I am trying to study the rates of convergence of various methods from Nocedal. They use Q convergence to quantify how fast certain sequences converge. I wanted some clarity on that,
Does this definition mean that after a particular $k_o$, this ratio is bounded by M?
For the sequence given, $x_k = 1 + {{0.5}^{2}}^{k}$ I am finding that M is 1. Am I correct?
Can anyone give me an example of a sequence that converges with power 3 or 4 in the denominator? i.e. order of covergence more than 2.
Adapted from the example sequence you've confirmed to be Q-quadratic, a sequence converging with power $n$ in the denominator would be the sequence
$$x_k = 1 + (0.5)^{n^k}$$
Confirmation: The limit $x^*$ is $1$, and $\|x_k - x^*\| = (0.5)^{n^k}$, so $$\|x_{k+1} - x^*\| = (0.5)^{n^{k+1}} = ((0.5)^{n^k})^n = \|x_k-x^*\|^n.$$