convergence towards multiple roots, with factor $1-\frac1m$ for multiplicity $m$.
Note that as $\frac\pi2>0=\sin(\pi)$ and $\frac\pi4<1=\sin(\frac\pi2)$, there are additional (double) roots inside $[\frac\pi2,\pi]$ and its mirrored interval.