= Solution
For iterates approaching $\mathbf x_*$, write $e_k=\|\mathbf x_k-\mathbf x_*\|$. Exact <order of convergence> $p$ means $e_{k+1}/e_k^p\to C$ for a finite constant $C>0$, with nonzero errors. Thus \b[<quadratic convergence> has $p=2$, and <cubic convergence> has $p=3$]. The weaker eventual bound $e_{k+1}\leq Ce_k^p$ means at least order $p$. Finite termination is a separate case.<NIST Digital Library of Mathematical Functions/3.8 Nonlinear Equations>[§3.8(i), equation 3.8.3 and its order-of-convergence discussion]{ref}
The assertion about being always faster is \b[false]. For example, the <fixed-point iterations> with errors $e_{k+1}=e_k^2$ and $d_{k+1}=50d_k^3$ have exact <quadratic convergence> and <cubic convergence>, respectively. Starting both at $0.1$, their next errors are $0.01$ and $0.05$: the <cubic convergence> method is worse on that step, although both converge. Error constants, starting points, and work per step all matter. The higher <order of convergence> describes asymptotic behavior within a convergence region, not a uniform comparison of all computations.
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